history of virtual work laws

Birkhauser

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Science Networks. Historical StudiesFounded by Erwin Hiebert and Hans WußingVolume 42

Edited by Eberhard Knobloch, Helge Kragh and Erhard Scholz

Editorial Board:

K. Andersen, AarhusD. Buchwald, PasadenaH.J.M. Bos, UtrechtU. Bottazzini, RomaJ.Z. Buchwald, Cambridge, Mass.K. Chemla, ParisS.S. Demidov, MoskvaE.A. Fellmann, BaselM. Folkerts, MünchenP. Galison, Cambridge, Mass.I. Grattan-Guinness, London

J. Gray, Milton KeynesR. Halleux, LiègeS. Hildenbrandt, BonnCh. Meinel, RegensburgJ. Peiffer, ParisW. Purkert, BonnD. Rowe, MainzA.I. Sabra, Cambridge, Mass.Ch. Sasaki, TokyoR.H. Stuewer, MinneapolisV.P. Vizgin, Moskva

Danilo Capecchi

History of VirtualWork LawsA History of Mechanics Prospective

Birkhauser

Danilo CapecchiUniversità La Sapienza, Rome (Italy)

ISBN 978-88-470-2055-9DOI 10.1007/978-88-470-2056-6

Library of Congress Control Number: 2011941587

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Preface

Lagrange, in the Méchanique analitique of 1788, identified three programs of re-search, or paradigms, in the history of statics: the lever, the composition of forces,and the principle of virtual work. The paradigm of the lever would have been inforce from antiquity until the early XVIII century, when Varignon was asserting theparallelogram law for composition and decomposition of forces. The principle ofvirtual work would have become dominant since the XIX century. This picture is inmy opinion quite realistic, although the final predicted by Lagrange was never fullyrealized because the principle of virtual work has never replaced the rule of the com-position of forces, but at most has outflanked it. Also the picture is too schematic.In fact, some form of law of virtual work has always existed in mechanics, alwayshowever with limited applications.

The law of virtual work, as usually presented in modern textbooks of mechanics,says that there is equilibrium for one or more bodies subjected to a system of forcesif and only if the total virtual work is zero for any virtual displacement. In Chapter2 of this book the meaning of the terms work and virtual is described in some detail;here I will only to mention that, since Lagrange in the second half of the XVIIIcentury, the law of virtual work had no appreciable changes in its formulation. Theview on its role in mechanics is instead still varying, passing from the enthusiasm ofthe XIX century to a modest presence in modern rational mechanics as well as, allconsidered, in the engineering field, albeit with some important exceptions.

The present book starts from the first documented formulations of laws of virtualwork. They usually have only a vague analogy to the modern ones and only mathe-matically. Attention is paid to Arabic and Latin mechanics of the Middle Ages. Withthe Renaissance there began to appear slightly different wordings of the law, whichwere often proposed as unique principles of statics. With Bernoulli and Lagrangethe process reached its apex. The book ends with some chapters dealing with thediscussions that took place in the French school on the role of the Lagrangian law ofvirtual work and its applications to continuum mechanics.

Even though the book takes a particular point of view, it presents an importantslice of history of mechanics. Essential reference is made to primary sources; sec-ondary literature is mainly used to frame the contributions of the scientists consid-

vi Preface

ered in their times. To allow a better understanding of the ideas of the authors studied,English translations are always accompanied by original quotations (Appendix). Nopre-conceived historical hypotheses have been explicitly assumed though. The mereexistence of the book suggests that I have in mind a continuous chain connectingconcepts from antiquity up to now. However the nature of the chain is complex andI leave it to the reader to unveil it.

The book is the result of a twenty year study of mechanics and its history andshould be of interest to historians of mathematics and physics. It should also arouseinterest among engineers who are now perhaps the most important witnesses of clas-sical mechanics, and with it, of the law of virtual work.

I want to acknowledge Giuseppe Ruta, Romano Gatto, Antonino Drago for con-tributing comments and suggestions to specific parts. Cesare Tocci for suggestionsregarding the whole book, and finally I want to acknowledge Raffaele Pisano for hisreading and the debates we have had.

Editorial considerationsFigures related to quotations are nearly all redrawn to allow a better comprehension.Symbols of formulas are always those of the authors, except in easily identifiablecases. Translations of text from French, Latin, German and Italian are as much aspossible close to the original.

Rome, September 2011 Danilo Capecchi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Virtual velocity laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Virtual displacement laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Virtual work laws as principles of mechanics . . . . . . . . . . . . . . . . . 51.4 Virtual work laws as theorems of mechanics . . . . . . . . . . . . . . . . . 91.5 Contemporary tendencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Final remarks. The rational justification of virtual work laws . . . . 12

2 Logic status of virtual work laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 The theorem of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Proofs of the virtual work theorems in the literature . . . . 232.1.1.1 Physics and rational mechanics treatises . . . . . . 232.1.1.2 Statics handbooks . . . . . . . . . . . . . . . . . . . . . . . . 242.1.1.3 Poinsot’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 The principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1 Force as a primitive concept . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1.1 Equilibrium case . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1.2 Motion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Work as a primitive concept . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2.1 Equilibrium case . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2.2 Motion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Greek origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Different approaches to the law of the lever . . . . . . . . . . . . . . . . . . 34

3.1.1 Aristotelian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1.1 Physica and De caelo . . . . . . . . . . . . . . . . . . . . . 353.1.1.2 Mechanica problemata . . . . . . . . . . . . . . . . . . . . 383.1.1.3 A law of virtual work . . . . . . . . . . . . . . . . . . . . . 43

3.1.2 Archimedean mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.2.1 Proof of the law of the lever . . . . . . . . . . . . . . . . 48

3.2 The mechanics of Hero of Alexandria . . . . . . . . . . . . . . . . . . . . . . . 51

viii Contents

3.2.1 The principles of Hero’s mechanics . . . . . . . . . . . . . . . . . 533.2.1.1 A law of virtual work . . . . . . . . . . . . . . . . . . . . . 553.2.1.2 Hero’s inclined plane law . . . . . . . . . . . . . . . . . . 58

3.3 The mechanics of Pappus of Alexandria . . . . . . . . . . . . . . . . . . . . . 593.3.1 Pappus’ inclined plane law . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Arabic and Latin science of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Arabic mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1 Weight as an active factor in Arabic mechanics . . . . . . . 684.1.1.1 Liber karastonis . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.1.2 Kitab al-Qarastun . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.2 Comments on the Arabic virtual work law . . . . . . . . . . . . 744.2 Latin mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.1 Weight as a passive factor in the Latin mechanics . . . . . . 804.2.2 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.2.1 Proposition I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.2.2 Proposition II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2.3 Proposition VI. The law of the Lever . . . . . . . . 864.2.2.4 Proposition VIII . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.2.5 Proposition X. The law of the inclined plane . . 88

4.2.3 Comments on the Latin virtual work law . . . . . . . . . . . . . 89

5 Italian Renaissance statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1 Renaissance engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1.1 Daniele Barbaro and Buonaiuto Lorini . . . . . . . . . . . . . . . 965.2 Nicolò Tartaglia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.1 Definitions and petitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.2 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.2.1 Proof of propositions I–IV . . . . . . . . . . . . . . . . . 1005.2.2.2 The law of the lever . . . . . . . . . . . . . . . . . . . . . . . 1015.2.2.3 The law of the inclined plane . . . . . . . . . . . . . . . 103

5.3 Girolamo Cardano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.1 De subtitilate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3.2 De opus novum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Guidobaldo dal Monte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.4.1 The centre of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4.2 The balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4.3 The virtual work law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5 Giovanni Battista Benedetti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.5.1 Effect of the position of a weight on its heaviness . . . . . . 1165.5.2 Errors of Tartaglia and Jordanus . . . . . . . . . . . . . . . . . . . . 118

5.6 Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.6.1 The concept of moment. A law of virtual velocities . . . . 1215.6.2 A law of virtual displacements . . . . . . . . . . . . . . . . . . . . . . 1275.6.3 Proof of the law of the inclined plane . . . . . . . . . . . . . . . . 131

Contents ix

6 Torricelli’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.1 The centrobaric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Galileo’s centrobaric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.3 Torricelli’s joined heavy bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3.1 Torricelli’s fundamental concepts on the centre of gravity 1416.4 Torricelli’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.4.1 Analysis of the aggregate of two bodies . . . . . . . . . . . . . . 1466.4.2 Torricelli’s principle as a criterion of equilibrium . . . . . . 148

6.5 Evolution of Torricelli’s principle. Its role in virtual work laws . . 1536.5.1 A restricted form of Torricelli’s principle . . . . . . . . . . . . . 154

7 European statics during the XVI and XVII centuries . . . . . . . . . . . . . 1577.1 French statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.1.1 Gille Personne de Roberval . . . . . . . . . . . . . . . . . . . . . . . . 1607.1.1.1 The inclined plane law . . . . . . . . . . . . . . . . . . . . 1607.1.1.2 The rule of the parallelogram . . . . . . . . . . . . . . . 161

7.1.2 René Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.1.2.1 The concept of force . . . . . . . . . . . . . . . . . . . . . . 1647.1.2.2 The application to simple machines . . . . . . . . . . 1677.1.2.3 The refusal of virtual velocities . . . . . . . . . . . . . 1707.1.2.4 Displacements at the very beginning of motion 1717.1.2.5 A possible precursor . . . . . . . . . . . . . . . . . . . . . . 173

7.1.3 Blaise Pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.1.4 Post Cartesians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.2 Nederland statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.2.1 Simon Stevin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.2.1.1 The rule of the parallelogram of forces . . . . . . . 1807.2.1.2 The law of virtual work . . . . . . . . . . . . . . . . . . . . 184

7.2.2 Christiaan Huygens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.3 British statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.3.1 John Wallis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.3.2 Isaac Nevton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8 The principle of virtual velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.1 The concept of force in the XVIII century . . . . . . . . . . . . . . . . . . . . 195

8.1.1 Newtonian concept of force . . . . . . . . . . . . . . . . . . . . . . . . 1958.1.2 Leibnizian concept of force . . . . . . . . . . . . . . . . . . . . . . . . 197

8.2 Johann Bernoulli mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998.2.1 Dead and living forces according to Bernoulli . . . . . . . . . 1998.2.2 The rule of energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.3 Varignon: the rule of energies and the law of compositionof forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2108.3.1 Elements of Varignon’s mechanics . . . . . . . . . . . . . . . . . . 2108.3.2 The rule of the parallelogram versus the rule of energies 213

x Contents

9 The Jesuit school of the XVIII century . . . . . . . . . . . . . . . . . . . . . . . . . . 2179.1 Vincenzo Angiulli and Vincenzo Riccati . . . . . . . . . . . . . . . . . . . . . 218

9.1.1 The principle of actions of Vincenzo Angiulli . . . . . . . . . 2189.1.1.1 The action of a force . . . . . . . . . . . . . . . . . . . . . . 2199.1.1.2 The principle of actions . . . . . . . . . . . . . . . . . . . 2219.1.1.3 The measure of actions . . . . . . . . . . . . . . . . . . . 2239.1.1.4 The principle of action and the principles of

statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2259.1.1.5 The applications to simple machines . . . . . . . . . 228

9.1.2 The principle of actions of Vincenzo Riccati . . . . . . . . . . 2309.2 Ruggiero Giuseppe Boscovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9.2.1 A virtual work law for Saint Peter’s dome . . . . . . . . . . . . 2349.2.1.1 The mechanism of failure and the forces . . . . . 235

10 Lagrange’s contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.1 First introduction of the virtual velocity principle . . . . . . . . . . . . . 240

10.1.1 The first ideas about a new principle of mechanics . . . . . 24010.1.2 Recherches sur la libration de la Lune . . . . . . . . . . . . . . . 242

10.1.2.1 Setting of the astronomical problem . . . . . . . . . 24510.1.2.2 The symbolic equation of dynamics . . . . . . . . . 24710.1.2.3 The virtual velocity principle . . . . . . . . . . . . . . . 250

10.1.3 The Théorie de la libration de la Lune . . . . . . . . . . . . . . . 25110.2 Méchanique analitique and Mécanique analytique . . . . . . . . . . . . 252

10.2.1 Méchanique analitique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25310.2.1.1 Constraint reactions . . . . . . . . . . . . . . . . . . . . . . . 258

10.2.2 Mécanique analytique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25910.2.2.1 Criticisms of Lagrange’s proof . . . . . . . . . . . . . 263

10.3 The Théorie des fonctions analytiques . . . . . . . . . . . . . . . . . . . . . . . 26410.4 Generalizations of the virtual velocity principle to dynamics . . . . 268

10.4.1 The calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . 27310.4.2 Elements of D’Alembert’s mechanics . . . . . . . . . . . . . . . . 274

10.4.2.1 D’Alembert principle . . . . . . . . . . . . . . . . . . . . . 277

11 Lazare Carnot’s mechanics of collision . . . . . . . . . . . . . . . . . . . . . . . . . 28111.1 Carnot’s laws of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

11.1.1 The first fundamental equation of mechanics . . . . . . . . . . 28711.1.2 Geometric motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28911.1.3 The second fundamental equation of mechanics . . . . . . . 291

11.2 Gradual changing of motion. A law of virtual work . . . . . . . . . . . . 29311.3 The moment of activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

12 The debate in Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29912.1 The criticisms on the evidence of the principle . . . . . . . . . . . . . . . . 300

12.1.1 Vittorio Fossombroni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30012.1.1.1 Invariable distance systems . . . . . . . . . . . . . . . . 301

Contents xi

12.1.1.2 The equation of forces . . . . . . . . . . . . . . . . . . . . . 30212.1.1.3 The equation of moments . . . . . . . . . . . . . . . . . . 304

12.1.2 Girolamo Saladini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30612.1.3 François Joseph Servois . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

12.2 The criticisms on the use of infinitesimals . . . . . . . . . . . . . . . . . . . . 31112.2.1 Gabrio Piola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

12.2.1.1 Piola’s principles of material point mechanics . 31212.2.1.2 System of free material points . . . . . . . . . . . . . . 31412.2.1.3 System of constrained material points . . . . . . . . 315

13 The debate at the École polytechnique . . . . . . . . . . . . . . . . . . . . . . . . . . 31713.1 One of the first professor of mechanics, Gaspard de Prony . . . . . . 319

13.1.1 Proof from the composition of forces rule . . . . . . . . . . . . 32013.2 Joseph Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

13.2.1 First proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32313.2.2 Second proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32513.2.3 Third proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

13.3 André Marie Ampère . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32813.4 Pierre Simon Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

14 Poinsot’s criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33514.1 Considérations sur le principe des vitesses virtuelles . . . . . . . . . . 33614.2 Théorie générale de l’équilibre et du mouvement des systèmes . . 339

14.2.1 Poinsot’s principles of mechanics . . . . . . . . . . . . . . . . . . . 34214.2.1.1 System of material points constrained by a

unique equation . . . . . . . . . . . . . . . . . . . . . . . . . . 34414.2.1.2 System of material points constrained by more

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34614.3 Demonstration of the virtual velocity principle . . . . . . . . . . . . . . . . 348

15 Complementary virtual work laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35315.1 Augustin Cauchy formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

15.1.1 Kinematics of plane rigid bodies . . . . . . . . . . . . . . . . . . . . 356

16 The treatises of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36116.1 Siméon Denis Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36216.2 Jean Marie Duhamel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36516.3 Gaspard Gustave Coriolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

17 Virtual work laws and continuum mechanics . . . . . . . . . . . . . . . . . . . . 37517.1 First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

17.1.1 Joseph Louis Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37517.1.1.1 Mono-dimensional continuum . . . . . . . . . . . . . . 37617.1.1.2 Three-dimensional continuum . . . . . . . . . . . . . . 377

17.1.2 Navier’s equations of motion . . . . . . . . . . . . . . . . . . . . . . . 38117.2 Applications in the theory of elasticity . . . . . . . . . . . . . . . . . . . . . . 383

xii Contents

17.2.1 Alfred Clebsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38317.3 The Italian school . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

17.3.1 Gabrio Piola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38817.3.2 Eugenio Beltrami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39017.3.3 Enrico Betti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

18 Thermodynamical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39518.1 Pierre Duhem’s concept of oeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . 396

18.1.1 Virtual transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39718.1.2 Activity, energy and work . . . . . . . . . . . . . . . . . . . . . . . . . 39818.1.3 Rational mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

18.1.3.1 Free systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40118.1.3.2 Constrained systems . . . . . . . . . . . . . . . . . . . . . . 402

Appendix. Quotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405A.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405A.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406A.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407A.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409A.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412A.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423A.7 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426A.8 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433A.9 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437A.10 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441A.11 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448A.12 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452A.13 Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454A.14 Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457A.15 Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463A.16 Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464A.17 Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467A.18 Chapter 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

1

Introduction

Hereinafter, the law of virtual work is not given its contemporary meaning. To doso would be misleading in that it would attempt to write a history based on unavoid-able recourse to categories of thinking that did not exist in the past. I use insteadthe broader meaning of law of equilibrium, where forces appear together with themotion of their points of application independently of the logical status assumed, beit a theorem, a principle or an empirical law. In this sense the laws of virtual workrepresent a particular historical point of view on mechanics.

Since the Greek origins of mechanics, there have been two alternative formu-lations of laws of virtual work (hereinafter VWL). The first, which dates back toAristotle’s school, today goes under the name of laws of virtual velocities, in whichthe effects of forces are assumed depending on the virtual velocities of their points ofapplication. The second, which has been known at least since the Hellenistic period,today goes under the name of laws of virtual displacements, in which instead theeffects of forces are assumed depending on virtual displacement of their points ofapplication. The two approaches, though conceptually different, are mathematicallyequivalent.

In the early days of VWLs, virtual motions were considered primarily as possi-ble motions, those which one would have imagined the body, or system of bodies,to assume within the respect of constraints, for example, following a disturbanceinduced by a small force that alters the equilibrium. If one imagines that a balancerotates around the fulcrum, at the same time one would imagine that the weights ofwhich it is burdened move. But with this type of ‘natural’ conception there coexistsanother, though not fully conscious at the beginning, in which the virtual motion isseen as purely geometric. On the one hand one sees the balance in equilibrium underassigned weights; on the other hand one imagines the unloaded balance moving witha motion that takes place with a time flowing in a super-celestial world. This wayof viewing virtual motions began to emerge from the ‘subconscious’ to become the‘natural’ one only in the XIX century with Poinsot and Ampère.

Capecchi D.: History of Virtual Work Laws. A History of Mechanics Prospective.DOI 10.1007/978-88-470-2056-6_1, © Springer-Verlag Italia 2012

2 1 Introduction

Below is a brief history of VWLs, almost a summary of the book, from which itis clear that the various formulations that have occurred during approximately twothousand years, from Aristotle to Galileo, showed no appreciable progress. AfterGalileo, there was instead an abrupt change of direction and in a few generationsvery sophisticated formulations were reached.

1.1 Virtual velocity laws

The reconstruction of the historical development of the laws of virtual velocitiesis currently very incomplete. It goes back to Aristotelian Mechanica problemataof the fourth century BC [12], with the law: “heavy bodies located at the end of alever are equilibrated when, in their possible motion, velocities are in inverse ratioto weights”. Its explicit formulation, however, is documented only by Galileo whointroduced it especially in Le mecaniche [119] and in the Discorsi sulle cose chestanno in sù l’acqua [115]; in the latter memoir he associated explicitly the law ofvirtual velocities to Aristotle. The law of ‘virtual velocities’ of Aristotle’s schooltook the functioning of the lever as the main reference. Velocity did not have its cur-rent quantitative meaning, but was rather the concept of the common man for whichthere was no well-defined measurement, and at most a formulation of a criterion ofmore or less. Moreover, even force – regardless of its metaphysical uncertainty –was a somewhat indefinite quantity. It could be measured by weight, and then in-troduced into the calculations, but uncertainties still remained. Its direction was notwell defined, or rather was defined tacitly: the force applied, for example, to the endof a lever was implicitly considered orthogonal to it. The law of virtual velocities,although formulated on the basis of magnitudes not well quantified, led to correctresults already in the Aristotelian school. The ‘velocity’ of the points of a lever thatrotates around its fulcrum can be said to vary in proportion to the distance from itand this was enough to determine a quantitative relationship between the forces andthe distances from the fulcrum. The idea of a VWL arose from the motion of pointson a circle which rotates around its centre:

Remarkable things occur in accordance with nature, the cause of which is unknown, andothers occur contrary to nature, which are produced by skill, for the benefit of mankind.[...] It is strange that a heavy weight can be moved by a small force, and that, too, when agreater weight is involved. For the very same weight, which a man cannot move without alever, he quickly moves by applying the weight of the lever. Now the original cause of allsuch phenomena is the circle; and this is natural, for it is in no way strange that somethingremarkable should result from some thing more remarkable [12].1

In their analysis of the motion of the circle, Aristotle’s followers concluded that thepoints which tend to move more easily require less force than those which tend tomove less easily. If to ‘more or less easily’ is given the meaning of ‘more or lessforce’, then one obtains a trivial tautology, but if it is given the meaning of ‘more orless quickly’, then a form of VWL is obtained.

1 pp. 133, 135.

1.1 Virtual velocity laws 3

Nowhere in the Mechanica problemata did Aristotle use the word or the conceptof equilibrium. At most equilibrium can be seen in dynamical key as the result of thecancellation of effects of opposing forces. Effect measured on the basis of virtualmotion. The higher the virtual velocity of the point of application of a force thegreater the effect. The study of equilibrium on the basis of possible motions seemeda contradiction in terms for those who could not conceive, with Aristotle, rest asmotion in power. And there were many who did not share the ideas of Aristotle. Buteven if this metaphysical difficulty is ignored, the Aristotelian law of virtual workwas too ‘complex’ from a logical point of view to be assumed as a principle, i.e. ithad to be demonstrated.

According to the mathematicians of the time a demonstration had to be basedon the existing model of geometry and had to consist of a derivation from evidentpropositions. The intuitive Aristotelian considerations had no probative value. Forthis reason in ancient Greece, the law of the lever among scientists but also amongtechnicians, followed a different approach, based on the concept of centre of grav-ity. Unfortunately we have few documents relating to the mechanical studies ofGreek mathematicians posterior to Aristotle. There are essentially the basic texts ofArchimedes on hydrostatics and centres of gravity, and some studies on the balanceatributed to Euclid. The most complete witness of Greek mechanics is contained intheMechanica of Hero of Alexandria, which had an applicative character. Howeverit can be said that Greek mechanicians assumed as their main conceptual model thelever and the law which regulates its behaviour was proved with considerations ‘be-yond any doubts’ from principles, fixed by Archimedes (see Chapter 3) which arealso ‘beyond any doubts’. Here equilibrium is the key concept, while motion is notconsidered, except to deny it.

In the modern era Galileo was the first to assume a VWL with a dynamical con-notation where equilibrium resulted from cancellation of opposing trends. The namehe gave to these trends was ‘momento’ (moment), a term which remained long inthe history of mechanics:

Moment is the propension of descending, caused not so much by the Gravity of the move-able, as by the disposure which divers Grave Bodies have in relation to one another; bymeans of which Moment, we oft see a Body less Grave counterpoise another of greaterGravity Moment is the propension to go downward, caused not so much on by severity ofthe gravity of a mobile, but by the mutual disposition of the different heavy bodies, by themoment of which you will see many times a less heavy body counterbalance another moreheavy [119].2 (A.1.1)

Galileo was not able to combine disparate magnitudes, such as weight and velocity,and the idea of momento was expressed in the language of proportions that remainsat a somewhat imprecise level. In the study of the lever, shown in Le mecaniche,Galileo saw virtual motion as that motion generated by altering the equilibrium witha small weight. He then retained for it a certain degree of reality.

2 p. 159. Translation in [121].

4 1 Introduction

1.2 Virtual displacement laws

The idea of virtual displacement is in principle simpler than that of virtual velocity,because a displacement could be detected unambiguously even in antiquity. So itwas natural that in addition to the law of virtual velocities, also the law of virtualdisplacements had emerged. It had developed along two completely different paths.The first, which is generally the most emphasised, took the functioning of devices forlifting and shifting – the machines – as the main reference. The beginning is foundin the writings of Hero of Alexandria, but it is present more clearly as a generallaw of mechanics in Thabit’s Liber karastonis in the IX century and in Jordanus deNemore’s De ratione ponderis in the XIII century. Jordanus assumed the law thatmoving a weight p at height h is equivalent to moving a weight q = p/k up to hk,whatever k. The logical status of Jordanus’ virtual displacement law is still disputed:is it a principle or a theorem derived from the Aristotelian laws of motion? It hadhowever a general character and was used in various demonstrations. Important isthat of the inclined plane, which for the first time was referred to correctly. Note thatJordanus’ is a law of equivalence, or conservation, but not of equilibrium. To obtainequilibrium it is necessary to present an ad absurdum argument. The examination ofthe proof of the law of the lever, reported in the De ratione ponderis, shows the way(see Chapter 4). Consider a lever with two weights P andQ placed at distances p andq in inverse proportion to P and Q respectively. For the law of equivalence, weightP can be replaced by a weight equal to Q placed at a distance q from the fulcrumof the lever, on the same side of P, since by hypothesis the relation Pp = Qq holdstrue. What is obtained in this way is a lever with equal arms and equal weights and,as such, in equilibrium, thus satisfying the principle of sufficient reason. This meansthat the balance was in equilibrium even before the change of weight P with theweight Q.

The ideas of Jordanus found their natural successor in René Descartes, who fo-cused on the concept of what we now call work, which he called ‘force’:

The same force that can lift a weight, for example of 100 pounds to a height of two feet, canalso lift 200 pounds to a height of one foot, or 400 pounds to a height of 1/2 foot, and other[96].3 (A.1.2)

But there was a second source of the law of virtual displacement that put equilib-rium in the spotlight. This is Torricelli’s principle, according to which the centre ofgravity of a system of bodies in equilibrium cannot sink for any virtual displacementcompatible with constraints. It is a generalisation of the ancient empirical principlethat the center of gravity of a heavy body moves necessarily down when there are noobstacles that prohibit it. Torricelli’s principle was already formulated by Galileo:

Because, as it is impossible for a heavy body or a mixture of them to move naturally upward,moving away from the common centre towards which all heavy things converge, so it isimpossible that it spontaneously moves, if with this motion its own centre of gravity doesnot approach the common centre [emphasis added] [118].4 (A.1.3)

3 vol. 2, p. 435.4 p. 215.

1.3 Virtual work laws as principles of mechanics 5

And it can be traced back to medieval times, but only with Evangelista Torricellicould it take the form of a physical law expressed in the language of mathematics.Torricelli’s principle was originally formulated for only two bodies: “Two joinedbodies cannot move by themselves, if their common centre of gravity does not sink”,but its extension to more bodies is straightforward. It had two great advantages overthe other formulations of VWLs: it was ‘convincing’ for it appealed to everydayexperience – and therefore no particular objection can be taken to assume it as thebasis of statics – and could be easily generalised to a system of bodies.

Starting from Torricelli’s ideas, John Wallis reworded the principle of Torricelli,saying that the sum of the products of forces times displacements of their points ofapplication in the direction of forces must be equal to zero. According to VarignonWallis was the man “who went farther than any other authors [before Bernoulli]”[238].5

1.3 Virtual work laws as principles of mechanics

However Torricelli’s principle was not received enthusiastically and was basicallyignored by nearly all other mechanicians. A good number of scholars (includingPardies, Lamy, Rouhalt and Borelli), acknowledged the truth of the fact of the an-nulment of the virtual work of forces, but no one considered it possible to take this asa principle of statics because it was not self-evident, as the epistemology of time re-quired for a principle. Moreover the principle, although very general, in many casesfailed. It was successful for simple machines (lever, inclined plane, wedge, etc.), inwhich the directions of force and motion remain constant during virtual motion. Itfailed where this condition did not occur, such as the motion of a body on a curvedprofile.

René Descartes was the first to realise that, for the validity of any VWL, it wasnecessary to consider not the actual motion of bodies but that it would progressalong straight lines or planes tangent to the constraints that limit the motion, i.e.the motion at the very beginning. This observation generalised the approach alreadyused in statics by Galileo and Roberval, which replaced the existing constraints withequivalent ones. For example (Galileo), the inclined plane with a lever perpendicularto it. Besides this important technical improvement, Descartes claimed clearly therole of a principle of mechanics for a VWL in the formulation he gave it, that movinga weight p at height h is equivalent to moving a weight q = p/k up to hk. For himit was a sufficiently clear and distinct proposition and was also enough to solve allproblems of statics.

Descartes’ idea of virtual motion was generalized further by Christiaan Huygenswho introduced the concept of infinitesimal displacements in Torricelli’s principle.His early works on the subject date back to 1667 (see Chapter 7) and concern theequilibrium of three or more ropes at the ends of which forces are applied. The mem-oirs of Huygens, related only to special cases, however were not published while he

5 Preface.

6 1 Introduction

was alive and it is unclear whether they came, indirectly, to the notice of JohannBernoulli.

It was Johann Bernoulli who refined the wording of the VWL in a systematicway by introducing explicitly the concept of infinitesimal displacements. The lawhe formulated is today known as the principle of virtual velocities and is commonlyconsidered as the prototype of the contemporary formulations of VWLs. The newedition of Johann Bernoulli’s works, which also includes unpublished letters, pro-vides a starting point for an interpretation of the origins of VWL in Bernoulli, slightlylowering the aura of mystery that until now it has been wrapped. In February 1714Johann Bernoulli published Manoeuvre des vaisseaux (see Chapter 8), a book ded-icated to the theory of sailing. The preparation of this book was stimulated by thepublication of another book by Renau of Elissgaray, a marine engineer, on the sametopic and the discussion that followed on the composition of forces. Bernoulli hadrecently embraced Leibniz’s ideas of dead and living forces. He distinguished be-tween the impulsive forces (living forces) and the forces that act continuously (deadforces), like the wind that pushes on the sails of ships. And the forces that act con-tinuously are characterized by their energies, i.e. the product of the force by thecomponent of the virtual infinitesimal displacement in the direction of the force,named by Bernoulli virtual velocities. In the end Bernoulli, as indeed did Huygens,tended to consider virtual velocities and virtual displacements essentially the same,and finally to consider the term ‘virtual velocity’ as a synonym for ‘infinitesimaldisplacement’. This fact created a never-ending controversy because velocities andinfinitesimal displacements are not perfectly matched, and while velocity was a wellestablished and accepted concept in the XVIII century, the infinitesimal displace-ment remained shrouded in an aura of mystery.

The formulation of Bernoulli’s VWL is commonly associated with a letter ofBernoulli to Pierre Varignon in 1715. Paradoxically, this letter appeared in the Nou-vellemécanique ou statique of 1725, a bookwhich presented Varignon’s rule of com-position of forces as the fundamental principle of statics alternative to any VWLs.Bernoulli’s statement affirmed that for a system of forces that maintains a point, asurface, on a body in equilibrium, the sum of positive energies equals that of nega-tive energies, considered with their absolute value. Bernoulli was well aware of theimportance of his principle. In his letter to Varignon he wrote that the composition offorces is not but a small corollary of his principle. Varignon of course did not sharethis enthusiasm and did not record in his book this part of Bernoulli’s letter. ForVarignon, Bernoulli’s law is at most a theorem, to be proved case by case. Thoughhe did not give a general proof, he devoted a large part of his Nouvelle mécanique[238] to prove it in ‘all cases’ where, using the rule of the composition of forces, itis known there is equilibrium.

Bernoulli’s VWL was not immediately accepted as a possible principle of me-chanics. Bernoulli himself seemed to have changed his attitude and, in his writings,referred to it only once, in 1728 in the Discourse sur le lois de la communication dumouvement [35]. Here he introduced again the virtual velocity, but as the velocitythat each element of a body gains or loses, over the velocity already acquired, in aninfinitely small time, according to its direction (see Chapter 8). The above defini-

1.3 Virtual work laws as principles of mechanics 7

tion is not equivalent to that contained in the letter to Varignon, because it explicitlynamed a velocity rather than a displacement, which is in general the variation dv of agiven motion. This new point of view is due to the fact that now Bernoulli’s interestis motion of bodies and not just their equilibrium. No reference or comment is madeto his earlier definition of the virtual velocity, as if he had never written anythingabout it.

After Bernoulli, probably the most significant contribution to the development ofVWLs was due to Vincenzo Riccati who introduced the principle of action in theDialogo di Vincenzo Riccati della compagnia di Gesù of 1749 and the De’ principidella meccanica of 1772. Vincenzo Angiulli moved in the wake of Riccati withhis Discorso sugli equilibri of 1770. Although the idea of the principle of actionwas essentially Riccati’s, the less original Angiulli was closer to the foundationalaspects of mechanics. Angiulli tried to prove his VWL not from other mechanicalprinciples but from ‘indubitable’ metaphysical principles, including the equivalenceof cause and effect. He began with the Leibnizian concept of dead force, which ispresented as an infinitesimal pulse, such as f ds (where f is the intensity of the pulse,identified with the ordinary force, and ds the infinitesimal displacement of the pointwhere the force is applied) continually renewed by gravity or some other cause andcontinuously destroyed by constraints. In the absence of constraints, the pulses canbe accumulated and the action of the dead force is that of cumulative pulses; theaction of the dead force generates then the living force and therefore the motion.

With the introduction of infinitesimals Angiulli could enunciate his principle ofactions, which he qualifies as a theorem because it is demonstrated with his meta-physical considerations:

The equilibrium comes from the fact that the actions of the forces which must be equili-brated, if born, would be equal and opposite, and therefore the equality, and opposition ofthe actions of the forces is the actual cause of equilibrium.[…]The equilibrium is nothing but the impediment of the motions, that is of the effects of theforces, to which it is not surprising if the prevention of the causes, i.e. of the actions them-selves is reached [4].6 (A.1.4)

The principle of action implies the relation ∑ f ds = 0, where f ds are the elementaryactions emerging in the infinitesimal displacements ds, compatible with constraints.It is therefore a possible formulation of VWL. For Angiulli, the ontological statusof the constraints was that of ‘hard bodies’, i.e. idealised bodies that absorb all thepulses and the living force. Constraints obey an economy criterion, acting only asmuch as it is needed. In practice Angiulli made the assumption of smooth constraintswithout being aware of the problematic nature of the fact. Note that the constraintshave only the effect of destroying the motions and do not exert any reactive force,as this is a foreign concept to Leibniz’s mechanics.

Half a century after the letter of Bernoulli to Varignon, Lagrange gave the VWL amore efficient form. Officially, he referred to Bernoulli, but its role was actually verydifferent. When in 1764, for the first time, Lagrange exposed Bernoulli’s principle

6 pp. 16–17.

8 1 Introduction

of virtual velocities, he recast it by talking about the equilibrium of bodies and not offorces and applied it considering all the motions compatible with constraints and notonly rigid motions. He did not conceive the law of virtual work as a theorem, derivedfor example in the context of Newtonian mechanics; it was rather an alternativeprinciple. This position should be clear from the introductory part of theMécaniqueanalytique, published more than twenty years later, where he presented the variousways of addressing the problems of equilibrium of bodies: the lever, the rule of theparallelogram and the principle of virtual velocities.

The first edition of theMécanique analytique [145] of Lagrange, with the promi-nence it gave to his VWL, was the genesis of a wide debate on its logic status andwas also the occasion for a critical analysis of the principles of statics. It was anoccasion that, in the history of classical mechanics, has an equivalent only in thedebate at the beginning of the XVIII century on the principles of dynamics, and ofwhich today one no longer understands the significance. The list of scientists whobecame interested in the problem should make us reflect on the extent of the effortthat was made and the opportunity to learn a lot by following their teachings: LazareCarnot, Lagrange, Laplace, Poinsot, Fourier, Prony, Ampère and then also Cauchy,Gauss, Poisson and Ostrogradsky. Lobachewsky too was involved, but the contentof his contribution has been lost. To have again such heated discussion of scientistson the fundaments of mechanics it will be necessary await up to the introductionof relativistic mechanics, one century later. To understand the reasons of the debateone needs to reflect that, though the logical status of dynamics was undoubtedlycontroversial, there was generally agreement that one could give statics a sharedformulation. But Lagrangian VWL seemed to many to not meet the assumptions ofepistemology of the times. Although one could say – but not everyone agreed evenon that – the VWL was prior to all the laws of mechanics in the sense that these lawscould be derived from it, one could not admit it was evident; in particular it seemedless simple and evident for example of the law of the lever. Lagrange also agreedand, in the second edition of the Mécanique [148], wrote:

And in general I can say that all the general principles that can be discovered in the scienceof equilibrium, will not be but the same as the principle of virtual velocities considereddifferently, and from which it differs only in form. But this principle is not only itself verysimple and general, it has, in addition, the precious and unique advantage of being translatedinto a general formula that includes all the problems that can be posed on the equilibrium ofbodies. […] As to the nature of the principle of virtual velocities, it is not so obvious that itcan be claimed as a primitive principle [emphasis added] [148].7 (A.1.5)

Young Lagrange was attracted by the “precious and unique advantage of being trans-lated into a general formula that includes all the problems that can be put on the equi-librium of the body” and did not hesitate to take an instrumental position. There isno doubt that he was essentially a mathematician and, in line with the times, stronglyattracted by the formal aspects of Calculus. Although this position is subject to crit-icism, credit must be given to Lagrange for an originality that allowed him to goagainst the perhaps too rigid epistemology of the times. His attitude certainly con-

7 pp. 22–23.

1.4 Virtual work laws as theorems of mechanics 9

tributed to the development of the more liberal epistemology of the XX century, bornto a large extent with the advent of non-Euclidean geometry.

Lazare Carnot reached a VWL for colliding bodies according to his laws of im-pact in a mechanical theory, in principle, without forces. A fundamental conceptdeveloped by Carnot which influenced the subsequent debate, Poinsot’s included,is that of geometric motion, that is motion considered in itself independently of anyforce (see Chapter 11).

1.4 Virtual work laws as theorems of mechanics

Concluding, one could not accept the statement of the Lagrangian VWL as a princi-ple and had to prove it by reduction to a theorem of another approach to mechanics.This question promoted, as already mentioned, a heated debate, especially in Francewhere the main contributions were those of Lazare Carnot, Fourier, Ampère andPoinsot. For Italy it is worth noting the contribution of Vittorio Fossombroni. Thereasons for the attention paid to Lagrange’s VWL were not only scientific, however.It was no coincidence that the interest was polarised in France. Here the Cartesiantradition was still alive and national pride was still an obstacle to a full acceptanceof Newtonian physics and metaphysics. VWLs seemed to offer the opportunity todevelop a completely ‘continental’ mechanics, freed from the concept of force of aNewtonian matrix.

Joseph Fourier tried several demonstrations. In the probably most successful one,Lagrange’s VWL is reduced to the law of the lever, replacing weights with ac-tive forces that exert their action by threads, rings and levers. André Marie Am-père, following Carnot, introduced the concept of virtual velocity as a vector tan-gent to the trajectories compatible with the constraints, where time “has nothingto do with”. Vittorio Fossombroni in a memoir of 1794, demonstrated Lagrange’sVWL in the case of a free rigid body starting from the cardinal equations of statics.Of some interest is Fossombroni’s attempt to replace infinitesimal virtual displace-ments, which created some embarrassment, with finite displacements of arbitraryvalue. He showed that if the forces are parallel to each other and if their points ofapplication are arranged along a line, the virtual work of these forces is zero for anyfinite rigid motion. This idea was generalised, to the case of forces in the space withapplication points lying on a plane, by Poinsot who felt the same embarrassment inthe use of infinitesimal quantities.

Louis Poinsot gave in my opinion the most successful proof of a VWL. Sincehis mechanical theory was based on the rule of composition of forces and reducedto mathematical formulas, his proof was and is still considered by mathematiciansand physicists, more interesting than the more geometric Fourier’s type, based onthe law of the lever, and has become a model for almost all textbooks of statics.Poinsot accepted the principle that a material point subject to a certain active forceis equilibrated on a surface if and only if the force is orthogonal to it. In additionhe considered other principles, among which the principle of composition of forcesand the principle of solidification, according to which if one adds constraints – both

10 1 Introduction

internal and external – to a system of bodies in equilibrium, the equilibrium is notaltered. On the basis of his principles Poinsot was able to fully characterise staticsand write the equations of equilibrium in which only the constraint equations and thecomponents of the active forces applied to various points of this system appear. Todemonstrate his VWL, Poinsot gave up the virtual displacement concept, to adoptthat of virtual velocity – contemporary meaning. According to Poinsot real time andvirtual time run on different universes:

It must be noted further that the system is supposed to move in any way, without referenceto forces that tend to move it: the motion that you give is a simple change of position wherethe time has nothing to do at all [197].8 (A.1.6)

By replacing virtual displacements with virtual velocities, it is then easy to provea VWL in the form ∑ f dv = 0, where v are the virtual velocities of the points ofapplication of the active forces f . One cannot stress enough the fact that Poinsot’svirtual velocity is purely geometric and his virtual work is only a mathematical def-inition. Poinsot thus closed the circle that had opened with Aristotle. The laws ofvirtual work were initially manifested as laws of virtual velocities, then the laws hadsplit into virtual displacement and virtual velocity laws. With Bernoulli there was apartial but ambiguous reunification; Poinsot brought everything back to the baselineby eliminating the laws of virtual displacements.

1.5 Contemporary tendencies

But not everyone followed Poinsot in dealing with VWLs as theorems of mechanicsand considering the virtual work as a purely mathematical concept. It is possible toidentify a line of thought that instead of diminishing the mechanical meaning of thevirtual work tended to enhance it. This line of thinking had its precursors in Descartesand Leibniz. Then it became precise with Lazare Carnot, who introduced the con-cept of work of a ‘power’ along an arbitrary path, named by him moment of activity,giving it the meaning of a physical magnitude and a key place in mechanics. A fewyears after Lazare Carnot’s contribution, Gaspard Coriolis established definitely in1829 the term ‘work’ to indicate the Bernoullian energies. This change of terminol-ogy also implied a change of the ontological status. Also referring back to the ideasof Lazare Carnot, virtual work began to take on the role of well-defined mechanicalmagnitudes. Coriolis adopted the molecular model of matter, where everything isreduced to material points treated as centres of force. In this mechanics, there are noconstraints in the classical geometrical sense: there are ‘material’ constraints com-posed of material points carrying out repulsion actions against the particles that wishto penetrate them. Since there are no constraints, the infinitesimal displacements arenot subject to any limitation and can be identified with – and indeed they were – realmotions. So next to virtual work, there was room for ‘real work’. Coriolis addressedfor the first time the thorny problem of friction. While in a traditional formulationof VWL it was difficult to consider the reactions, without which it is impossible

8 p. 13, part II.

1.5 Contemporary tendencies 11

to introduce friction, for Coriolis there was no difficulty. Friction is represented bythe tangential components of the interaction between two bodies resulting from thesuperposition of forces exerted by the material points constituting the bodies. Thefollowing passage serves to illustrate the idea:

We are led to realize that the principle of virtual velocities in the equilibrium of a machine,composed of more bodies, cannot take place without considering first the sliding friction,where the virtual displacements cause the slipping of the bodies, one on others, and finallythat the rolling when bodies cannot take that virtual motion without deformation near thecontact points.Frictions are recognized always, for experience, able to maintain equilibrium in a certaindegree of inequality between the sum of the positive work and the sum of the negative work,here taking as negative the elements belonging to the smaller sum. It follows that the sumof the elements to which they give rise has precisely the value that can cancel the total sumand is equal to the small difference between the sum of the positive and negative elements[79].9 (A.1.7)

Parallel to the discussion on the concept of virtual or real work, a new science, ther-modynamics, was developing, where real work had a physical meaning in everyrespect.

Sadi Carnot put the work that he indicated with the term ‘engine power’ at thecentre of his Reflexions sur la puissance du feu of 1824 [62]. Work moved from ther-modynamics to mechanics with Rankine, Helmholtz and Duhem in the XIX century.In Duhem’s mechanics, a VWL came from the principle of conservation of energy,basically in its variational version. The connection of Duhem’s VWL to ‘real’ workor energy marked in some way a reconciliation with the principle of the impossi-bility of perpetual motion. Until then the two principles were kept strictly separate.Lagrange in particular, in his writings, never referred to the perpetual motion.

The role of VWLs in contemporary classical mechanics is not well defined, it ishowever not essential. In theoretical treatises on rational mechanics, which takes astrong axiomatic point of view, VWLs are often not even mentioned, even thoughthe axioms upon which mechanics is erected, such as Lagrange or Hamilton equa-tions, could be derived from them. In the applied mechanics of rigid bodies, VWLsare present but not important. They are used to solve some particular problems, inwhich for the presence of constraints it would be difficult to use other methods. Butwhen considering the mechanical theory as a whole, it is generally preferred to startfrom the cardinal equations. The constraints are taken into account by introducingauxiliary unknowns such as the reactive forces which are then removed in the so-lution of the single problems. There are no conceptual difficulties in dealing withconstraints friction; it is enough to provide the appropriate ‘constitutive’ relation-ships. In continuum mechanics the role of VWL is instead rather important. Butthis does not depend on its ability to address the various conditions of constraints,but rather on the mathematical expressions the virtual work law takes, that makes iteasier for approximate solutions in many cases, for example with the finite elementmethod.

9 p. 117.

12 1 Introduction

Table 1.1. Various versions of virtual work laws

Scholar Century Real mot. Geom. mot. Logic status Real work

Aristotle IV BC v pHero I d tThabit X d pJordanus XIII d d tGalileo XVI v pStevin XVI d tDal Monte XVI d tTorricelli XVII d pDescartes XVII d p •Wallis XVII d p ?Bernoulli XVIII d p •Riccati XVIII d p •Angiulli XVIII d p •Lagrange XVIII d p ?Fossombroni XVIII d tCarnot L XVIII v ?Fourier XIX d tAmpère XIX v tPoinsot XIX v tPiola XIX d ? ?Servois XIX d p •

In the table above the main characteristics of the various VWL formulations arereported. It is distinguished if the virtual motions are real or fully geometric, in thesense that if they run as the time of the forces or not, if displacement (d) or velocity(v) is concerned. The logic status is distinguished, i.e. the law is considered a prin-ciple (p) or a theorem (t) and if the virtual work is a physical magnitude (bullet) orinstead a pure mathematical expression.

1.6 Final remarks. The rational justification of virtual work laws

The history of the various forms that VWLs have taken also focused on attemptsthat have been made to give them a rational justification. The degree of satisfactionachieved was different from period to period. Together with a certain agreementthere was however always a tension towards overcoming the law, searching for amore powerful expression.

Aristotle seemed at first sight convincing enough to justify the law according towhich the efficacy of weights placed on the arms of a balance depends on their dis-tance from the fulcrum. He was considered persuasive by many mathematicians,but not by his contemporaries, accustomed to the high standards of rigor exempli-fied by Euclidean geometry. The justification of Jordanus de Nemore’s propositionthat what can lift p to h can also lift p/n to nh was criticized by his immediatesuccessors. It was perfected by Tartaglia, but Tartaglia’s arguments were the sub-

1.6 Final remarks. The rational justification of virtual work laws 13

ject of severe criticism by Archimedean mathematicians of 1500, in particular byGuidobaldo dal Monte. Galileo did not attempt any proof, but justified his law ofmoments on an intuitive level; the same approach was followed by Johann Bernoulliwith his rule of energy. Descartes considered self evident the simple virtual worklaw as expounded by Jordanus de Nemore. Riccati and Angiulli tried an externaljustification of Bernoulli’s rule of energy, from ‘certain’ metaphysical principles,such as the equality between causes and effects; principles not accepted by mostmathematicians. Lagrange at the beginning considered as evident a virtual work lawsubstantially coincident with Bernoulli’s rule of energy, which he called the princi-ple of virtual velocities. Then he presented a very simple and elegant justification,which however did not meet completely the standards of rigor of the times accordingto which any recourse to geometric intuition was not allowed, that instead Lagrangehad introduced, albeit without the use of figures. Fourier, Ampère, Laplace, Poinsotand many other scientists attempted to reduce their laws of virtual work, substan-tially coincident with the principle of virtual velocities of Bernoulli and Lagrange,to the elementary principles of statics, essentially the law of the lever and the rule ofthe parallelogram of forces. These attempts were followed by others who felt themas not entirely satisfactory.

From the above, must it be concluded that the laws of virtual work have neverbeen rationally justified?Orwith a term that has amore restricted, but strongermean-ing, have they never been proved? The answer is not simple. To understand why justrecall that the other fundamental laws of mechanics such as the law of the lever andthe rule of the parallelogram of forces followed the same fate. Many explanationswere proposed but always something was found to complain about.

Even contemporary scholars have dedicated themselves to attempt to justifyVWLs, albeit with less passion and strength [283]. The problem has a bit shiftedand transformed itself into the question: Is the mechanics L – statics and dynam-ics – resulting from the adoption as a founding principle the most advanced law ofvirtual work equivalent to the mechanics N resulting from the adoption of the mostadvanced version of Newtonian mechanics? The problem is a little bit easier thanthat to justify a VWL because the acceptance of various principles assumed for theadvanced version of Newtonian mechanics can be object of a less strict scrutiny.

However the problem absorbs more philosophers of science than specialists ofclassical mechanics. To the latter the problem seems not difficult to solve and in apositive way, following the reasoning of Chapter 2 of the present text. On the otherhand, Poinsot in hisMémoire sur la théorie générale de l’équilibre et du mouvementsdes systèmes of 1806 had given the problem a fully satisfactory response accordingto the modern standards of evidence, by proving the equivalence between a VWLand a Newtonian mechanics enlarged with a series of principles to take into accountthe presence of constraints.

Concluding from the historical path and also from a modern logical analysis itcan be concluded that the laws of virtual work have been considered justified in afairly satisfactory way in the past and today, no less than many other fundamentallaws of mathematical-physics.

2

Logic status of virtual work laws

I use the term law of virtual work to mean any rule of equilibrium that includes bothforces and possible displacements of their points of application. In this chapter I willgive a more restricted meaning by referring only to the most (modern) advanced for-mulations and I will consider their logical status, i.e. whether they are principles of anautonomous mechanics or theorems of ‘another’ mechanics. To demonstrate a law,a proposition, means in the broadest sense, to bring it to laws ‘assumed as known’,with a sense that varies with the epistemological frame of reference. Until the de-velopment of modern axiomatic theories and their application to physical science bythe neopositivists of the XX century, a principle was considered as acceptable if ithad an intuitive nature of evidence, possibly established a priori. Currently there is amore liberal view and one does not require evidence of the principles, only that theymust have sufficient strength and do not lead to logical contradictions.

The problem of provability of a VWL clashes immediately with the fact that eventoday there does not exist a reference theory of mechanics that is fully defined anduniversally accepted. This is true even for systems of material points, although theredo exist some axiomatizations [382, 360, 390]. Amajor difficulty encountered in var-ious formulations of VWL and mechanics concerns the ontological status attributedto constraints and reactive forces. Before the XVIII century, constraints had beentreated only as passive elements not able to act. Only after studying elasticity andaccepting models of matter based on particles considered as centres of forces, haveresearchers begun to think of constraints as capable of administering active forces.In dynamics, according to Lagrange, the first scholars to assimilate constraint reac-tion to active forces were the Bernoullis, Clairaut and Euler, in the period 1736 to1742. “The use of these forces dispensed from taking account of the constraints andallows one to make use of the laws of motion of free bodies” [145].1 In statics, con-straint reactions are less problematic; they can be considered as the forces necessaryto maintain the constraint. The first to introduce them in calculations was probablyVarignon in his Nouvelle mécanique ou statique of 1725 [238].

1 p. 178–179.


16 2 Logic status of virtual work laws

It was simply the difficulty of incorporating the reactive forces in a consistent me-chanical theory that led Johann Bernoulli to the formulation of an effective law ofvirtual work, known after Lagrange as the principle of virtual velocities, which pro-vides a criterion of equilibrium without intervention of these undesirable ‘beings’.But his statement that "the sum of power each multiplied by the distance traveledfrom the point where they are applied, in the direction of this power will always bezero", always questioned its nature; in this regard the following comment by Fos-sombroni is of interest:

That common faculty of primitive intuition, so everyone is easily convinced by a simpleaxiom of geometry, as for example, that the whole is greater than the part, certainly doesnot need to agree on the aforementioned mechanical truth, which is much more complicatedthan that one of the common axioms, as the genius of the great Men who have admitted theaxiom, exceeds the ordinary measure of human intelligence, and it is therefore necessaryfor those who are not satisfied to obtain a proof resting on foreign theories […] or to rest onthe faith of chief men despising the usual reluctance to introduce the weight of authority inMathematics [109].2 (A.2.1)

The success of Lagrange’s Mécanique analytique [145] which assumed Bernoulli’sprinciple of virtual velocities, suitably reformulated, as the source of all mechanics,opened a heated debate on its plausibility. In this chapter I do not refer to these ef-forts, nor to the preceding others, but try mainly to clarify in what sense one canprove a law of virtual work, be it Lagrange’s or otherwise. Attempts to demonstratecan be divided into two categories. In the first, which I refer to as foundational, onetries to deduce a VWL without reference to existing criteria of equilibrium; in thesecond category, which I refer to as reductionist, one tries to deduce a VWL by apre-existing criterion of equilibrium of a pre-existing mechanics. It should be saidhowever that there is a certain arbitrariness in this dichotomy, because any reduc-tionistic attempt can be reformulated as a foundational one, as will be clear in thefollowing.

Attempts were made in the first direction by Vincenzo Riccati and VincenzoAngiulli, Johann Bernoulli, Lazare Carnot and Lagrange himself. The first twothought they could demonstrate the law of virtual work with metaphysical consid-erations, using a reference mechanics of Leibnizian type but without a pre-existingcriterion of equilibrium. Carnot tried to reach a VWL starting from the law of impactusing a mechanics of reference in principle without force; Lagrange made use of thelaw of the pulley. Attempts in the second direction were made by French scientistsof the École polytechnique. Summing up and using the categories of Lagrange’smechanics, they assumed as reference mechanics those derived from the law of thelever and the rule of the parallelogram. The demonstration of Poinsot was the onethat most influenced subsequent treatises of statics.

In the recent scientific literature, the problem of the logic status of VWL is onlyaddressed in the manuals of statics, where the author gives his idea in a few pageson the subject, usually referring to a limited number of ‘basic versions’ [283]. Giventhe predominantly teaching character of the manuals, problematic aspects tend to behidden to provide greater certainty. To my knowledge, there are no recent theoretical

2 pp. 13–14.

2.1 The theorem of virtual work 17

works on basic aspects of VWLs, as there are no recent theoretical works on thefoundation of the mechanics of a material point, even if the argument is far fromexhausted. It seems further that there are only a few recent studies on the historyof VWL [197], and that it is treated marginally in the numerous monographs onLagrange, Laplace, etc.

In this chapter I will try to highlight the logical status of VWLs focusing mainlyon the reductionist approach that is more widespread. Just because there is no gen-erally accepted formulation of classical mechanics I will not consider the situationin its generality and assume only what is more consolidated, in particular I will as-sume a mechanics of material points and forces applied to them – i.e. corpuscularmechanics. The wording of the VWL will emerge in a natural way along the lineof least resistance, avoiding inessential complications. I will simplify the constraintconditions, limiting myself to dealing with holonomic, bilateral and independent-of-time constraints, that can be represented mathematically by an algebraic equationonly of the position variables, because I think they equally capture the essence ofthe problem and an extension to more general constraints is possible involving onlytechnical complications. The reductionist approach to VWLs examined, assumingan already given mechanics, presupposes the concept of force. However, it is possi-ble to tackle the problem from a different point of view, in which it is not necessaryto posit the concept of force, giving as a primitive the concept of work. This viewwill be discussed briefly at the end of the chapter. According to this approach also,the VWL will be stated as a principle or deduced from the most fundamental laws,always related to the concept of work, which will now be virtual in a different way.

2.1 The theorem of virtual work

A constrained system S of a finite number n of material points is a system the con-figuration of which is defined by a number m of degrees of freedom less than the 3nthat would be needed to describe the configuration of the system as supposedly free.I will indicate with M the space, or better the manifold, of the possible configura-tions of dimensionm for S and with N the space of configurations of dimension 3n inthe absence of constraints. Each space of configuration M and N is associated witha space of tangent vectors indicated below with MT and with NT respectively. Forexample, for a material point P constrained to move on a surface, N has dimension 3and M has dimension 2. The vector space NT is the space associated with the ordi-nary three-dimensional vector space; the space MT is the set of vectors that lie on thetangent plane to the surface in the position occupied by P. In the case of two materialpoints, constrained to keep a constant distance, the space of all configurations N hasdimension 6, corresponding to the 6 degrees of freedom of two free material pointsin three dimensional space. The space M of compatible configurations has insteaddimension 5, because the degrees of freedom are reduced by a constraint equationwhich expresses the invariance of the distance between the two points. The vectorspace NT consists of pairs of vectors representing the displacements of the twomate-rial points which can be any, and the space MT is represented by the pairs of vectors


MT

MC

virtual displacement

possible displacement

Fig. 2.1. Tangent manifold

that have the same component in the line joining the two material points. The spaceMT of vectors u tangent to M is called the tangent manifold of M and vectors u arecalled virtual displacements. It is clear from the definition that the virtual displace-ments in general are not possible displacements, which are motions taking place onM. The difference between possible and virtual displacements is shown in Fig. 2.1.The virtual displacements coincide with the possible motion only for infinitesimalvalues. If it is considered that the virtual displacements occur in the direction tan-gent to the constraints and that the possible velocities are tangent to the constraints,it is instead possible to identify the virtual displacements with possible velocities,considering time as an arbitrary parameter.

There are essentially two ways to study the equilibrium of a constrained system.In the first way it is assumed that there are known external forces, named activeforces, and forces due to the constraints, named reactive forces or constraint reac-tions, the presence of which should be inferred indirectly from the empirical evi-dence that motions of the material points of a constrained systems are different fromthose registered without constraints. The value of constraint reactions is not given,depending on the geometry of constraints and the active forces. In the second modethere are only active forces while the constraints are characterized exclusively bytheir geometry; in this paragraph I will examine the first situation.

On the system S of material points there are active forces fi, with a given lawof variation in time and space and reactive forces ri, associated to the constraints, apriori unknowns. Collecting the active forces in the vector f and the reactive forcesin the vector r, assume the following principle of equilibrium:

P1. A system of material points constrained to a manifold M starting withzero velocity, is in equilibrium in a given configuration C if and only if thefollowing relation is satisfied at any time:

f + r = 0. (2.1)

Notice that the sufficient part of the principle (i.e. if f + r = 0 then there is equilib-rium) calls for this other principle:

P∗1. If constraints can furnish reactive forces r such that f + r = 0 then theyactually furnish them.


MT

MC

MH

ΠH ( f )

ΠT ( f )

f

Fig. 2.2. Orthogonal projections

In the following, in all considerations relating to equilibrium, I will assume implicitlya certain configuration C and any instant of time while the initial condition is rest.

Denote by ΠT the projection operator from NT to MT, and by ΠH the projectionoperator from NT to MH, the complementary vector space of NT orthogonal to MT .Then equation (2.1) is equivalent to the two relations:

ΠT ( f + r) = ΠT ( f )+ΠT (r) = 0ΠH( f + r) = ΠH( f )+ΠH(r) = 0.

(2.2)

Fig. 2.2 clarifies the meaning of (2.2) on a two-dimensional space. The space ofadmissible configurations is defined by the curve M, the tangent space MT is the linetangent to M in C – the position occupied by the material point P – the orthogonalspace is the line MH orthogonal in C to MT.

Define now the virtual work of forces acting on S as the linear form on NT :L(u) = ( f + r) · u, where u is a vector of NT and dot denotes the inner, or scalar,product. Then consider the two other linear forms La(u) = f · u and Lr(u) = r · u,respectively called virtual work of active forces and reactive forces. Note that thevirtual work coincides with the classical definition of work but it refers to a virtualdisplacement and not to a possible displacement. If the virtual displacements areidentified with velocities, then the virtual work has the mechanical significance ofpower. The following theorem of virtual work can easily be proved:

T1. A system of material points on a manifold M is equilibrated if and only ifLf (u)+Lr(u) = 0 for any u in NT .

Indeed Lf (u)+Lr(u) = ( f +r) ·u= 0 ∀u∈NT ↔ f +r = 0, for the same definitionof scalar product.

To check the balance, with theorem T1, it is necessary to specify the manner inwhich the reactive forces vary on the manifold M. A traditional way to characterizethe reactive forces is to introduce the concept of smooth constraint, which can beexpressed as:

D1*. A system of constraints associated to a manifold M and a system of ma-terial points S is smooth if and only if it is able to furnish reactive forces r suchthat ΠT (r) = 0.


That is in a system of smooth constraints the reactive forces belong to the space MH

orthogonal to the tangent space MT in C. This mean that if ‖ΠT ( f )‖> 0, i.e. if thereis at least a force fi that has a non-zero component in the direction of the displacementui allowed by constraints, the equilibrium is not possible and the system moves,however small is the force fi. This corresponds to the intuitive concept of smoothconstraints as constraints without friction.

The characterization of smooth constraint can also be given, equivalently, refer-ring to the linear form Lr(u), reaching the definition:

D1. A system of constraints associated to a manifold M and a system of mate-rial points S is smooth if and only if Lr(u) = 0 for any u in MT .

Or, alternatively, in a less formal way, using the definitions of virtual displacementand work:

D1. A system of constraints associated to a manifold M and a system of mate-rial points S is smooth if and only if the virtual work of reactive forces is zerofor any virtual displacement.

Usually the characterization of smooth constraints assumes only the condition thatr belongs to MH . But it is equally important to stress that constraints are able toexercise all the forces belonging to MH regardless of their intensity. So if constraintsare smooth their reactions could be any values in a known direction, and it is possibleto apply the criteria of balance P1 or T1, to state the two theorems:

T∗2. If the constraints are smooth, a system of material points on the manifoldM is equilibrated if and only if ΠT ( f ) = 0.T2. If the constraints are smooth, a system of material points on the manifoldM is equilibrated if and only if Lf (u) = 0 for any u in MT .

Proof of T∗2 is simple and is implicitly contained in equations (2.2). Necessary part:if a system of material points is in equilibrium for P1 it is f + r = 0, then equa-tions (2.2) hold, and from the first of them, because constraints are smooth andΠT (r) = 0, it is ΠT ( f ) = 0. Sufficient part: assume ΠT ( f ) = 0, because for smoothconstraints ΠT (r) = 0, the first relation of (2.2) is satisfied. The second relationΠH( f )+ΠH(r) = ΠH( f )+r = 0, is also satisfied because the constraints (smooth),by definition, can provide all the reactions orthogonal to MT , and therefore alsor = −ΠH( f ). It follows that f + r = 0, and then the system of material points is inequilibrium. The demonstration of T2 immediately follows from T∗2 for the proper-ties of scalar product.

It is worth noting that in the case of constraints that are not smooth, to check theequilibriummay not be easy. As an example consider the material point of Fig. 2.3 inwhich there is also a tangential component of the constraint reaction, due to friction.If the point is in equilibrium it is certainly f = −r, but for an arbitrary value of fit is not said that there will be equilibrium because for example the friction is notenough and the constraint is not able to provide r =− f .

T2 is a theorem of virtual work as is T1; although commonly only T2 is calledtheorem of virtual work. It would thus appear to have solved the problem of the logic


ΠH ( r )

ΠT ( r )

f

r

Fig. 2.3. Not smooth constraint

status of the law of virtual work, at least as formulated above: if properly formulated,it is a theorem of statics. Unfortunately, such a conviction is no longer anything butan illusion, disguised in the words with which the concept of smooth constraintshas been introduced. In fact, it is given only a definition but it does not provideany ‘decisions’ criterion and the definition leads to circularity: if the constraint issmooth, reactive forces are orthogonal to virtual motion and if the reactive forcesare orthogonal to virtual motion then the constraint is smooth.

To justify the usefulness of the theorem of virtual work, and then the opportunityof referring to T2 as a VWL, an operating criterion is necessary to determine inadvance whether a constraint is smooth or not, and this criterion cannot exist becausethe constraints are usually defined analytically only by the variety M and are notobservable, i.e. they are not entities on which to have a priori reasoning. The onlyway to use T2 (and T∗2) it seems is to assume the following principle:

P2. All constraints are smooth.

Then from P2, by applying modus ponens to T∗2 and T2, two theorems are obtained:

T∗3. A system of material points on the manifold M is equilibrated if and onlyif ΠT ( f ) = 0.

T3. A system of material points on the manifold M is equilibrated if and onlyif Lf (u) = 0 for any u in MT .

Today theorem T3 is usually called principle of virtual work; for historical reasonseven here it is a theorem. It derives from a principle of the mechanics of materialpoints (P1) and a principle (P2) that seems external to it. Given the critical role ofP2 in the proof of T3 it itself is often called the principle of virtual work. In thefollowing I will not accept this use and with the term virtual work principle I alwaysrefer to T3.

T3 may be a theorem of the reference mechanics only if P2 holds good. It isthen clear that the problem of provability of the virtual work principle is closelyrelated to the problem of the characterization of constraints and, ultimately, of thereference mechanics, so that it be complete. If in the reference mechanics there are


no assumptions about the constraints it does not make sense to think seriously aboutthe provability of the virtual work principle.

Is it possible to say anything more about the constraints within a reference par-ticle mechanics in which ‘solids’ are assumed to consist of material points whichact as centres of forces that underlie the cohesion? Rather than the locus of pointsexpressed by an algebraic equation a constraint can be associated, and it normallyis, with a body sufficiently ‘hard’ to be considered impenetrable. When a particleapproaches the body that acts as a constraint, forces awake – the reactive forces –which are opposed to opening up the parts of that body. Knowing the laws of forcesas functions of distance of the centres, the laws of interaction between the bodyand the particle could be determined, at least in principle. In practice this is notpossible and recourse to an approximate description is necessary with an empiricalcharacter, in the broadest sense, which will provide the necessary characterizationof constraints.

In this way there would be no problem to determine whether a particular con-straint system is smooth or not on the basis of its constitutive relationships and defi-nitions D1 andD∗1 and the theorem of virtual work T2 then wouldmake sense becausethere is an operational criterion to be applied case by case to decide on the basis ofempirical observations if T2 can be applied or not. Note however that in mechanics,in fact, one tends to apply the principle of virtual work T3; generally, the assumptionof smooth constraints is not object to scrutiny because it is not practically possibleto do so.

Assuming that the constraints are formed of bodies, in the past it was thought, andsometimes it is still thought, to prove the principle P2 and the theorem T3 showingfirst a seemingly weaker assumption, namely that:

P3. All the surfaces of the material bodies are smooth constraints for materialpoints.

It is clear that this principle expresses ideals; in practice constraints are never smoothand there are horizontal forces, or friction. Ignoring this fact and accepting the idealnature of P3, is it possible to accept it? It seems doubtful that criteria of symmetry andsufficient reason – in the sense that the reaction forces must be ‘always’ orthogonalbecause there is no reason that they are not – can be applied in the particle model inwhich the very concept of the surface of a body presents difficulties.

But even by accepting P3, P2 cannot be proved without any other assumption. Infact, P3 does not say anything about the internal constraints between material points,where the attribute smooth appears unintuitive. So, if the reference mechanics isnot changed, even assuming P3, P2 is not certified and therefore the virtual workprinciple is not a theorem.

Before concluding this section I would like to briefly refer to the extension ofVWLs to dynamics, extension that was made for the first time by Lagrange in 1763(see Chapter 10) [142]. Though VWLs can be extended quite easily to dynamics,of course they will no longer provide a criterion for equilibrium, but a criterion ofbalance that leads to the equations of motion. Including among forces also the forcesof inertia equal to −ma, the theorem takes the form T4:


T4. For smooth constraints a motion of a system of material points movingon the manifold M is such that the virtual work of all forces, the inertial onesincluded, is zero for any virtual displacement.

The proof is immediate, because the equation of motion can be written as −ma+f + r = 0 and by assuming I =−ma in the form (I+ f )+ r = 0, the same assumedin the proof of theorem T2.

2.1.1 Proofs of the virtual work theorems in the literature

In the technical and teaching literature the provability of the virtual work theorems,T1, T2 or T3 is addressed some ways differently than the one reported above. Thedifference in presentation depends on the audience to which it refers. There are the‘proofs’ described in treatises of physics and mechanics, those of specific texts ofstatics and those of texts of continuum mechanics. For the last area, the virtual workprinciple is usually presented for systems that are either unconstrained or subject toconstraints that require simply the vanishing of the displacements of certain points,so it becomes a theorem which can be easily derived from the principles of contin-uum mechanics, generally described by partial differential equations. More difficultand interesting is the approach in the other two types of texts.

2.1.1.1 Physics and rational mechanics treatises

As regards the presentation of the classic texts onmechanics the work by Capriglioneand Drago [283, 301] which I sum up briefly, seems conclusive. The goal is usuallyto demonstrate the virtual work principle, the hypothesis of smooth constraints isimplicit and there is always the consciousness of the author of the manual that he isproving the theorem of virtual work in the form T2 but not the virtual work principleT3. It can be said that although there is no complete agreement on how to definethe virtual displacement, in most cases the infinitesimal displacement du instead ofvelocity are adopted, but the ‘degree of virtuality’ of this displacement is not alwaysclear. This problem did not appear at the beginning of the present chapter whereit was assumed that the motions have only a virtual geometric characterization, inwhich time does not intervene. The concept of virtual displacement has, however,developed historically with reference to a magnitude that evolves over time; thevirtual velocities are obtained by the derivatives of displacements with respect totime. Traces of this historical development have remained in the demonstrations inphysics textbooks. Sometimes there is distinction between the time with which theforces change – referred to as the ‘real time’ – and the time with which the motionvaries – referred to as the ‘virtual time’ – flowing independently of each other. Inthis case it may be that the real time is frozen and only the virtual time flows; virtualdisplacements have in this case only a purely geometric characterization, consistentwith the presentation in § 2.1.

The proof of the necessary part of the virtual work principle, that is if a system isin equilibrium then the virtual work of active forces is zero, takes place essentiallyas presented in the previous pages where the criterion of equilibrium was provided


by the annulment of the forces acting on individual material points. If the system isin equilibrium, then the balance of forces f + r = 0 subsists. Multiplying both sidesby the virtual motion du, it is:

L = ( f + r) ·du = 0. (2.3)

If the constraints are smooth r · du = 0 = 0, so (2.3) provides the necessary part ofthe virtual work principle, L = Lf = f ·du = 0.

The proof of the sufficient part, i.e. the virtual work of active forces is alwayszero, then the system is in equilibrium, is usually treated in a manner substantiallydifferent from what has been done in previous paragraphs:

The proof is by reduction to the absurd. It is assumed that despite being valid (*) [Lf = 0],the system is put in motion, namely that at least one of its points, say the i-th, is affected,in the time dt subsequent to t, by a displacement dri, compatible with constraints. Since thematerial point under consideration starts from rest, it is necessarily: Fidri > 0, then the sumof all partial work relating to other parts of the system that actually moves, it is also:

∑Fidri > 0, (1)

since the sum is made up entirely of non-negative terms and at least one of them, by as-sumption, is not null.But Fi = F(a)

i +Ri, for which we rewrite (1) as:

∑(F(a)i +Ri) ·dri > 0. (2)

At this point one makes the assumption of smooth constraints and the absurd is obtained∑F(a)

i ·dri > 0 [Lf > 0], because against the hypothesis [283].3 (A.2.2)

The demonstration is taking place assuming that true motions exist, then consideringthese as virtual motions and assuming that the constraints are smooth. Rather thanto show that Lf = 0 for all virtual displacement is equivalent to f + r = 0, and thenan existing equilibrium criterion is fulfilled, it is shown that to admit the motionis in contradiction with Lf = 0 for all virtual displacements, using an argument ofdynamic type.

The asymmetry between the demonstration of the necessary and sufficient condi-tion is not convincing for me. Besides, the use of virtual displacements taking placein real time does not permit the direct extension of the proof to the case of time-dependent constraints and the case of an impulsive force. This form of proof of thesufficient part of the virtual work principle was probably introduced for the first timeby Poisson in his Traité de mécanique of 1833 [200]. The central point of the prooflies in taking the dynamic assumption that the resultant forces Fi and displacementsdri, which are generated by the absurd, necessarily have the same sense, and thisassumption is not at all obvious, as will be explained further in Chapter 16.

2.1.1.2 Statics handbooks

Generally theorem T1 or at most the necessary part of T2, is proved, which is suffi-cient for applications. Instead of a system of particles, reference is made to a system

3 pp. 331–348.


of rigid bodies connected to each other and with the outside by a system of rigidhinged rods, thus tacitly admitting that any kind of constraint can be reproduced byan appropriate system of rods. Normally the following additional assumptions aremade:

1) constraints (connecting rods) exert forces Rh (reactive forces) that have the sameontological status of the active forces Fk;

2) the direction of the reactive forces Rh is that of the rods;3) for each rigid body the equilibrium is defined by the satisfaction of the cardinal

equations of statics between Rh and Fk;4) an infinitesimal displacement field is assumed, i.e. virtual displacements are co-

incident with virtual velocities (unless an inessential constant);5) all forces and displacements are independent of time.

Assumption 2 is a principle analogous to P2 because the rod imposes the point intouch with a body to move on a sphere and then the reactive force, being collinearto the rod, is orthogonal to the surface.

The proof below differs a little from those normally presented in statics hand-books [285], because it avoids any recourse to matrix calculus which, making theproof automatic, the proof hides the nature of assumptions, implicit and explicit.

As first, at least in many handbooks, it is shown that the work of a system of forcesapplied to a rigid body can be estimated from the resultants F and static momentMO – about a point O fixed – of forces, in the form:

L = FuO +MOθ, (2.4)

where uO is the virtual displacement of a reference point O and θ the rigid bodyrotation, according to Fig. 2.4.

Having proved (2.4), indicating respectively with Fa and MaO the resultant of

forces and moments of active forces and with Fr and MrO the corresponding quanti-

ties of reactive forces, it is easily shown that there is equilibrium for the body if andonly if the relation holds true:

L = (Fa +Fr)uO +(MaO +Mr

O)θ= 0, (2.5)

O

F x

y

x

y

MO

O

uO

θ

x

Fig. 2.4. Force and virtual displacement for a rigid body


for any virtual displacement uO,θ, i.e. that a generalization of theorem T1 is true, ascan be proved into two steps:

a) the condition L= 0 is necessary for the equilibrium. Indeed if there is equilibriumthe cardinal equations of statics are satisfied:

Fa +Fr = 0

MaO +Mr

O = 0(2.6)

and then L = 0;b) the condition L= 0 is sufficient for the equilibrium. Indeed if L= 0 for any values

of uO,θ, from (2.5) it is easy to see that the cardinal equation of statics (2.6) aresatisfied.

In the case of smooth constraints and inextensible rods, the virtual work of thereactive forces is zero and from (2.5) and (2.6), verified at equilibrium, one can derivethe necessary part of the theorem T2:

L = FauO +MaOθ= 0. (2.7)

Usually nothing is said on the demonstration of the sufficient part of the theorem T2,which is a bit more complicated.

2.1.1.3 Poinsot’s proof

The examination of Poinsot’s proof, which will be discussed in more detail in Chap-ter 14, has presently a considerable interest because it influencedmuch of the demon-strations of statics handbooks and is considered as the best one ever given. The rea-sons for this success are essentially two: (a) Poinsot considers a reference mechanicswhere the equilibrium is determined by the balance of forces which can be expressedby simple analytical relations – equations of equilibrium of a particle – and (b), pri-marily, he uses virtual velocities, obtained considering a non-physical time, insteadof virtual infinitesimal displacements.

Poinsot takes for granted the assumption P3, or more precisely its modified ver-sion, that does not require the concept of constraint reaction, whereby a particle isin equilibrium on a surface if and only if the applied force is perpendicular to it. Inthe words of Poinsot:

Indeed, it is shown that if a point has no freedom in space other than to move on a fixedsurface or line, there may not be equilibrium unless the resultant of forces which press it isperpendicular to the surface or the curved line [197].4 (A.2.3)

In fact he does not prove what he says and does not even have the opportunity todo so because his constraints have the ontological status of relations between thepositions of points and are not bodies. Poinsot is not the only one to think it logicallynecessary that a constraint cannot resist tangential forces. Laplace is also convincedof this:

4 p. 467.

2.2 The principle of virtual work 27

The force of pressure of a point on a surface perpendicular to it, could be divided into two,one perpendicular to the surface, which would be destroyed by it, the other parallel to thesurface and under which the point would have no action on this surface, which is against thesupposition [156].5 (A.2.4)

The reasoning is not conclusive, in fact it reduces to the trivial tautology that theconstraint is not acting in a tangential direction because it does not act in the tangentdirection. Lagrange also expressed similar ideas:

Now if one ignores the force P, and assuming that the body is forced to move on this surface,it is clear that the action or rather the resistance that the surface opposes to the body cannotact but in a direction perpendicular to the surface [149]. 6 (A.2.5)

But he seems to realize the problematic nature of the concept of smooth constraintbecause, often, in his writings he associates the constraints, expressed by mathemati-cal relations, to constraints made of inextensible wires deprived of bending stiffness.In this case the orthogonality of the reaction to the surface – for example the spher-ical surface described by a material point with a wire connected to a fixed point –is more convincing, even if this evidence has in fact an empirical rather than logicbasis, making reference to our everyday experiences.

In addition to P3, Poinsot considers other principles. The first principle is that ofsolidification, for which if one adds constraints – both internal and external – to asystem of bodies in equilibrium, the equilibrium is not altered. The second principle,presented by Poinsot as the fundamental property of the equilibrium, states that nec-essary condition for the equilibrium of an isolated system of particles is that all theforces applied at various points can be reduced to any number of pairs of equal andopposite forces. The third principle is required by the second, even if not explicitly,and concerns the possibility to decompose a force into other forces using the ruleof the parallelogram. A fourth principle concerns the mechanical superposition forconstraints, for which if on a system of points there act more constraints, they areable to absorb the sum of the forces that each constraint is able to absorb separately.Based on these principles, he quite convincingly proves the principle P2, with ref-erence to the definition D∗1; more precisely he proves that if L = 0, M = 0,etc. arethe relations among the coordinates x,y,z, x′,y′,z′, etc., which define constraints, thereactive forces are orthogonal to the resulting constraints. The demonstration of thevirtual work principle follows the same reasoning of the first part of § 2.1 and in myopinion is free from any criticism.

2.2 The principle of virtual work

The term principle has a meaning not entirely unique, as is the case with all im-portant concepts. It is the foundation of a science, which in turn may allow morethan one set of principles. Among the principles, even at the time of Lagrange, there

5 vol. 1, p. 9.6 vol. 9, p. 378.


were axioms, principles and theorems otherwise proved. Today this distinction nolonger holds and there is the tendency to group all the principles under the term ax-ioms. Compared to the views of the XIX century, essentially Aristotelian, now thepremises, i.e. the explicans of the theories, are neither required to be certain nor thepremises of the conclusions, i.e. the explicandum, to be better known. The first re-quirement is ignored because it would exclude almost all scientific laws, for whichinstead of a certain knowledge plausible conjectures are considered. The second re-quirement is usually ignored, and there are theories – such as atomic and quantum,for example – that explain relatively well-known phenomena with beings of inde-cipherable nature. Moreover, the boundary between what seems known and whatdoes not, is largely the result of metaphysical and epistemological conceptions ofthe times and does not reside only in the object. With the use, things that were notknown or obvious, become of public domain, an example of this are the concepts offorce, atom, energy that sparked diffuse controversy when they were introduced.

According to the modern epistemology therefore the principle of virtual workmay also be accepted as a principle of mechanics if it proves to have sufficient logicstrength to describe all the mechanics – of course combined with other axioms – andto produce results in agreement with the experimental evidence. In the following Iwill try to analyze in more detail the consequences of taking a VWL as a principlewithout any reference to another mechanical theory.

There can be considered essentially three formulations, which gradually moveaway from what was presented as a theorem in the previous pages. It is posssible:

a) to assume forces as the primitive quantities and virtual displacements that takeplace in a virtual time;

b) to assume forces as the primitive quantities and virtual displacements that takeplace in the real time (in the same time with which the forces vary);

c) to assume work as the primitive quantity that takes place in real time.

In the following I will refer only to cases a) and c), being easy to extrapolate tocase b) the considerations valid for the first two. With the usual ambiguity, whichthat should not bother us, the virtual displacements are treated as virtual velocitiesand virtual work as virtual power.

2.2.1 Force as a primitive concept

2.2.1.1 Equilibrium case

Assuming some concept of force, even a pre-Newtonian one, that in principle canalways be replaced by a weight attached to a rope. I introduce only the active forces,while the reactive forces do not appear explicitly, in the limit the concept can also bemissed, which avoidsmany problems of both logical and ontological type. In the caseof a system of nmaterial points, with the symbols used in the preceding paragraphs,if f is the vector of the active forces and u the vector of the virtual velocities/virtualdisplacement – the virtual work is defined as the product f · u, and the principle ofvirtual work could be enunciated in the form of T3:


PP1. A system of particles constrained to move on a manifold M is in equilib-rium if and only if the virtual work of the active forces is zero for any virtualdisplacement.

Note however that now, because there is no reference mechanics and a priori crite-rion of equilibrium, equilibrium is not intended as a balance of forces, but simply asrest. The principle can also be seen with Poincaré, as a methodological principle, astipulation. If the applications to an empirical case do not work, it is always possibleto say: it is because there are hidden forces, for example frictions.

One might ask whether a mechanical theory based on PP1 is acceptable as fol-lows: does it provide satisfactory results from an empirical point of view? Has itsomething different from Newtonian mechanics? The first thing that catches the eyeis the extraordinary simplicity of the principle in the case of constrained systems.The idea of constraint reaction that creates difficulties should not be formulated. Allbreaks down in the examination of only the active forces. All the rules of simplemachines and the rule of the parallelogram, which can be used for alternative for-mulations of mechanics, become simple theorems.

The necessary part of the principle is falsified by every experience, however.That is, if a system is in equilibrium under a system of active forces f , it is nottrue that the virtual work is zero for every possible virtual displacement; it couldbe both positive and negative. An example of this is obtained by considering theequilibrium of a heavy object on an inclined plane. It is found empirically that, fora very rough surface there is equilibrium for a material point even when the planeis tilted several degrees, but the weight force can make a positive virtual work – fordownward virtual displacements – and a negative virtual work – for upward virtualdisplacements. The sufficient part of the virtual work principle PP1, i.e. if the virtualwork is zero for each value of virtual velocities, then the system of particles is inequilibrium, seems instead to be always empirically verified. This is somehow aconsequence of the principle P∗1, in the sense that if the reactive forces r associatedwith smooth constraints are sufficient to maintain in equilibrium a system of points(L = 0), then the actual non-smooth constraints furnish effectively the reaction r andthe system is in equilibrium.

To treat the case with friction it is necessary to reformulate the virtual work prin-ciple by involving the forces of friction, that is the forces in the direction of thevirtual displacement:

PP2. A system of particles constrained to move on a manifold M is in equilib-rium if and only if the virtual work of the active and reactive forces is zero forany virtual displacement.

2.2.1.2 Motion case

The question of applicability of the virtual work principle to motion arose also whenit was considered as a theorem of a mechanics of reference, but in that case, justbecause there is a mechanics of reference, one could think that motion would be oth-erwise faced within this mechanics. Adopting the virtual work principle as a properprinciple one should instead investigate whether it is possible to study the motion of


bodies and how. To use the virtual work principle in the case of motion it is neces-sary to operate in a mechanics where it is at least possible to define the mass and alsoto talk about inertial reference frames; in fact it seems necessary to assume most ofthe concepts of Newtonian mechanics. Supposing that these difficulties have beenovercome, a possible statement of the virtual work principle could be the following:

PP3. The virtual work of the active and inertia forces for a system of materialpoints constrained on a manifold M is always zero in any inertial referencesystem.

(Here the term inertia forces must be considered merely a nominal definition of−ma.) Statement PP3 may seem asymmetrical with that represented by PP1. In factit can be divided into two parts, one necessary and the other sufficient. To calculatethe virtual work in a factual situation, with a and f ‘real’, one gets L = 0; vice versaif one requires L = 0 for any virtual displacements, one obtains the real value of a.

The falsification of the virtual work principle in the presence of constraints thatare not smooth was not taken very seriously by scientists from the XVI to the XIXcentury. The problem of the correspondence of scientific theories to physical real-ity is probably as old as science, but it was put into evidence by Galileo Galilei.Guidobaldo dal Monte’s polemic on the law of isochronism of the pendulum pro-posed by Galileo is well known. Dal Monte, along with his contemporaries, arguedthat the law of isochronism was not verified in practice; Galilei argued instead thatthe ideal pendulum would obey the law. Then Dal Monte replied that physics mustrelate to the real world and not an imaginary ideal world, a ‘paper world’ [86].7 To-day the position of Galileo is generally accepted, but it is also clear that it is notpossible to abstract from accidents the essence of phenomena, mainly because onecannot always tell in advance which attributes are accidental and which ones are sub-stantial. If in many situations, the friction is presented as an accident, which masksthe substantial reality of the problem, in other situations this is no longer true. Andthe justification that the cases in which the presence of friction are important only totechnology and not to science is senseless.

Without entering the merits of theories for the study of the motion of constrainedbody systems, think of how strange would appear a world in which the virtual workprinciple holds true strictly; in this world without friction, life itself would be im-possible. And though in some calculations friction can be neglected, in describingthe substance of the world it must be considered. A theory that does not have theconceptual tools to address important issues must be considered unsatisfactory andits basic principles as incomplete. Then a mechanics with a rigid axiomatic structurewith PP1 incorporated, cannot be the ‘mechanics’. The only solution to solve theaporia that appears – PP1 should be taken as an axiom but it cannot be taken as anaxiom – is to adopt a liberal epistemology, not rigidly axiomatic, which may allowacceptance of PP1 in some cases and non-PP1 in others, without being able to decidewhich option to choose. The choice whether to apply PP1 or non-PP1 is so delegatedto ‘moral’ considerations, in charge of the i, though not explicitly verbally.

7 Preface.


2.2.2 Work as a primitive concept

The interpretation of work as a primitive concept is perhaps the most interesting in-terpretation of the virtual work principle that has been revived in recent years in theinternational literature [328]. Taking, perhaps without a clear understanding of thehistorical reference, considerations of the late XIX century, it involves a completerevision of classical mechanics where the concept of work, which is taken as a scalarquantity susceptible of measurement, is accepted as primitive while the force is sim-ply a definition. So there is a reversal from what happens in Newtonian mechanics,in which it is the force that is primitive and the work is a defined magnitude.

This possibility is not completely counterintuitive. The idea of work and fatigue asphysical magnitudes surely already appeared in Galileo [119] though it was only theenergetic movement of the late XIX century that captured the attention of physicists.The mathematical formulation of this idea could take the following line: consider asystem of material points S and a vector space V that contains the virtual motions(velocities or displacements) of S that are considered to be in real time. The impo-sition of an element of V to S gives the virtual work L, to be treated as a physicalquantity that is in principle measurable. The assumption, empirical in nature, is thatL is a linear form defined on V. The vector space F dual of V is the space of theforces. In other words, the components of the forces acting on a system of mate-rial points, whose motions are defined by the virtual components uk with respect toa fixed coordinate system, are those numerical values fk that determine the virtualwork, according to the relation:

L = ∑ fkuk.

This formulation is valid for both discrete and continuous systems. In the case of adiscrete system, in which V is a finite dimensional space, the foregoing considera-tions are substantially equivalent to those developed for the first time in Lagrange’sMécanique analytique, when the generalized forces are introduced, with the impor-tant difference that there the generalized forces are defined in terms of other forces,regarded as primitive and known quantities.

2.2.2.1 Equilibrium case

In this new formulation of mechanics the virtual work principle can be expressedeither directly, without reference to forces, or indirectly, considering the forcesthat now are defined quantities. Moreover it can be a principle or can be obtainedfrom a more fundamental law and therefore to appear as a theorem. In the follow-ing I treat only the first point of view, postponing the second to Chapter 18, dedi-cated to energetism. Considering the work of friction, it is possible to formulate theprinciple:

PP4. The equilibrium is possible if and only if the virtual work of all forcesapplied to the system is never positive, whichever the virtual displacementassumed.


Here, as for PP1, there is no reference to an a priori equilibrium criterion, and equi-librium is simply rest. In this formulation of the virtual work principle, the possibilityof also considering the virtual work done by the reactive forces, i.e. the frictions, isnot ruled out, even though in practical application it has to rely on the assumptionof smooth constraints. The same difficulties in studying the problem of constrainedbodies in the context of Newtonian mechanics are found but with an important dif-ference: the virtual work is a primitivemagnitude and therefore it seemsmore naturalto characterize smooth and non-smooth constraints:

PP5. The virtual work made by the reaction forces cannot be positive.

This characterization is implicit in the commonly accepted principle of the impossi-bility of perpetual motion.

Consider, for example, a heavy material point that is bilaterally constrained tomove on a rough plane, slightly sloped, and which is in equilibrium. The work Lr

made by the reactive forces (frictions) is always negative, the work La of the weightcan be both positive and negative. For a virtual downward motion it is Lr < 0 andLa > 0, so it is possible to have L = Lr +La = 0; for a virtual upward motion it isLr < 0 and La < 0 so that L = La +Lr < 0.

2.2.2.2 Motion case

It is not entirely clear how to generalize principle PP4 to dynamics. In [328] it is sug-gested to consider the forces of inertia as ordinary forces. This suggestion, however,is controversial because, if work is the primitive concept, the concept of force, albeitof inertia, should be eliminated. Probably the only satisfactory way, from a logi-cal and epistemological point of view, is to define kinetic energy and to add kineticpower (the time derivative of kinetic energy) to static power. This is what Duhemdid, as in discussed in Chapter 18, somehow solving also the difficulty of introduc-ing the concept of mass. With this addition it is possible to enunciate the followingprinciple (only valid for smooth constraints):

PP6. In each moment and for any system of material points the virtual work,measured as the sum of that of all the efforts applied to the system and thekinetic power, is zero whatever is the considered virtual displacement.

3

Greek origins

Abstract. This chapter explains the meaning of the partition of Greek mechanicsinto Aristotelian and Archimedean. In the first part the Aristotelian mechanics isconsidered that identifies as a principle the following VWL based on virtual velocity:The effectiveness of a weight on a scale or a lever is the greater the greater its virtualvelocity. In the central part the Archimedean mechanics is considered where there isno reference to any VWL. In the final part devoted to the late Hellenistic mechanics,the VWL of Hero of Alexandria is considered for which the possibility of raising aweight is determined by the ratio of its virtual displacement and that of the power.The law is presented not as a principle but as a simple corollary of equilibrium.

In ancient Greece, mechanics was the science that dealt primarily with the study ofequipment or machines (in Greek mhqan†), to transport and lifting weights, also asa response to other technological problems of the times. The search for equilibriumwas not of practical interest – excluding the case of weighing bymeans of a balance –and mechanics, at least at the beginning, did not take care of it. From this point ofview mechanics was very different from modern statics which is instead seen as thescience of equilibrium.

Pappus wrote that “the mechanician Heron and his followers distinguished be-tween the rational part of mechanics (involving knowledge of geometry, arithmetic,astronomy, and physics) and its manual part (involving mastery of crafts such asbronze-working, building, carpentry, and painting)” [181].1 In the following, me-chanics is only used in the sense of rational mechanics with Pappus’ meaning; assuch its status was neither well-defined for its social appreciation nor for its episte-mology.

Regarding the social appreciation there were contrasting valuations. Mechanicsconcerned problems of everyday life, connected to manual work and as such con-sidered negligible by the intellectual aristocracy. But its applications gave rise towide interest. To cite the famous attribution to Archytas of Tarentum of a dove“which flew according to the rules of mechanics. Obviously it was kept suspended

1 p. 447.


34 3 Greek origins

by weights and filled with compressed air” [201].2 Aristotle appreciated the activ-ity of mechanicians, as is clear from the prologue to his Mechanica problemata. Tothe contrary Plutarch (46–127 AD) in his Vitae parallelae, wrote that Archimedesfelt ashamed for his interest in mechanics and wanted to be remembered only forhis mathematical works. Even though Plutarch attributes his own conception toArchimedes, his opinion indicates the low consideration mechanics held in somecircles [192].3

As far as the epistemological status is concerned, mechanics was considered byAristotle as a mixed science:

These are not altogether identical with physical problems, nor are they entirely separate fromthem, but they have a share in both mathematical and physical speculations, for the methodis demonstrated by mathematics, but the practical applications belong to physics [12]. 4

This classification was used through all the Middle Ages. Archimedes probably didnot share this opinion and considered mechanics as a branch of pure mathematics.

The mechanics of Greek, as has happened in many areas of Western knowledge,is the basis of modern conceptions. Available sources are not numerous, but theyare important. The earliest references are to the pythagorean Archytas of Taren-tum (c 428–350 BC). For sufficiently precise written documentation, however, ref-erence to Aristotle (384–322 BC), Euclid (fl 365–300 BC), Archimedes (287–312BC), Hero (I century AD) and Pappus of Alexandria (fl 320 AD) is needed.

In the following I first present the ideas of Aristotle, who is usually creditedfor a mechanics based on a law of virtual work. Then I introduce the principles ofArchimedes’s mechanics, alternative to the Aristotelian and where there is no useof any virtual work laws. Finally, a hint of the mature Hellenistic mechanics, thatalthough influenced more by Archimedes is also influenced by Aristotle.

3.1 Different approaches to the law of the lever

3.1.1 Aristotelian mechanics

The principal Aristotelian treatises on mechanical arguments are the Physica, Decaelo and the Mechanica problemata. They were largely studied and commentedupon, both for their philological aspects and for their content. In the following Iwill give a very concise summary. Firstly I will consider the Physica and De caelowhich describe motion according to nature (free motion) and motion against nature(forced motion) with both qualitative (causes) and quantitative (mathematical lawsof motion) considerations. We could say that the context is quite general, it concernsall kinds of forces and bodies and can be defined as ‘theoretical physics’. I shallthen consider the Mechanica problemata. In this treatise, which can be defined as‘engineering based’, the approach is less systematic than the Physica and De caelo.

2 vol. I, p. 483.3 Marcellus.4 p. 331.

3.1 Different approaches to the law of the lever 35

3.1.1.1 Physica and De caelo

The first Aristotelian thesis developed in the Physica and De caelo refers to themotion according to nature of a heavy body: it is downward along the line connectingthe body with the centre of the world. The space traversed in the fall, in a given time,is directly proportional to the weight and inversely proportional to the resistance ofthe medium:

Further, the truth of what we assert is plain from the following considerations. We see thesame weight or body moving faster than another for two reasons, either because there isa difference in what it moves through, as between water, air, and earth, or because, otherthings being equal, the moving body differs from the other owing to excess of weight or oflightness [14].5

A, then, will move through B in time Γ, and through Δ, which is thinner, in time E (if thelength of B is equal to Δ), in proportion to the density of the hindering body. For let B bewater and Δ air; then by so much as air is thinner and more incorporeal than water, A willmove through Δ faster than through B. Let the speed have the same ratio to the speed, then,that air has to water [14].6

A given weight moves a given distance in a given time; a weight which is as great and moremoves the same distance in a less time, the times being in inverse proportion to the weights.For instance, if one weight is twice another, it will take half as long over a given movement[13].7

The Aristotelian law onmotion according to nature is assumed bymost scholars withthe mathematical relation v = p/r, where v is the velocity, p the weight and r theresistance of the medium. There are however some objections to this view [287],8

[349].9 Perhaps the main objection is that of regarding velocity as a definite kine-matic quantity, summarizing space and time with their ratio, which to a modern isjust velocity. This assumption is clearly anachronistic. Not only because in Greekmathematics there was no sense in the ratio between two heterogeneous quantities,like space and time, but also because there was no use for the quantification of ve-locity, which was, in fact introduced only as an intuitive concept, something thatallowed one to say something is greater or lesser [14]10 [287].11

To the community of mechanics scholars, the first known writings on the quan-tification of velocity, which were presented within the mathematics of proportions,and on the systematic use of this quantification is commonly considered that by Ger-ardus de Brussel, in the first half of 1200, moreover expressed in a form not com-pletely explicit as ‘petitiones’ of his famous book Liber de motu, where it is saidthe velocity (motus) is measured by the space traversed in a given time [127]. AfterGerardus de Brussel many medieval and all Renaissance scholars read Aristotle asmost modern scholars do. It must however be said that Aristotle in some places con-

5 IV, 8, 215a.6 IV, 8, 215b.7 I, 6, 274a.8 Chapter 7.9 Chapter 9.10 VI, 2; VII, 4.11 Chapter 3.

36 3 Greek origins

ceives of velocity as a well-defined kinematical quantity. This occurs, for example,in the preceding quotation of Physica, where velocity and resistance are consideredas inversely proportional to each other, and in the following passage:

For since the distinction of quicker and slower may apply to motions occupying any periodof time and in an equal time the quicker passes over a greater length, it may happen that itwill pass over a length twice, or one and a half times, as great as that passed over by theslower: for their respective velocities may stand to one another in this proportion [emphasisadded] [14].12

where it is said quite clearly that velocity can be measured with space covered in agiven time. But Aristotle does not develop this reasoning and any time he refers tomathematical laws he speaks about space and time separately and not about velocity.

The second Aristotelian thesis on motion, clearly stated in the Physica, concernsthe motion against nature of a heavy body: it occurs along a straight line and thespace covered, in a given time, is directly proportional to the ‘force’ applied to thebody and inversely proportional to its weight:

Then, A the movent have moved B a distance Γ in a time Δ, then in the same time the sameforce A will move 1/2B twice the distance Γ, and in 1/2 Δ it will move 1/2B the wholedistance for Γ: thus the rules of proportion will be observed. Again if a given force movea given weight a certain distance in a certain time and half the distance in half the time,half the motive power will move half the weight the same distance in the same time. LetE represents half the motive power A and Z half the weight B: then the ratio between themotive power and the weight in the one case is similar and proportionate to the ratio in theother, so that each force will cause the same distance to be traversed in the same time [14].13

The difficulty for the modern reader in the interpretation of Aristotle’s writing liesmainly in giving sense to ‘force’. Aristotle, at the beginning of book VII of the Phys-ica explains which are precisely the kinds of forces to consider, but this is not enoughto achieve a complete understanding:

Themotion of things that are moved by something else must proceed in one of four ways: forthere are four kinds of locomotion caused by something other than that which is in motion,viz. pulling, pushing, carrying, and twirling [14].14

[…]Thus pushing on is a form of pushing in which that which is causing motion away fromitself follows up that which it pushes and continues to push it; pushing off occurs when themovent does not follow up the thing that it has moved: throwing when the movent causes amotion away from itself more violent than the natural locomotion of the thing moved, whichcontinues its course so long as it is controlled by the motion imparted to it [14].15

Most scholars maintain that ‘force’ had the same meaning as it has today, though notin the Newtonian sense of cause of motion variation, but in the less demanding senseof muscular activity [287, 171, 305]. Expressed in modern terms, and synthesizingspace and time into velocity, this position assumes the direct proportionality between

12 VI, 2, 233b.13 VII, 5, 249b.14 VII, 5, 243a.15 VII, 5, 243b.


force ( f ), velocity (v) and weight (p); with a formula f = pv. Other scholars considerthe modern concept of work as being the closest to ‘force’ [370, 295, 136]. Thisinterpretation is strongly advised in the case of a thrown object, where it is difficult tosee a force in the preceding sense. Even the fact that a force must act together againsta resistance seems to confirm this position. Finally, there are those who think that tointerpret Aristotle’s writings it is enough to make references to the common senseof an uneducated person. Some scholars indeed think that the learning process ofa person resumes the historical process (the ontogenesis resumes the phylogenesis)and the scientific conceptions of classical Greece could be understood by assumingthe identity of a youth who has not yet studied Newtonianmechanics [373, 359, 369].This last position has the merit of averaging the two preceding, because sometimesit is more straightforward to translate ‘force’ with force, sometimes with work, andsometimes more with static moment.

Before taking a position it must be said that the precise differentiation betweenforce and work will occur only in the XVIII century, and as late as the XIX century‘force’ will ambiguously be used to mean both force and work [129]. Moreover, itmust be noticed that the various interpretations cannot be decided upon empirically.The experimental context needed to verify the Aristotelian laws of motion is differ-ent from that foreseen by the modern paradigm, the Newtonian for example. WithNewton one has to observe the motion of a material point in a void space under aforce with assigned direction and intensity. With Aristotle one has to study the mo-tion of an extended body, which moves against resistances of the external mediumwhich tends to oppose the applied force andmaintains the body in a status of constantvelocity. For Aristotle it is implicit that a resistance opposes a force, otherwise therewould be nomotion, or an infinity velocity motion would occur, which is impossible.

The causes of resistance to a body’s motion are not made explicit by Aristotle; ina passage of De caelo, he attributes them to weight:

Again, a body which is in motion but has neither weight nor lightness, must be moved byconstraint, and must continue its constrained movement infinitely. For there will be a forcewhich moves it, and the smaller and lighter a body is the further will a given force moveit. Now let A, the weightless body, be moved the distance CE, and B, which has weight, bemoved in the same time the distance CD. Dividing the heavy body in the proportion CE : CD,we subtract from the heavy body a part which will in the same time move the distance CE,since the whole moved CD: for the relative speeds of the two bodies will be in inverse ratioto their respective sizes. Thus the weightless body will move the same distance as the heavyin the same time. But this is impossible. Hence, since the motion of the weightless body willcover a greater distance than any that is suggested, it will continue infinitely [13].16

In other passages resistances are attributed even to the medium [14]17 and one wouldnot be mistaken by assuming friction too as a resistance.

When results furnished by Aristotelian laws of motion are compared with thoseof modern mechanics (Newtonian or Lagrangian), one sees that the Aristotelian lawsare ‘true’ whatever the interpretation of ‘force’, when the parameter time is not con-sidered. They are in general ‘false’ when this parameter is considered. For example,

16 III, 2, 301b.17 IV, 8, 215b.

38 3 Greek origins

by interpreting ‘force’ as force one gets agreement only for equal intervals of time;by interpreting ‘force’ as work one gets agreement only if the parameter time isexcluded completely.

In what follows, for both motions, according to nature and against nature (naturaland violent motions), I will refrain as much as possible from adopting a preconceivedposition on the ontological status of the various mechanical concepts, because oftenthere is no need to do this and different interpretations do not necessarily conflict.When I have to choose I will opt for the (pre-Newtonian) common sense.

3.1.1.2 Mechanica problemata

In the Mechanica problemata (known also as Mechanical problems, Mechanica)there is no explicit affirmation of the general theoretical principles contained in thePhysica and De caelo, in particular no reference to the laws of natural and violentmotion. Also for this reason and for its practical contents, the attribution of this trea-tise to Aristotle is still debated.18 In the following I will not enter into the merit ofthis attribution and, for the sake of simplicity, I will talk about Mechanica prob-lemata as an Aristotelian work, instead of, as frequently seen, a pseudo-Aristotelianone.

The writing is largely dedicated to the solution of problems, some mechanical innature; they are referred to in Table 3.1 [296].19 The object of the mechanical prob-lems is mainly the study of the shifting of heavy bodies. Nowhere in the text do theconcept and the word equilibrium (isorropein) occur. The functioning of machinesor devices, among them the wedge, pulley and winch, is reconnected to the lever.The validation of the law of the lever is suggested and may be the first in the historyof mechanics. In the following I will comment on this validation.

At the beginning Aristotle refers all the mechanical effects to the properties ofthe circle:

Remarkable things occur in accordance with nature, the cause of which is unknown, andothers occur contrary to nature, which are produced by skill for the benefit of mankind.Among the problems included in this class are included those concerned with the lever. Forit is strange that a great weight can be moved by a small force, and that, too, when a greaterweight is involved. For the very same weight, which a man cannot move without a lever,he quickly moves by applying the weight of the lever. Now the original cause of all suchphenomena is the the circle; and this is natural, for it is in no way strange that somethingremarkable should result from something more remarkable, and the most remarkable fact isthe combination of opposites with each other [12].20

The cause of the farthest points of a circle moving more easily than the closest undera given force is identified by Aristotle in the fact that in circular motion the compo-

18 During the Middle Ages and Renaissance the attribution of the Mechanica problemata to Aris-totle was substantially undisputed. For the attributions in more recent periods see [15]. It is worthnoticing that Fritz Kraft considers the Mechanica problemata to be an early work by Aristotle,when he had not yet fully developed his physical concepts [348, 378]. A recent paper by Winterconsiders Archytas of Tarentum as the author of Mechanica problemata [398].19 pp. 136–137.20 pp. 331–333.


Table 3.1. Problems of Mechanica problemata

1 Why larger balances are more accurate than smaller ones.2 Why the balance seeks the level position when supported from above, but not when supported

from below.3 Why small forces can move great weights by means of the lever, despite the added weight of

the lever.4 Why rowers in the middle of ships contribute most to their movement.5 Why the rudder, though small, moves the huge mass of the ship.6 Why ships go faster when the sail yard is raised higher.7 Why in unfavourable winds the sail is reefed aft and slackened afore.8 Why round and circular bodies are most easy to move.9 Why things are drawn more easily and quickly by means of greater circles.10 Why a balance is more easily moved when without weights than when weighted.11 Why heavy weights are more easily carried on rollers than in carts.12 Why a missile thrown from a sling moves farther than one thrown by the hand.13 Why larger handles move windlasses more easily.14 Why a stick is more easily broken over the knee when the hands are equally spaced, and

farther apart.15 Why seashore pebbles are rounded.16 Why timbers are weaker the longer they are, and bend more when raised.17 Why a wedge exerts great force and splits great bodies.18 Why two pulleys in blocks reduce effort in raising or dragging.19 Why a resting axe does not cut wood, and a striking axe splits it.20 How a steelyard can weigh heavy objects with a small weight.21 Why dentists use forceps rather than the hand for extraction.22 Why nutcrackers operate without a blow.23 Why the lines traced by points of a rhombus are not of equal length.24 Why concentric circles trace paths of different length when rolled jointly on this or that cir-

cumference.25 Why beds are made in length double the width.26 Why long timbers are most easily carried by their centres.27 Why longer timbers are harder to raise to the shoulder.28 Why swing beams at wells are counterweighted.29 Why of two men carrying a beam, the man nearer the centre of the beam feels more of the

weight.30 Why men move feet back and shoulders forward to rise from sitting.31 Why objects in motion are more easily moved than those at rest.32 Why objects thrown ever stop moving.33 Why objects move at all when not accompanied by the moving power.34 Why bodies thrown cannot move far, but related to the thrower.35 Why objects in a vortex finish at the centre.

nent against nature of the farthest points is proportionally less than that accordingto nature. Indeed the farthest points describe a larger circle and the motion in thiscircle is closest to the linear one, which is considered to be natural (for the meaningof this term in the Mechanica problemata see below).

Let ABΓ be a circle, and from the point B above the centre let a line be drawn to Δ; it isjoined to the point Γ if it travelled with velocities in the ratio of BΔ to ΔΓ it would movealong the diagonal BΓ. But, as it is, seeing that it is in no such proportion it travels along thearc BEΓ. Now if of two objects moving under the influence of the same force one suffers

40 3 Greek origins

more interference, and the other less; it is reasonable to suppose that the one suffering thegreater interference should move more slowly than that suffering less, which seems to takeplace in the case of the greater and the less of those radii which describe circles from thecentre. For because the extremity of the less is nearer the fixed point than the extremity ofthe greater, being attracted towards the centre in the opposite direction, the extremity of thelesser radius moves more slowly. This happens with any radius which describes a circle; itmoves along a curve naturally in the direction of the tangent, but is attracted to the centrecontrary to nature. The lesser radius always moves in its unnatural direction for because itis nearer the centre which attracts it, it is the more influenced. That the lesser radius movesmore than the greater in the unnatural direction in the case of radii describing the circlesfrom a fixed centre is obvious from the following considerations [12]21.

Shortly after Aristotle ‘proves’ his assertion by simple geometric considerations thatrefer to Fig. 3.1.

In reality it must be said that Aristotle’s arguments make difficult reading. Forinstance the interpretation of the locution “according to nature” presents some prob-lems. For Bottecchia [15]22 the motion according to nature is what happens alongthe circumference, while the motions along the tangent and the radius would be bothagainst nature. But this makes the text unintelligible. The most common interpreta-tion, which avoids this problem, assumes the motion according to nature along thetangent and the motion against nature towards the centre of the circle. Moreover tofollow the Aristotelian classification of natural motion, Duhem suggests consideringthe natural downward motion from the horizontal radius, as shown in Fig. 3.2.23 Formore considerations about natural and violent motions see the work by ChristianeVilain [394].

In the Aristotelian text there are also some other ambiguities, as for example therole played by the force, for which there is no precise indication on the direction ofapplication and nature (muscular force or weight). For this purpose it is interesting tonote the comment by Giuseppe Vailati, for whom any translation of the Aristoteliantext is problematic because of the ambiguities of the Greek language:

A

B

E

Δ

Γ

Fig. 3.1. Motion against nature in the circle

21 pp. 341–343.22 p. 67.23 Rotated ninety degrees clockwise, [15] p. 67.


AME B

Ψ

PH

ΘΩ

Γ

Δ

Z X K Y

Fig. 3.2. Motion of points at different distances from the centre in the circle

‘Ambiguities’ that could be considered as a linguistic document of the belief in a primordialconnection between the different compatible velocities of the various points the positions ofwhich depend on each other, and the ease with which different points can be moved, otherthings being equal [391].24 (A.3.1)

For this reason I prefer not to comment on the wording of Aristotle because minewould be only one of the possible interpretations. In any case, perhaps, more inter-esting of the real intentions of Aristotle are the possible interpretations of a readerof the Mechanica problemata subsequent to Aristotle. I will return to this point inthe following paragraphs and in the chapter on Medieval mechanics. Here I merelystate the interpretation of Aristotle’s proof shared by most commentators. For themAristotle proves two things:

a) for a given amount of natural (vertical) motion, the motion against nature isgreater for points closer to the centre of the circle (XZ against BP);

b) for a given time, with the most distant points that describe longer arcs, the mo-tion according to nature is greater for points further away from the centre (HKagainst ΘZ).

This is considered enough to attribute to Aristotle the idea that a force applied(tangentially-vertically) to the points of the circle farthest from the centre has agreater effect because these points move faster then the closest, or, which is thesame, that the effect of a force depends on the velocity of its point of applicationin a possible motion. But this is a law of virtual work, according to the meaningassumed in the book. It is however a qualitative law and as such useless to obtainmathematical expressions for equilibrium or motion.

Aristotle transformed this qualitative law into a quantitative one in the attempt tosolve the problem related to the law of the lever: “Why is it that small forces canmove great weights by means of a lever?”.

24 p. 10.

42 3 Greek origins

Table 3.2. Two versions of the lever law (problem 3) proof in Mechanica problemata

A. Quoniam autem ab aequali pondere celerius movetur maior earum, quae a centro sunt; duo veropondera quod movet et quod movetur. Quod igitur motum pondus ad movens, longitudo patitur adlongitudinem; semper autem quanto ab hypomoclio distabit magis, tanto facilius movebit. Causaautem est, quae retro commemorata est: quoniam quae plus a centro distat, maiorem describit cir-culum. Quare ab eadem potentia plus separabitur movens illud, quod plus ab hypomoclio distabit[162].25

B. But since under the impulse of the same weight the greater radius from the centre moves themore rapidly, and there are three elements in the lever, the fulcrum, that is the cord or centre, andthe two weights, the one which causes the movement, and the one that is moved; now the ratio of theweight moved to the weight moving is the inverse ratio of the distances from the centre [emphasisadded]. Now the greater the distance from the fulcrum, the more easily it will move. The reasonhas been given before that the point further from the centre describes the greater circle, so that bythe use of the same force, when the motive force is farther from the lever [sic! correct: fulcrum], itwill cause a greater movement.

ΔΑ Β

Ε Γ

ΔΗ

ΓΚ

Fig. 3.3. The equilibrium in the lever

Let AB be the bar, Γ the weight, and Δ the moving force, E the fulcrum; and let H be the point towhich the moving force travels and K the point to which the weight moved travels [12].26

The main part of the solution Aristotle suggested for this problem is reported inTable 3.2 in two different translations: the first translation fromGreek into Latin (A);a modern translation into English (B): A noteworthy aspect clearly stated in the Aris-totelian text – “and there are three elements in the lever, the fulcrum, that is the cordor centre, and the two weights, the one which causes the movement, and the onethat is moved” – is the assimilation of weight to a motive power, at least as far asthe effects are concerned. And this is independent of the fact that the tendency of aheavy body to direct itself towards the centre of the world is associated with a force,interior or exterior, or it simply depends on the nature of the body. It is true that intheoretical treatises like the Physica and De caelo, Aristotle is reluctant to considera weight as a motive power, but many of Aristotle’s successors, Latin and Arabic,perceived weight as a special kind of force. And it is certain that although weightwas not put completely on the same footing as force, it was often treated as such[321, 186, 246, 343].

In the proof of the law of the lever, according to which “the ratio of the weightmoved to the weight moving is the inverse ratio of the distances from the centre”,Aristotle assumes that the effect of a force or aweight vary linearly with its virtual ve-locity, i.e. linearly with its distance from the fulcrum, transforming so its qualitative

25 p. 4.26 pp. 353, 355.


law into a quantitative one. This seemed quite natural to many later commentators,simply because it was usually enough to turn any laws into a linear proportion. It wasnot the same view of Bernardino Baldi (and Guido Ubaldo dal Monte), who whileappreciating a lot of the Mechanica problemata denounces the lack of evidence ofthe quantitative law:

Thus when Aristotle discloses the reason for which the lever moves a weight more easily,he says that this happens because of the greater length on the side of the power that moves;and this [accords] quite well with his first principle, in which he assumes that things at thegreater distance from the centre are moved more easily and with greater force, from whichhe finds the principal cause in the velocity with which the greater circle overpowers thelesser. So the cause is correct, but it is indeterminate; for I still do not know, given a weightand a lever and a force, how I must divide the lever at the fulcrum so that the given forcemay balance the given weight. Therefore Archimedes, assuming the principle of Aristotle,went on beyond him; nor was he content that the force be on the longer side of the lever, buthe determined how much [longer] it must be, that is, with what proportion it must answerthe shorter side so that the given force should balance the given weight [19].27 (A.3.2)

Fig. 3.4. Simple machines of the Mechanica problemata [18]29 (reproduced with permission ofBiblioteca Alessandrina, Rome)

3.1.1.3 A law of virtual work

Some scholars, historians and scientists believe that the Aristotelian mechanics con-tains in a nutshell the modern principle of virtual work, at least as regards a formu-lation restricted to weights. In my opinion they attributed to much to Aristotle.

Galileo was among the first to associate a law of virtual work, in the form ofvirtual velocities, to the Mechanica problemata:

The second principle is that the moment and the force of gravity is increased by the speed ofmotion, so that absolutely equal weights, but combined with unequal speed, have strength,moment and virtue unequal, the faster and more powerful, according to the proportion of itsspeed to the speed of the other. […] Such equivalence between gravity and speed is foundin all the mechanical instruments, and was considered to be, by Aristotle, a principle inhis Mechanica problemata; so we can still take as a quite true assumption that absolutely

27 pp. 54–55. Translation by [298], p. 14.29 p. 6.

44 3 Greek origins

unequal weights alternately be counterbalanced and to make equal moment, every time theirgravitas are in inverse proportion with the speed of motions, that is one is less heavy thanthe other, as much it is able to move more quickly than the other [115].30 (A.3.3)

He was however preceded by Giuseppe Moletti (1531–1588), professor of math-ematics at Padua, just before Galileo. Moletti during his stay in Padua lectured onmechanics and used theMechanica problemata as an authoritative text. Below somequotations from his lectures edited by Laird [351] are reported, related to the mar-vellous properties of the circle on the footsteps of Aristotle:

But in order not to confuse us I shall first discuss the principle that Aristotle states for boththe machines we have given as examples, that is, for the pulley, and the lever. [...] All theoperations ofmachines, then, consist in their movement, and consequently the samemachinewill have a greater or a lesser effect to the extent that the movement that it makes is nearer toits own, proper movement. [...] Thus it is clear from the preceding demonstrations that theless a weight is constrained to move in a circle, or the farther a force is from the centre, withsomuchmore speed it will move and somuchmore effect the forcewill have [351].31 (A.3.4)

Pierre Duhem goes further and sees the principle of virtual work also in the Aris-totelian law of violent motion, for which force, speed, and weight are linked bymathematical relationships expressed by means of proportion. For example, Duhembelieves he can derive the law of conservation of the product of weight for the heightby the law of forced motion (see § 4.2). Like almost all interpretations of historicaltexts Duhem’s is subject to criticism, not because it is incompatible with the Aris-totelian mechanics, but because theMechanica problemata has such ambiguities thatmany interpretations are possible and it seems unlikely that the one based on mod-ern categories, like those used by Duhem, may be the right one. For example, it isdoubtful that Aristotle could have considered homogeneous the speed of a free bodywith that of a body hanging from the end of a scale that moves partially of naturalmotion and partially of violent motion.

Giovanni Vailati shares, with some cautions, Duhem’s view limited toMechanicaproblemata [391]. Ernst Mach is once again more cautious and says verbatim:

Let us now consider some details. The author of the Mechanical problems mentioned onp. 511 remarks about the lever that the weights which are in equilibrium are inversely pro-portional to the arms of the lever or to the arcs described by the endpoints of the arms when amotion is imparted to them. With great freedom of interpretation we can regard this remarkas the incomplete expression of the principle of virtual displacements [355].32

Note that the main feature of the laws of virtual work, including that of Aristotle,is that the effect of a force is somehow considered a posteriori in the sense that itdepends not only on the force in itself, but also on the motion which is determined bythe presence of constraints. It is possible that this idea may have come to Aristotle,or who for him, for his metaphysical conceptions of motion and rest, which identifythe rest as motion in power and therefore somehow ‘aware’ of its future. But it isalso possible, and it seems more likely, that the dependence on the effect of a force

30 pp. 72–73.31 pp. 86–87; 122–123.32 pp. 6–7.


by its motion was obtained without any ideological mediation as an empirical intu-ition. The same goes for the laws of natural and violent motions. They are classifiedin the general Aristotelian metaphysics, but do not necessarily flow from it, rather itis more likely to be the result of empirical observations which were then absorbedin some way by Aristotle.

3.1.2 Archimedean mechanics

Developments in Greek mechanics after Aristotle are poorly documented. There re-main a few of the writings of Archimedes (287–212 BC) and Euclid (fl 300 BC).Then the writings of Hero (fl I century AD) and Pappus of Alexandria (fl 320 AD),which have educational purpose. In their treatises, Archimedes and Euclid mainlyfocused on theoretical foundations of mechanics, with particular emphasis on thedemonstration of the law of the lever. The approach of the Mechanica problematacould not satisfy the mathematicians of Hellenistic periods, used to standards of noless rigor than the modern ones. They could not accept such a leap of logic that ledfrom Aristotle’s empirical evidence, not problematic, of the increased effectivenessof the applied forces farther from the fulcrum of a lever, to the mathematical formu-lation of the law that this different effect is proportional to the virtual velocity andthen the distance.

Though equilibrium was not a central problem for applications of mechanics itbecame a central theoretical problem for Archimedes. He realized that once the prob-lem of equilibrium was solved, the problem of rising was also solved. Indeed if aweight p equilibrates a weight q in a lever, a weight only slightly heavier than pwill lift q. But there is an advantage of this shifting of the theory from transport toequilibrium, because the equilibrium is much easier to study in a rigorous way. Inthe following I will refer to only the approach of Archimedes, reserving the right toreturn to Hero and Pappus in the following paragraphs and to Euclid in Chapter 4,devoted to Arabic mechanics.

Archimedes set his mechanical theory on a few suppositiones (suppositions, prin-ciples), partly empirical in nature, which certainly appear more convincing than theAristotelian. His goal was not only to formulate the law of the lever, but also to ad-dress the equilibrium of extended bodies, which are found in his theory of hydrostat-ics. The equilibrium of a body, or a set of bodies, was reduced to the determinationof its centre of gravity and to make sure it is well constrained. Today there remainsonly a text where Archimedes treats the problem of equilibrium, the Aequiponder-anti (in English,On the equilibrium of plane figures). However, there are indicationsthat he wrote some other texts, or that the Aequiponderanti is only a part of a largertreatise. See for example the writing by Hero referred to in the following.

Archimedes was the first scientist to set rational criteria for determining centresof gravity and his theory was the first known physical theory formalised on a mathe-matical basis. For my purpose, I shall mainly examine Book I of the Aequiponderanti[10, 11, 291, 287] where Archimedes, besides studying the rule governing the law ofthe lever, also evaluates the centres of gravity of various geometrical plane figures.He gave the basic elements of the theory of the centre of gravity, establishing seven

46 3 Greek origins

suppositiones, shown in Table 3.3 [11].33 Using them Archimedes is able to provethirteen propositiones (propositions, theorems), shown in Table 3.4 which allowsus to calculate the centre of gravity of composed bodies, starting by knowing thecentres of gravity of the simple bodies from which they are formed.

Tables 3.3 and 3.4 show that the organisation of Archimedes’ theory differedfrom that used by Euclid. For example, Archimedes does not intend to develop acomplete theory of mechanical science and write a detailed treatise. Instead he con-centrates on facing a problem, important but unique: the determination of the centresof gravity for bodies of any shape. Archimedes draws on Euclid’s Elements, which

Table 3.3. The suppositiones of Archimedes’ centre of gravity theory

S1 Equal weights at equal distances are in equilibrium, and equal weights at unequal distancesare not in equilibrium but incline towards the weight which is at the greater distance.

S2 If, when weights at certain distances are in equilibrium, something be added to one of theweights, they are not in equilibrium but incline towards that weight to which the addition wasmade.

S3 Similarly, if anything be taken away from one of the weights, they are not in equilibrium butincline towards the weight from which nothing was taken.

S4 When equal and similar plane figures coincide if applied to one another, their centres ofgravity similarly coincide.

S5 In figures which are unequal but similar the centres of gravity will be similarly situated. Bypoints similarly situated in relation to similar figures I mean points such that, if straight linesbe drawn from them to the equal angles, they make equal angles with the corresponding sides.

S6 If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them willalso be in equilibrium at the same distances.

S7 In any figure whose perimeter is concave in (one and) the same direction the centre of gravitymust be within the figure.

Table 3.4. The first seven propositiones of Archimedes’ centre of gravity theory

P1 Weights which balance at equal distances are equal.P2 Unequal weights at equal distances will not balance but will incline towards the greater

weight.P3 Unequal weights will balance at unequal distances, the greater weight being at the lesser

distance.P4 If two equal weights have not the same centre of gravity, the centre of gravity of both taken

together is at the middle point of the line joining their centres of gravity.P5 If three equal magnitudes have their centres of gravity on a straight line at equal distances,

the centre of gravity of the system will coincide with that of the middle magnitude.Cor. 1. The same is true of any odd number of magnitudes if those which are at equal

distances from the middle one are equal, while the distances between their centres of gravityare equal.

Cor. 2. If there be an even number of magnitudes with their centres of gravity situated atequal distances on one straight line, and if the two middle ones be equal, while those whichare equidistant from them (on each side) are equal respectively, the centre of gravity of thesystem is the middle point of the line joining the centre of gravity of the two middle ones.

P6,7 Two magnitudes, whether commensurable [Prop. 6] or incommensurable [Prop. 7], balanceat distances reciprocally proportional to the magnitudes.

33 pp. 189–192.


were widely accepted at the time. The order of the propositions follows a basic logic,similar, although not identical, to those of Euclid. It advances from the simpler casesto those more complex. In Archimedes’ work however, the axiomatic structure andthe concatenation of the propositions are less evident. The first four propositions forexample are demonstrated directly from the suppositions and are not consequent toone or the other; the second after the first, the third after the second, etc. The maindifference between the organisation of Euclidean and Archimedean theories can befound in the epistemological status of the principles. Indeed not all of Archimedeanprinciples are ‘necessary’ as Euclid’s ones; some of them are indeed simply true orempirically evident [276, 374].

A fundamental supposition is the empirical assertion according to which if equalweights are positioned on a lever, at different distances from a fulcrum, there is noequilibrium but instead a tendency for the more distant weight to move downward.At first glance such an assertion could be interpreted in a dynamical key, holdingthat the greater weight has the higher tendency to move downward, thus breakingthe horizontal equilibrium. But it could also be seen as a matter of fact which doesnot necessarily need to be explained.

Some Archimedes’ suppositions seem to have a level of evidence more or lessanalogous to that of certain postulates of Euclid’s Elements. For example, the firstpart of S1 and S4 could derive from the principle of sufficient reason; the epistemo-logical state of S5 and S6 is instead difficult to determine. Note that the suppositionsS4, S5, S6 and S7 refer to the centre of gravity concept which was not introducedexplicitly by Archimedes in the Aequiponderanti.

When one considers the order of suppositions and propositions, a degree of organ-isational coherence is evident between the first four suppositions and propositions,as they complete each other. It seems that Archimedes wanted to reduce as muchas possible the content of the suppositions, declaring only the parts impossible todemonstrate – either because self-evident or because empirically evident – leavingto the propositions the role of making precise the whole concept. In particular, thefirst part of supposition 1 and the whole of proposition 1 refers to the two sides ofthe implication: equal weight↔ equilibrium. And that Archimedes considers sup-position 1 as evidently known (equal weights → equilibrium), which provides thesufficient condition for equilibrium and not proposition 1 (equal weights ← equi-librium), which provides the necessary condition is a questionable choice. From apurely logical point of view, Archimedes could have chosen proposition 1 as the firstpart of supposition 1. But Archimedes’ actual choice is more convincing because itcan be considered as self evident, or when this is not the case, supposition 1 is easierto be verified experimentally than proposition 1.

The nature of the suppositions being not completely self evident, it seems morenatural to deny their opposites than to affirm them, leading naturally to the use of thereduction ad absurdum, rarely practiced by Euclid, as the preferred kind of proof. Inwhat follows I report the demonstration of the first proposition to show the typicalway of Archimedean argument by reduction to the absurd, then I will take back thesixth proposition on the law of the lever.

48 3 Greek origins

Table 3.5. Formal expressions of proof for proposition 1

Assumptions and suppositions (S) Propositional logic Content

1 – Negation of proposition 1 ¬(E→U) or ¬U ∧E Weights are not equal2 – 1 + S3 U ∧¬E or ¬(U → E) Equal weights are not in equilibrium3 – S1 U → E Equal weights are in equilibrium4 – Absurdum from 2 and 3 E→U Weights are equal

Weights which balance at equal distances are equal. For if they are unequal, take away fromthe greater the difference between the two. The remainder will not then balance [supposi-tion 3], which is absurd [supposition 1]. Therefore the weight cannot be unequal [11].34

In order, I shall use the formalism of classic propositional logic which enhances theunderstanding of Archimedes’ assumptions. Proof of proposition 1 can be synthe-sised using the following Table 3.5 [276]:35

In the table the following symbols are used, U: equal weights, E: equilibrium,while ¬(E→U)↔¬U∧E andU∧¬E↔¬(U→ E) are trivial theorems of propo-sitional logic.

3.1.2.1 Proof of the law of the lever

Here the proof of proposition 6 is reported, according to which two heavy bodieswith commensurable weights (the ratio is a rational number) balance when they aresuspended at distances inversely proportional to weights. The proof of proposition 7,the case of not commensurable weights, which makes use of the exhaustion method,is not presented, because both less interesting and more problematic.

The main reason to concentrate on the original Archimedean proof is that inmany texts in the history of science, the demonstrations of the lever attributed toArchimedes, are actually often only a rough paraphrase. This is also due to the in-terpretation of Ernst Mach, which I will report shortly thereafter.

Propositions 6, 7.Two magnitudes, whether commensurable [Prop. 6] or incommensurable [Prop. 7], balanceat distances reciprocally proportional to the magnitudes.

I. Suppose the magnitudes A, B to be commensurable, and the points A, B to be theircentres of gravity. Let DE be a straight line so divided at C that:

A : B = DC : CE.

We have then to prove that, if A be placed at E and B at D, C is the centre of gravity ofthe two taken together.

Since A, B are commensurable, so are DC, CE. Let N be a common measure of DC, CE.Make DH, DK each equal to CE, and EL (on CE produced) equal to CD. Then EH = CD,since DH = CE. Therefore LH is bisected at E, as HK is bisected at D. Thus LH, HK musteach contain N an even number of times. Take a magnitude O such that O is contained as

34 p. 190.35 p. 89.


L KE C H D

A B

O

Fig. 3.5. The lever

many times in A as N is contained in LH, whence

A : O = LH : N.

ButB : A = CE : DC = HK : LH.

Hence, ex aequali, B : O = HK : N, or O is contained in B as many times as N is containedin HK. Thus O is a common measure of A, B.Divide LH, HK into parts each equal to N, and A, B into parts each equal to O. The parts ofA will therefore be equal in number to those of LH, and the parts of B equal in number tothose of HK. Place one of the parts of A at the middle point of each of the parts N of LH,and one of the parts of B at the middle point of each of the parts N of HK. Then the centreof gravity of the parts of A placed at equal distances on LH will be at E, the middle pointof LH (Prop. 5, Cor. 2), and the centre of gravity of the parts of B placed at equal distancesalong HK will be at D, the middle point of HK. Thus we may suppose A itself applied atE, and B itself applied at B. But the system formed by the parts O of A and B together is asystem of equal magnitudes even in number and placed at equal distances along LK. And,since LE = CD, and EC = DK, LC = CK, so that C is the middle point of LK. Therefore C isthe centre of gravity of the system ranged along LK. Therefore A acting at E and B actingat B balance about the point C [11].36

The demonstration opens with the assertion that to prove proposition 6 it is enough toshow that the centre of gravity of the two weights A and B coincides with the fulcrumof the lever. The first part of the proof is purely geometric. Given the extremes E andD of the lever with fulcrum C, extends to the right of DK = EC and to the left of EL= CD, so that C is the midpoint between L and K. Then choose the point H thatsatisfies the ratio CD : CE = LH : KH, involving HD = DK, HE = EL. Since CD andCE are commensurable, LH and KH are also commensurable. Let N be the measurein common between CD and CE, and let n = CE / N and m = CD / N, it will bealso LH / N = 2n, HK / N = 2m. Divide then HL into 2n parts and HK in 2m and atthe midpoint of each part put a weight O = A/2n = B/2m. We have so many equalweights evenly distributed on LK: 2n of them centred on E and 2m centred on D, asclear from Fig. 3.6 for the case of n = 4, m = 3.

There are now two situations, a lever 1 (DE), with weights A and B hanging fromD and E and a lever 2 (LK), with 2(n+m) equidistant weights O. At this pointArchimedes can apply his suppositions, showing first that the lever 2 is balanced,then that lever 1 is equivalent to lever 2 and thus balanced too. Lever 2 is balanced

36 pp. 192–193.

50 3 Greek origins

BA

C

HE K L

O

N D

2n 2m

Fig. 3.6. The lever

because for corollary 2 of proposition 5, 2(n+m)weights have their centre of gravityin C. As far as lever 1 one can say that the 2nweights O on the left (with 2nO= A) ofthe lever 1, have E as centre of gravity, they are so ‘equal’ to A, and the 2m weightsO on the right, have D as their centre of gravity and are so equal to B. They can thusbe replaced by the two weights A and B to obtain a lever that is in equilibrium forsupposition 6.

The demonstration of Archimedes has been criticised since ancient times. La-grange enumerates the scientists who, in modern time, tried to improve the demon-stration: Stevin, Galileo, Huygens, Daniel Bernoulli. Lagrange’s position is almostclear; he believes that the law of the lever cannot be entirely deduced a priori but it isalso based on empirical principles. Moreover he believes that the demonstration byArchimedes, is, all considered, to be preferred to those proposed by other theorists,because “It should be said that by altering the simplicity of this proof it is addedquite nothing of exactitude”[148].37

In modern times the criticism that had the greatest success is that of Ernst Mach,who accused the proof of Archimedes of circularity (that is the proof implicitly as-sumes what needs to be proved) [355].38 I do not quote here Mach’s considerations,I simply point out that only recently it has been shown clearly that they are incon-sistent by Toeplitz, Dijksterhuis, Stein, Goe, Suppes [317].39 For Galletto, who hasconducted an in-depth study of its logical status, the demonstration of Archimedesis correct in the sense that it follows deductively by his suppositions without anylogic error. If there should be be any criticism, then it might concern the plausibilityof the suppositions, but this is another story. Galletto, however, acknowledges somegaps in the Archimedean text, including the failure to introduce the concept of cen-tre of gravity. Together with other authors, including Vailati, Galletto believes thatthe concept of gravity was defined by Archimedes in some other work, now lost. Hebelieves that the concept used is that reported by Pappus of Alexandria, for whichthe centre of gravity is the point of suspension of neutral equilibrium.

37 p. 3.38 pp. 9–11; 512–513.39 pp. 470–472.

3.2 The mechanics of Hero of Alexandria 51

3.2 The mechanics of Hero of Alexandria

About the life of Hero of Alexandria, mathematician and engineer, author of funda-mental treatises of mathematics, mechanics, optics, etc., almost nothing is known,there is only widespread consensus that he lived in the first century AD [330].40

Hero was among the first researchers to combine theoretical knowledge with thetechnical; to make a modern parallel one can compare him to the engineers of theÉcole polytechnique in the early XIX century. He is the heir and successor of a ma-jor scientific revolution that occurred in the Hellenistic period, the representativesof which in mechanics are Ctesibus (fl 285–222 BC) and Philo of Byzantium (c.280 - c. 220 BC), which sees its climax with Archimedes (c. 287 - c. 212 BC) andthe sunset with Pappus of Alexandria (c. 290 - c. 350 AD) [379]. Although Hero’soriginality is not comparable to that of Archimedes, in him there is a more completesummary of theoretical and practical knowledge, in particular between mathematicsand mechanical practice. Hero’s writing that contains a theoretical study of mechan-ics is the Mechanica [330]. The work has been received in its entirety only in theArabic version of Qusta ibn Luka. According to Carra de Vaux the manuscript wascarried in Europe by Jacob Glolius (1596–1667) at the beginning of the XVII cen-tury and translated by him into Latin; unfortunately this translation was lost [131].41

A summary of Golius work was published by Anton Brugmans in 1785 [52].Of Qusta ibn Luka’s manuscript there are today two Western translations,42 one

into French by Carra de Vaux [131], to which I shall refer, and another into Germanwith a few fragments in Greek added [132], the examination of which gives theimpression that the Arabic text is not faithful to the original Greek. The Mechanicais generally regarded as a compilation and dissemination text, but it is not unlikelythat Hero has made his way; in any case the text is a testimonial of an impressiveaccumulation of knowledge of which no other documents remains. It is divided intothree books, the first two of a theoretical nature, the third more applicative, thatbrings considerably complex war machines and lifting of weights. Some of them areshown in Fig. 3.7.

In the first book, problems of general mechanics are dealt with, also of kine-matics, including the problem of the wheel of Aristotle, issue 24 of the Mechanicaproblemata. The book ends with an analysis (incorrect) of the inclined plane (seesubsequent sections), the weight distribution of a beam on supports and some con-siderations on the centres of gravity:

He said: the centre of gravity or point of inclination is such a point that, when a load issuspended at it, it is divided into two equal parts. Therefore Archimedes and his follow-ers in mechanics have particularized this theorem and distinguished between the point ofsuspension and the centre of gravity [131].43 (A.3.5)

40 p. 25.41 p. 8.42 There is also an English version by Jutta Miller [133] on the web page of the Max Plank Institute,Berlin.43 p. 73. Translation in [133].

52 3 Greek origins

Fig. 3.7. Hero’s machines for the lifting of large weights and a press [131]44 (reproduced withpermission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

The second book is dedicated to the so-called Hero’s five machines, shaft with thewheel, lever, block and tackle, wedge and screw (the screw was not named in Aris-totle’s Mechanica problemata and probably it is an invention of the Hellenistic pe-riod). Missing from the list is the inclined plane which is treated separately. Herorefers to them as the powers (d‘namic). The fact that from all the devices of the tech-nology of the times Hero concentrated only on five of them, very different in formfrom each other, could be explained because with their combination all the machinesused in practice can be obtained. The introduction of the wedge, generally not usedin combination with other machines, could be explained by the fact that the screw isreducible to it. The exclusion of the inclined plane is not so easy to explain.

In both the first two books the influence of Aristotle and Archimedes is evident.Aristotle and theMechanica problemata were never mentioned in the text, but thereis the reference of the operation of all machines to the circle, at least in principle,

44 a) p. 167; b) p. 169; c) p. 172; d) p. 182.


and the idea of increasing the effectiveness of forces with the increasing of speed.Towards the end of the book there is also a list of seventeen problems, which havea formulation very close to those of the Mechanica problemata. Archimedes wasnamed many times as the author of contributions not present in his extant writings.For examples to Archimedes, as well as the demonstration of the law of the lever, isalso attributed that of the angular lever:

Some have thought that inverse proportionality is not present in irregular scales. Let ustherefore also imagine a differently heavy and dense scale beam of any material that is inequilibrium when it is suspended at point γ. Here, we understand with equilibrium the restand standstill of the scale beam, even if it is inclined to any side [emphases added]. Then wesuspend weights at random points, namely δ and ε, and we let the beam again be in balanceafter their suspension. Now Archimedes has proven that also in this case weight to weightis inverse to distance to distance [131].45 (A.3.6)

η ξ θ

γ

δ

ε

Fig. 3.8. The angular lever

Hero attributes to Archimedes also a book on the supports:

It is now urgently needed to give some explanations concerning pressure, transport andsupport with regard to quantity, as are suitable for an introduction. For Archimedes hasalready adopted a reliable procedure on this part in his book with the title Book of Supports[131].46 (A.3.7)

Of course, given the large interval of time between Hero and Archimedes, Hero’squotations are not first-hand and therefore must be considered with caution.

3.2.1 The principles of Hero’s mechanics

Hero declares continuously that all the wondrous and paradoxical effects of all thesimple machines can be explained by means of the lever, the law and reason ofwhich are attributed exclusively to Archimedes. The circle is also assumed as anexplanatory model, but it is submitted to the lever.

45 pp. 88–89. Translation in [133].46 p. 77. Translation in [133].

54 3 Greek origins

ηζ θ

γδβ α ε

Fig. 3.9. Two circles with he same centre

Let us assume two circles around the same centre [Fig. 3.9], namely point α, whose twodiameters are the lines βγ and δε. Let the two circles be mobile around the point α, theircentre, and let them be perpendicular to the horizon. If we now suspend at the two points βand γ equal weights, namely η and ζ, then it is clear that the circles do not incline to anyside, since the weights ζ and η are equal and the spaces βα and αγ are also equal, so βγis a scale beam that can be set in motion around the point of suspension, namely point α. Ifwe now shift the weight at γ and suspend it at ε, then the load ζ will sink and set the circlesin rotation. If we however increase weight θ, then it will again keep the balance of weight ζand load θ then relates to load ζ like the distance βα to the distance αε and we thus imaginethe line βε as a balance that can be set in motion around the point of suspension, namelypoint α. Archimedes has proven this in his work on the balancing of inclination. From thatit is evident that it is possible to move an immense bulk with a small force [131].47 (A.3.8)

That the five powers that move a load are similar to circles around one centre is proven bythe figures that we have designed in the preceding; but it appears to me that they look moresimilar to the balance than to the circles [emphasis added], because in the preceding thebases of the proof for the circles resulted from the balance. For it was proven that the loadsuspended from the smaller side relates to the one suspended from the larger side like thelarger scale beam to the smaller one [131].48 (A.3.9)

But Hero’s claims to reduce all the machines to the lever seems to me in some casesonly a rhetorical artifice. In fact, besides the law of the lever Hero uses other prin-ciples to explain the functioning of the block and tackle and the wedge, at least thisis my opinion and I am not convinced of the frequent attempt of historians to justifythem somehow with the lever. The block and tackle is explained simply by assum-ing the additivity of equal parallel forces due to different pieces of ropes. Considerfor example the block of pulleys of Fig. 3.10a [131].49 Hero affirms that the ratiobetween the power and resistance (weight) is 1 : 6 because each of the six pieces ofthe rope sustaining the weight is tied equally by the power and there are six pieces ofrope. To explain why the rope has a constant tension it would be possible to see eachpulley as a balance with fulcrum in its centre and equal arms which is equilibrated

47 pp.106–107. Translation in [133].48 p. 127. Translation in [133].49 p. 99.


η

κ

ζ

Fig. 3.10. System of pulleys (reproduced with permission of Biblioteca Guido Castelnuovo, Uni-versità La Sapienza, Rome)

only if the tension of the rope at its extremities are equal, but it seems to me morenatural to assume as a matter of fact that the tension of the rope is constant.

In the following quotation Hero explains the functioning of a simple block of twopulleys, one fixed and one mobile, as shown in Fig. 3.10b.

Let us now imagine a different weight at ζ and fasten to it the pulley η, pull over this pulleya rope and tie its two ends to a firm crossbeam, so that the weight ζ floats, then each of thetwo tightened parts of the rope lifts the weight of half the load. If somebody now unties theone end of the rope tied at k and stands there himself and holds the rope, then he carries halfthe load and the whole load is twice the force that holds it [131].50 (A.3.10)

For the wedge Hero’s reasoning is not completely clear to me and thus I prefer notto report it. It seems however that he assumed as an explanatory principle a law ofvirtual work according to which the efficacy of the wedge depends on the ratio oftransversal and longitudinal displacements or velocities.

Hero’s failure to reduce all the principles to the lever can be considered as a defectfor those who assumed that science should have an axiomatic structure. But it canalso be seen as the effect of a pragmatic epistemology, that allows a prolific approachto mechanics.

3.2.1.1 A law of virtual work

While there is no doubt that Hero uses a quite advanced form of virtual displacementlaw, it is not easy to decide on its logic status. There is no doubt that Hero, to explainthe operation of machines (with some reservations for the block and tackle, wedgeand the screw), refers to the law of the lever. But to this explanatory principle Herojoins a kinematical analysis, which suggests a law of virtual work. Notice that thekinematical analysis is not a simple geometric exercise; it is required by the nature

50 pp. 115–116. Translation in [133].

56 3 Greek origins

itself of mechanics which is the science of lifting and shifting of heavy bodies, andthe relative displacement of the weight to be lifted and the applied power has a tech-nological relevance. This remains true even when, mainly for the sake of simplicity,the search of the moving power is replaced by the search for the equilibrating power.And what was said above about Hero’s approach is valid also for his followers inthe Renaissance, as for example Guidobaldo dal Monte and Simon Stevin.

To compare Hero’s approach with that of Aristotle, one should refer to the lawof the lever. In the Aristotelian text the law of the lever is a theorem derived fromthe principle of virtual velocities. From the inverse relationship between weight andspeed, with a simple geometrical reasoning, the inverse relationship betweenweightsand distances follows. In Hero’s text, instead the law of the lever is a principle. Fromthe inverse relationship between weights and distances it is possible with simplegeometric reasoning to obtain the law of virtual velocities or, which is the same,of virtual displacements as a theorem. The same reasoning applies to all machines.On this matter it is of some interest to confront points of view different from mine,i.e. the comments of Clagett who sees Hero’s virtual work as a principle [287],51

Duhem who consider Hero as substantially Aristotelian [305]52 and Giusti [333]who considers Hero as essentially Archimedean.

In the following quotation, which deals with how to raise 1000 talents with only5 talents, Hero sets out clearly what is stated above:

A delay occurs however with this tool and those similar to it of great power, because thesmaller the moving force is in relation to the load to be moved, the more time we need[emphasis added], so that force to force and time to time are in the same (inverse) ratio. Anexample for this is the following: Since the force in wheel β was two hundred talents andit moved the load, one requires one rotation for the rope wound around α to wind up, sothat the load through the motion of wheel β moves the amount of the circumference of α.

Fig. 3.11. A series of shafts with the wheel connected in series (reproduced with permission ofBiblioteca Guido Castelnuovo, Università La Sapienza, Rome)

51 Chapter 1.52 vol. 1, p. 186.


If it is moved, however, through the motion of cogwheel δ, the wheel on γ has to move fivetimes for the axle a to move once, because the diameter of β is five times the diameter ofthe axle γ. Thus five times the amount of γ is equal to a single β, if we make the respectiveaxles and wheels equal to one another. But if not, then we find a proportionality similar tothis one. The cogwheel δ moves at β and the five revolutions of δ take fivefold the time of40 talents. Thus the ratio of the moving force to the time is inverse. The same shows withmultiple axles and multiple wheels and is proven in the same way [131].53 (A.3.11)

Notice that Hero in the previous quotation makes reference to time, but Clagett con-tends that it is quite clear from the text that one can identify time with covered space[287]. So one can read a virtual work law according to which the ratio betweenthe moving force and the moved weight is inverse to the ratio of the correspond-ing covered spaces. In the following quotations Hero refers with some emphasis to‘ralentissement de la vitesse’ which occurs in all the machine where with a smallforce a heavy body is raised.

A delay occurs however with this tool and those similar to it of great power, because thesmaller the moving force is in relation to the load to be moved, the more time we need, sothat force to force and time to time are in the same (inverse) ratio.[…]That the delay also occurs with this tool [131].54 (A.3.12)

The interpretation of time with space sustained by Clagett is contrasted by MarkSchiefsky [380], who sustains – really with some oscillations – that Hero does notmake reference to a single machine and compares the space covered by power andresistance in a given time, as Clagett thinks. He should instead make reference to twodistinct machines with different mechanical performance for which it is a matter offact, easily verifiable in practice, that the machine with a lower performance willtake less time than a machine with a greater performance to lift the same weight atthe same height. If the interpretation of Schiefsky were correct it would be moredifficult to see in Hero some form of virtual work law. It remains the fact that areader subsequent to Hero can read him as Schiefsky, but also as Clagett – and me –and derive from Hero a virtual law. The law of virtual work is as a matter of factcontained in Hero’s work, independently of his intentions.

In any case there is at least a circumstance where Hero uses clearly the virtualwork displacement law and as a principle: to explain why a lesser force is neededto displace a weight hung by a wire when one presses farther from the hinge. Thiscan lead to the conclusion that somehow Hero could consider the virtual work lawas an explanatory principle, where the use of the circle or the lever becomes difficultto handle. It should be stressed however that he did not apply, unfortunately, anyvirtual work law to the inclined plane.

Let us, for instance, imagine the firm support that the load is suspended from at point α andlet the rope be the line αβ. Let us now draw the line αγ perpendicular to αβ and let usassume on line αβ two randomly positioned points, δ and ε. […] Thus if we pull the loadfrom ε, it comes to κ; if we pull it, however, from point δ, then it reaches η, so that the load

53 p. 132. Translation in [133].54 pp. 131–132, 134. Translation in [133].

58 3 Greek origins

is lifted higher from point δ than from point ε. The load, however, that is lifted to a higherpoint, strains the force more than the one lifted to a lower point, because the one lifted to ahigher point takes more time [131].55 (A.3.13)

3.2.1.2 Hero’s inclined plane law

In the following I report the treatment of the inclined plane by Hero, that even thoughnot associated in any way to virtual work laws is important because the explanationof its operation affects the entire history of mechanics until at least to Galileo. Herospeaks briefly of the inclined plane apart from other machines in the first book, with-out illustrations and without any explicit quantification. The reasoning is quite sim-ple. Consider Fig. 3.12 [219]56 in which a cylinder is placed on the inclined planeBC. The vertical plane FD divides the cylinder into two parts. Hero argues that theright side of FED is balanced by an equivalent part FHD of the left side, and thereforethe only part that must be supported by an external force is the lunula highlighted ingray in Fig 3.13.

The explanation of Hero is clearly wrong according to modern conceptions ofstatics, because the bodyDHFE although symmetric with respect to the vertical is notbalanced. However, it is clever, interesting for its high rhetoric value and providesresults that are not easily contestable by experience. In particular, it provides a zeroforce if the plane is horizontal and a force equal to the weight of the entire cylinder ifthe plane is vertical. However it should be said that the text contains some ambiguity,as Hero does not speak explicitly of a force parallel to the inclined plane, but rathera force needed to pull the cylinder up:

We will make recourse to some power or weight applied to the other side, to equilibrate thegiven weight, so that an excess of power prevails on the weight and pulls it upward [131].57

(A.3.14)

ε

δ

α γ

ζ

θ

η

κ

β

Fig. 3.12. Force applied transversely to a pendulum

55 pp. 149–150. Translation in [133].56 p. 41.57 p. 71.

3.3 The mechanics of Pappus of Alexandria 59

A B

KG

I

H

F

E C

D

Fig. 3.13. Hero’s proof of the inclined plane law

3.3 The mechanics of Pappus of Alexandria

The last contribution of Greekmechanics is that of Pappus of Alexandria (IV centuryAD). Of the great work of Pappus, in eight books and entitled Synagoge or Math-ematical collections, we possess only an incomplete part, the first book being lost,and the rest having suffered considerably. The last part, Book VIII, treats principallyof mechanics, the properties of the centre of gravity, and some mechanical powers.Interspersed are some questions of pure geometry. Of Book VIII with the Greek ver-sion, translated into Latin by Commandino and Hultsch [181, 182] there exists alsoan Arabic version translated into English by David Jackson [183].58 The Arabic ver-sion is more complete than the Greek one and probably closer to the original; on theother hand the Greek version contains fragments of Hero’s Mechanics.

Pappus’ book 8 describes the five Hero’s simple machines and explains how theywork, referring for the theory to Archimedes and Hero.

The sum of what concerns the knowledge of the centre of gravity is then for the most part aswe have given it. You can learn the basic principles through which this science is acquiredif you look at Archimedes’ book ‘On Aequilibria’, and at Heron’s work ‘On Mechanics’,while here we shall set out in order those points connected with this that most people findunclear, among which is the following [183].59

In this way we learn how to move a given weight with a given force. It is said that thissection of Mechanics is one of Archimedes’ discoveries and that when he discovered it hesaid, “Give me a place to stand that I may move the world for you!” Heron of Alexandriagave a most clear exposition of this operation in his book called “The Drawing ofWeight” inwhich he makes use of a lemma proven in his books on Mechanics where he also mentionsthe five powers which are: wedge, lever, screw, compound pulley, and shaft with wheels.These are the things by which, in general, a given weight is moved by a given power, I meaneach of these powers [183].60

The importance of the text of Pappus is not so much in its content, which is essen-tially an epitome of the mechanics of Hero:

58 To point out an edition by John Wallis in 1688.59 p. 9.60 p. 23.

60 3 Greek origins

Fig. 3.14. A complex mechanical device. The baroulkos or weight hauler (reproduced with per-mission of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome)

That is something which has been explained in the book On Drawing Weight. As regardsthe five powers which we mentioned and which we said had been mentioned by Hero, weshall treat of them briefly as an, aide-memoire for lovers of knowledge [183].61

as the fact that in the Renaissance, along with the De architectura of Vitruvius, itwas the only witness to the Greek-based mechanical technology.

Important aspects of the mechanical theory in the text of Pappus are the famousdefinition of the centre of gravity as the suspension point of equilibrium (see Chap-ter 6), the law of the inclined plane, less convincing than Hero’s and still wrong.Finally, perhaps most importantly, the implicit reference to the law of virtual work,reported in the case of machines where the transmission is by wheels, as shown inthe following two quotations:

In his book “Barulcus”, however, he explains how a given weight is moved by a given powerderived from positioning toothed wheels, when the ratio of the wheel’s diameter to that ofthe axle is five to one, and when the weight to be moved is 1000 talents and the motivepower 5 talents [183].62

I say that the ratio of the speed of movement of wheel A to the speed of movement of wheelB is as the ratio of the number of teeth on wheel B to the number of teeth on wheel A [183].63

3.3.1 Pappus’ inclined plane law

The treatment of the inclined plane of Pappus of Alexandria has even a greater inter-est than that of Hero, not so much for its quality, which is not excellent, but becauseit was well known in the Renaissance, it was the object of praise before and heavycriticism after. Before any comments I refer to Pappus’ analysis, which in reality isalso not easy to decipher.

61 p. 62.62 p. 23.63 p. 56.

3.3 The mechanics of Pappus of Alexandria 61

NM

B

A

H

K D C

FG E X

L

Fig. 3.15. Pappus’ proof of the inclined plane law

With reference to Fig. 3.15:

a) a weight A requires a non-zero forceC to be carried on a horizontal plane;b) the weight A is balanced on the inclined plane by a weight B determined consid-

ering the angled lever with fulcrum L, with A supposed concentrated in E and Bin G;

c) to carry this weight B on the inclined plane, a force D proportional toC given bythe proportionC : D = A : B is necessary;

d) the force necessary to move A on the inclined plane is obtained by adding to Dthe forceC.

Some mathematics gives the relationship: D :C = GE : FG, that for the horizontalplane (GE/FG = 1) gives as it should be D =C, but for the vertical plane (FG = 0)furnishes an infinitive value for D, which is absurd. This fact probably could nothave impressed Pappus who could have said that in practice there is never a verticalplane. Note that Pappus’ formula is deprived of any practical value becauseC is notgenerally known. Moreover as in Hero the direction of the force to move the weightalong the inclined plane is not defined.

It is not easy to justify Pappus’ assumptions, especially c) and d) that to our sensi-bility have little sense. Inmy opinion the only way in which Pappus and his followersaccepted this proof was that it was at the moment the only way to link the inclinedplane with the lever and so to succeed in the reductionist purpose to reduce the wholeof mechanics to the lever.

4

Arabic and Latin science of weights

Abstract. In this chapter Latin and Arabic Middle Ages mechanics are compared,both based on virtual displacements VWLs. In the first part Arabs are consideredwho, with Thabit ibn Qurra, use as a principle of equilibrium a VWL for whichthe effectiveness of a weight on a scale is proportional to its virtual displacementmeasured along the arc of the circle described by the arm from which the weightis suspended. In the second part Latin scholars are considered who, with Jordanusde Nemore, assume as principles two distinct VWLs. A VWL is associated with theconcept of gravity of position for which the efficacy of a weight on a scale is thegreater the more its virtual displacement is next to the vertical. Another VWL isassociated with the resistance of a weight to be lifted, which depends on the liftingentities in a given time. In formulas: What can raise a weight p of a height h canraise p/n of nh.

It was and still is an axiom of historiography that, since its origins, mechanics hasfollowed two main routes classified as Aristotelian and Archimedean [287, 125].The Aristotelian route is associated with Mechanica problemata by Aristotle. Itslaws are proved ‘dynamically’, as balance of tendencies of weights going downward.The level of formalization and rigor varies from author to author but it is usually notexcellent. The Archimedean route is associated with the Aequiponderanti (and to alesser extent with Euclid’s book on the balance) and dynamics is, instead, reduced toa minimum.Weights are considered as plane figures (geometrical instead of physicalentities) with the main concern being evaluation of the center of gravity; moreoverthere is more attention given to rigorous proof than to physical aspects. The level offormalization and rigor is usually excellent.

This dichotomy is a bit simplistic, I think. It forgets for example, that in ancientmechanics – and even up to the XVIII century – the basic law was in any case that ofthe lever. The Aristotelian and Archimedean approach differed only in the mannerof its proof. One could say they differed in the ‘meta-mechanics’, in which differentprinciples were used – such as a VWL or the theory of centres of gravity – to demon-strate as a theorem the law of the lever, then taking it as the principle in dealing withit later in ‘mechanics’.


64 4 Arabic and Latin science of weights

Recent studies, as for example those by Jaouiche [136] and Knorr [345] suggesta different interpretation. According to these authors, while it is certain that thereare two different approaches in mechanics, the assumption that one is derived fromAristotle and the other from Archimedes, is yet to be proved. For example Knorrobserves that the principles of the so-called Aristotelian mechanics express nothingbut a diffuse knowledge which is not unique to Aristotle [345]. This holds true alsofor the violent law of motion according to which a displacement of a body is pro-portional to the force applied. This law was formulated by Aristotle in his De caeloand Physica, as shown in Chapter 3, but it is not difficult to accept that he simplyformalized what was but a diffuse kind of knowledge. According to Knorr most ofthe so-called Aristotelian mechanics could be found in Archimedes’ work and also,may be in lost treatises [345]. Archimedean mechanics would represent simply themore formalized approach adopted by a mature Archimedes, avoiding, in the proofs,physical concepts like force for example, whose meaning is difficult to grasp withcertainty.

Indeed there is a general change in attitude among the historians of ancient sci-ence, especially those educated in mathematics and physics. They are becoming con-vinced that the development of mathematics and mechanics was quite independentof that of philosophy, and that at the most they influenced each other. Therefore, thelabels ‘Aristotelian’, ‘Platonic,’ etc. for ancient scientists are probably less importantthan usually supposed [327].

The theses by Jaouiche and Knorr would explain why, in nearly all the medievaltechnical writings on mechanics, both Latin and Arabic, there is no explicit refer-ence to the name or to the works of Aristotle. Not even in the historical periodswhen diffusion of the theoretical works, Physica and De caelo, was at large. Thisholds true also for those medieval mathematicians who were familiar with Aristo-tle’s philosophical works, among them Thomas Bradwardwine. They seem to bemoving on two levels. On the one hand the mathematician, working on a technicaltext and organizing it more geometrico; on the other hand the philosopher workingon a philosophical text, involving more or less the same arguments. In the chapter,for the sake of simplicity and according to tradition, I will continue to use the labels‘Aristotelian’ and ‘Archimedean’, the former where weight is treated as a motivepower (active factor) or a resisting effect (passive factor), kinematics concepts areintroduced (i.e. virtual motion) and the concept of centre of gravity is not relevant,the latter when the centre of gravity and the rules of their evaluation are dominantconcepts and kinematics has no role.

The Greek concept of mechanics is revived in the Renaissance, with the synthe-sis of Archimedean and Aristotelian routes. This is best represented by Mechanico-rum liber by Guidobaldo dal Monte [86] who reconsiders the mechanics by Pap-pus Alexandrinus, maintaining the original purpose that was to reduce all simplemachines to the lever. During this period mechanics was considered a theoreticalscience and it was mathematical, although its object had a physical nature and hadsocial utility [350].

Texts in the Arabic Middle Ages diverged from the Greek and Renaissance onesmainly because they divide mechanics into two parts. In particular al-Farabi (c 870–

4 Arabic and Latin science of weights 65

950) established the epistemological status of mechanics by differentiating it in thescience of weights and in the science of devices, both considered part of mathe-matics, divided into seven disciplines: arithmetics, geometry, perspective, music,science of weight and science of devices [248].1 The science of devices referred topractical use and construction of machines. The science of weights, probably alsobecause centered on the balance, was a science of equilibrium and not of transportas was the Greek mechanics.

Besides translations of Aristotle’s theoretical works, Physica2 and De caelo,3

available since the IX century, the scholars of Islamic lands had surely access tomechanical writings by Pappus and Hero written in Greek. Also circulating weretwo treatises on the balance attributed to Euclid (known as the Euclid book on thebalance4 andDe ponderoso et levi). It seems instead that of Archimedes’ mechanicalwork, only that on floating bodies was known. Regarding the AristotelianMechanicaproblemata “it can now definitively be established that this text was known to Arabicauthors, thanks to the rediscovery of a significant passage of a partial Arabic epitomefound in the Fifth Book of the Kitab mizan al-hikma [by al-Khazini]” [247, 249].

In the Latin world a process similar to that registered in theArabicworld occurred.Even here a science of weights was constituted, named Scientia de ponderibus.5

Besides this there was a branch of learning called mechanics, sometimes consideredan activity of craftsmen, sometimes of engineers (Scientia de ingeniis).

Texts on mechanics available in the Latin Middle Ages were: Liber de canonio,translated anonymously fromGreek into Latin in an unknown period; Liber karasto-nis, translated into Latin by Gerardo da Cremona from a Thabit ibn Qurra’s Arabictreatise;De ponderoso et levi, from Arabic, attributed to Euclid; Aequiponderanti byArchimedes, translated fromGreek byMoerbeke in 1269;De insidentibus aquae, byArchimedes, translated from Greek in 1269, also by Moerbeke; Liber Archimedis deinsidentibus in humidum or Liber Archimedis de ponderibus, from Arabic and Latinsources, uncorrectly attributed to Archimedes; and Excertum de libro Thabit de pon-deribus, an epitome of the Liber karastonis. There are also indications that in someway Mechanica problemata by Aristotle circulated in some form. Finally there arethe various treatises attributed to Jordanus de Nemore [171, 50, 345], which I willdiscuss in the following sections.

On the whole there are few works where a deep comprehension of Middle Agesmechanics has been attempted from logical, epistemological and ontological pointsof view. The most widely known are [305, 287, 171, 297, 50, 51], who studied the

1 p. 12.2 The Arabic translation of Physica has a long and complicated history. The first translation isattributed to Ibn-an-Nadima (786–803). The best, and only extant today, is by Isahaq-ibn-Hunayn,at the end of the nineth century [334].3 De caelo was translated by al-Kindi circle during the 9th century.4 The attribution of this text to Euclid is controversial. It is known only recently in an Arabic versionby Franz Woepcke [399].5 The expression Scientia de ponderibus comes from the translation from Arabic into Latin ofal-Farabi’s work (Science of devices was instead translated as Scientia de ingenii) by DominicusGundissalinus [248], p. 17.


ancient treatises in their native language. The present chapter follows these attempts,particularly those of Duhem. The objective is not to discuss new sources but to putknown sources under a different light. The study, carried out in some detail, on theway the principles of mechanics evolved, leads to the examination of the evolutionof the various proofs of the law of the lever. Among these proofs, I only considerthose in the Aristotelian route traced in the scientia de ponderibus of the Arabic andLatin Middle Ages, because only they are related to virtual work laws, which is myhistorical point of view. In the first part of the chapter I refer to the most relevantanalyses of the principles of scientia de ponderibus developed in the Islamic lands,by referring mainly to Ibn Qurra Thabit’s treatises because they are the most an-cient available texts. I also mention the contributions by al-Muzaffar al-Isfizari, whopartially followed Thabit, and to al-Kazini. In the second part, devoted to the LatinMiddle Ages, reference is mainly made to Jordanus de Nemore’s (XIII century) trea-tises because they are the first comprehensive texts left to us.

4.1 Arabic mechanics

Both the science of weights and science of devices (machines) were relevant for Ara-bic technology. An important reason for the attention paid to the science of deviceswas the need to solve problems of water lifting in the Iranian plateau, where therewere numerous underground aqueducts [309]. The science of weights was insteadmotivated by a more diffuse need, though apparently less demanding, that of precisebalances.The interest in the balance in Islamic scientific learning was culturally nur-tured by its role as a symbol of good morals and justice. Considered the tongue ofjustice and a direct gift of God, the balance was made a pillar of society and a toolof good governance. But probably the most important reasons should be sought inthe eminent importance of balances for commercial purposes. In a vast empire withlively commerce between culturally and economically fairly autonomous regions,more and more sophisticated balances were, in the absence of standardization, keyinstruments governing the exchange of currencies and goods, such as precious met-als and stones. Abd ar-Rahman al-Khazini, around 1120, wrote the Kitab mizan al-hikma, dedicated to the description of an ideal balance conceived as a universal toolof a science at the service of commerce, the so-called balance of wisdom. This wascapable of measuring absolute and specific weights of solids and liquids, calculatingexchange rates of currencies, and determining time [248].6

As mentioned in the introduction besides Aristotle’s theoretical and technicalworks, Arabic scholars had no doubt access to two treatises on the balance attributedto Euclid (The Euclid book on the balance and De ponderoso et levi), a book onfloating bodies by Archimedes. These Greek texts were joined by Arabic texts, byvarious authors as for example Thabit ibn Qurra (or Qorrah) (836–901), Al Muzaffaral-Isfizari (1048–1116) and Abd ar-Rahman al-Khazini (fl. 1115–1130). They alsoknew mechanical writings by Pappus and especially Hero of Alexandria. In partic-

6 pp. 3–4.

4.1 Arabic mechanics 67

Fig. 4.1. The balance of wisdom (modified from [343]7)

ular Hero’s Mechanica was the object of many comments and translations [309].Interesting for this purpose is the book Kitab al-Hiyal (The book of ingenious de-vices) of the three brothers Banu Musa, scholars from IX century Baghdad [259].Their book is an outstanding contribution in the field of mechanical sciences. In theform of a catalogue of machines, it is a large illustrated work on mechanical de-vices including automata. The book described a total of a hundred devices and howto use them. It was based partly on the work of Hero of Alexandria and Philon ofByzantium and contained original work by the brothers. Some of these inventions in-clude: valve, float valve, feedback controller, automatic flute player, a programmablemachine, trick devices, and self-trimming lamp. Noteworthy is also the work by al-Karaji (c. 953–c. 1029) about hydraulics [310].

But, as it does not seem that Arabs introduced new elements about virtual worklaws in the Heronian texts, in the following I will concentrate on the science ofweights, where instead there was an important Arabic contribution. Particularly Iwill discuss only a treatise on the balance attributed to Thabit, to which I will referin the following with its Latin name Liber karastonis which is on the Aristotelianroute; for works on the Archimedean route see [246].

Al-Sabi Thabit ibnQurra al-Harrani (836–901)was a native of Harran and amem-ber of the Sabian sect. In his youth Thabit was a money changer in Harran. Themath-ematician Muhammad ibn Musa ibn Shakir, one of three sons of Musa ibn Shakir,who was traveling through Harran, was impressed by his knowledge of languages –his native language was Syriac but he knew perfectly Greek and Arabic also – andinvited him to Baghdad; there, under the guidance of the brothers, Thabit became agreat scholar in mathematics and astronomy. Here he translated and revised manyof the important Greek works; particularly all the works of Archimedes that havenot been preserved in the original language, and Apolonius’ Conics. Manuscriptsof Euclid’s Elements which were translated by Hunayn ibn Ishaq were revised by

7 p. 93.


Thabit. Although he contributed to a number of areas, the most important of hiswork was in mathematics where he played an important role in preparing the wayfor such important mathematical discoveries as the extension of the concept fromnatural number to real numbers, ‘integral calculus’, theorems in spherical trigonom-etry, analytic geometry, and non-euclidean geometry. In astronomy Thabit was oneof the first reformers of the Ptolemaic system, and in mechanics he was a founder ofthe science of weights [290].

The Liber karastonis is one of the most important treatises in the science ofweights of Arabic origin.

The medieval science of weights owed an extraordinary debt to the production of a sin-gle work, a treatise on the balance, Kitab al Qarastun, by the ninth century mathematician-astronomer Thabit ibn Qurra. It retained a prominent place within the theoretical section onmechanics in the rich compendium compiled by al-Khazini, Kitab mizan al-hikma, two cen-turies later. Beginning from the 12th century, it exercised a major influence on mechanicalstudies in the LatinWest, through the translation as the Liber karastonis made by Gerardo daCremona. Four centuries later, writings onmechanics still clearly betrayed their proveniencethrough elaborations and commentaries on this work [345].8

It was translated into Latin during the XII century by Gerardo da Cremona [171].9

A large number of manuscripts exist, all derived from a unique progenitor. In whatfollows I will refer to the text edited byMoody and Clagett, which is derived partiallyfrom a manuscript conserved in Paris and partially from a text edited by Bucher,based on a manuscript conserved in Milan [171],10 [264]. It will be enough for mypurpose that is not to present a philological correct version of the text, but only toevidence some particular aspects that would have emerged in any way independentlyof the interpretation of particular words or phrases.

Besides the Liber karastonis, at least three manuscripts in Arabic exist, with anal-ogous subjects [246] [136],11 called Kitab al-Qarastun, controversially attributedto Thabit [345]12[136],13 one conserved in London, one in Kraków and another inBeirut.14 The first manuscript was edited, translated into French and commented onby Khalil Jaouiche [136]. The second, while in Berlin, was edited and translatedinto German by Eilhard Wiedmann [397], and subsequently studied by MohammedAbattouy [246]. The third one has been studied by Wilbur Richard Knorr [345].

4.1.1 Weight as an active factor in Arabic mechanics

In Arabic mechanics the main dynamic concept is the motive power associated withweight. The fact that weight plays the role of an active factor, like a ‘force’ is stressed

8 p. 5.9 pp. 77–118.10 pp. 84–85.11 pp. 2–3.12 p. 47.13 p. 31.14 Mohammed Abattouy registers a recent hitherto unknown copy in Florence [247], p. 17.


over the fact that it opposes resistance to upward motion. From this idea it followsa virtual work law for which the efficacy of a weight on a balance depends on thevirtual displacement of the point at which the weight is hung. This will be clear evenwith a superficial reading of Thabit’s book.

4.1.1.1 Liber karastonis

The Liber karastonis is composed of a prologue followed by eight propositions andfinally a comment. They all relate to the karaston, that is the steelyard or Roman bal-ance, which is a straight-beam balance with arms of unequal length. It incorporatesa counterweight which slides along the calibrated longer arm to counterbalance theload and indicates its weight. The exposition of the theory, though it is classifiedin the Aristotelian route, has a high standard of rigor, not far from the texts of theArchimedean route, with the exception of the first propositions, where the reader isasked to accept much more than in the Archimedean route.

Immediately after the prologue the following propositio (proposition) I is pre-sented:

I. I say, therefore, in the case of two spaces which two moving bodies describe in the sametime, that the proportion of the one space to the other is as the proportion of the power ofthe motion of that which cuts the one space to the power of the motion of that which cutsthe other space.15

I posit the following example for this proposition. In the case of two walkers, one walksthirty miles and the second walks sixty miles in the same time. It is noted, therefore, thatthe power of the motion [emphasis added] of he who walks the sixty miles is double thepower of the motion of he who walks the thirty miles, just as the space sixty miles is doublethe space thirty miles. This proposition is admitted per se and is immediately evident to theintellect [171].16 (A.4.1)

The term proposition in ancient texts usually means theorem; but what is writtenjust after “This proposition is admitted per se and is immediately evident to the in-tellect”, qualifies it rather as a principle. The assertion of evidence, not completelyshared by the modern reader, suggests that in the cultural climate of the period, theproportionality between force and displacement were part of common knowledgeand Arabic natural philosophy, be it derived from the Aristotelian texts or not.

Before attempting to comment on proposition I, let me clarify its content. In theproposition, the motion of two bodies, whose shapes are not specified, is discussed.The distances covered are introduced also (plane in the Arabic version), withoutspecifying the kind of pattern. Because in the core of the Liber karastonis the arcsof a circle are considered, it can be presumed an affirmation of general character ispresent and then the paths can be any thing.

By entering the merit of proposition I, two things should be stressed. First, thereis no distinction between natural and violent motions here. This confirms the atti-tude of Arabic scholars to consider weight as an intrinsic mover and consequently to

15 A similar statement is found also in the Liber Euclidi de ponderosi et levi: “bodies are equal instrength whose motions through equal places, in the same air or the same water, are equal in times”and in some following propositions [171], p. 27.16 p. 90.


consider both natural and violent motions as a consequence of a force. For the associ-ation of a force to weight see for example the references listed in [246].17 Secondly,reference is made to “virtus motus” (power of motion), and not “virtus”. This createsinterpretation problems, which can be overcome, as Jaouiche does, by consideringthe Latin version to be affected by an error of translation from Arabic into Latin byGerardo da Cremona [136].18 According to Jaouiche the Arabic version of propo-sition I suggests “virtus of mobile (force du mobile)” instead of “virtus of motion(virtus motus)” [136].19 To confirm his thesis, Jaouiche considers the presence of thesentence which specifies the equality of times, simply to pay homage to the traditionand then not essential [136].20 I prefer an interpretation which consists in looking atproposition I as a reinterpretation of the ‘Aristotelian’ laws of motion. In these lawsthe measure of force was known a priori, independently of motion; Thabit insteadsuggests measuring force a posteriori by the effects it produces; more precisely, bythe space covered in a given time: the greater the space covered the greater the act-ing force. It then seems correct to speak about “virtus motus” instead of “virtus”.The suggested interpretation of proposition I, which is so considered as a virtualwork law formulation, makes it easier to understand the proof of the law of lever(proposition III) whose statement is given below:

III. Since this is manifest now, then I propose [the following with respect to] every linewhich is divided into two different segments and imagined to be suspended by the dividingpoint and where there are suspended on the respective extremities of the two segments twoweights, and the proportion of the one weight to the other, so far as being drawn downwardis concerned, is inversely as the proportion of the lines. [I say that in these circumstances]the line is in horizontal equilibrium [171].21 (A.4.2)

The Arabic version is analogous. In what follows I refer to Abattouy’s translation:

This being proved, I say that if the line AB is suspended from point G and there are set at itsends, at point A and B, two weights proportional to its two parts and inversely proportionalto them, the line AB will be parallel to the horizon [246].22

The proof of proposition III, has to relay, besides proposition I, on the followingproposition II:

II. Then I say that in the case of every line which is divided into two parts and fixed at thedivision point and where the whole line is moved with a movement not directed to its naturalplace, then such a movement produces two similar sectors of two circles. The radius of oneof these circles is the longer line and the radius of the second is the shorter line. And theproportion of the arc which the point of the extremity of one of the two lines describes tothe arc which the point of the extremity of the other line describes is as the proportion ofthe line whose revolution produces the one arc to the line producing the other arc [171].23

(A.4.3)

17 pp. 33–35.18 p. 120.19 pp. 146–147.20 pp. 50–63.21 pp. 92, 94. Translation in [171].22 pp. 37–38.23 p. 90. Translation in [171].


baA G

E B

T

D

Fig. 4.2. The lever with different arms in the Latin manuscript

which in itself only expresses a theorem of plane geometry, according to which for anassigned angle arcs and radii of a circle are proportional, for an assigned angle. Butmainly it needs the following comment Thabit added to proposition II after havingproved it:

We have already said that in the case of two spaces which two moving bodies describe inthe same time, the proportion of the power of the motion of one of the bodies to the powerof the motion of the other is as the proportion of the space which the first motion cuts tothe other space. And point A with the motion of the line has already cut arc AT and pointB with the motion of the line has already cut arc BD, and this in the same time. Therefore,the proportion of the power of the motion of point B to the power of the motion of point Ais as the proportion, one to the other, of the two spaces which the two points describe in thesame time, evidently the proportion of arc BD to arc AT. This proportion has already beenshown to be the same as the proportion of line GB to line AG [171].24 (A.4.4)

From the previous passage it is convenient to distinguish between what Thabit saysand why he says it. Thabit clearly affirms that the “power of motion” of the point Bof the longest arm of the balance is greater than that of the point A, or more generallythat the power of motion of a point of a balance is directly proportional to its distancefrom the fulcrum. Note that displacement is measured according to arcs of circlesthat the weights describe in their motion; this is not peculiar to Thabit, but can befound also in the works by al-Isfizari:

Since the two weights A and B were supposed to be equal, the motion took place becausethe arc BO, along which the weight B moves with a natural motion is greater than the areAS, along which A moves with violent motion [246].25

and by Galileo himself [119].26

Thabit justifies his affirmation by saying “We have already said”, which can onlymake reference to proposition I. But this induces, at least formodern readers I think, aserious interpretation problem. Indeed proposition I when adapted to weight seemsto make sense only for downward motions, but in the previous passage Thabit isconsidering both upward and downwardmotions. One (Thabit?) could overcome thisdifficulty by assuming that if a weight suspended from one side of a balance moves

24 p. 92. English translation, my accommodation.25 p. 44.26 p. 164.


upward it could move downward too the same distance in the same amount of time,when the rotation of balance is imagined to revert and then one can always makereference to a possible downward motion. The same problem occurs in Galileo’sdemonstrations about equilibrium with the use of the concept of ‘momento’ (seeChapter 5).

A translation into modern concepts of Thabit’s reasoning, not consistent for mod-ern mechanics, could be based on the following assumptions:

a) there are two bodies A and B of equal weight;b) if the two bodies A and B are suspended from a balance with different arms, with

B on the longest arm, the balance will sink on the side of B;c) because B describes a greater arc than A, in the same amount of time, in point B

there is a greater power of motion (VWL);d) the power of motion of B is proportional to the distance of B from fulcrum.

Here is Thabit’s proof of proposition III:

The demonstration of this follows: I cut from BG the longer segment an amount equal toAG the shorter segment. This cut off line is GE. If then, two equal weights [a and b] aresuspended at points A and E, the line AE will be in horizontal equilibrium, since the powerof motion at the two points is equal as we have demonstrated. So that if I incline point A topoint T, the weight [b in E] there suffices for its return to a position of horizontal equilibriumthrough arc AT. And when we change the weight from point E to point B, and if we wishthe line to remain in horizontal equilibrium, it is necessary for us to add something extrato the weight at A, so that the proportion of its total to the weight which is at B is as theproportion of BG to AG. Since the power of the point B exceeds the power of point A by theamount that BG exceeds AG, as we have shown, hence the weight which is at the point ofthe stronger power is less than the weight which is at the point of weaker power accordingas is the proportion of arc to arc. Therefore, when there is a weight at point B and a secondweight at point A and the proportion of weight a to weight b is as the proportion of GB toAG, the line is in horizontal equilibrium [171].27 (A.4.5)

The proof unfolds into two steps. In the first step a symmetry situation is considered,equal weights being located at the same distance from the fulcrum. The equilibriumis not considered as being self-evident, but is justified by means of proposition I.The two weights compensate each other because powers of motion of their points ofsuspension are equal as they pass equal arcs (for proposition II) in the same amount oftime. This part of the proof is followed by the comment that if one weight is inclineddownward the other will force it back and the horizontal position is recovered. Thatis, in modern terms, the horizontal position is stable (which today is known to befalse).

In the second step Thabit proposes to lengthen the arm by moving the weightfrom E to B so that GB>GE. Thabit says two things: a) the balance inclines towardthe side B; b) to resume equilibrium the weight A must be increased until the inverseproportionality between weights and distances holds good. The first assumption isnot justified by Thabit, perhaps because its justification is contained in the second.The second assumption is justified by the comment added to proposition II, previ-ously referred to where it was concluded that the power of motion of a point on a



b

gd

d

ab

a

Fig. 4.3. The lever with different arms in the Arabic manuscript

balance arm is directly proportional to its distance from the fulcrum. So when thetwo weights are inversely proportional to the distances their powers are equal andthe balance is in a status of equilibrium.

4.1.1.2 Kitab al-Qarastun

Arabic manuscripts are quite different from those in Latin. In what follows I willuse essentially the version edited by Jaouiche, with a few references to the othermanuscripts.The order of propositions, indeed not numbered, in the Arabic versionsis different from the Latin one. Jaouiche uses the name postulate for the propositioncorresponding to proposition I (which, moreover, is not located at the beginning).He uses the name lemma for the proposition corresponding to proposition II, whileproposition III is named theorem 1. The texts of propositions are virtually the same asthose in the Liber karastonis, except for secondary aspects. The texts of explanationsand/or proofs are instead very different; shorter andmuch less satisfactory than thoseof the Latin version.

Postulate I is not followed by any comment; similarly the lemma is not followedby the dynamical comment where the proportionality between power and distancefrom fulcrum is affirmed. Thus the proof of theorem 1 (Jaouiche nomenclature) isincomplete, as clear from the following piece which refers to it in full:

I say if ab is suspended [Fig. 4.3] at g and if at both ends, a and b, two weights proportionaland equivalent to these two segments are applied, [ab] is parallel to the horizon. Indeed,taking on the longest side ag a segment gd equal to gb, if one applies to d a weight equal tothe weight applied to b, [ab] is parallel to the horizon. If the weight which is in d is tilteddown, the weight which is in b will rise and pass the arc dd equal to the arc bb because gdis equal to gb. If then the weight is moved from point d to point a, the latter being in thelower position and one wants to raise it up to the higher position a, one must increase theweight in b such that the ratio of total [weight] [in b] to the weight in a is equal to the ratioof the arc aa to the arc dd, which are passed in the same time though they are uneven. Butthis ratio is equal to the ratio of one of two segments of the straight line to the other [136].28

(A.4.6)

The version of Berlin’s manuscript is substantially equivalent:

We cut from the longer AG [a segment] like GB and that is GD. If a weight equal to theweight at B is suspended from point D, AB will be parallel to the horizon, so that if it isinclined from the higher D to the lower D, the weight at B would move it and raise it up tothe higher D, making it traverse the arc DD. But the arc DD is equal to the arc BB, for GD

28 p. 149.


is as GB. Nevertheless, the arc DD and the arc AA are traversed in the same time. Henceif we move the weight from D to the lower A and we wish that it is raised up to the higherA, we will need to add to the weight at B an addition such that the ratio of the whole to theweight at A will be as the ratio of the arc AA to the arc DD, if these two arcs are traversedin the same time even though they are different. This ratio is the ratio of one of the two partsof the line to the other [246].29

A similar reasoning is developed by Al-Isfizari:

Therefore, we have here two distinct notions each one of them requiring the sinking, namelythe weight and the distance. The excess of one of them over the other in weight is as theexcess of this latter over the former in distance. The equality between them required thecounterbalance and that the beam is extended in parallelness to the horizon, so that the lineAB remains parallel to the horizon. The ratio of the arc BO to the arc AS is as the ratio ofthe line GB to the line GA, as it was demonstrated by Euclid in his book [246].30

and the explanation of the reason for equilibrium of the balance with equal weightand arms is more detailed. As compensation for its shortness, the proof of theorem1 is followed by a reference to a case where the balance arm is deprived of weight:

If the axis is heavy and it is divided into two unequal segments, we increase the thickeningof the shortest segment until the axis is parallel to the horizon. […] We are then reduced tothe case already treated in the axis free of weight [136].31 (A.4.7)

The obvious conclusion is that Arabic manuscripts do not add anything of impor-tance to the lever law interpretation, at least from the point of view of the role playedby weight.

4.1.2 Comments on the Arabic virtual work law

Thabit’s argument to prove the law of the lever is based on the assumption thatequilibrium is determined by the equity of causes of motion, that is, of the motivepowers of weights. This is an axiom of ancient philosophy, which, however, haslittle meaning when translated into the precise language of physics. Indeed Thabit’sreasoning is successful because there is a shift in the meaning of the term power(virtus). According to proposition I, power should be interpreted in the usual way, i.e.as amuscle force. But this positionwhen carried out coherently leads to a paradoxicalconsequence. If two equal powers – equivalent to two equal muscle forces – areapplied to the extremities of a balance with different arms, the balance cannot bein equilibrium whatever the ratio of the arms might be. To overcome this paradox,power should be given a different meaning, that of the efficacy of the power or thecapacity to produce a rotation of the balance; in modern terms the meaning of staticmoment. This is a rhetorical artifice which has a relevant heuristic role but no valuefrom a logical point of view. The same problem arises for many proofs of the leverlaw, that of Galileo included.

29 p. 38.30 p. 44.31 p. 15.

4.2 Latin mechanics 75

Anyway the choice of Thabit to measure by the ease with where power its point ofapplication moves is a form of virtual work law. By comparing this form with thoseof modern virtual work principles, some similarities and differences are found. Oneof the differences is that in the modern laws, displacements are evaluated along thevertical direction, and are consequently straight, while Thabit makes reference tocurvilinear paths. Regarding the role of time, which is implicit in Thabit’s principle,it can be said that there is not a substantial difference. Actually, even in the modernprinciple, though it does not appear in the enunciation, time plays an essential role;indeed various virtual displacements occur in time; they are contemporary (in respectof the congruency of the system moved) and then occur in the same given time.Anyway, neither Thabit nor adherents of the modern principle have any interest inthe effective measurement of time.

4.2 Latin mechanicsAccording to many historians of science, the reasons for interest in the mechanicsof Latin Middle Ages is different from that of Arabic Middle Ages, less oriented tothe needs of society. The development of mechanics in Europe in the XIII and XIVcenturies should be referred to the general revival of interest in the texts from theGreek and Arabic worlds, and then somehow separated from applications. It must besaid though that if in the Latin Middle Ages there was no need to study the scienceof weights in order to design appropriate scales for trade, there was a stimulus toimprove the general knowledge of statics required by construction of the Gothiccathedrals, very daring buildings that saw their heyday in the XIII century. So it islikely that it was not just a cultural interest to date from the XIII century the writingsof Jordanus de Nemore, which I will discuss below, which represented a significantimprovement compared to those of Thabit, especially because it covered a widerrange of problems.

In the Latin Middle Ages various treatises on the scientia de ponderibus circu-lated, as already noted. Some were Latin translations from Greek or Arabic, a fewwere written directly in Latin. An outstanding scholar was Jordanus of Nemore orJordanus Nemorarius. Practically nothing is known about his life. He appeared atthe beginning of the XIII century. Besides writings about mechanics he was the au-thor of many mathematical works [344, 337, 305]. Treatises attributed to Jordanusare: Elementa Jordani super demonstratione ponderum (version E), Liber Jordanide ponderibus (cum commento) (version (P), Liber Jordani de Nemore de rationeponderis (version R, discovered by Duhem). They used to be commented upon upto the XVI century; recently they have been studied by historians of science withvarious tendencies.

For the medieval comments there are manuscripts of the XIII century, classi-fied by Moody and Clagett as Corpus Christi [50]; the manuscript of the XIV cen-tury published by Petrus Apianus in 1533, referred to as Aliud commentum [50];the manuscripts of the XIV century named Commentum Henrici Angligena [50]; theQuestiones super tractatum de ponderibus, by Blasius of Parma, of the XIV-XVcenturies [50]. For modern comments there are essentially those by Duhem, Clagett,Moody and Brown, in the already cited works.


Fig. 4.4. A modern view of the Erfurt cathedral

Duhem’s studies are the only ones which exhibit a deep understanding of me-chanical concepts, and notwithstanding some justified criticisms on the historicalapproach, they remain still fundamental. In what follows I will mostly refer back toDuhem, seeking to get an understanding of concepts rather than to attempt a histor-ical and philological reconstruction of treatises by Jordanus, which may be foundin the secondary literature32 giving only a few hints about this reconstruction, toshow that its results are in evident contrast to those obtained by a scrutiny of thefundamental concepts of the various treatises.

The law of the lever, or more in general a form of virtual work law, could bederived by a contemporary physicist with the Aristotelian violent motion law alone,for which if A move B to Γ then Amoves 1/2 B to 2 Γ (in a given time), in a differentand easier way than that carried out by the Arabic mathematicians. This has beenclearly shown by Duhem:

Consider a lever with power α and resistance β; the resistance is at a certain distance fromthe fulcrum. If the power α can move β so that it describes in a time δ the arc γ, it wouldmove the weight β/2, located at a double distance from the fulcrum, in the same time δ andpass an arc 2γ. It needs so the same power to move a certain weight, located at a certaindistance from the fulcrum, and a half weight to a double distance. From this we can easilyjustify the theory of the lever as given in the Mechanica problemata [305].33 (A.4.8)

32 There are various hypotheses about the roots of Jordanus’ mechanical works. Quite convincingis the hypothesis of the Arabic roots: [248], p. 17; [312], pp. 4, 12; [50, 287].33 vol. 2, p. 122.


Table 4.1. The suppositions by Jordanus

S1 The movement of every weight is toward the center (of the world), and its strength is a powerof tending downward and of resisting movement in the contrary direction.

S2 That which is heavier descends more quickly.

S3 It is heavier in descending, to the degree that its movement toward the center (of the world)is more direct.

S4 It is heavier in position when in that position its path is of descent is less oblique.

S5 A more oblique descent is one which in the same distance, partakes less of the vertical.

S6 One weight is less heavy in position, than another, if it is caused to ascend by the descent ofthe other.

S7 The position of equality is that of equality of angles to the vertical, or such that these are rightangles, or such that the beam is parallel to the plane of the horizon.

Duhem’s argument leads to a mathematical relation analogous to that found with theArabic version of the virtual work law as referred to in the preceding section: the keyfactor for equilibrium is the product of weight times virtual displacement (in a giventime). It must be said that Duhem keeps his argument at a superficial level, cavalierlyconfusing concepts of force and static moment or work, more or less the same thatThabit did. If he had taken seriously Aristotle’s law of violent motion which wasformulated only for free bodies he would have never applied it to bodies suspendedfrom a lever.

In what follows I will try to understand whether Duhem’s argument is similar ornot to that carried out by Latin medieval scholars, who since the XIII century wrotetreatises on mechanics, or more precisely on the scientia de ponderibus.

I will take as the basic treatise the Liber Jordani de Nemore de ratione ponderis(version R, in the following De ratione) as edited by Ernest Moody and MarshallClagett [171]. They tried to get the most plausible version from a mixing of variousmanuscripts. A more philologically accurate reconstruction of the text can be foundin Joseph Edward Brown [50].34 I will also refer to some comments and especially tothe version ofDe ratione edited by Nicolò Tartaglia [224] and toQuesiti et inventionidiverse, also by Tartaglia [223], which largely represents a paraphrase of it. For morecomments see [273].

The De ratione is quite a complex treatise, divided into four books. In the firstbook there are the principles and theorems of the science of weights. The second andthird books are more technical and concern the solutions of some of the problemsof the balance, with arms endowed or not with natural weight. The fourth book isabout various issues, among which the fall and breaking of bodies.

The first book, the one concerning the principles and theorems, starts with sevensuppositions (suppositiones) – referred to in Table 4.1 (A.4.9). The suppositionshave different logical status. Some look like principles in the modern sense (S2, S3),some look like definitions (S5), some others are difficult to classify. Supposition S1is the most complex one. It contains:

34 p. 75.


a) a principle in the modern sense (“Omnis ponderosi motum esse ad medium”);b) a definition (that of “virtu”) (“virtutemque ipsius esse potentia ad inferiora ten-

dendi virtutem ipsius et motui contrario resistendi”).

Suppositions from S3 to S6 introduce the gravity of position concept. In suppositionS3 Jordanus makes a generic assertion, for which a body weighs more the moredirectly it goes toward the centre of the world. It implies that ‘heaviness’ dependsnot only on the body, but also on its possible, or virtual, motion, assuming so aform of virtual work law. In supposition S4 the meaning of S3 is specified, with theintroduction of the locution gravity according to position. A body is heavier thananother, by position, when its descent is less oblique. It is then stated precisely whena motion is less or more oblique in supposition S5: a direction is more oblique thananother when it is closer to the horizon. Which is in clear contrast to the modernuse of the term obliquity, but coherent with Jordanus’ ideas for which the referencedirection is the vertical one.

Supposition S6 on the one hand can be seen as a definition of ‘less heavy’, on theother hand it describes a factual situation, the rising of a less heavy body caused bya more heavy body. The same holds for supposition S7, which on the one hand canbe seen as a definition of equilibrium and on the other hand as a factual situationrepresenting equilibrium. Suppositions S6 and S7 make sense only for two weightsbelonging to a balance. This holds true for S1 as well, because it refers to contrary orupward motions. Indeed considering contrary motions would require implicitly theassumption of a force causing them; but De ratione concerns only weights and thencontrary motion cannot be due but by the weight placed on the side of the balancewhich is opposite to that to be raised. Supposition S6 makes it clear that Jordanuswould consider a weight to be able to rise another weight and then to act as a motivepower. However in Jordanus’ treatise it is never explicitly stated that both weightssuspended from the end of a balance tend to go down. Rather it seems that as a bodyis pushed up it loses its heaviness. It is not clear if this corresponds to a Jordanus’philosophical conception or if it is simply due to his difficulty in quantifying thetendency of bodies to move downward.

Jordanus’ suppositions contain certain keywords which would deserve a com-ment because their meaning is not so easy to grasp: “gravis”, “ponderosus”, “veloc-itas”, “virtus”, “gravitas secundum situm”. In what follows, for the sake of space, Ishall limit myself to commenting on the last two keywords which have a particularimportance. The interpretation of “virtus” is quite a delicate question. One is temptedto associate it with the meaning of force. There are however reasons not to do this.The most important is that virtus, besides the tendency to go downward, representsthe resistance to go upward.

In the De ponderoso et levi, virtus was connected to velocity, at least for themotion according to nature: “Bodies are equal in virtus when their motions are equalin equal times and equal spaces in the same air or water” [171].35 Nothing is insteadsaid for the motion against nature. Nicolò Tartaglia wrote in the Quesiti et inventionidiverse: “Definition four. Bodies are of the same virtus or power when in equal time

35 p. 26.


they pass equal spaces” [223]36 and added to the supposition I of the De rationehe edited this interesting comment, not appearing in the text edited by Moody andClagett:

The motion of every body is toward the centre and its virtus is a power of tending downward,and we can understand the power from the arm length or from its velocity which is deter-mined by the length of the balance arms and to resistance to the contrary motion [224].37

(A.4.10)

The final part of Tartaglia’s supposition P1 explicitly asserts that the weights are notfree but are suspended from a balance and proposes a method to evaluate the virtus:virtus is measured by velocity.

Jordanus does not explain what causes virtus, but his use of a unique term forboth motion against and according to nature, should indicate he is thinking of aunique cause. A modern term to translate virtus could be ‘heaviness’, but this createsambiguities. For this reason in what follows virtus will often not be translated, or insome cases it will be translated as strength.

Concerning the concept of gravity of position, it can be said that there iswidespread agreement among historians [287, 305] that it is partially derived fromMechanica problemata. This could be evident enough from suppositions, particu-larly from supposition S3, and is suggested by the preface of version P which doesnot start directly with the suppositions, as the other treatise attributed to Jordanusdoes, but presents an ample discussion from which I refer to the outstanding points:

It is therefore clear that there is more violence in the movement over the longer arc, thanover the shorter one; otherwise the motion would not become more contrary (in direction)Since it is apparent that in the descent (along the arc) there is more impediment acquired,it is clear that the gravity is diminished on this account. But because this comes about byreason of the position of the heavy bodies, let it be called positional gravity in what follows.For in reasoning in this way about motion, as if the motion were the cause of heavinessor lightness, we conclude, from the fact that a motion is more contrary (in direction) thatthe cause of this contrariety is more contrary – that is, that it contains a greater element ofviolence. For if a heavy body descends, this occurs by nature; but that its descent is along acurved path, is contrary to its nature, and hence this descent is compounded of the naturaland the violent. But since, in the ascent of a weight, there is nothing due to its nature, wehave to argue as we do in the case of fire, because nothing ascends by nature. For we reasonconcerning the ascent of fire, as we do concerning the descent of a heavy body; fromwhich itfollows that the more a heavy body ascends, the less positional lightness it has, and thereforethe more positional gravity [171].38 (A.4.11)

Beside the consideration of motion along an arc of a circle with different radii, oneshould make note of the explicit introduction of the locution “gravitas secundumsitum”. Of course the preface of version P does not demonstrate the influence of theMechanica problemata on the derivation of this concept. Perhaps the preface wasadded for the sake of ‘completeness’ by the editor.

36 p. 82r.37 p. 3.38 p. 150. Translation in [171].


It is not easy to express with a unique term the gravity of position concept. Fordownward motion, with a little forcing, the gravity of position can be represented bythe product of the weight (p), considered as an active force multiplied by the (vir-tual) velocity of sinking (v), mathematically pv (this is essentially what the Arabicmechanics did). It is difficult to say whether Jordanus would recognize himself inthis representation. In effect he never gave a mathematical expression to gravity ofposition. For him it remains a qualitative concept, defined by the more or the less,which is useful to prove certain assertions but not to furnish numerical laws. Jor-danus used the concept of gravity of position, for example, to show that a balancewith different arms and equal weights is not in a horizontal equilibrium but sinkstoward the longer arm, or to show that a balance with equal weights and arms is ina stable equilibrium configuration. When he needed a mathematical law he used adifferent approach, described in the following.

Another issue is raised by historians of science on Jordanus’ concept of gravityof position. If gravity is supposed to be a quality, a form of the body, it is possibleto think the gravity could change by varying the disposition of the body with respectto another body. If instead the gravity is conceived as a force (internal or external)independent of the position of the body, absolute gravity, it effectiveness can begreater or lower depending on the resistance of the constraints.

4.2.1 Weight as a passive factor in the Latin mechanics

In the Latin mechanics, the two dynamical concepts of virtus and gravitas secundumsitum associated with weight, make sense for both upward and downward motions.In the first case they appear as passive factors because the weight for its virtus orgravitas opposes a resistance when an applied force tends to raise it; in the secondcase they appear as active factors, because the weight for its virtus or gravitas isresponsible for a motive power directed downward. Jordanus uses the concept ofgravitas secundum situm mainly as an active factor and the concept of virtus (re-sistendi) as a passive factor. The passive factor is the only one used to formulatein mathematical terms a virtual work law, and from this point of view it is the keydynamical concept. This will be clear from the examination of a few theorems of theDe ratione.

4.2.2 Propositions

After the suppositions, the De ratione continues with forty three propositiones (the-orems); Table 4.2 refers to the first ten (A.4.12). Among them the propositionesP1, P2, P6, P8 and P10, dealing expressly with the principles of mechanics, have aparticular relevance. Though the reference treatise is the De ratione (R version), Iwill consider also the versions E and P and some comments, by Middle Ages andRenaissance scholars.


Table 4.2. The first ten propositions by Jordanus

P1 Between any heavy bodies, the strengths are proportional to the weights.P2 When the beam of a balance of equal arms is in the horizontal position, then, if equal weights

are suspended from its extremities, it will not leave the horizontal position; and if it is movedfrom the horizontal position, it will revert to it. But if unequal weights are suspended, thebalance will fall on the side of the heavier weight until it reaches the vertical position.

P3 In whichever direction a weight is displaced from the position of equality, it becomes lighterin position.

P4 When equal weights are suspended from a balance of equal arms, inequality of the pendantsby which they are hung will not disturb their equilibrium.

P5 If the arms of the balance are unequal, then, if equal weights are suspended from their ex-tremities, the balance will be depressed on the side of the longer arm.

P6 If the arms of a balance are proportional to the weights suspended, in such manner that theheavier weight is suspended from the shorter arm, the weights will have equal positionalgravity.

P7 If two oblong bodies, wholly similar and equal in size and weight, are suspended on a balancein suchmanner that one is fixed horizontally onto one arm, and the other is hung vertically, andso that the distance from the axis of support to the point from which the vertically suspendedbody hangs, is the same as the distance from the axis to the mid point of the other body thenthey will be of equal positional gravity.

P8 If the arms of a balance are unequal, and form an angle at the axis of support, then, if theirends are equidistant from the vertical line passing through the axis of support, equal weightssuspended from them will, as so placed, be of equal heaviness.

P9 Equality of the declination conserves the identity of the weight.P10 If two weights descend along diversely inclined planes, then, if the inclinations are directly

proportional to the weights, they will be of equal strength in descending.

4.2.2.1 Proposition I

Table 4.3 refers to different accounts of the proposition I for version E (italic), P(small caps) and R (A.4.13). Notice that proposition I, at least for versions E and P,is somehow equivalent to proposition I of the Liber karastonis. Its logical status ishowever different; there it was a principle, here it is a theorem. In short Jordanus ismore prudent than Thabit; instead of assuming the proportionality between weightand velocity – notice that now velocity is considered like a well-defined kinemati-cal quantity (see § 3.1.1.1) – he assumes a weaker statement which asserts only themonotony between weight and velocity as expressed by the supposition S2, accord-ing to which the greater the weight, the greater the velocity.

Table 4.3. Different accounts of Jordanus de Nemore’s proposition I

The proportion of the velocity of descent, among heavy bodies, is the same as that of weight, takenin the same order, but the proportion of the descent to the contrary ascent is the inverse proportion.

BETWEEN ANY TWO HEAVY BODIES, THE PROPER VELOCITY OF DESCENT IS DIRECTLY PROPOR-TIONAL TO THE WEIGHT, BUT THE PROPORTION OF DESCENT AND OF THE CONTRARY MOVEMENT OF

ASCENT IS THE INVERSE.

Between any heavy bodies, the strengths are proportional to the weights.


The delicacy of proposition I is highlighted by the fact that it is given differentaccounts; the statements of versions E and P are substantially the same but differfrom that of version R in two important aspects. They refer to the relation betweenweight and velocity rather than to weight and virtus and they consider explicitlyboth the downward and upward motions. It is possible that the substitution of theterm “virtus” (strength) in version R to the term “velocitas” was made to allow aunitary treatment of upward and downward motions, because the concept of virtusis independent of the direction of motion.

Here is the proof of proposition I of version R:

Let there be two weights, ab and c, of which c is the lighter. And let ab descend to d, andlet c descend to e. Again, let ab be raised to f , and c raised to h. I then say that the distancead is to the distance ce, as the weight ab is to the weight c; for the velocity of descending isas great as the power of the weight. But the power of the combined weight consists of thepowers of its components. Let the weight a then be equal to the weight e, so that a’s poweris the same as that of e. If then the ratio of the weight ab to the weight c is less than the ratioof the power of ab to the power of e, the ratio of the weight ab to (its component weight)a will likewise be less than the ratio of the power of ab to the power of a. And thereforethe ratio of the power of ab to that of b will likewise be less than the ratio of the weight abto the weight b. Consequently the ratio of the same weights will be both greater and lessthan the ratio of their powers. Since this is absurd, the proportion must be the same in bothcases. Hence the weight ab is to the weight c, as the distance ad is to the distance ce, andconversely as the distance ch is to the distance a f [171].39 (A.4.14)

h

e

c

f

d

a b

Fig. 4.5. Downward and upward motions

The first part of the above passage proves proposition I as given in version R; thesecond part proves what is added in versions E and P.

The text makes quite a direct reference to suppositions S1 and S2 and an indirectreference to S3, by assuming vertical paths of weights instead of circular, as Thabitdid. According to suppositions S1 and S2 Jordanus can assume that virtus growswith weight; he goes ahead and assumes also the additivity with respect to weight.Additivity is assumed explicitly “But the power of the combined weight consists ofthe powers of its components”. It is assumed implicitly when Jordanus affirms thatthe strength of the portion of ab equal to c equals that of c; this means also that posit

39 pp. 174, 176. English translation adapted.


c = a, the residual part of the virtus is that of ab−c = b. The final part of the quotedpassage, “Hence the weight ab is to the weight c, as the distance AD is to the distanceCE, and conversely as the distance CH is to the distance AF”, is a simple corollaryand, by relating strength and velocity, states the proportionality between weight andvelocity for the downward motion: “the weight ab is to the weight c, as the distanceAD is to the distance CE”, and the inverse proportionality for upward motion: “asthe distance CH is to the distance AF”.

The proof of the first part of proposition I appears clearly circular to a modernreader and then inconsistent, because it assumes what is to prove. This has beennoticed even by Brown [50].40 The fact that Jordanus did not consider additivity andproportionality as equivalent notions, as they would be by modern mathematicians,is probably due to his lack of familiarly with the algebraic calculus.

The proof consists of a reductio ad absurdum. Suppose, says Jordanus, the propor-tionality between strength and weight is not direct but the ratio of weight to weightis less than the ratio of strength to strength. Then, with p(.) that means strength, itfollows: (a+b)/a < p(a+b)/p(a) = [p(a)+ p(b)]/p(a), but, Jordanus continues,then (a+ b)/b > [p(a) + p(b)]/p(b) = p(a+ b)/p(b). In short, at the same timethat the ratio of weight to weight is both less and greater than the ratio of strengthto strength, which is absurd; then the assumption that the ratio of weight to weightis less than the ratio of strength to strength should be denied. The proof is clearlytoo hasty; it is made explicit in the version of the De ratione edited by Tartaglia andin some writings of Middle Ages commentators, with the aid of proposition 30 ofEuclid’s Elements book V.41

Even the conclusion, weight and velocity (space) are proportional, is too hasty,probably because Jordanus had modified the enunciation of proposition I in versionsE and P to arrive quickly at R and he may have not finished his work, deferringthe discussion of the ratio of strength to velocity to a subsequent (not yet existing)proposition. To note that in the final and initial parts of the proof of the R version,distances of descent are associated with velocities, with time implicit. It looks as if ametric for the velocity has been introduced bymeasuring it against the space coveredin a given time.

Concerning the upward motion, Jordanus’s text leaves one still more bewilderedbecause of its terseness. Indeed, upward motion is only mentioned in the final sen-tence: “Hence the weight ab is to the weight c, as the distance ad is to the distance ce,and conversely as the distance chis to the distance a f ”, where ch and a f are upwardmotions.

Though the proof of proposition I leaves one unsatisfied, its conclusion is clear.In the downward motion distance ad and ce covered in an assigned time, are pro-portional to weights ab and c respectively; in the upward motion, distance a f and ch

40 p. 208.41 This proposition states that given four quantities, A, B, H, K, if (A+B)/A > (H+K)/H, then(A+B)/B< (H+K)/K [221], p. 104, 105. So assumed A = a, B = b, H = p(a); K = p(b), c = afrom (a+b)/c < p(a+b)/p(c) i.e. (a+b)/a < [p(a)+ p(b)]/p(a) it follows (a+b)/b> [p(a)+p(b)]/p(b) = p(a+b)/p(b).


covered in an assigned time are inversely proportional to weights ab and c respec-tively. I repeat that these conclusions, particularly the one concerning upward mo-tion, make sense only when the weights are thought to be suspended from the armsof a balance, where the weight which sinks from one side raises the weight on theother side. Moreover, the sinking weight which acts as a motive power, must betaken to be unchanged, at the same distance and with constant velocity. In this waythe result of proposition I can be formulated by asserting that what can raise p at theheight h can raise p/n a at the height nh. This is exactly the formulation of the virtualwork law Duhem considered at the beginning of Section 4.1; the argumentation ishowever much more articulated and convincing.

4.2.2.2 Proposition II

Propositio IIWhen the beam of a balance of equal arms is in the horizontal position, then, if equal weightsare suspended from its extremities, it will not leave the horizontal position; and if it is movedfrom the horizontal position, it will revert to it. But if unequal weights are suspended, thebalance will fall on the side of the heavier weight until it reaches the vertical position [171].42

f

rzm

y

e

th

dc

xsl

n

bg

k

a

Fig. 4.6. The lever with equal arms

This proposition was carefully considered in the Renaissance, and its conclu-sion, in Thabit’s footsteps, that the balance returns to its horizontal position whenremoved (stable equilibrium) will be according, to the various authors, confirmed ordenied. For instance Tartaglia agrees with Jordanus; Benedetti claims for unstableequilibrium (balance assumes the vertical position under perturbation of the hori-zontal one). Dal Monte is for indifferent equilibrium (balance stays where it is left).This last position is that accepted by modern mechanics. The problem could not besolved empirically in the Middle Ages and the Renaissance for various reasons: theuse of systematic experiments to verify a theory was not accepted, the presence of



imperfection (inequality on masses, friction) made it difficult to read any conclu-sions, etc.

The first part of the proposition, equal weights hanging from a balance with equalarms are equilibrated, rather than being taken as a postulate, is demonstrated in thesame manner as Thabit did, arguing that the two weights are moving with the sameobliquity, so they have the same gravity of position and equilibrate themselves. Thesecond part is proved by showing that when assuming a position different from thehorizon, the gravity of position of the weight that is lower (b in Fig. 4.6) is less thanthe weight that is higher (c in Fig. 4.6) because in a virtual rotation of the balance,the higher c is lowered more than the lower b, so its gravity of position is greater andthe scale returns horizontally:

Let it now be supposed that the balance is pushed down on the side of b, and elevatedcorrespondingly on the side of c. I say that it will revert to the horizontal position. for thedescent from c toward the horizontal position is less oblique than the descent from b towarde. For let there be taken equal arcs, as small as you please, which we will call dc and bg;and let the lines czl and dmn, and also bkh and gyt, be drawn parallel to the horizontal, andlet fall, vertically, the diameter f rzmakye. Then zm will be greater than icy, because if anare cx, equal to cd, is taken in the direction of f , and if the line xrs is drawn horizontally,then rz will be smaller than zm; and since ri is equal to ky, zm will be greater than ky. Sincetherefore any arc you please, which is beneath c, has a greater component of the verticalthan an arc equal to it which is taken beneath b, the descent from c is more direct: than thedescent from b; and hence c will be heavier in its more elevated position, than b. Thereforeit will revert to the horizontal position [171].43 (A.4.15)

Note that in the proof Jordanus assumes, rightly in my view, arbitrarily small arcsbecause the motion is to be considered at the very beginning. But he does not makethe passage to the limit and consider them as infinitesimals – the times were notright – and then he fails to notice that in the limit, for infinitesimal arcs, verticaldisplacements of A and B are equal, then the gravity of their position are equal, thenequilibrium is indifferent. However the reduction to infinitesimal motion would leadto an evaluation of the gravity of position different from that proposed by Jordanus. Ifthe motion on a given circle with infinitesimal displacements is assumed, everythingis going as for finite displacements; gravity of position is maximum at the horizontalposition of the balance and is zero in the vertical position; in an intermediate positionthe gravities of the weights are equal and the balance is in equilibrium. But if circlesof different radius are concerned, the consideration of infinitesimal displacementsdoes not attribute the greater gravity, with the same inclination of the lever arm, tothe weights that are on the larger circle. Considering finite displacements insteadenables this attribution. The concept of gravity of position, although interesting andsuggestive, seems to take more than a simple infinitesimal reinterpretation in orderto be adopted by modern statics.

43 pp. 176, 178. Translation in [171].


4.2.2.3 Proposition VI. The law of the Lever

On the basis of the virtual work law implicit in proposition I, which if my interpre-tation is correct is a theorem of statics, it was not difficult for Jordanus to prove thelaw of the lever:

Proposition VIIf the arms of a balance are proportional to the weights suspended, in such manner that theheavier weight is suspended from the shorter arm, the weights will have equal positionalgravity. Let the balance beam be ACB, as before, and the suspended weights a and b; andlet the ratio of b to a be as the ratio of AC to BC. I say that the balance will not move ineither direction. For let it be supposed that it descends on the side of B; and let the line DCEbe drawn obliquely to the position of ACB. If then the weight d, equal to a, and the weighte equal to a are suspended, and if the line DG is drawn vertically downward and the lineEH vertically upward, it is evident that the triangles DCG and ECH are similar, so that theproportion of DC to CE is the same as that of DG to EH. But DC is to CE as b is to a;therefore DG is to EH as b is to a. Then suppose CL to be equal to CB and to CE, and let lbe equal in weight to b; and draw the perpendicular LM. Since then LM and EH are shownto be equal, DG will be to LM as b is to a, and as l is to a. But, as has been shown, a andl are inversely proportional to their contrary (upward), motions. Therefore, what suffices tolift a to D, will suffice to lift l through the distance LM. Since therefore l and b are equal,and LC is equal to CB, l is not lifted by b; and consequently a will not be lifted by b, whichis what is to be proved [171].44 (A.4.16)

GA

dl

e

H B

E

bM

D L

Ca

Fig. 4.7. The lever

The proof is clear enough, except for some prolixity when showing the similitudeof triangles. Substantially Jordanus says: suppose, for argument’s sake, the balanceis not equilibrated and rises on the left, but this is impossible (absurd) because, forproposition I, a weight d in D is equivalent to a weight l = b in L symmetric to B,and the balance should behave as a balance with equal arms and weight, which wasproved in a preceding proposition (proposition P2) to be equilibrated.

4.2.2.4 Proposition VIII

Propositio P8If the arms of a balance are unequal, and form an angle at the axis of support, then, if theirends are equidistant from the vertical line passing through the axis of support, equal weightssuspended from them will, as so placed, be of equal heaviness [171].45

44 pp. 182, 184. Translation in [171].45 pp. 184, 186. Translation in [171].


e

g

m

p

h

b r

yx

otk

f l

ndk

dc

z

a

Fig. 4.8. The angled lever with equal weights

The proof is laborious due to the need of Jordanus to take finite displacements. It isconducted by reduction ad absurdum, assuming that the balance is moving and byshowing that in this case there is the negation of proposition P1, whereby a weightin its descent cannot lift a weight equal to itself, i.e. the absurd.

Note that on this occasion Jordanus compares directly the descent of a heavybody with its ascent. That is, he rewrites proposition P1 by asserting that if a weightp descends a distance of h, a weight np will rise a distance of h/n. This fact, notusually commented on by historians of science, should be kept in mind for a correctinterpretation of the proofs, such as that of proposition P6, in which one is reducedto the comparison of raising weights. Here Jordanus’ reasoning: with reference toFig. 4.8, assume by absurdly that the angled lever with two equal weights on theends, and placed at the same distance (measured from the vertical) from the fulcrum,is not in equilibrium, but rotate for example clockwise. The weight at b describesthe arc bm, while the weight at a the arc ax. Then the ascent mp of b would begreater than the descent tx of a, which is impossible given the equality of a and band the proposition P1. The same applies if the lever rotates counterclockwise, inthis case the ascent would be ln and the descent rh. The reasoning would be easier ifinfinitesimal displacements could be considered as made in modern formulations, inwhich case bwould have the same vertical displacement of a, and this is precisely thecondition required for equilibrium. However the arguments developed with the useof finite displacements enable recognition of both equilibrium and stability, althoughthis was probably not completely clear to Jordanus.

Let the axis be c, the longer arm ac, and the shorter arm bc and draw the vertical line ceg;and let the lines ac and be, perpendicular to this vertical, be equal.[…]For let ag and be be extended by a distance equal to their own length, to k and to z; and onthem let the arcs of circles, mbhz and kxal, be drawn; and let the arcs ax and al be equal toeach other, and similar to the arcs mb and be and let the arcs ay and a f also be equal andsimilar. If then a is heavier in this position than b, let it be supposed that a descends to x andthat b is raised to m. Then draw the lines zm, kxy, k f l; and let mp be erected perpendicularlyon zbp, and xt and ed on kad. And because nt is equal to ed, while ed is greater than xt –


on account of similar triangles – mp will also be greater than xt. hence b will be liftedvertically more than a will descend vertically, which is impossible since they are of equalweight [171].46 (A.4.17)

There are hints that Jordanus also knew the general law of the angled lever with un-equal weights for which the weights should be in inverse proportion to their distancefrom the vertical, but he did not provide the demonstration [171]47.

4.2.2.5 Proposition X. The law of the inclined plane

Proposito P10If two weights descend along diversely inclined planes, then, if the inclinations are directlyproportional to the weights, they will be of equal strength in descending [171].48

eg

nm

x z yh

bk

lrt

d

ca

Fig. 4.9. The inclined plane

The proof of the law of the inclined plane is preceded by proposition P9 (see Ta-ble 4.2), for which the gravity of position is constant along an inclined plane. This isnot clear and also it is not clear to me the meaning of the proposition P9. The propo-sition seems self-evident. Perhaps a reason could be the assertion that the gravity ofposition depends only on the ratio between the length of the plane and its verticalprojection.

The proof is very similar to the one given for the lever. It proceeds by reductioad absurdum, replacing the situation of equilibrium of weights e and h placed onopposite inclined planes dc and dk, to the lifting of weights g = e and h locatedon the same side of inclined planes da and dk. Suppose by absurdity that h and eare not balanced and that, for example, e descends a distance of er and h ascendsa distance of xm. For proposition P1, g is equivalent to h because the two weights

46 p. 186. Translation in [171].47 Proposition R 3.01, p. 20448 p. 190. Translation in [171].


are inversely proportional to the obliquity of the planes and then there is the inverseproportion between g and h and their ascents zn and xm for an assignment descentof e, so h on the side ad can be replaced by g on side kd. But e and g have the samegravity of position and then are balanced, therefore there cannot be motion. Hencethe absurdum.

Let there be a line abc parallel to the horizon, and let bd be erected vertically on it; andfrom d draw the lines da and dc, with dc of greater obliquity. I then mean by proportion ofinclinations not the ratio of the angles, but of the lines taken to where a horizontal line cutsoff an equal segment of the vertical. Let the weight e, then, be on dc, and the weight h on da;and let e be to h as dc is to da. I say that those weights are of the same virtus in this position.For let dk be a line of the same obliquity as dc, and let there be on it a weight g, equal to e.If then it is possible, suppose that e descends to l, and draws h up to m and let gn be equal toit, which in turn is equal to el. Then let a perpendicular on db be drawn from g to h, whichwill be ghy; and another from l, which will be tl. Then, on ghy, erect the perpendiculars nzand mx; and on lt, erect the perpendicular er. Since then the proportion of nz to ng is as thatof dy to dg, and hence as that of db to dk, and since likewise mx is to mh as db is to da, mxwill be to nz as dk is to da – that is, as the weight g is to the weight h, but because e doesnot suffice to lift g to n, it will not suffice to lift h to m. Therefore they will remain as theyare [171].49 (A.4.18)

4.2.3 Comments on the Latin virtual work law

Proposition I at first sight could seem to be derived in a straightforward way fromAristotle’s law of violent motion as expounded in the Physica or De caelo and onecan assume that Jordanus used these treatises as reference. There are reasons how-ever to doubt this thesis. Firstly, there is no mention of Aristotle in Jordanus’ writ-ings, with the exception of the preface to version P of De ratione, which in any caseis related to the Mechanica problemata only. Secondly, the setting of De ratione isdifferent from that of Physica and De caelo, because the weights are not free in thespace but suspended from a balance.

In short, it is possible Jordanus followed a different line of thinking than that sug-gested by Duhem at the beginning of the section – a line of thinking which is notAristotle’s. Proposition I when interpreted as suggested previously is a theorem ofstatics, stating that which can raise p to the height h can raise np to the height h/n.More precisely it is a form of the virtual work law and presents strong analogies withmodern virtual work principles, at least in the versions considering them as a balanceof work. The main difference is that in the modern laws the work of the two weights,rising and sinking, have a different algebraic sign. In Jordanus, works of differentsituations of rising are instead equated to a unique work of sinking. That makes Jor-danus’ law useful only indirectly as an equilibrium criterion. The proof of the law ofthe lever, for example, is indeed obtained only by a reductio ad absurdum, while withthe modern principle it is sufficient to write an algebraic equation. With his law, Jor-danus was able to prove easily, and for the first time correctly, the inclined plane lawby assuming direct proportionality between the raised weight and the vertical com-ponent of displacement along the inclined plane; but this will not be considered here.


5

Italian Renaissance statics

Abstract. This chapter deals with Italian Renaissance mechanics in which the Aris-totelian approach with WVLs is joined to the Archimedean without VWLs. The firstpart presents the mechanics of Nicolò Tartaglia, who takes as a principle the VWLassociated with Jordanus de Nemore’s concept of gravity of position. The final partshows the mechanics of Galileo Galilei, who uses as a principle the VWL based onvirtual velocities with the concept of moment for which the efficacy of a weight ona scale is the greater the greater its virtual speed. And shows as a corollary the VWLbased on virtual displacements according to which anything that can lift a weight pof a height h can raise p/n of nh. In the central part the contributions of GirolamoCardano, Guidobaldo dal Monte and Giovanni Battista Benedetti are presented, allof which somehow refer to a VWL.

In the Middle Ages it was possible to identify in Europe two distinct traditions ofmechanics; the science of weights, in particular that of Jordanus Nemorarius, andthe philosophy of motion. Alongside these theoretical traditions there was the taskof practical ‘mechanicians’ somehow continuing the tradition of the Roman period.In the XVI century there was a recovery of the ancient knowledge and Jordanus’tradition is seconded by the Hellenistic, Aristotelian (Mechanica problemata) andArchimedean traditions [298, 296].

The postclassical tradition of theMechanica problemata is a typical phenomenonof Renaissance, as it was practically unknown during the Middle Ages. Howeverat least a Greek manuscript dating from the twelfth century survived, which testi-fies that the text was potentially accessible to Medieval philosophers. Few of theHellenistic writings reached Europe. The Mechanica by Hero of Alexandria wasknown for sure in an Arabic translation only in the XVII century. Renaissance math-ematicians had access to it through some epitome contained in Pappus’ Book 8 ofthe Mathematical collections (which remained in manuscript form until 1588) andBook 10 of Vitruvius’ De architectura. There is however at least a clue that someknowledge of Hero’s text should exist in the Renaissance; indeed Nicola Antonio (orColantonio) Stigliola referred to in his book De gli elementi mechanici a treatment


92 5 Italian Renaissance statics

of the inclined plane substantially equivalent to that of Hero [219].1 Hero’s othertwo manuscripts were also known, the Pneumatica and the Automata, that were ob-jects of translations and comments. Very important from a technological point ofview, they were lacking theoretical arguments on mechanics. For the Archimedeantradition, considerations similar to that of the Mechanica problemata hold good, forsome of his manuscripts were known in the Middle Ages, in Greek and Latin, butthey had no impact in mechanics and mathematics. Archimedes’ ideas spread in theRenaissance thanks to Tartaglia’s editions of Moerbeke’s Latin translation of bookson the centre of gravity and on floating bodies [222].

According to Drake [298, 296] in Italy, the leading nation of the period, therewere twomain schools inmechanics, sharing one ormore of the traditionsmentionedabove. The North one, formed by Giovanni Battista Benedetti (Venice, 1530–1590),Nicolò Tartaglia (Brescia, 1499?-1557), GirolamoCardano (Pavia, 1501–1576). TheCentre one, formed by Federico Commandino (Urbino, 1509–1575), Guidobaldo dalMonte2 (Pesaro, 1545–1607), Bernardino Baldi (Urbino, 1553–1617). The school ofthe North would be more interested in practical issues, which require the study ofmotion and that is why they focused on Jordanus’s orMechanica problemata insteadof Archimedean mechanics devoid of any reference to kinematics.

There are, however, differences and difficulties in this classification scheme.Tartaglia was critical to the setting of the Aristotelian Mechanica problemata, andappreciates Jordanus. Cardano followed theMechanica problemata, andmore gener-ally the physics of Aristotle. Benedetti did not accept the approach of theMechanicaproblemata, nor that of Jordanus, but tended to follow Archimedes, at least for whatconcerns the study of equilibrium. In the Centre, next to the strictly Archimedean ofCommandino approach, one must register the approach of dal Monte that showed acertain appreciation to Mechanica problemata and that of Bernardino Baldi, whoalso was a follower of Mechanica problemata but perhaps less attentive to theArchimedean approach, though he made use of the theory of centres of gravity instatics.

This partition in schools has to be mitigated and integrated considering the suc-cession of generations and the dissemination of the various texts on mechanics andmathematics (see Tables 5.1, 5.2). It is undeniable that there was a close correla-tion between the professional or mathematical culture and the preference towardArchimedean mechanics. In the North, for various reasons, there was less atten-tion paid to Archimedes, not only because of a greater emphasis devoted to prac-tical aspects and motion, but also because of the lake of cultural tools (mathemati-cal) to appreciate the mechanics of Archimedes. Tartaglia was certainly a talentedmathematician, but more for his intelligence and originality than for education.The same applies to Cardano. Benedetti, a generation after, was able to read more

1 p. 41. According to Romano Gatto it is probable that during theMiddle Ages at least a Greek copyof theMechanica survived [326] because in Montfaucon’s Bibliotheca bibliothecarum manuscrip-torum [172], p. 472, it is attested the presence – among Libri Greci – of the titleHeronisMechanica,& alia multi quae rare reperiuntur. According to Gatto however Stigliola had no direct access toHero, but he read Leonardo da Vinci [325], p. 300.2 For the spelling of dal Monte’s name see [305], vol. 2, p. 351.

5 Italian Renaissance statics 93

easily Archimedean texts, which were published in Italian too, and to appreciateArchimedes’ mechanical side. In central Italy the samewas true of Bernardino Baldi,who certainly did not have the mathematical training of Commandino or dal Monte,although he was essentially a contemporary of Galileo.

From the schema outlined above, scholars of southern Italy, Francesco Mau-rolico (Messina, 1494–1575), Nicola Antonio Stigliola (1546–1623) and Luca Va-lerio (Napoli, 1553–1618) remain on the outside. All were expert mathematicians,followers of Archimedes in mechanics and as such less interesting from my pointof view. An in depth study of the southern Italy school is due to Romano Gatto[323, 324, 325, 125]; see also [367].

The Aristotelian textMechanica problemata already presented in Chapter 3, wasof considerable importance in the Renaissance. By its nature it was able to mobilizepeople of different backgrounds, humanists interested in the philosophical aspect andmathematicians and engineers interested in its theoretical and technological content.There is agreement that the Mechanica problemata as such remained without directinfluence from the decline of Hellenistic science until the Greek revival of the Re-naissance. Latin writers of the Middle Ages who encountered the Greek text wereinsufficiently impressed by it to continue the discussion.

Table 5.1. Hero, Jordanus, Archimedes’ texts

Heronian texts

1501 De expetendis et fugientis rebus. Valla

1521 Di Lucio Vitruvio Pollione de architectura libri dece traducti de latino in vulgare affigurati.Cesariano

1550 De subtilitates. Cardano

1575 Spiritalium liber. Commandino

1588 Mathematica collectiones. Commandino

1589 Gli artificiosi et curiosi moti spirituali. Aleotti

1589 Automata. Baldi

1581 Pneumatica. Baldi

1592 Spiritali di Herone Alexandrino, ridotte in lingua volgare. Giorgi

Jordanus’ texts

1533 Liber de ponderibus. Apianus

1546 Quesiti et inventioni diverse. Tartaglia

1565 Jordani opuscolorum de ponderositate. Tartaglia

Archimedean texts

1543 Opera Archimedis. Tartaglia

1551 De insidentibus aquae. Tartaglia (in Italian)

1558 Archimedis opera non nulla. Commandino

1570? Momenta omnia mathematica. Maurolico (published 1685)

1565 Archimedis De iis quae vehuntur in aqua libri duo. Commandino

1588 In duos Archimedis aequeponderantium libros paraphrasis. Dal Monte


Table 5.2. Editions of Mechanica problemata [15, 378]

Year Author Title

1517 Vittore Fausto Aristotelis Mechanica. Parisiis

1525 Niccolò Leonico Opuscola. Venetia

1565 Alessandro Piccolomini In mechanicas quaestiones Aristotelis paraphrasis. Romae

1570 Girolamo Cardano Opus novum de proportionibus numerorum. Basileae

1573 Antonio Guarino Le mechanice d’Aristotile trasportate di greco in volgar idioma,con le sue dimostrazioni nel fine. Modena

1582 Vannocci Biringucci Parafrasi di monsignor Alessandro Piccolomini sopra le meca-niche d’Aristotile. Roma

1585 Giovanni B. Benedetti De mechanicis in diversarum speculationum mathematicarum etphysicarum liber. Taurini

1599 Henri de Monanthenil Aristotelis mechanica, graeca, emendata, latina facta et commen-tariis illustrata. Parisiis

1613 Francesco Maurolico Problemata mechanica cum appendice, et a magnetem, et a pirox-idem nautica pertinentia. Messane

1581 Giuseppe Biancani Aristotelis loca mathematica ex universis ipsius operibus collecta.Bononiae

1621 Bernardino Baldi In mechanica Aristotelis problemata exercitationes. Moguntiae

1627 Giovanni de Guevara In Aristotelis mechanicas commentarij. Rome

The XV century saw the rapid multiplication of Greek copies. The beginning ofthe XVI century saw two important Latin translations by two humanists. The firstwas due to Vittore Fausto (1480–1511), but the most largely circulating copy wasthe second translation by Niccolò Leonico Tomeo (1456–1531). Table 5.2 reportsa quite exhaustive list of the translations and commentaries of the Mechanica prob-lemata.

In the course of later development, theMechanica problemata gave way to moresophisticated mathematical treatment of the problems discussed qualitatively in it.There were also vernacular versions, a very important one being by Oreste VannocciBiringucci (1558–1585), the nephew of the homonymous author of De la pirotech-nica), encouraged in the translation enterprise by Alessandro Piccolomini (1508–1579) who felt an Italian translation of Mechanica problemata to be necessary sothat also engineers could profit from it. Highly original additions were offered lastlyby Bernardino Baldi. The background of merging of humanist with practical con-cerns is traceable primarily to the emergence of architecture as a distinguished pro-fession in the XV century, and particularly with the revival of interest in the textof Vitruvius, and the engineering treaties by (Taccola and) Francesco di GiorgioMartini.

5.1 Renaissance engineering 95

5.1 Renaissance engineering

The XV and XVI centuries, especially in Italy, saw the emergence of a considerablenumber of skilled professional engineers. The contribution of these engineers, whohave not yet even been fully counted, is known only very superficially [305]. A study,which will require the work of a large research group, is essential to reconstruct theirrole with some accuracy. A first examination of the sources consulted3 leads to theconclusion that the widespread opinion that the professional Italian engineers had norelevant direct influence in development of the mechanical theory should be shared[364, 293, 1, 368, 89, 45, 27, 92, 163, 108]. They however had an indirect influence instimulating mathematicians and philosophers to develop theories which could helpthe solution of the technological problems connected to the enormous developmentof industry and architecture of the Renaissance [272, 331, 288, 251, 352]. A majorlimitation to the possibility that professional engineers could contribute to the devel-opment of mechanical theory, in addition to the characteristics of their necessarilypractical business, is due to the fact that almost all scientific texts were written inGreek or Latin. There was however some important contamination especially withthe traditions of the Problemata mechanica and Hero’s writings. Giuseppe Ceredaand Vittorio Zonca made reference to Aristotle when they spoke of their machines.Giambattista Aleotti, an engineer at the court of Ferrara, quoted Archimedes, Aristo-tles and Hero [342]. Antonio Guarino, an engineer at the court of Modena, translatedinto Italian from the Greek theMechanica problemata. Daniele Barbaro in his para-phrase of Vitruvius’s book showed a really noteworthy knowledge of mechanicaltheory [21].

A different story is about the most famous engineer of the Italian Renaissance,Leonardo da Vinci (1452–1519), who left hundreds of drawings and pages devotedto mechanics.4 It is difficult to give a full account of the opinions of historians onhis role for science in general and mechanics in particular. One goes from an en-thusiastic vision of the early XIX century, especially on the side of historians ofscience educated in literature, to a more mature appreciation of Duhem and finallyto a fierce criticism of Truesdell, whominimises both the originality and contributionto the subsequent science development of the work of Leonardo. Eduard Dijkster-huis eventually considers studying Leonardo not for his contributions to science, but

3 Including the writings of Taccola (Siena, 1381–1458), Leon Battista Alberti (Genova, 1404–1472), Francesco di Giorgio Martini (Siena, 1439–1501), Leonardo da Vinci (Vinci, 1452–1519), Vannoccio Biringuccio (Siena, 1480–1539), Francesco de’ Marchi (Bologna, 1504–1576),Giovanni Battista Bellucci (San Marino, 1506, 1554), Daniele Barbaro (Venezia, 1513–1570),Bonaiuto Lorini (Firenze, 1540–1611), Domenico Fontana (Ticino, 1543–1607), Giuseppe Ceredi(Piacenza, f. 1560), Camillo Agrippa (Milano, fl 1570), Vittorio Zonca (Padova, 1568–1602), Gi-ambattista Aleotti (near Ferrara 1546–1636), Antonio Guarino (1504–1590).4 The many interests of Leonardo were previously considered in the early 1400, by the SieneseMariano Taccola interested in the writings of mechanical and technical military of Pneumaticaby Philo of Byzantium (280–220 BC). In recent times Giambattista Venturi published in 1797 afamous essay on the scientific work of Leonardo [393] and in the years 1880–1936 his notebooksand manuscripts were published in facsimile, and today all Leonardo’s works are printed witha diplomatic transcription. Leonardo was also studied in depth by Pierre Duhem [306], CliffordTruesdell and Roberto Marcolongo [357, 388].


for the opportunity offered by his copious notes to follow the maturation of variousscientific concepts [292]. The difficulty of analyzing the role of Leonardo is also dueto the nonexistence of an organic edition of his works. In this situation I preferred tonot refer to Leonardo’s contribution to the development of the law of virtual work,which is however probably also important.

5.1.1 Daniele Barbaro and Buonaiuto Lorini

As examples of the level of knowledge of mechanical science and the laws of virtualwork, I recommend excerpts from the Dieci libri di architettura di M. Vitruvio [21]by Daniele Barbaro [21] and the Fortificationi by Buonaiuto Lorini [163].

In his commentary on Book X of Vitruvius’s Dieci libri di architettura, Barbarorefers to Aristotle’sMechanica problemata and tries to bring to the lever the varioussimple machines. The explanation of the operations is quite brief, reflecting the textof Vitruvius. However I feel it to be of a certain interest, considering the time ofpublication, the explanation of how the system of pulleys, i.e. the block and tackle,works:

There is no doubt that if a weight is attached to a simple rope, let’s say a thousand pounds,all the work and force is supported by the rope, then if that rope will be doubled and to thata pulley is suspended where to hang the weight, the rope is to get half of fatigue, and a halfforce is enough to lift that weight. And if there are more pulleys? [...] If the first doublingtakes away half of the weight, the second doubling to which a half remains, will take awayhalf of that half and the whole weight will be taken away by the fourth part of the forcewhich lifted the first weight [21].5 (A.5.1)

The explanation does not refer to the lever and is substantially the same as Hero’s,based on simple considerations of equilibrium. Barbaro however did not know Heroand probably not even the works of Pappus of Alexandria that had not yet beenpublished by Commandino.

Buonaiuto Lorini is at least a generation younger than Barbaro, and then he wasable to read the latest developments in mechanics, of dal Monte certainly, but per-haps also of Cardano and Benedetti, of whom I shall speak below. In Book V ofhis Fortificationi Lorini shows both the mechanics of simple machines, with sometheoretical considerations, and the complex construction equipment for lifting heavyweight, earth and water. He quotes Hero, Archimedes and Guidobaldo dal Monte.In particular, unlike Barbaro he refers the operation of the pulley to the lever, asdal Monte did. An interesting reference is to a law of virtual work for which, to agreater ratio of weight and power, there corresponds a greater ratio of the motion ofthe power with respect to that of the weight:

The secret of all the inventors of mills and other machines is to look for, just like you said,to accompany force with speed, a really difficult thing, because since the same power has tomultiply into many, which one after another may lift, or carry a load, it is necessary that thetime it is multiplied likewise, as for example it would be if you were to carry a weight ofone thousand pounds from one place to another, through the sheer force of one man, whichshall take only a part, that will be fifty pounds [163].6 (A.5.2)

5 p. 446.6 p. 238.

5.2 Nicolò Tartaglia 97

Fig. 5.1. The simple machines of Daniele Barbaro (reproduced with permission of Biblioteca Cen-trale della Facoltà di Architettura of Università La Sapienza, Rome)

5.2 Nicolò Tartaglia

Nicolò (or Niccolò) Tartaglia was born in Brescia probablyin 1499 and died in Venice in 1557. He received no formaleducation, except for a period of fifteen days in a “scuola perscrivere” when he was fourteen. He learned to read Latin but,with a single exception, he wrote only in a not very elegantItalian [294]. In 1537 he published his first book, the Novascientia, inspired by practical problems of gunnery. In 1543his editions of Euclid (in Italian) and Archimedes (in Latin)were published, see Table 5.1. In 1551 he published in Ital-

ian the first book of Archimedes’ De insidentibus in aquae, in 1546 the Quesiti etinventioni diverse, where his version of the science of weights is reported. Althoughthe book was largely a paraphrase of Jordanus’ De ratione ponderis first book, ofwhich he possessed a copy published posthumously in 1565 [224], Tartaglia did notcite the fact, and this brought upon him the accusation of plagiarism.

Of Tartaglia’s writings on mechanics I will refer only to Book VIII of Quesiti etinventioni diverse, because it is the only one related to the virtual work laws.


5.2.1 Definitions and petitions

Book VIII of Quesiti et inventioni diverse starts with some definitions (seventeen)and petitions (six). Table 5.3 reports the statement of the main definitions and pe-titions [298, 223]. (A.5.3) Examination of the table makes evident the more formalapproach of Tartaglia with respect to Jordanus de Nemore. Definitions III and IVrelate to virtue (see § 4.2). The third definition has a qualitative character and ap-plies to both the upward and downward motions. The fourth definition, which is stillabout virtue, is more problematic. Meanwhile it identifies the measure of virtue –that below he often names power – with speed, it seems to apply only to the down-ward motion. Of some importance seems to me Definition XIV, which takes awayany ambiguity to the introduction of weight.

Table 5.3. Tartaglia’s definitions and petitions

Definition III By virtus of a heavy body is understood and assumed that power which it has totend or go downward, as also to resist the contrary motion which would draw itupward.

Definition IV Bodies are said to be of equal virtus or power when in equal times they runthrough equal spaces.

Definition XIII Abody is said to be positionally more or less heavy than another when the qualityof the place where it rests and is located makes it heavier [or less heavy] than theother, even though both are simply equal in heaviness.

Definition XIV The heaviness of a body is said to be known when one knows the number ofpounds, or other named weight, that it weighs.

Definition XVII The descent of a heavy body is said to be more oblique when for a given quantityit contains less of the line of direction, or of straight descent toward the centreof the world.

Petition II Likewise we request that it be conceded that that body which is of greater powershould also descend more swiftly; and in the contrary motion, that is, of ascent,it should descend more slowly - I mean in the balance.

Petition III Also we request that it be conceded that a heavy body in descending is so muchthe heavier as the motion it makes is straighter toward the centre of the world.

Petition VI Also we request that it be conceded that no body is heavy in itself.

After definitions, petitions follow, which to Tartaglia are those propositions thatshould be asked the opponent being accepted for the conduct of the demonstrations(they are then postulates). Notice that the second petition is linked to the fourthdefinition, comparing power with speed, both for downward and upward motions.Here he used the word power, which in the fourth definition was identified withvirtue.

5.2.2 Propositions

Tartaglia considers fifteen propositions (theorems), some of them are shown in Ta-ble 5.4 [298, 223]. (A.5.4)


Table 5.4. Tartaglia’s propositions

I The ratio of size of bodies of the same kind is the same as the ratio of their power.

II The ratio of the power of heavy bodies of the same kind and that of their speeds (in descent)is concluded to be the same; also that of their contrary motions (that is, of their ascents) isconcluded to be the same, but inversely.

III If there are two bodies simply equal in heaviness, but unequal positionally, the ratio of theirpowers and that of their speeds will necessarily be the same. But in their contrary motions(that is, in ascent) the ratio of their powers and that of their speeds is affirmed to be inverselythe same.

IV The ratio of the power of bodies simply equal in heaviness, but unequal in positional force,proves to be equal to that of their distances from the support or centre of the scale.

V When a scale of equal arms is in the position of equality, and at the end of each arm thereare hung weights simply equal in heaviness, the scale does not leave the said position ofequality; and if it happens that by some other weight [or the hand] imposed on one of thearms it departs from the said position of equality, then, that weight or hand removed, thescale necessarily returns to the position of equality.

VI Whenever a scale of equal arms is in the position of equality, and at the end of each arm arehung weights simply unequal in heaviness, it will be forced downward to the line of directionon the side where the heavier weight shall be.

VII If the arms of the scale are unequal, and at the ends thereof are hung bodies simply equal inheaviness, the scale will tilt on the side of the longer arm.

VIII If the arms of the balance are proportional to the weights imposed on them, in such a waythat the heavier weight is on the shorter arm, then those bodies or weights will be equallyheavy positionally.

XIV The equality of slant is an equality of [positional] weight.

XV If two heavy bodies descend by paths of different obliquities, and if the proportions of incli-nations of the two paths and of the weights of the two bodies be the same, taken in the sameorder, the power of both the said bodies in descending will also be the same.

Before going into the validity of the proof of the various propositions, I want tostress Tartaglia’s ideas. He found in Jordanus’s writings two possible principles ofstatics, one based on the concept of gravity of position, the other on the capabilityof a weight to lift another. According to my interpretation, Jordanus used both, thefirst for qualitative proof, the second to establish mathematical relations. Tartagliamakes instead a choice and decides to base his mechanics only on the gravity ofposition. This notwithstanding, he maintains a trace of Jordanus’ ideas and to statethe equilibrium of a lever or an inclined plane considers the equivalence of weightdisposed on the same side and not on the opposite.

Tartaglia reconsiders Jordanus’ original proposition II by splitting it into three‘propositions’ and modifying in part the conclusion. In the first proposition he‘proves’ that greater weights have greater power. In the second that speed and powerare in the same proportion in downward motion and in the inverse proportion in up-ward motion. In the third proposition that speed and weight are in direct proportionin downward motion and in inverse in upward motion. In the fourth propositionhe proves that the power of weights is proportional to their distances from the ful-crum.


A B C

D E F

Fig. 5.2. Ratio of sizes (A, B, C) and powers (D, E, F)

5.2.2.1 Proof of propositions I–IV

Proposition IThe ratio of the size of bodies of the same kind is the same as the ratio of their power.Let there be the two bodies AB and C of the same kind; let AB be the greater, and let thepower of the body AB be [represented by the line] DE, and that of the body C [by the line]F. Now I say that that ratio which the body AB bears to the body C is that of the power DEto the power F. And if possible (for the adversary), let it be otherwise, so that the ratio ofthe body AB to the body C is less than the ratio of the power DE to the power F. Now letthe greater body AB include a part equal to the lesser body C, and let this be the part A, andsince the force or power of the whole is composed of the forces of the parts, the force orpower of the part A will be D, and the force or power of the remainder B will necessarilybe the remaining power E; and since the part A is taken equal to C, the power D (by theconverse of Definition 7) will be equal to the power F, and the ratio of the whole body ABto its part A (by Euclid V.7, 2) will be as that of the same body AB to the body C (A beingequal to C), and similarly the ratio of the power DE to the power F will be as that of the saidpower DE to its part D (D being equal to F). Therefore [by the adversary’s assumption] theratio of the whole body AB to its part A will be less than that of the whole power DE to itspart D. Therefore, when inverted (by Euclid V.30), the ratio of the body AB to the residualbody B will be greater than that of the whole power DE to the remaining power E, whichwill be contradictory and against the opinion of the adversary, who wants the ratio of thegreater body to the less to be smaller than that of its power to the power of the lesser body.Thus, the contrary destroyed, the proposition stands [223].7 (A.5.5)

In his proposition I, Tartaglia assumes bodies of the same material but different size,so there is no doubt on the meaning of the proposition. He takes for granted, evenif not explicitly stated in his petitions, that a heavier body has more power than alighter. Tartaglia reproduces the framework of proof of proposition II of Jordanus,making it more clear. But there are still some points not acceptable to a modernreader. Without specifying exactly what it is and how to measure the power of abody, Tartaglia accepts additivity: the power of a body is given by the sum of thepower of its parts. Like Jordanus, he does not notice, however, that in this way hetakes for granted what he wants to prove (see § 4.2.2.2). A modern reader is baffledby the almost miraculous demonstration such as Tartaglia’s, as would be that ofJordanus. There is the impression that with this way of reasoning one can proveanything, for example, that beauty is proportional to size.

7 p. 88r. Translation in [298].


Plate 1. Tartaglia’s books on Scientia de ponderibus (reproduced with permission, respectively, ofBiblioteca Guido Castelnuovo, Università La Sapienza, Rome, and of Max Planck Institute for theHistory of Science, Berlin)

Tartaglia’s proof of his proposition II is based on the same reasoning. This timethings are slightly clearer because the definitions third and fourth and petition sec-ond, connect somehow power and speed, in particular they argue that it has a higherspeed if there is a higher power. The first part of proposition II, that bodies falldown with speeds proportional to their size, is proved with arguments similar to thatused in proposition I. Additivity of speed with power is assumed and proportional-ity demonstrated. To demonstrate the inverse relationship between power and speedTartaglia assumes that the resistance to upward motion is proportional to the powerof the body. So that power that will barely fit in the other arm to lift the body AB,will be sufficient to lift faster the body C and the relationship of speed of C to ABis that of ED to F (Fig. 5.2). From propositions I and II follows the proportionality(direct or inverse) between weight (size) and speed.

The logical status of proposition III is not clear; to amodern reader it seems an im-mediate consequence of propositions I and II, however, a demonstration is proposedby following exactly the arguments of proposition I. In proposition IV Tartagliaaims to quantify the concept of gravity of position, at least for bodies connected tothe arms of a balance. The proof again follows the same line of argument, with somemore difficulty. Tartaglia seems to make the assumption that the sum of distancescorresponds to the sum of weights.

5.2.2.2 The law of the lever

With proposition IV the demonstration of the law of the lever should be immediate,it would suffice to argue that the two weights at each end of the lever are equal


CH

LD

AF M

B

E

Fig. 5.3. Equilibrium of the lever with different arms

in gravity of position and therefore balanced. Tartaglia, however, prefers to repeatJordanus’s approach, where instead of the equilibrium of opposing tendencies it isconsidered the equivalence of weights that tend to move in the same direction. In thepassage below, Tartaglia refers the lever with weights E and D to the lever in whichthe weights areD and L= E, on the same side. Through his proposition IV he arguesthat they are equally heavy for position and D may be replaced by L arriving at abalance with equal arms (LC = EC) and equal weights, and as such, in equilibriumfor the proposition V (not commented here).

Proposition VIIIIf the arms of the balance are proportional to the weights imposed on them, in such a waythat the heavier weight is on the shorter arm, then those bodies or weights will be equallyheavy positionally.[…]First let there be the bar or balance ACB and the weights A and B hung thereon, and let theratio of B to A be as that of the arm AC to the arm BC. I say that this balance will not tiltto either side. And if (for the adversary) it is possible for it to tilt, let us assume it to tilt onthe side of B and to descend obliquely as the line DCE in place of ACB, and [let us] take Das A and E as B; and the line DF falls perpendicularly, and the line EH rises similarly. […]and put L equal in heaviness to B and descending along the perpendicular LM, then, sinceit is manifest that LM and EH are equal, the proportion of DF to LM will be as the simpleheaviness of the body B to that of the body, or as the simple heaviness of the body L tothat of D […]. Whence if the said two heavy bodies, that is, D and L, were simply equal inheaviness, standing then in the same positions or places at which they are presently assumedto be, the body D would be positionally heavier than the body L (by the Fourth Proposition)in that ratio which holds between the whole arm DC and the arm LC. And since the body Lis simply heavier than the body D (by our assumption) in the same ratio as that of the armDC to the arm LC, then the said two bodies D and L in the level position would come to beequally heavy, because by as much as the body D is positionally heavier than the body L,by so much is the body L simply heavier than the body D; and therefore in the level positionthey come to be equally heavy. […] Therefore if the body B (for the adversary) is able to liftthe body A from the level position to the point D, the same body B would also be able andsufficient to lift the body L from the same level position to the point where it is at present,which consequence is false and contrary to the Fifth Proposition […]. Thus, the adversary’sposition destroyed, the thesis stands [223].8 (A.5.6)

8 pp. 92v–93r. Translation in [298].


At the end of his proposition Tartaglia refers to the demonstration of Archimedes,stating that since the matter of his treatise is quite different from the Archimedean,he has considered to demonstrate the law of the lever with other principles as moreappropriate. Note that Tartaglia did not study the angular lever (Jordanus’ proposi-tion VIII). This is because the use of the concept of gravity of position does not workin such a case [305].9

5.2.2.3 The law of the inclined plane

The proof of the law of the inclined plane is preceded by a lemma similar to thatreported by Jordanus, according to which the gravity of position along an inclinedplane is constant. Tartaglia does not make the step that it would seem natural toexplicitly state that the gravity of position is inversely proportional to the obliquity(with the meaning he gave to this term). The lack of this step is critical because in theproof of the law of the inclined plane Tartaglia uses it effectively. From this pointof view the demonstration of Tartaglia is less satisfactory than that of Jordanus. Theproof is developed as in the case of the lever, bringing the equilibrium to an equiv-alence. But the reasoning is less strict, because it asserts without explanation thatthe two heavy bodies H and G are equally heavy for position as they have weightsinversely proportional to their obliquities, which although intuitive, has not yet beendemonstrated by Tartaglia. In his beautiful work, Storia del metodo sperimentale,Raffaello Caverni [284] considers Tartaglia’s demonstration as the first truly exem-plary proof, of higher value than that of Jordanus, of whom Caverni seems howeverto not know the De ratione. Caverni reports and comments on the demonstration ofTartaglia, justifying it with the statement of the proposition XIV [284],10 which forme is a logical gap.

EG

NM

X Z YH

BK

LR

T

D

CA

Fig. 5.4. Equilibrium on the inclined plane

9 Vol. 1, p. 121.10 vol. IV, pp. 321–232.


Proposition XVIf two heavy bodies descend by paths of different obliquities, and if the proportions of incli-nations of the two paths and of the weights of the two bodies be the same, taken in the sameorder, the power of both the said bodies in descending will also be the same.[…]Then let the letter E represent a heavy body placed on the line DC, and the letter H anotheron the line DA, and let the ratio of the simple heaviness of the body E to that of the bodyH be the ratio of DC to DA I say that the two heavy bodies in those places are of the samepower or force. And to demonstrate this, I draw DK of the same tilt as DC, and I imagineon that a heavy body, equal to the body E, which I letter G, in a straight line with EH, thatis, parallel to CK. […] Also the ratio of MX to NZ will be as that of DK to DA; and (byhypothesis) that is the same as that of the weight of the body G to the weight of the bodyH, because G is supposed to be simply equal in heaviness with the body E. Therefore, byhowever much the body G is simply heavier than the body H, by so much does the body Hbecome heavier by positional force than the said body G, and thus they come to be equalin force or power. And since that same force or power that will be able to make one of thetwo bodies ascend (that is, to draw it up) will be able or sufficient to make the other ascendalso, [then], if (for the adversary) the body Eß is able and sufficient to make the body Hascend to M, the same body E would be sufficient to make ascend also the body G equal toit, and equal in inclination. Which is impossible by the preceding proposition. Therefore thebody E will not be of greater force than the body H in such place or position; which is theproposition [223].11 (A.5.7)

5.3 Girolamo Cardano

Girolamo Cardano was born in Pavia in 1501 and died inRome in 1576. He was educated at the university of Pavia,and subsequently at that of Padua, where he graduated inmedicine. He was, however, excluded from the College ofphysicians at Milan on account of his illegitimate birth, andit is not surprising that his first book should have been an ex-posure of the fallacies of the College. In 1547 he accepted achair of medicine at Pavia university. The publication of hisworks on algebra and astrology had gained for him a Euro-

pean renown. In 1551 his reputation was crowned by the publication of his greatwork, De subtilitate rerum, here after De subtilitate, which embodied the soundestphysical learning of his time and simultaneously represented its most advanced spiritof speculation [294].

Cardano’s writings onmechanics are only a small part of his interests, which weremainly medical and astrological, for he wrote more than 200 works on medicine,mathematics, physics, philosophy, religion, and music. His role in mechanics iscontroversial. Duhem is convinced that he borrowed abundantly from Leonardo daVinci [305], but Drake is doubtful on the purpose [298].12 Though Cardano citesArchimedes, Hero, and Ctesibius, he was strongly influenced by Aristotle.

11 pp. 97r, 97v. Translation in [298].12 p. 26.

5.3 Girolamo Cardano 105

A

OP

N

Q

DC

R

FG

H

E

S

U

M

B

Fig. 5.5. The balance with equal arms

Statics is dealt with in the last part of the De subtilitate [54] first book, withsome other considerations scattered elsewhere. Other considerations on statics arein De opus novum de proportionibus [56]. In the following I will comment mainlyon Cardano’s considerations of the balance.

5.3.1 De subtitilate

The text of the De subtilitate where Cardano discusses the effectiveness of theweights placed on the arms of a balance is difficult to read because it is not alwaysentirely consistent. The aim is to comment on issue 2 of Mechanica problemata, inwhich Aristotle discusses the stability of the scale with fulcrum above or below thebeam. Cardano speaks about a balance CD hanging in A, as shown in Fig. 5.5:

Next we must consider weights that are placed upon a balance. Let there be a balance whosepoint of suspension is at A, let the point where the arms of the beam are joined be B, and letthe beam be CD. It is clear that CD moves about B as a fixed centre, because CD cannot beseparated from B. Let the angles ABC and ABD be right angles [54].13 (A.5.8)

But then he develops all his considerations as if the balance were hanging in B. Hecompares the effectiveness of the same weight p placed respectively in F and C toconclude that it is heavier in C. He argues the conclusion in two ways, for the firstway he refers to qualitative considerations based on common experience, mainly onthe evidence that weights more distant from the fulcrum are more effective:

I say that a weight placed at C (the beam being in the horizontal position CBD) will beheavier than if the beam were put in any other position, as, for instance, with the end of thebeam at F [...], therefore, I shall show by two arguments that this happens when the weightand beam are placed at C rather than at F […]. The first of these arguments may be explainedin this way. It is clear that, in steelyards and in those instruments which raise weights, thefarther the weight is from the point of suspension, the heavier it seems […] it is also clearthat, the farther the balance-arm descends toward C from A, the heavier the weight becomes

13 p. 23.


and, therefore, the more swiftly it moves; but, for the opposite reason, in the movementfrom C toward Q, the weight is lighter and the motion is slower – a fact which is proved byexperience [54].14 (A.5.9)

The second way of argument is more complex, it refers to the principles of Aris-totelian physics and is quantitative in nature. To bring coherence to the argumentof Cardano one should refer instead of a balance in two different positions (FR andCD), to points C and F of a circle, describing equal arcs in equal times. With simplegeometrical considerations Cardano shows that for a given rotation of the circle, thepoint F describes a path OP, measured on the vertical line, less than that BM of C,and then it moves with a less vertical velocity. The greater velocity of C with respectto F leads to the conclusion that the weight p is more effective (heavier) in C thenin F.

Cardano’s reasoning is different from that of all his predecessors, particularlyof Tartaglia, although it is similar. He seems to refer to the law of virtual work ofThabit, reported in Chapter 4, for which the effectiveness, or force, of a weight ismeasured by (is proportional to) its virtual velocity. The difference is that Thabitconsiders motions along arcs, Cardano vertical motions.

The second argument may be demonstrated as follows: Let the arc CH be laid off equal tothe arc CE […] therefore, BN is greater than OP, and, because of this, BM is greater thanOP. Now, while the end of the beam is moved from C to E, the weight descends throughthe distance BM and is thus brought closer to the centre than it was at C. While the beam ismoved through the length of the arc FG, the weight descends through OP. And BM is greaterthan OP. Now, supposing that in equal times this weight passes from C to E and from F toG, it descends still more quickly from C than from F; therefore, it is heavier at C than at F[emphasis added] [54].15 (A.5.10)

Cardano closes his argument by asserting that from above it is not difficult to seehow the balance is stable with the fulcrum over and unstable with the fulcrum below[54].16 In fact Cardano’s conclusion is not clear, although it can be inferred easilyfrom the conclusions reached by him when the balance has the fulcrum in B. In fact,taking for example the fulcrum in A one sees that if the two weights are not at thesame level, the higher will be more effective than the lower and the balance willreturn with weights at the same level.

Cardano returns to problems of statics in other books of the De subtilitate, par-ticularly interesting are the considerations on the block and tackle:

The fourth example of subtilitates is the block and tackle. But because the ratio of times isas that of powers, [the boy] will pull four times more slowly with two pulleys, six times withthree pulleys [...] so it will happen that the boy in a hour will pull just the same weight withthe pulley that a man, six times more strong, being above, can pull on the spot with a singlerope [54].17 (A.5.11)

Notice that also Cardano, like Hero, speaks about time instead of space.

14 Liber primus, pp. 23–24. Translation in [57].15 Liber primus, p. 24. Translation in [57].16 p. 24.17 Liber XVII, p. 467–468.

5.3 Girolamo Cardano 107

5.3.2 De opus novum

In De opus novum Cardano addresses topics of statics on several occasions. In par-ticular, he gives a personal demonstration (wrong) of the inclined plane, which isdifferent from that of Pappus (wrong too) and Jordanus Nemorarius (correct), whichwere probably both known by him. In essence Cardano believes that the force re-quired to move a weight on an inclined plane is proportional to the angle of the planewith the horizon, instead of the sinus as it should be [56].18

Of some interest is the demonstration of the law of the balance, in which Cardanomeasures the effectiveness of the weights based on the virtual displacements, as hedid in De subtilitate, except that now the motion instead of being measured on thevertical seems to be measured along the arc, as Thabit did.

h

e bm

p on

f

gi

d

ck

a

Fig. 5.6. The law of balance

Proposition forty fiveShow the law of balanceIf the beam bd is put in e and f and if the ratio of eb to b f is as that of g to h, I say there willbe equilibrium. Otherwise hwould move to k, until it reaches the line ad. If hwere not fixed[at the beam bd] it would move along [the vertical] eh; but because it is fixed it will movealong the curve hk. Take a point [m] near [to b] in be and n at equal distance in b f . Becauseall eb is moved in any part with a same force, i.e. the weight h, and because the point in hmoves along hk and the point in m along mp, the ratio of hk to mp equals that of the forcein mp to the force in hk, and so the force will be nearly infinite in b [56].19 (A.5.12)

18 p. 63.19 p. 34.


5.4 Guidobaldo dal Monte

Guidobaldo dalMonte was born near Pesaro in 1545 and diedin Pesaro in 1607. He entered the university of Padua in 1564having as a companion Torquato Tasso, studied mathematicswith Federico Commandino and was teacher of BernardinoBaldi. He was one of the greatest mathematicians and me-chanician of the late XVI century. He was also a highly com-petent engineer as well as a director of the Venice arsenal andlast but not least the brother of a prominent cardinal. He wasthe mentor of Galileo and secured for him his first academic

position as a lecturer in mathematics at Pisa and Padua universities [294]. In 1577he published the Mechanicorum liber [86, 377], translated into Italian by FilippoPigafetta in 1581 as Le mechaniche [88]. The book had an enormous editorial suc-cess and was read for the whole XVII century. In 1588 he published the Archimedisaequeponderantium, a paraphrase of Archimedes’ Aequeponderanti [87], in 1600 animportant book on perspective.

Dal Monte was one of the major critics of the approach of Jordanus de Nemore.According to him those of Jordanus and his followers, among which he includesTartaglia, are not valid demonstrations and goes so far as to say that Jordanus shouldnot even be counted among the true mathematicians. Bernaldino Baldi went still fur-ther and considered as paralogisms the demonstrations of Jordanus [18].20 Criticismsof dal Monte must be placed in his time to be understood. Scholars of mathematicsof the period, particularly those of central and southern Italy, could not fail to becharmed by the elegance and rigor of geometry as it was revealed by the recentlypublished Greek translations of Euclid and Archimedes. Archimedes, moreover tohis mathematical theory flanked a consistent mechanical theory and with the samestandards of rigor. It was therefore natural to accept the argument of Archimedes inmechanics and reject those by Jordanus. Although to a modern observer the full re-fusal of Jordanus seems unjustified because the De ratione ponderis has a Euclideanapproach based on definitions, axioms and theorems. It is certainly the ancient textin which the Euclidean approach is extended further outside geometry. It is all in alla very modern text. Dal Monte, however, could hardly accept to reason with con-cepts such as gravity of position which remained a bit undefined and made recourseto empirical intuition.

Given that Jordanus’ theses were then quite common in Italy, dal Monte some-how felt the need to re-establish the ‘truth’, by writing the Mechanicorum liber andArchimedis aequeponderantium that can be seen as the natural completion of thework of spreading Archimedes’s mechanical thought. Commandino indeed had onlypreviously published his text on floating bodies and his anxiety over the rigor led dalMonte to make criticisms that today seem ungenerous, such as those that considerwrong the demonstrations based on the parallelism of descent lines of heavy bod-ies. The hostility towards the approach of Jordanus also led dal Monte to refuse the

20 p. 32.

5.4 Guidobaldo dal Monte 109

correct proof of the inclined plane for the incorrect one by Pappus of Alexandria.Which brought upon the blame, among others, of Evangelista Torricelli [233].21

Although the Mechanicorum liber on the one hand had given up the fertility ofJordanus’s approach, based on the concept of gravity of position and a law of virtualwork, playing in some way a conservative role, it expanded the scope of mechanics.The medieval science of weights, in which attention was focused on demonstratingthe law of the lever, is led back to the Greek tradition of mechanics as a science ofmachines, influenced in this by theMechanica problemata, but especially by Hero’sapproach, then known only through the work of Pappus of Alexandria.

5.4.1 The centre of gravity

In the following, instead of the Mechanicorum liber I will refer to its Italian trans-lation Le mechaniche, which was more diffuse. The text begins with the definitionof the centre of gravity, which is worthy to be reported because of the great weightthis concept will have in formulating the principle of Torricelli. Dal Monte takes thedefinition of Pappus, with the addition of a definition due to Commandino:

The centre of gravity of any body is a certain point within it, from which, if it is imaginedto be suspended and carried, it remains stable and maintains the position which it had atthe beginning, and is not set to rotating by that motion. This definition of the centre ofgravity is taught by Pappus of Alexandria in the eighth book of his Collections. But FedericoCommandino in his book On Centres of Gravity of Solid Bodies explains this centre asfollows: The centre of gravity of any solid shape is that point within it around which aredisposed on all sides parts of equal moments, so that if a plane be passed through this pointcutting the said shape, it will always be divided into parts of equal weight [88].22 (A.5.13)

Is still unclear the role that dal Monte gives to the centre of gravity for a system ofbodies. On the one hand the bodies are taken individually, subject to gravity con-verging toward the centre of the world, on the other hand, the gravity is consideredto be concentrated in the centre of gravity of the whole, which is determined bythe Archimedean rules. It should be noted that dal Monte, with many other math-ematicians of the time, will definitely realize that, from a practical point of view,to consider the lines of action of gravity parallel to each other or to consider themconverging to the centre of the world did not matter much, nevertheless he believedthat, to establish the ‘reality’ of things, one could not accept this approximation.

5.4.2 The balance

After studying the balance with equal weights and arms and with suspension pointsabove and below the centre of gravity, correctly recognizing the stability in the firstcase and instability in the second, dal Monte then goes on to study the still con-troversial case, the quality of the equilibrium of a balance when it was suspended

21 vol. 3, p. 439.22 p. 1. Translation in [298]. Notice that the second definition is the same as that referred to in § 3by Hero.


Plate 2. The two editions of dal Monte’s Mechanics (reproduced with permission, respectively,of Biblioteca Guido Castelnuovo, Università La Sapienza, Rome, and of Biblioteca Alessandrina,Rome)

for his centre of gravity C as shown in Fig. 5.7a, clearly stating that it is in neutralequilibrium:

A balance parallel to the horizon, having its centre within the balance and with equal weightsat its extremities, equally distant from the centre of the balance, will remain stable in anyposition to which it is moved.[…]I say, first, that the balance DE will not move and will remain in that position Now sincethe weights A and B are equal, the centre of gravity of the combination of the two weights Aand B will be at C. Hence the same point C will be the centre of gravity of the balance andof the whole weight. And since the centre of gravity of the balance, C, remains motionlesswhile the balance AB together with the weights moves to DE, the centre of gravity is notmoved [88].23 (A.5.14)

According to dal Monte, Jordanus, Cardano and Tartaglia, who assumed a stablestate of equilibrium for the horizontal scale, were wrong and even went againstArchimedes:

Now since they say that the weight placed at D is heavier in that position than is the weightplaced at E in its lower position, then, when the weights are at D and E, the point C will nolonger be their centre of gravity, inasmuch as they would not be stable if suspended fromC. But that centre will be on the line CD, by Archimedes, On Plane Equilibrium, 1.3. It willnot be on CE, the weight D being heavier than the weight E; let it therefore be at H, fromwhich, if they were suspended, the weights would remain stationary. And since the centreof gravity of the weights joined byAB is at the point C, but that of those placed at D and E



F

H

G

B

D

E

AK

C

a) b)

N L

G TK V

E

F

D

M

OH

C

Fig. 5.7. Equilibrium in the balance with equal arms and weights

is the point H, when the weights A and B are moved to DE, the centre of gravity C wouldbe moved toward D and would approach closer to D, which is impossible. For the weightsremain the same distance apart, and the centre of gravity of any body stays always in thesame place with respect to that body [88].24 (A.5.15)

His searches were picky, but not as rigorous as he claims, in an attempt to refute theviews of Jordanus. With reference to Fig. 5.7a he starts by underlining the weak-ness of Jordanus’ claims that the weight is heavier in D than in E, recovering andimproving the limit analysis of the gravities of position in D and E, when the spacescovered become very small.

Things being taken as before, and from the points D and E the lines DH and EK being drawnperpendicular to the horizon, let there be taken another equal circle LDM, with centre N,which is tangent to the circle FDG at the point D.[…] But the ratio of angle MDH to HDG issmaller than any other ratio that exists between greater and smaller quantities; therefore theproportion of the weights at D and E will be the smallest of all possible ratios, or, rather, willnot be a ratio at all. […] we shall find ratio diminishing ad infinitum, and it follows thus thatthe ratio of the weight placed at D to that at E is not so small that one infinitely less cannotbe found. And since the angle MDG can be divided in infinitum, so also one may divide ininfinitum the excess of weight which D has over E [88].25 (A.5.16)

He sets out very clearly the view of the school of Jordanus that the weight in D, inFig. 5.7a, is heavier than that in E but of a very small amount. Indeed with referenceto Fig. 5.7b where the portion of circle passing from E is redrawn in D as LDM,it is possible to see that the mixed angles HDG (the obliquity of D) and MDH (theobliquity of E) differ by an angle that is small as one likes, in particular, smaller thanany angle bounded by straight segments. The challenge of this statement is showingthat there are countless angles whose difference with the angle HDG tends to zero. Sothe gravity of the weight in D is equal to that of the weight in E, and the equilibriumis indifferent.

24 pp. 11–12. Translation in [298].25 pp. 13–14. Translation in [298].


Fig. 5.8. Difference in gravity of position

Then dal Monte shows, paradoxically, that if one followed the reasoning of Jor-danus, assuming that the forces of weights converge toward the centre of the world(or earth), then the weight in E would be heavier than the weight in D, because theangle ODS (the obliquity of D) is greater than TES (the obliquity of E), and then thebalance should assume a vertical direction rather than return to a horizontal position(Fig. 5.8a).

The arguments of dal Monte against the concept of gravity of position continue;for him it is an ill defined concept and its quantification depends on the size andarrangement of the arcs that are considered, a hiatus with the preceding argumentwhere infinitesimals were spoken of:

For the weight placed at L would move freely toward the centre of the world along LS, andthe weight at D along DS. But since the weight at L weighs wholly on LS, and that at Don DS, the weight at L will weigh more on the line CL than that at D on DC. Therefore theline CL will more sustain the weight than the line CD; and in the same way, the closer theweight is to F, it will be shown for this reason to be more sustained by the line CL, sincethe angle CLS is always less, which is obvious. For if the lines CL and LS should cometogether, which would happen at FCS, then the line CF would sustain the whole weight thatis at F and would render it motionless, nor would it have any tendency to descend [gravezza]along the whole circumference of the circle [88].26 (A.5.17)

Dal Monte maintains with Jordanus, that the gravity position is greater where thedescent path is closer to the line joining this point with the centre of the world.According to this logic and with reference to Fig. 5.8b the point where the gravityof position is greater is O and not A. Dal Monte justifies this greater propensitybecause the arm of the balance offers the least resistance. That is, in modern terms,



L

AC

DM

B

E

H

S K

G

F

Fig. 5.9. Indifferent equilibrium of the balance with equal arms and weights

instead of directly recognizing that there is a greater tendency to fall down becausethe component of weight is greater, he says there is a greater tendency because thereis minor resistance.

With an argument closely related to his idea of gravity, dal Monte proves onceagain that the equal weights placed in E andD at the ends of the balance of Fig. 5.9 areequally heavy and that the balance remains stationary (indifferent equilibrium) in theposition DE. It is one thing, he says, to consider the weights in D and E separately,in which case they would move to S along DS or ES respectively; the other is toconsider them together, so their centre of gravity would move to S along CS, whilethe weights in D and E along DH and EK, as shown in Fig. 5.9. But since C cannotsink, the weights remain at their place, D and E.

If the weight placed at E is heavier than the weight placed at D, the balance DE will neverremain in that position, as we have undertaken to maintain, but it will move to FG. To whichwe reply that it makes a great deal of difference whether we consider the weights separately,one at a time, or as joined together; for the theory of the weight placed at E when it is notconnected with another weight placed at D is one thing, and it is quite another when theweights are joined in such a way that one cannot move without the other. For the straight andnatural descent of the weight placed at E, when it is without connection to another weight,is made along the line ES; but when it is joined with the weight D, its natural descent willno longer be along the line ES, but along a line parallel to CS. For the combined magnitudeof the weights E and D and the balance DE has its centre of gravity at C, and, if this werenot supported at any place, it would move naturally downward along the straight line drawnfrom the centre of gravity C to the centre of the world S until C reached S. […] But if theweights E and D are joined together and we consider them with respect to their conjunction,the natural inclination of the weight placed at E will be along the line MEK, because theweighing down of the other weight at D has the effect that the weight placed at E mustweigh down not along the line ES, but along EK. The same is true of the weight at E; that is,the weight at D does not weigh down along the straight line DS, but along DH, both of them


being prevented from going to their proper places [...].Thus the descent of the weight at Dwill be equal to the rise of the weight at E, and the weight at D will not raise the weight atE. From which it follows that the weights at D and E, considered in conjunction, are equallyheavy [88].27 (A.5.18)

The first chapter ends with a study of unequal arm balance treated according to theArchimedean approach. In the sense that the equilibrium of the balance is guaranteedif it is suspended from its centre of gravity, which as demonstrated by Archimedesdivides the beam of the balance in parts inversely proportional to the weights.

In the second chapter,Della leva, dal Monte examines the lever as if it were some-thing different from the balance. Probably the reason for this apparent duplicationof treatment, in addition to the somewhat pedantic writing, comes from the fact that,besides the weight that is to be raised, it is considered also a muscle force. This forcein the limit might not be vertical, but simply perpendicular to the lever arm; an ex-amination of the text does not completely dissolve the ambiguity, made even greaterby the fact that in the drawings the points of the lever to which the force is appliedare identified only by letters and the direction of the force is never highlighted.

Fig. 5.10. The simplest example of block of pulleys (reproduced with permission of BibliotecaAlessandrina, Rome)

A particularly interesting chapter is the third, dedicated to the block and tackle whichwere of great significance in applications for the lifting of heavy weights, especiallyin construction. The theory of a block of pulleys had never been treated with clarity inthe modern era, in particular, their study was not addressed by Jordanus. Dal Monte,following the suggestion of theMechanica problemata refers the operation block ofpulleys to the lever. I will return to this point in Chapter 7, which shows Descartescriticisms of dal Monte’s demonstration.

In the last two chapters, devoted respectively to the wedge and the screw, dalMonte refers to the famous Pappus’ demonstration of the inclined plane, by fully ac-



cepting it. The reason for this acceptance must be sought in the fact that this demon-stration fitted with his reductionist attempt to reduce all the machines to the lever.The proof of Jordanus would have been ignored by dal Monte even had he consid-ered it as correct, but this was not the case, because he assumed a different principle.

5.4.3 The virtual work law

Dal Monte complements his static analysis with a kinematic analysis also therebyarriving at a statement of a law of virtual displacements. For example, after studyingthe statics of the lever he switches to analyze the relationships between force, weightand motion.

F

G

H

B

D

A

E

KC

Fig. 5.11. The shaft with the wheel

Let there be the lever AB with its fulcrum C, and let the weight D be attached at the pointB, and let the power at A move the weight D by means of a lever AB. Then the space of thepower at A is to the space of the weight as CA is to CB.But let there be the lever at AB, whose fulcrum is B, and the moving power is at A and theweight at C; I say that the space of the moved power to the space of the weight carried is asBA to BC [88].28 (A.5.19)

But dal Monte does not give importance to this fact and he is ready to deny it is trueand therefore not worthy of being engaged in laws of mechanics.

CorollaryFrom these things it is evident that the ratio of the space of the power which moves to thespace of the weight moved is greater than that of the weight to the same power. For thespace of the power has the same ratio to the space of the weight as that of the weight to thepower which sustains the same weight. But the power that sustains is less than the powerthat moves; therefore the weight will have a lesser ratio to the power that moves it than tothe power that sustains it. Therefore the ratio of the space of the power that moves to thespace of the weight will be greater than that of the weight to the power [88].29 (A.5.20)

28 pp. 76–77. Translation in [298].29 pp. 77–78. Translation in [298].


Dal Monte basically says that although the kinematic analysis indicates that the dis-tances traveled by the weight and power in equilibrium are inversely proportional tothem, in practice one must take into account that when the weight is moving it is notin balance and the power must be a bit greater than that necessary for the equilibriumand thus the ratio of the distance traveled by the weight and that of power will begreater than the ratio between strength and weight.

5.5 Giovanni Battista Benedetti

Giovanni Battista Benedetti was born in Venice in 1530 and died in Turin in 1590.He received his first and only systematic education in philosophy, music and mathe-matics from his father. Though never mentioned by Tartaglia he was nevertheless ahis pupil for a short time. In 1558 Benedetti became court mathematician for DukeOttavio Farnese in Parma. In 1567 he was invited by the Duke of Savoy, EmanuleFiliberto, to the court in Turin, where until his death he remained an important ad-viser to the court [294]. In his first book, the Resolutio of 1553 [28] he exposed, inthe letter of dedication, the theory of falling bodies, according to which bodies ofsame density fall with equal speed, independently of the weight. In mechanics hischief work was the Diversarum speculationum of 1585 [29]. The book deals largelywith questions of dynamics; there were however fundamental contributions to stat-ics, where a quite modern concept of static moment of a force is referred to. Thoughthe Diversarum speculationum may be considered a commentary of the Mechanicaproblemata, Benedetti’s approach was essentially Archimedean. He criticises bothTartaglia and Jordanus de Nemore for their kinematic approaches.

In the following I will comment shortly upon the law of the lever and the efficacyof a force applied to a arm of a balance.

5.5.1 Effect of the position of a weight on its heaviness

In the Diversarum speculationum Benedetti is critical toward the exposition on thematter of both Mechanica problemata and De ratione ponderis. He begins his re-marks by stating that the effect of a weight on the end of an arm of a balance dependson the inclination of the arm:

The ratio of [the effect of] the weight at C to [the effect of] the same weight at F will beequal to the ratio of the whole arm BC to the part, BU [29].30 (A.5.21)

To justify this fact Benedetti refers not to the greater ease of motion and thus togreater virtual velocity, but rather to a greater or lesser resistance offered by the arm,considered as a constraint for the motion. The same argument used by dal Monte inhis Mechanicorum liber. For Benedetti the more the slope, the greater the effect ofthe constraint:

30 Chap, 2, p. 142. Translation in [298].

5.5 Giovanni Battista Benedetti 117

A

B D

QMO

C u

e

F

Fig. 5.12. A weight pending from an arm

To make this clearer let us imagine a string Fu perpendicular [to the horizontal], with theweight that had been at F now hanging at the extremity u of the string. It will be clear fromthis that the weight will produce the same effect as if it had been at F. […] And I wouldmake the same assertion if the arm were in position eB. […] For the weight hanging by astring from u is the same [in its effect] as that which had hung freely from a string at pointE of arm BE. And this would be due to the fact that in part it hung from the centre B. And ifthe arm were in position BQ, the whole weight would be suspended from the centre B, justas in position BA it would rest wholly upon that centre [29].31 (A.5.22)

Chapter 2 of the Diversarum speculationum ends with a comment of some interest,which states that if it is true that the heavy bodies tend toward the centre of theworld according to straight lines with the direction dependent on their position onthe balance, it may be assumed that these directions differ only slightly and then thelines, such as CO and BQ of Fig. 5.12, can be considered as parallel.

Now I call side BC horizontal, supposing that it makes a right angle with CO, whence angleCBQ is less than a right angle by the size of an angle equal to that which the two lines COand BQ make at the centre of the region of the elements.Yet this makes no difference, sincethat angle is too small to be measured [29].32 (A.5.23)

In Chapter 3 Benedetti changes to make considerations of quantitative character andclearly states that the effect of a force – led by a weight attached to a rope or a muscle– on an arm of a balance, however inclined, is proportional to the distance of the lineof action of the force from the fulcrum. First this statement is referred to in a notproblematic case of vertical forces:

From what we have already shown it may easily be understood that the length of Bu[Fig. 5.12], which is virtually perpendicular from centre B to the line of inclination Fu,is the quantity that enables us to measure the force of F itself in a position of this kind, i.e.,a position in which line FU constitutes with arm FB the acute angle BFu [29].33 (A.5.24)

then it is referred to also to forces or weights which act in inclined directions.Benedetti’s argument, even if unequivocal, in the substance does not appear entirelyconvincing.31 Chapter 2, p. 142. Translation in [298].32 Chapter 2, p. 143. Translation in [298].33 Chapter 2, p. 143. Translation in [298].


a

a

t

t

ib

b

ee

oo

c

c

i

Fig. 5.13. The ‘static moment’ of weights and forces

To understand this better, let us imagine [Fig. 5.13] a balance boa fixed at its centre o, andsuppose that at its extremities two weights are attached, or two moving forces, e and c, insuch a way that the line of inclination of e, that is be, makes a right angle with ob at pointb, but the line of inclination of c, that is ac, makes an acute angle [Fig. 5.13a] or an obtuseangle [Fig. 5.13b] with oa at point a Let us imagine, then, a line ot perpendicular to the lineof inclination ca […]. Imagine, then, that oa is cut at point i, so that oi is equal to ot, and thata weight is suspended at i, equal to c and with a line of inclination parallel to that of weighte. But we assume that the weight or force c is greater than e in proportion as bo is greaterthan ot. Obviously, then, according to Archimedes, De ponderibus, boi will not move fromits position. Again, if in place of oi we imagine ot rigidly connected [in the same line] withob and subjected to force c acting along line tc, the result will obviously be the same – botwill not move from its position [29].34 (A.5.25)

Curious, to say the least, is the way in which Benedetti explains in Chapter 4 of theDiversarum speculationum the reason of the different effectiveness of two forceswhich are at different distances from the fulcrum of a lever. Somehow repeating thereasoning above, and at the same time denying it. In essence he argues that one mustconsider the lever as a solid body and not as a rod, as shown in Fig. 5.14. The forceapplied in n shall weigh more on the fulcrum compared to the force applied in u,because the line ni is closer to the vertical than ui, it will therefore be less effective.

n

x i t s

ueo

Fig. 5.14. A three-dimensional lever

5.5.2 Errors of Tartaglia and Jordanus

Benedetti in Chapter 7 of the Diversarum speculationum exposes those he believedwere errors made by Tartaglia (and Jordanus de Nemore). Among his criticisms one

34 Chapter 3, p. 143. Translation in [298].

5.5 Giovanni Battista Benedetti 119

of the most interesting is relative to the analysis of the balance with equal weightsand arms. For Tartaglia the balance has the horizon as position of stable equilibrium.For Benedetti instead, the equilibriumwas unstable. First he ‘proves’ that Tartaglia’sargumentation is fallacious given his premises, i.e. the parallelism of gravity actions.In such a case the gravity of position for the weights a and b of Fig. 5.1535 shouldbe the same:

e

y

zc

w

x

n

l

i

f

p

a

a

r

t

k

m

o

q

s

v

j

d

b

b

Fig. 5.15. Tartaglia’s fallacious reasoning

And in the second part of the fifth proposition he [Tartaglia] fails to see that no difference inweight is produced by virtue of position in the way in which he argues. For if body b mustdescend on arc il, body amust ascend on arc vs, equal and similar to arc il and placed in thesame way. Therefore, just as it is easy for body a to ascend on arc vs it is easy for body bto descend on arc vs. And this fifth proposition is the second proposed by Jordanus [29].36

(A.5.26)

Benedetti believes that the reasoning of Tartaglia (and Jordanus) is not consistentbecause he regards the two weights a and b as if they were independent and bothcould move downward. They are in fact constrained by the beam ab and when oneof them falls the other rises. So it should be compared the paths il and vs, which havethe same vertical projections xy and d j respectively.

But the reasoning of Tartaglia, according to Benedetti, is wrong in the merit too,because the lines of action of gravity are not parallel but converging toward the centreof the world u. With reference to Fig. 5.16, consider with Benedetti the balanceaob displaced from the horizontal position. Assuming the two weights a and b havethe same absolute gravity, their efficacy depends on the distances of their line ofaction, respectively ao and bo from the fulcrum o. For the weight a the distance isrepresented by ot for the weight b by oe, and it is simple to prove that ot > oe. Thenthe gravity of weight a is greater than that of weight b, and the balance is going totilt up the vertical position, i.e. the horizon is an unstable equilibrium position.

Benedetti closes the chapter criticizing the (correct) solution given by Tartagliaon the inclined plane, saying it is worthless without specifying the reason.

35 The figure in not in Benedetti’s text, it is derived from [223], p. 89r.36 Chapter 7, p. 148. Translation in [298].


ns

o

u

a c

et

b

Fig. 5.16. Unstable equilibrium for the balance with equal arms and weights

5.6 Galileo Galilei

Galileo Galilei was born in Pisa in 1564 and died in Florence(Arcetri) in 1642. In Pisa he undertook the study of mathe-matics under the guidance of Ostilio Ricci, a pupil of NicolòTartaglia. Of this period are the first contact with ChristopherClavius (1538–1612) and Guidobaldo dal Monte. In 1589,Galileo obtained a professorship of mathematics at Pisa. In1592 he moved to Padua as a professor of mathematics. In1610 he published the Sidereus nuncius [114], a work thatmade him famous around the world; still in 1610 he was

named Philosopher and Mathematician of the Grand Duke of Tuscany. In 1616 theHoly Office condemned the Copernican theory and Galileo was warned (is it true?)not to defend it. In 1632 he published his Dialogo sopra i due massimi sistemi [116]on Ptolemaic and Copernican systems. In June 1633, in the guise of penance andkneeling, in front of catholic cardinals, Galileo was forced to pronounce the solemnrecantation and admission of guilt [294]. In 1638 he published, still incomplete, theDiscorsi e dimostrazioni matematiche sopra due nuove scienze [118, 347, 297, 296].

The contribution that Galileo provided to statics is far less decisive than that todynamics, nonetheless it is important. Though there may be doubts on the originalityof some of his writings, it is certain that no one before him had formulated and solvedhis own problems with extraordinary clarity.

The main works of Galileo, which specifically concern the equilibrium are:Le mecaniche (1593–1594 early manuscripts, first printed in a French version byMersenne in 1634 and in Italian, in 1649, after the death of Galileo [119], the Dis-corso intorno alle cose che stanno in su l’acqua e scritture varie, printed in 1612[115] and the already cited Discorsi e dimostrazioni matematiche sopra due nuovescienze [118].

5.6 Galileo Galilei 121

5.6.1 The concept of moment. A law of virtual velocities

In Le mecaniche Galileo introduces a concept and a term, that of moment (“mo-mento”), that will be of great fortune and adopted, at least in Italy, until the earlyXIX century:

Moment is the propension of descending, caused not so much by the Gravity of the move-able, as by the disposure which divers Grave Bodies have in relation to one another; bymeans of whichMoment, we oft see a Body less Grave counterpoise another of greater Grav-ity: as in the Steelyard, a great Weight is raised by a very small counterpoise, not throughexcess of Gravity, but through the remotenesse from the point whereby the Beam is up held,which conjoyned to the Gravity of the lesser weight adds thereunto Moment, and Impetusof descending, wherewith the Moment of the other greater Gravity may be exceeded. MO-MENT then is that IMPETUS of descending, compounded of Gravity, Position, and the like,whereby that propension may be occasioned [119].37 (A.5.27)

The concept is taken up and elaborated in the Discorso intorno alle cose che stannoin su l’acqua:

Moment for mechanics, means that virtue, that force, that effectiveness with which themotormoves and the mobile resists [emphasis added], a virtue which depends not only on thesimple gravity, but on the speed of motion, from the different angles of the spaces overwhich the motion is made, because a heavy body makes more impetus in a very inclinedspace than in one less inclined.The second principle [the first was that equal weights with equal speed have equal forces andmoments] is, that the moment and the force of gravity is increased by the speed of motion sothat absolutely equal weights, but combined with unequal velocities, are of force, momentand virtue unequal, and the fastest is more powerful, according to the proportion of its speedto the speed of the other. Of this we have a very suitable example in the balance with unequalarms, where absolutely equal weights do not press and are not equally strong, but that whichis at the greatest distance from the center, around which the balance moves, sinks and raisesthe other, and it is the motion of ascending fast, the other slow: and such is the force andvirtue that the speed of motion gives to the mobile that receives, and it can compensate asmuch weight as added to the other mobile; so that if one arm of a balance were ten timeslonger than the other, in order to move the balance around its middle, the end of that passedten times more space than the end of this, a weight placed at the greater distance can sustainand equilibrate another ten times heavier than itself, and this because, moving the balance,the lower weight will move ten times faster than the other [115].38 (A.5.28)

From the reading of passages quoted above it is clear as Galileo espouses the viewthat the downward velocity of a heavy body increases its efficacy or ‘force’ to godown while the upward velocity increases its resistance to be lifted. His conceptionis rather uncommon in statics and differs from dal Monte and Benedetti’s who in-stead believed that there was no increase of ‘force’ due to velocity, but only a greatervelocity due to lower resistance of constraints. It also differs from Jordanus’s con-cept of gravity of position, measured by the rate of possible descent, i.e. a purelygeometric motion therefore not increasing the ‘force’ or the resistance of weights.Galileo tried without success to provide a measure of the static force equivalentto the increase of the effectiveness of a weight with his speed in the last section

37 p. 159. Translation in [121].38 pp. 68–69.


of Le mecaniche and during the sixth day added to the Discorsi e dimostrazionimatematiche sopra due nuove scienze in the Florentine edition of 1715. Only in theXIX century, when the difference between force and energy was fully clarified, wasit recognized that the force due to the speed (kinetic energy) is incommensurablewith the static force.

Paolo Galluzzi [321]39 attaches great importance to the use of the term momentand identifies in the Lemecaniche the place where Galileo first introduced the defini-tion in a technical sense. He preferred it to the more generic medieval term gravitas,used in the De motu [113] a few years before, which gave rise to ambiguity becausesometimes it pointed to the sheer weight.

In the Le mecaniche, after having proved the law of the lever according toArchimedes and similarly to what he will do in the first day of the Discorsi (with areasoning similar to that of Stevin, that probably he did not know) Galileo examinesthe equilibrium of the lever using the concept of moment:

A BC

D

E

Fig. 5.17. The lever

Now being that Weights unequall come to acquire equall Moment, by being alternatelysuspended at Distances that have the same proportion with them; I think it not fit to overpasse with silence another congruicy and probability, which may confirm the same truth; forlet the Ballance AB, be considered, as it is divided into unequal parts in the point C, and lettheWeights be of the same proportion that is between the Distances BC, and CA, alternatelysuspended by the points A, and B: It is already manifest, that the one will counterpoise theother, and consequently, that were there added to one of them a very small Moment ofGravity, it would preponderate, raising the other, so that an insensible Weight put to theGrave B, the Ballance would move and descend from the point B towards E, and the otherextream A would ascend into D, and in regard that to weigh down B, every small Gravityis sufficient, therefore not keeping any accompt of this insensible Moment, we will put nodifference between one Weights sustaining, and one Weights moving another [emphasisadded]. Now, let us consider the Motion which the Weight Bmakes, descending into E, andthat which the other A makes in ascending into D, we shall without doubt find the SpaceBE to be so much greater the Space AD, as the Distance BC is greater than CA, formingin the Center C two angles DCA, and ECB, equall as being at the Cock, and consequentlytwo Circumferences AD and BE alike; and to have the same proportion to one another, ashave the Semidiameters BC, and CA, by which they are described: so that then the Velocityof the Motion of the descending Grave B cometh to be so much Superiour to the Velocityof the other ascending Moveable A, as the Gravity of this exceeds the Gravity of that; andit not being possible that the Weight A should be raised to D, although slowly, unless theother Weight B do move to E swiftly, it will not be strange, or inconsistent with the Order of

39 p. 199–221.


Nature, that the Velocity of theMotion of the Grave B, do compensate the greater Resistanceof the Weight A, so long as it moveth slowly toD, and the other descendeth swiftly to E, andso on the contrary, the Weight A being placed in the point D, and the other B in the point E,it will not be unreasonable that that falling leasurely to A, should be able to raise the otherhastily to B, recovering by its Gravity what it had lost by it’s Tardity of Motion. And bythis Discourse we may come to know how the Velocity of the Motion is able to increaseMoment in the Moveable, according to that same proportion by which the said Velocity ofthe Motion is augmented [emphasis added] [119].40 (A.5.29)

Galileo’s reasoning is not very clear. He starts with a balance with two weights Aand B inversely proportional to the arms that are assumed to be equilibrated. Thenhe imagines a motion led by a small weight added on one side, which because ofits smallness does not alter the ratio of the weights of the two bodies. As a result ofthe motion the heavy bodies A and B acquire velocities proportional to the distancesfrom the fulcrum, then velocities and weights are in inverse relationship, then themoments are the same and A and B are in equilibrium.

But the reasoning is circular because the equilibrium is proved after it was as-sumed. Galileo’s reasoning would not be circular, and perhaps this could be his in-tention, if he had not specified at the beginning that the equilibrated weights were ininverse relationship to distance, as follows: consider a balance in equilibrium withthe two weights A and B and upset the balance by adding a small weight on theside of B. In the following motion there will be determined two moments that willcompensate only if the weights are in inverse proportion to distances.

The principle Galileo invokes for the equilibrium is the equality of moments, i.e.a law of virtual work, expressed by means of velocity. He states that this principlecan be deduced from the Mechanica problemata of Aristotle:

This equality between gravity and speed is found in all the mechanical instruments, andwas considered by Aristotle in his Mechanical questions; so we can still take for grantedthat absolutely unequal weights alternatively counterweight and make themselves of thesame moment, once their gravities have contrary proportion with the speed of their motions[115].41 (A.5.30)

By considerations similar to those developed for the lever with straight arms, Galileodemonstrated that for the angular lever the magnitude that defines the equilibrium isthe distance from the fulcrum of the vertical line through the weight.

There is also another thing, before we proceed any farther, to be considered; and this istouching the Distances, whereat, or wherein Weights do hang: for it much imports how weare to understand Distances equall, and unequall; and, in sum, in what manner they oughtto be measured. […] But if elevating the Line CB, moving it about the point C, it shall betransferred into CD, so that the Ballance stand according to the two Lines A C, and CD, thetwo equall Weights hanging at the Terms A and D, shall no longer weigh equally on thatpoint C, because the distance of the Weight placed in D, is made lesse then it was when ithanged in B. For if we confider the Lines, along [or by] which the said Graves make theirImpulse, and would descend, in case they were freely moved, there is no doubt but that theywould make or describe the Lines AG, DF, BH: Therefore the Weight hanging on the pointD, maketh it’s Moment and Impetus according to the Line D F: but when it hanged in B, it

40 pp. 163–164. Translation in [121].41 p. 275.


made Impetus in the Line BH: and because the Line DF is nearer to the Fulciment C, then isthe Line BH Therefore we are to understand that the Weights hanging on the points A andD, are not equidistant from the point C, as they be when they are constituted according totheir Right Line ACB [119].42 (A.5.31)

B

D

C A

F

H

Fig. 5.18. The angled lever

In the passages above, Galileo leaves a certain ambiguity in the way the equalityand moments should be understood. A modern reader based on the notion of staticmoment, and maybe after having read Jordanus, would be tempted to see a balanceof two trends to go down. But some doubt remains because Galileo speaks of raisingof a body and lowering of the other.

In hisDiscorso intorno alle cose che stanno in su l’acqua Galileo specifies, or per-haps decides, that moment is also the resistance to gain speed up. So the equilibriumis not from the equality of two trends to go down, but from the balance of the impetusto go down and the resistance to go up, increased both by the speed (for commentson this shifting of meaning see [321]). Galileo treats his law of virtual work – theequality of moments – as a principle to study the equilibrium of fluids. For example,to justify the height of the fluid on both sides of a siphon of very different sectionsis the same. In reality, Galileo does not prove the equality, he limits himself onlyto substantiating the plausibility of the thing. The equality will be demonstrated byPascal with the use of Torricelli’s principle (see Chapters 6 and 8).

The justification, making reference to Fig. 5.19, assumes, by absurdity, that thefluid of the largest side of the siphon drops of the amount HO and at the same timethe fluid of the small end of the siphon will rise of the height LA greater than HO –so that the volume ABL is equal to the volume QOH – and therefore much faster. Itis plausible, therefore, that the motion of the water of the largest side (a great weight)be offset by the large velocity of the water on the smaller side (a small weight) andit is therefore justified that the water on both sides has the same level.

And for very large confirmation, and clearer explanation of this, consider this figure (and,unless I am mistaken, it could be used to tear out fault some practical mechanicians, whoon a false basis sometimes try impossible enterprises), in which the wide vessel EIDF, iscontinued by the thin barrel ICAB, and fill them with water up to LGH, which in this stateis at rest, but not without some wonder, for who will not understand as soon as it is seen thatthe heavy burden of the large amount of water GD, pressing down does not raise and drive



Fig. 5.19. The siphon (reproduced with permission of Biblioteca Guido Castelnuovo, UniversitàLa Sapienza, Rome)

away the small amount of the other contained within the barrel CL, by which the descent isdisputed and prevented? But this will no longer be a wonder, if we begin to pretend to havelowered the water GD only up to QO, and then consider what water CL has made, which, togive place to the other that has diminished from the level GH to the level QO, must have inthe same time been lifted up from the level L to AB, and the ascent LB is as much greaterthan the descent GQ, as much the amplitude GD of the vessel is greater than the width LCof the barrel, which in sum is what the water GD is more than the LC. But since the momentof the speed of motion in a mobile compensates for the gravity of another, what wonder if itwill be the fastest ascent of a little water CL to resist the very slow greater quantity of waterGD? [115]43. (A.5.32)

It is worthwhile even to notice an exchange between Salviati and Sagredo, takenfrom Dialogo sopra i due massimi sistemi del mondo where the idea is emphasizedthat the velocity increases the effectiveness of the weights:

SAGR. But do you think the speed compensates precisely the gravity? That is, that both themoment and the force of a mobile of four pounds of weight are as that of a hundred, whenthis had one hundred degrees of speed and that only four degrees?SALV. Of course yes, as I could show with many experiences, but for now it suffices to con-firm this one of the balance, where you will see the little heavy roman be able to support andcompensate the very heavy bale, when its distance from the centre over which the balanceis sustained and rotates will be greater than the other lesser distance from which the balehangs, as the absolute weight of the bale is greater than that of the roman. And the cause forwhich the bale cannot lift the roman, much less heavy, cannot be other but the difference ofthe movements of this and that. When the bale with the lowering of one finger did lift theroman of one hundred fingers [116].44 (A.5.33)

In the Discorsi e dimostrazioni matematiche Galileo adds other meanings to theterm moment [321]. He does it for example on the second day where the strength of

43 pp. 77–78.44 p. 241.


materials is considered. Here Galileo uses the concept of moment for the equilibriumof straight and angular levers with a language very close to that of modern textbooksof statics and the reader is tempted to assume the Galilean moment as the staticmoment (i.e. the product of force by its arm). In the evaluation of the resistanceof a cantilever, Galileo introduces the moment of the strength to breaking, i.e. themoment of a ‘force’ with an ontological status similar to that of reactive forces andfor which probably the concept of moment as propension to motion could hardly beapplied. For details see [272].

Particularly interesting is the observation Galileo makes the fourth day, the effi-cacy of a very small weight h in lifting a very large one as illustrated in Fig. 5.20.A simple kinematic analysis shows that the relationship between the lowering e f ofthe weight h to the raising f i and f l respectively of the weights c and d is as great asyou like. This implies that whatever the weight h, its velocity will compensate theweights c and d, however big they are, raising them.

According to Duhem [305]45 the above considerations were suggested to Galileoby the reading of the Traité de mechanique of 1636 by Roberval, in particular bythe demonstration of the rule of the parallelogram by means of a law of virtual work(see Chapter 7). This is possible, as the fourth day of Discorsi was added after thefirst edition in 1638 and Galileo before his death would certainly have got to knowthe ideas of Roberval.

Fig. 5.20. Large weights raised by a very small weight (reproduced with permission of BibliotecaGuido Castelnuovo, Università La Sapienza, Rome)

SAGR. You are quite right; you do not hesitate to admit that however small the force of themoving body be, it will overcome any resistance, however great, provided it gains more invelocity than it loses in force and weight. Now let us return to the case of the cord. In theaccompanying figure ab represents a line passing through two fixed points a and b; at theextremities of this line, as you see, two large weights e and d hang, which stretch it withgreat force and keep it truly straight, being it merely a line without weight. Now I wishto remark that if from the middle point of this line, which we may call e, one suspends anysmall weight, say h, the line abwill yield toward the point f and on account of its elongationit will compel the two heavy weights c and d to rise. This I shall demonstrate as follows:with the points a and b as centres describe the two quadrants, eig and elm; now since the twosemidiameters ai and bl are equal to ae and eb, the remainders f i and f l are the excessesof the lines a f and f b over ae and eb; they therefore determine the rise of the weights cand d, assuming of course that the weight h has taken the position f , which could happenwhenever the line e f , which represents the descent of h had greater proportion than the linef i – associated to the rise of the weights c and d – of the heaviness of both the two weights to

45 vol. 1, p. 324.


the heaviness of the weight h. But this will necessarily occur however large be the heavinessof weights c and d and little that of the weight h. Even when the weights of c and d are verygreat and that of h very small this will happen [118].46 (A.5.34)

5.6.2 A law of virtual displacements

Galileo in the early stages of his studies considered the law of virtual work as a prin-ciple, that of virtual velocities. However, he did not disdain to also consider virtualdisplacements, coming to present the law of virtual work, as a theorem, proved forall simple machines.

At the beginning of Le mecanicheGalileo seems to assign the law of virtual workbased on displacements a fundamental role, given its immediate evidence, althoughbased on every day experience. This passage anticipates in a form strikingly similarthe reasoning of Descartes in his letters to Constantin Huygens and Mersenne in1637–38:

Of which mistakes I think I have found the principal cause to be the belief and constantopinion these Artificers had, and still have, that they are able with a small force to moveand raise great weights; (in a certain manner with their Machines cozening nature, whoseInstinct, yea most positive constitution it is, that no Resistance can be overcome, but by aForce more potent then it:) which conjecture how false it is, I hope by the ensuing true andnecessary Demonstrations to evince [119].47

[…]Now, any determinate Resistance and limited Force whatsoever being assigned, and anyDistance given, there is no doubt to be made, but that the given Force may carry the givenWeight to the determinate Distance; for, although the Force were extream small, yet, bydividing the Weight into many small parts, none of which remain superiour to the Force,and by transferring them one by one, it shall at last have carried the whole Weight to theassigned Term: and yet one cannot at the end of the Work with Reason say, that that greatWeight hath been moved, and trans ported by a Force lesse then it self, howbeit indeed itwas done by a Force, that many times reiterated that Motion, and that Space, which shallhave been measured but only once by the whole Weight. From whence it appears, that theVelocity of the Force hath been as many times Superiour to the Resistance of the weight, asthe said Weight was superiour to the Force; for that in the same Time that the moving Forcehath many times measured the intervall between the Terms of theMotion, the saidMoveablehappens to have past it onely once: nor therefore ought we to affirm a great Resistance tohave been overcome by a small Force, contrary to the constitution of Nature. Then onelymay we say the Natural Constitution is overcome, when the lesser Force transfers the greaterResistance, with a Velocity of Motion like to that wherewith it self doth move; which weaffirm absolutely to be impossible to be done with any Machine imaginable. But becauseit may sometimes come to passe, that having but little Force, it is required to move a greatWeight all at once, without dividing it in pieces, on this occasion it will be necessary to haverecourse to the Machine, by means whereof the proposed Weight may be transferred to theassigned Space by the Force given [119].48

[…]And this ought to passe for one of the benefits taken from the Mechanicks: for indeed itfrequently happens, that be ing scanted in Force but not Time, we are put upon moving

46 pp. 311–312.47 p. 155. Translation in [121].48 pp. 156–157. Translation in [121].


great Weights unitedly or in grosse: but he that should hope, and at tempt to do the sameby the help of Machines without increase of Tardity in the Moveable, would certainly bedeceived, and would declare his ignorance of the use of Mechanick Instruments, and thereason of their effects [119].49 (A.5.35)

The proof of the law of virtual work based on the displacements is exhibited for allsimple machines after their operation has been explained by the law of the lever. Forexample, for the lever Galileo writes:

G

B

CL

M

D

I

Fig. 5.21. Virtual displacement law for the lever

And here it is to be noted, which I shall also in its place remember you of, that the benefitdrawn from all Mechanical Instruments, is not that which the vulgar Mechanicians do per-suade us, to wit, such, that there by Nature is overcome, and in a certain manner deluded,a small Force over-powring a very great Resistance with help of the Leaver; for we shalldemonstrate, that without the help of the length of the Leaver, the same Force, in the sameTime, shall work the same effect. For taking the same Leaver B C D, whose rest or Fulci-ment is in C, let the Distance C D be supposed, for example, to be in quintuple proportion tothe Distance C B, & the said Leaver to be moved till it come to I C G: In the Time that theForce shall have passed the Space D I, the Weight shall have been moved from B to G: andbecause the Distance D C, was supposed quintuple to the other C B, it is manifest from thethings demonstrated, that the Weight placed in B may be five times greater then the movingForce supposed to be in D: but now, if on the contrary, we take notice of the Way passed bythe Force from D unto I, whilst the Weight is moved from B unto G, we shall find likewisethe Way D I, to be quintuple to the Space B G. Moreover if we take the Distance C L, equalto the Distance C B, and place the same Force that was in D, in the point L, and in the point Bthe fifth part onely of the Weight that was put there at first, there is no question, but that theForce in L being now equal to this Weight in B, and the Distances L C and C B being equall,the said Force shall be able, being moved along the Space LM to transfer the Weight equallto it self, thorow the other equall Space B G: which five times reiterating this same action,shall transport all the parts of the said Weight to the same Term G: But the repeating of theSpace L M, is certainly nothing more nor lesse then the onely once measuring the Space DI, quintuple to the said L M. Therefore the transferring of the Weight from B to G, requirethno lesse Force, nor lesse Time, nor a shorter Way if it wee placed in D, than it would needif the same were applied in L: And, in short, the benefit that is derived from the length of theLeaver C D, is no other, save the enabling us to move that Body all at once, which would nothave been moved by the same Force, in the same Time, with an equall Motion, save onely inpieces, without the help of the Leaver [emphasis added] [119].50 (A.5.36)

In the previous quotation Galileo highlights two things: on the one hand that whilea load p/n covers a path h, the load p will cover the path h/n. On the other hand

49 p. 157. Translation in [121].50 pp. 166–167. Translation in [121].


100 200

100

100 100

1

2

1 1

100a) b)

Fig. 5.22. Separate raising of a weight of 200 pounds to different heights

that the load p would be brought up to h/n by carrying n times a load p/n at h/n, asshown in Fig. 5.22, where it is clear that lifting a weight of 100 pounds to a heightof two foot (Fig. 5.22a) is equivalent to lifting 200 pounds to a height of one feet(Fig. 5.22b). This operation can be done without the help of machines, from whichthe conclusion that the machines make the work of man more comfortable but donot allow any savings in work, or fatigue.

In the case of the inclined plane it would seem that the law of virtual displace-ments is not true, because a weight E is shifted by a lower weight F of the sameamount. This is true, but, says Galileo, the largest displacement of F should not bemeasured along the plane but on the vertical because “heavy bodies make no resis-tance to translational motions”. If the distance traveled on the vertical is measured,it can be seen that weights are in inverse ratio of changes in altitude.

A

E

DC

F

B

Fig. 5.23. Virtual displacement law for the inclined plane

Lastly, we are not to pass over that Consideration with silence which at the beginning hathbeen said to be necessary for us to have in all Mechanick Instruments, to wit, That what isgained in Force by their assistance, is lost again in Time, and in the Velocity: which perad-venture, might not have seemed to some so true and manifest in the present Contemplation;nay, rather it seems, that in this case the Force is multiplied without the Movers moving alonger way than the Moveable: In regard, that if we shall in the Triangle ABC suppose theLine AB to be the Plane of the Horizon, AC the elevated Plane, whose Altitude is measuredby the Perpendicular CB, a Moveable placed upon the Plane AC, and the Cord EDF tyed toit, and a Force or Weight applyed in F that hath to the Gravity of the Weight E the sameproportion that the Line BC hath to CA; by what hath been demonstrated, the Weight Fshall descend downwards, drawing the Moveable E along the elevated Plane; nor shall theMove able E measure a greater Space when it shall have passed the whole Line AC, thanthat which the said [119].51 (A.5.37)



But here yet it must be advertised, that al though theMoveable E shall have passed the wholeLine AC, in the same Time that the other Grave F shall have been abased the like Space,nevertheless the Grave E shall not have retired from the common Center of things Gravemore than the Space of the Perpendicular CB. but yet the Grave F descending Perpendicu-larly shall be abased a Space equal to the whole Line AC. And because Grave Bodies makeno Resistance to Transversal Motions, but only so far as they happen to recede from theCenter of the Earth; There fore the Moveable E in all the Motion AC being raised no morethan the length of the Line CB, but the other F being abased perpendicularly the quantityof all the Line A C: Therefore we may deservedly affirm that Way of the Force E main-taineth the same proportion to the Force F that the Line AC hath to CB; that is, the WeightE to the Weight F . It very much importeth, therefore, to consider by [or along] what Linesthe Motions are made, especially in examine Grave Bodies, the Moments of which havetheir total Vigour, and entire Resistance in the Line Perpendicular to the Horizon; and inthe others transversally Elevated and Inclined they feel the more or less Vigour, Impetus, orResistance, the more or less those Inclinations approach unto the Perpendicular Inclination.[119].52 (A.5.38)

Galileo verifies the law of virtual displacements even for a block of pulleys. Re-ferring to Fig. 5.24, after having demonstrated using the law of the lever that therelationship between the weight in H and force in I is 3 : 1, he concludes by notingthat the shift of I is 3 times that of H, and the law of virtual displacement is verified.

E F

ID

GA B

C

H

Fig. 5.24. Virtual displacement law for the pulley

Which being demonstrated, we will pass forwards to the Pulleys, and will describe the in-feriour Gyrils of ACB, voluble about the Center G, and the Weight H hanging thereat, wewill draw the other up per one E F, winding about them both the Rope DFEACBI, of whichlet the end D be fastned to the inferiour Pulley, and to the other I let the Force be applyed:Which, I say, sustaining or moving the Weight H, shall feele no more than the third part ofthe Gravity of the same. For considering the contrivance of this Machine, we shall find thatthe Diameter A B supplieth the place of a Leaver, in whose term B the Force I is applied,and in the other A the Fulciment is placed, at the middle G the Grave H is hanged, and an-other Force D applied at the same place: so that the Weight is fastned to the three Ropes IB,FD, and EA, which with equal Labour sustain the Weight. Now, by what hath already been



contemplated, the two Forces D and B being applied, one, to the midst of the Leaver A B,and the other to the extream term B, it is manifest, that each of them holdeth no more butthe third part of the Weight H: Therefore the Power I, having a Moment equal to the thirdpart of the Weight H, shall be able to sustain and move it: but yet the Way of the Force in Ishall be triple to the Way that the Weight shall pass; the said Force being to distend it selfaccording to the Length of the three Ropes I B, F D, and E A, of which one alone measureththe Way of the Weight H. [119].53 (A.5.39)

5.6.3 Proof of the law of the inclined plane

The problem of the inclined plane did not appear, neither in Aristotle’s Mechanicaproblemata nor in Archimedes’ Aequiponderanti; the first knownwriting on the sub-ject was reported on Hero’s Mechanica where was proposed a clever solution butonly roughly approximated to the ‘real’ one.

After Hero, Pappus of Alexandria (see Chapter 3), Leonardo da Vinci, GirolamoCardano (with Nicola Antonio Stigliola in the wake of Hero) and some others did in-deed formulate their own solutions different from each other but different also fromthe ‘correct’ one. At the end of the XVI century, the people who have stated ‘cor-rectly’ this law could be counted on the fingers of one hand: Jordanus de Nemoreand later Nicolò Tartaglia (see Chapter 4), Michel Varro (see Chapter 7), SimonStevin, and Galileo Galilei. It is nearly impossible to evaluate the influence of thevarious scholars on each other. While there are no doubts that Tartaglia and Varrowere inspired by Jordanus, Festa and Roux say instead that there are no externalclues to affirm that Galileo knew his (or Tartaglia’s) writings [311], and also proofsof the contact between Stevin and Galileo are lacking.54 It must be noted howeverthat it is very difficult to find a precise precursor to Galileo. For this there are alsopsychological reasons that a historian dissuade the possibility of crititizing a greatscientist.

Today the inclined plane is seen as a conceptual model different from that rep-resented by the lever and essentially not reducible to it. The inclined plane is repre-sentative of virtual displacement laws and is somehow its geometric representation;the lever is representative of the virtual velocity laws. In the past, as it should ap-pear clear from the short historical notes listed above, things however were not seenin this way. That the inclined plane had its peculiarities was understood by Aristo-tle who did not treat it and by Hero who treated it apart from the other machines.However after Pappus of Alexandria had reduced it to the lever, the difficulties inthe study of the inclined plane seemed to vanish. In the Renaissance the problemreappeared because some scholars did not accept Pappus’ solution, because theyconsidered both logically unconvincing and inadequate empirically. For example itfeatured an infinite value of the force required to lift a weight on a vertical plane,and this is patently absurd. Other scholars did not accept it because in contrast withJordanus de Nemore’s solution, whose demonstration seemed more consistent, theprinciples adopted could appear not very obvious.

53 p. 172. Translation in [121].54 pp. 202–203


With Galileo the reductionist project, started with Pappus and strongly supportedby Guidobaldo dal Monte, to reduce all simple machines including the inclinedplane to the lever, was perfected. Note that Galileo’s attempt to reduce the inclinedplane to the lever was accepted not because verified empirically – with the con-ceptions of experiment of the times also the results of Hero or Cardano were veri-fied – but because he finally presented a rigorous reasoning and employed reason-able assumptions. Moreover Galileo’s result coincided with that of Jordanus andwith that of Stevin more or less of the same period, very elegant and based on dif-ferent assumptions as referred to in Chapter 7. Note again that if the reasoning ofGalileo was corroborated by the result of Jordanus and Stevin, the reasoning of Jor-danus and Stevin was corroborated by that of Galileo and from now on the problemof the inclined plane was considered by all the mathematicians to be definitivelysolved.

In the section devoted to the mechanics of the screw, Galileo shows how theinclined plane can be reduced to the lever and furnishes a simple mathematical law.The proof reproduces the one that he had reported in De motu [113],55 differingmainly for the use of the word moment instead of gravitas.

The present Speculation hath been attempted by Pappus Alexandrinus in Lib. 8. de Collec-tion. Mathemat. but, if I be in the right, he hath not hit the mark, and was overseen in theAssumption that he maketh […]. Let us therefore suppose the Circle AIC, and in it the Di-ameter ABC, and the Center B, and two Weights of equal Moment in the extreams B and C;so that the Line AC being a Leaver, or Ballance moveable about the Center B, the Weight Cshall come to be sustained by the Weight A. But if we shall imagine the Arm of the BallanceBC to be inclined downwards according to the Line B F, but yet in such a manner that thetwo Lines AB and BF do continue solidly conjoyned in the point B, in this case the Momentof the Weight C shall not be equal to the Moment of the Weight A, for that the Distance ofthe point F from the Line of Direction, which goeth accord ing to BI, from the Fulciment Bunto the Center of the Earth, is diminished: But if from the point F we erect a Perpendicularunto BC, as is FK, the Moment of the Weight in F shall be as if it did hang by the Line KF[119].56 (A.5.40)

A B C H

N G

M K

O

E

D

FLI

Fig. 5.25. The law of inclined plane

55 pp. 297–298.56 p. 181. Translation in [121].


See therefore that the Weight placed in the extream of the Leaver B C, in inclining down-wards along the Circumference CFLI, cometh to diminish its Moment and Impetus of goingdownwards from time to time, more and less, as it is more or less sustained by the Lines BFand BL.[…]If therefore upon the Plane HG the Moment of the Moveable be diminished by the totalImpetus which it hath in its Perpendicular DCE, according to the proportion of the LineK B to the Line BC, and BF, being by the Solicitude of the Triangles KBF and KFH thesame proportion betwixt the Lines KF and FH, as betwixt the said KB and BF, we willconclude that the proportion of the entire and absolute Moment, that the Moveable hath inthe Perpendicular to the Horizon to that which it hath upon the Inclined Plane HF, hath thesame proportion that the Line HF hath to the Line FK; that is, that the Length of the InclinedPlane hath to the Perpendicular which shall fall from it unto the Horizon. So that passingto a more distinct Figure, such as this here present, the Moment of Descending which theMoveable hath upon the inclined Plane CA hath to its total Moment wherewith it gravitatesin the Perpendicular to the Horizon CP the same proportion that the said Line PC hath toCA. And if thus it be, it is manifest, that like as the Force that sustaineth the Weight in thePerpendiculation PC ought to be equal to the same, so for sustaining it in the inclined PlaneCA, it will suffice that it be so much lesser, by how much the said Perpendicular CP wantethof the Line CA: and because, as sometimes we sce, it sufficeth, that the Force for movingof the Weight do insensibly superate that which sustaineth it, therefore we will infer thisuniversal Proposition, [That upon an Elevated Plane the Force hath to the Weight the sameproportion as the Perpendicular let fall from the Plane unto the Horizon hath to the Lengthof the said Plane] [119].57 (A.5.41)

The key assumptions to demonstrate the law of the inclined plane are:

a) for static purposes, moving on the inclined planes like NO or GH is the same asmoving on the circumference described by the lever arms BL or BF of Fig. 5.25;

b) the effectiveness of a heavy body on an angled lever is determined by the hori-zontal distance from the fulcrum.

In essence, a weight p hanging in F from the angular lever ABF is balanced by aweight q = p BK/BA in A. But the q in A is also balanced by a force in F equal toq orthogonal to BF and then parallel to GH, because BA = BF. Therefore q is theforce parallel to the plane necessary to support the weight. The second assumptionis an accepted theorem of statics, but the first has a logic status not completely clear.It indeed appears quite intuitive, at least after its formulation, because to study theequilibrium it seems sufficient to verify that also very small displacements cannotoccur. In this way the displacements at the extremity of the lever and on the inclinedplane are the same, the two kinds of constraints are locally equivalent and can bereplaced the one with the other. But this intuitive character stems more from empir-ical than logical considerations; it would be then a postulate which could even notbe accepted, as will be shown further in Chapter 13.


6

Torricelli’s principle

Abstract.This chapter is devoted entirely to Evangelista Torricelli who formulated aprinciple of equilibrium without referring explicitly to dynamic aspects. In the firstpart some elements of centrobarica are introduced. In the central part Torricelli’sprinciple is introduced: The centre of gravity of an aggregate of heavy bodies cannotlift by itself. Following this law, which in a first reading does not seem to be a VWL,Torricelli and his successors derived the VWL based on virtual displacements forwhich any force that can lift a weight p to a height h can raise p/n of nh. In the finalpart generalization and simplification of Torricelli’s principle are presented.

In the common interpretation, Torricelli’s principle is a criterion of statics whichclaims that it is impossible for the centre of gravity of a system of bodies in equilib-rium to sink from any virtual movement of the bodies. This criterion had a vital rolein the history of mechanics. It represents a generalisation of the ancient principle thata single body is in equilibrium if its centre of gravity cannot sink. The generalisationdevised by Torricelli states that if the centre of gravity of an aggregate of rigid bod-ies, considering the aggregate as a whole, is evaluated according to Archimedeanrules, then this point has effectively the physical meaning of a centre of gravity.1

6.1 The centrobaric

The idea of the centre of gravity in antiquity can be found both in the writings ofphilosophers and mathematicians with a different value in the two cases. The firstconcentrate on a dynamic conception that can be traced back at least to Aristotle who,in De caelo, addressed issues relating to the shape of the earth. Here he specifiedthe basic principle of his mechanics, that bodies move with natural motion towardthe centre of the world, coincident, only accidentally, with the centre of the earth.According to Aristotle, it is clear that it is not enough that only the extreme part of

1 The first part of this chapter is taken from [275] which is summarized and largely revised.


136 6 Torricelli’s principle

a body reaches the centre, but the heavier parts must outweigh to the rest, to includewith its centre the centre of the world:

A short consideration will give us an easy answer, if we first give precision to our postulatethat any body endowed with weight, of whatever size, moves towards the centre. Clearly itwill not stop when its edge touches the centre. The greater quantity must prevail until thecentre of the body occupies the centre [of the world]. For that is the goal of its impulse [12].2

In fact the part of a body which should be closer to the centre of the world is itscentre. What then is this centre, Aristotle did not specify; only for the sphere andother regular solids can one sense that it coincides with the centre of the figure.

The XIV century philosopher Albert of Saxony (c. 1316–1390), nicknamed Al-bertus Parvus by the Italian scholastics, pointed to the Aristotelian thesis. He as-sumed it is the centre of gravity and not simply the geometrical centre of a bodythat moves toward the centre of the world. Moreover he assumed that the gravityis concentrated in the centre of gravity [305].3 Thus he denied that the tendency tomove downward belongs to the component parts of the body, because in his opinionthis case would imply an interference between the parts and a slowdown in the freedescent.

Albert of Saxony argued that even in the case of an aggregate of heavy bodies itis the centre of gravity of the whole which tends toward the centre of the world. Buthe did not specify what type of constraint makes a series of bodies an aggregate. Inparticular, a heavy body tends to come down until the centre of gravity of the entireaggregate formed by it and the rest of earth is the centre of the world [305].4 Fromwhich the disturbing conclusion that the earth moves relentlessly because its centreof gravity moves toward the centre of the world for any shifting of bodies on thesurface.

A not very different approach was pursued by Arabic scholars; see for exam-ple [343]. Very interesting are the considerations by Muzaffar al-Isfizari about theway joined bodies tend to reach the centre of the earth, resulting in one of the mostinteresting and original ‘proofs’ of the law of the lever [247].

The idea that gravity is concentrated in the centre of gravity was taken up bymathematicians during the Renaissance. In particular Guidobaldo dal Monte wrote:

All we said about the centre of gravity allows us to understand that a heavy body weighs,properly speaking, in its centre of gravity. The name itself seems to evoke this truth. All theforce, all the gravity of the weight is concentrated in the centre of gravity; it seems to flowtoward this point from all sides.Because of its gravities, indeed, the weight wants naturally to reach the centre of the world,but we said that what actually tends toward the centre of the world is the centre of gravity.Thus, when any weight is sustained by a whatever power from its centre of gravity, thenthe weight remains in equilibrium, and the whole gravity is perceived by senses. The sameoccurs if a weight is sustained from a point such that the straight line joining it to the centreof gravity passes from the centre of the world. In such a case, in effect, all is as if the weightwere sustained from the centre of gravity.

2 II B 14, 297 b.3 vol. 2, p. 20.4 vol. 2, p. 25, in note 6.

6.1 The centrobaric 137

It is not the same if the weight is sustained from another point. In such a case the weightdoes not remain in equilibrium; before one could perceive its gravity, it turns, until, as in theprevious case, the line joining the centre of gravity goes toward the centre of the universe[87].5 (A.6.1)

In the second part of the quoted passage it was derived almost as a logical conse-quence of the definition of the centre of gravity the criterion according to which aheavy body is in equilibrium if and only if its centre of gravity is prevented fromlowering.

It is however possible that historically the concept of gravity was directly de-rived from experiences of statics where attention is focused on equilibrium, easierto interpret, and only then was the concept extended to the case of motion. Greekmathematicians, engaged in research of the laws of equilibrium, reconnected to thestatic conception of centre of gravity and related the physical concept of centre ofgravity to the geometrical concept of centre of a figure. The most remote terms andconcepts which have been handed down to us on the centres of gravity are those,very far apart in time, of Archimedes and Pappus of Alexandria. With regard toArchimedes we received, unfortunately, neither the definition of centre of gravitynor the reasons for its introduction; we did however receive the axioms that deter-mine the centre of gravity of composite bodies from elementary bodies of knowncentres of gravity (see Chapter 3). From Pappus we received instead the definitionthat fared well in the Middle Ages and the Renaissance. Here it is in the lesson ofFederico Commandino:

We say still centre of gravity of any body at its point, such that if the heavy body is imaginedsuspended from it and left at rest, it remains in the same position it had at the beginning, anddoes not tilt [75].6 (A.6.2)

After Pappus’ definition, Commandino adds one of his own that was then recoveredby most of the mathematicians of the XVI century, including Guidobaldo dal Monte[86]:7

We can also define the centre of gravity of any solid body is one of its points, around whichits parts have the same moment. If indeed a plane is drawn through this centre the planesplits the body in to two parts equally heavy [75].8 (A.6.3)

According to this last definition, the centre of gravity is identified by reference to thecauses of motion which are in balance and not only through an empirically verifiabledefinition.

On the idea of centre of gravity was based the centrobarica, a science in whichthe equilibrium problem of a body is reduced to the search for its centre of gravityand assurance that it is securely tied. One problem that was pressing scholars ofthe XVI century centrobarica, such as Commandino, Luca Valerio, Guidobaldo dalMonte, and Galileo was the difficulty of reconciling the rules to evaluate the centres

5 p. 10.6 p. 1.7 p. 1.8 p. 1.


of gravity according to the Archimedean criteria, with the views of gravity of thetime. Regardless of how the causes of gravity are conceived, all scholars agreedthat bodies tend to reach the centre of the world. In other terms, using the moderncategories to be understood, they believed that the weight forces were convergingto the centre of the world and not just vertical and parallel. While the Archimedeanrules of composition implicitly required parallelism.

Simon Stevin (see Chapter 7) was the first in the history of mechanics to havesome clear ideas on the subject. He did not accept the idea that gravity was con-centrated in the centre of gravity of a body and thought instead it acted on all itscomponents. On the one hand he showed that since the action of gravity convergestoward the centre of the world, the centre of gravity in the sense of Pappus cannotexist for a body other than the sphere. On the other hand, however, he argued thatthis conclusion is only a theory and in practice because the action of gravity differsby a very small angle, the centre of gravity determined by the Archimedean rulesmeets the demand of Pappus to be the point of suspension of neutral equilibrium.

The problem of the existence of centres of gravity was discussed again in the XVIIcentury by René Descartes (1596–1650),9 Pierre de Fermat (1601–1665), Gilles Per-sonne de Roberval (1602–1675) and Blaise Pascal (1623–1662), using a concept ofdistributed force [305].10 The end of the debate took place only when the distinctionbetween mass and weight was clarified by Isaac Newton and the centre of gravitywill no longer be the centre of weights but the centre of masses. Also interesting wasthe contribution by Girolamo Saccheri (1667–1683), who in his Neo-statica [212]11

showed that the centre of gravity also exists for central forces provided they vary ininverse proportion to their distance from the centre.

6.2 Galileo’s centrobaric

To understand the thought of Torricelli it is useful to summarize the ideas of Galileoon the centres of gravity referred to mainly in Le mecaniche and in an Appendix toDiscorsi e dimostrazioni matematiche [112].12

The various texts of Galileo show the insistence on the propensity to move toward“the common centre of heavy things”, while the drawings and the arguments onlysuggest lines of descent parallel of heavy bodies. He usually represents weights sus-pended from the lever by means of ropes, as also Luca Valerio did [236, 366], andthe ropes are clearly parallel with each other, pointing out the implicit assumptionof parallelism of the lines of descent of heavy bodies. Only a few times, the weights

9 In addition, Descartes will show that in the sphere, according to the theory of gravity force inhis day, although there is a centre of gravity, it does not coincide with its centre but is lower [96],vol. 2, p. 245.10 vol. II, pp. 156–186.11 pp. 74–75.12 pp. 187–208. It seems that the Appendix was prior to Le mecaniche. The composition of thework according to some historians dates from the period between 1585 and 1588 [123], vol. 1,pp. 181–182.

6.2 Galileo’s centrobaric 139

are applied directly from the ends of the lever so the parallelism of the weight de-scent is not made explicit. This occurs, for example, in a manuscript by VincenzoViviani13 essentially reproducing the reasoning of dalMonte [86] for which the leverwith equal arms and weights behaves as a single body with the gravity concentratedin the centre of gravity, coinciding with the fulcrum, and as such, in equilibriumregardless of its inclination.

There are some aspects of the work of Galileo, whichmay have directly suggestedthe formulation of his principle to Torricelli, even if the conditional is a must. On thethird day of the Discorsi e dimostrazioni matematiche, in the second edition of thework, the Medieval thesis is resubmitted that the centre of gravity of an aggregate,subject only to the weight, tends to move toward the centre of the world:

Because, as it is impossible for a heavy body or a compound of heavy bodies to move nat-urally upward, departing from the common centre where all things tend, so it is impossiblethat it spontaneously moves, if with this motion his own centre of gravity does not approachto the said common centre [118].14 (A.6.4)

The interpretation of the text is not immediate, because it also depends on the mean-ing Galileo gives to the term compound. If it were a single body (more or less rigid)one could say that there was nothing new compared to the medieval view. If insteadthe compound or aggregate, were understood as a set of bodies also disconnectedwith each other, one would be in the presence of a new fact, i.e. it would be possibleto identify the geometrical idea of centre of an aggregate of bodies unrelated to eachother with a physical entity, the centre of gravity, as the point of application of thewhole gravity.

This is very important background knowledge for the further development of theideas of Torricelli. But one must say that probably Torricelli did not know the overreferred quotation, which appeared only in the edition of Discorsi e dimostrazionimatematiche of 1656 [120], before his arrival in Arcetri at the end of 1641. Thoughthere is still the possibility that Torricelli knew the ideas of Galileo on the centreof gravity through Benedetto Castelli, to whom Galileo in 1639 sent a trace of thearguments given in the passage quoted above.

In Le mecaniche, when the equilibrium of a body on an inclined plane is con-cerned, Galileo compared the weights and the motions of two bodies connected bya rope, one that moves on the inclined plane and another that moves on the verti-cal, to conclude that at equilibrium the lowering of both measured on the verticalare inversely proportional to their weights (see § 5.5.2). Note the proximity of thisconclusion to the one given by Torricelli when studying the motion of two bodies onan inclined plane, to conclude that the centre of gravity of the whole does not loweror rise. Even here, however, it should perhaps be added that Torricelli did not knowLe mecaniche.

13 This paper is considered as authentic of Galileo for sure by Favaro [123], vol. 8, p. 439 and withsome doubt by Caverni [284] vol. 4, pp. 164–166.14 p. 215.


6.3 Torricelli’s joined heavy bodies

Evangelista Torricelli was born in Rome in 1608 and diedin Florence in 1647.15 As a boy he lived in Faenza with hismaternal uncle Don Jacopo Torricelli of the order of Camal-dolese. His uncle became his second father, giving him thepossibility to study: humanities topics with him, sciences atthe Jesuit school. Important were the years when he studiedin Romewith Benedetto Castelli (1577–1644) andMichelan-gelo Ricci (1619–1682), in turn a pupil of Castelli. Galileoin 1641, thanks to Castelli, was able to read a Torricelli’s

manuscript on the motion of bodies. He was so impressed that he invited him, thesame year, to Arcetri as his disciple. Torricelli reached Arcetri October 1641 andremained there until the death of the master, which occurred in January 1642. Torri-celli was then an effective Galilean disciple only for a few months. On the death ofGalileo, the Grand Duke of Tuscany, Ferdinand II, appointed Torricelli official suc-cessor of Galileo in the Studio Fiorentino for the reading of mathematics. He wasalso appointed member of the Accademia della Crusca. In addition to the famousexperiences around the barometer, fundamental are Torricelli’s works on geometry,Calculus, mechanics and optics [261, 320]. He organized a workshop [319]16 for theproduction of lenses and telescopes with a ‘secret’ method of work [294].

Torricelli alive had the satisfaction of seeing published only his greatest effort,the Opera geometrica [227]. Subsequently the Lezioni accademiche were printed[231, 232]. Other writings are published in [233, 234, 329, 392] and extensiveexcerpts from the manuscripts in [284]. The manuscripts of Torricelli are cur-rently stored at the National Library in Florence in the collection of the Galileanmanuscripts – volumes 21 to 44 of fourth division. His unpublished manuscriptshad not an easy life. His premature death prevented the preparation and publicationof his latest works that remained in the form of notes and memos. Not even his clos-est friends, able mathematicians, were able to continue his work as they also diedshortly thereafter. For example, Bonaventura Cavalieri died in 1647 and Michelan-gelo Ricci in 1682. The historian Gino Loria mentions an amusing incident at thesame time unforgivable. The cabinet with the manuscripts of Torricelli was sold to agrocer; one day the seller of sausage wrapped something into a sheet of a manuscriptand sold it to Giovanni Battista Clemente Nelli. He recognized the importance of thequotations on the sheet and hastened to buy in bulk all them [392].17 Only in 1861did Torricelli’s manuscripts find a safe home in the Libreria Palatina of the NationalLibrary of Florence.

Evangelista Torricelli devoted much space to the study of mathematics and thisfavored a geometric Archimedean approach to mechanics. Most of his contributions,

15 It is now known with certainty that Rome was the birthplace of Torricelli. On Torricelli’s birthone can see the recent works by Bretoni [258, 263].16 pp. 84–95.17 pp. 31–33.

6.3 Torricelli’s joined heavy bodies 141

of puremathematics, related to the determination of the centre of gravity of plane andsolid figures, particularly complex. The most important conclusions are contained inthe Opera geometrica, but results are scattered in manuscripts. Other contributions,perhaps less important quantitatively, but fundamental from my point of view, con-cern the physical aspects of the theory of gravity; in the following I will focus onlyon this part. The texts which I will refer to are the Opera geometrica, correspon-dence and published manuscripts. I consider these sufficient to provide an adequatefull physical understanding of the theory of centres of gravity of Torricelli.

6.3.1 Torricelli’s fundamental concepts on the centre of gravity

The Opera geometrica can be divided into three main parts, of which, the first twoare divided in turn into books. A brief index follows: Part 1. De sphaera et solidissphaearalibus, first book, pp. 3–46; second book, pp. 47–94. Part 2. De motu grav-iun naturaliter descendetium et proiectorum, first book, pp. 97–153; second book,pp. 154–243. Part 3. De dimensione parabolae solidique hyperboloci, pp. 1–84, notsubdivided in books, which contains: Quadratura parabolae, De dimensione cy-cloidis, De solido acuto hypebolico, De dimensione cochlea.

The fundamental ideas of Torricelli on the centres of gravity are brought back inthe first book of the De dimensione parabolae. Though this book was written afterDe motu graviun naturaliter descendetium et proiectorum where Torricelli formu-lated his famous principle, the main concern of this chapter, it can be consideredrepresentative of young Torricelli’s conceptions. They are expressed with six items,definitions, theorem and postulates, under the undifferentiated name of Suppositionsand Definitions. I refer to only those which seemed more important to me for thisstudy.

Suppositions and DefinitionsI. Suppose that the nature of the centre of gravitas is such that a magnitude freely suspendedfromwhatever its part will be never at rest unless it centre of gravity reaches the lowest pointof its sphere [228].18

VI. Equal heavy bodies, suspended at equal distances, are equilibrated both when the balanceis parallel to horizon andwhen it is tilted. Andweights having inverse proportion to distancesare equilibrated both for the balance parallel to horizon and when it is tilted [228].19 (A.6.5)

Assertion I seems more a definition than a postulate because it is the first time thecentre of gravity is named. If this is true, Torricelli’s definition is different from thetraditional ones given by Pappus and Commandino and in some ways is more com-plete; with it the ‘empiric’ rule that a body is in equilibrium if its centre of gravityis on the vertical line from the centre of suspension becomes a trivial theorem. Theassertion however deserves some comments. It can be written formally and synthet-ically with the following implication:

Centre of gravity not at bottom→ not equilibrium

18 p. 11.19 pp. 13–14.


i.e. equilibrium → centre of gravity at bottom. However in the explanation of theproposition Torricelli uses also the implication: centre of gravity at bottom→ equi-librium. It is then possible to argue that he could imply:

Centre of gravity at bottom↔ equilibrium

Assertion VI is presented as a theorem. In fact, as clear from below, it is alsoa postulate which implicitly asks the reader to admit that the actions of gravity arevertical and parallel. At a first reading Torricelli seems to take a position on thethorny problem of the nature of the equilibrium of the lever with the same arms andweights: it would be neutral equilibrium. An analysis of the proof of the theoremshows that it is not exactly what Torricelli does.

Torricelli was aware of the discussions in the international arena and especiallyin France, about the manner of variation of gravity with the assumption that it orig-inates from the centre of the world [305].20 As clear from his correspondence withBenedetto Castelli and Antonio Nardi (1589–1649 ca.) [362],21 which date back atleast to 1635 [284],22 [322]. But he shared with Galileo the idea that in physics itis possible to idealize the situation, and in the particular case of the lever, abstract-ing from the convergence of the actions of gravity. In this way it is possible to de-velop ex-supposizione exact reasoning that could also apply to reality, less than forsmall imperfections widely acceptable in practice. Torricelli exacerbates the reason-ing of the substantial parallelism of the actions of gravity, by postulating the trueparallelism, imagining a balance currently at infinite distance from the centre of theworld:

It is accepted even by sound scholars the objection that Archimedesmade a false assumption,when he considered the wires through which the magnitude are suspended from a balance,as if they were parallel, when in fact they converge toward the centre of the earth. But I (paceof some illustrious men) believe that one must consider the principles of Mechanics in anentirely different way. I grant well that if physical magnitudes are suspended freely [with-out external forces] from a balance, the material wires of suspension will be converging, asthey all tend toward the centre of the earth. However, if the same balance, even material isconsidered not on the surface of the earth, but high in regions beyond the orb of the sun, thenthe wires, although there also tend toward the centre of the earth, will be much less conver-gent with each other, and will be almost parallel. Imagine we take our mechanical balance,beyond the Libra, to an infinite distance. Who does not understand that now the wires ofsuspension will not be converging, but exactly parallel to each other? [228].23 (A.6.6)

Assuming the wires with which the weights hang from the balance as ‘really’ in-finitely long appears a choice of infinity in act in the mathematical theory. UnlikeGalileo, Torricelli, most interested in mathematical aspects, shows all his skills ofabstraction and does not hesitate to claim daring theoretical positions:

20 vol. 2, pp. 24–26.21 p. 10.22 vol. 4, pp. 156–212.23 pp. 9–10.

6.3 Torricelli’s joined heavy bodies 143

So it can be said that it is false the mechanical principle for which the ropes of the balanceare parallel, where the magnitudes suspended from the balance are material and real andtend toward the centre of the earth. It will not instead be false in the case the magnitude, arethey abstract or real, tend not to the centre of the earth or a point near to the balance, but toan infinitely distant point [228].24 (A.6.7)

Although Torricelli is dealing with the lines of descent as parallel, when he writesthe Opera geometrica he has not yet made completely clear what were the issues in-volved in centrobarica if the lines of descent are assumed as convergent. One symp-tom of this lack of clarification can be found still in the proof of proposition VI ofDe dimensione parabolae, where after proving that a balance with weights inverselyproportional to the arms is in equilibrium whether it is horizontal or tilted, he speci-fies:

I don’t miss that in the dispute among the authors about the tilted balance, i.e. whetherit comes back or it remains in its position, the centres of magnitudes are located into thebalance. But because in the booklet we assume the magnitudes located below the balance[emphasis added], we prefer to follow our purpose instead to adapt our demonstrations to acontroversy among other people [228].25 (A.6.8)

Torricelli says explicitly that when the centre of gravity of the weights is below thefulcrum, as it occurs when they are suspended by wires from the end of the arms, thebalance is in equilibrium whatever its inclination. He avoids entiring the ‘dispute’that still exists in the case of balances with weights attached to the ends where thecentre of gravity of the weights coincides with the fulcrum. Thus revealing someuncertainty.

After 1644 Torricelli comes back to consider the possibility of actions converg-ing toward the centre of the world. They are only sporadic observations that showhowever a definitive understanding of the problem. For completeness, I consider ituseful to report some considerations on the equilibrium of the lever. Torricelli firsttreats the lever with different arms and weights:

When we admit that the weights of the balance have inclination towards the centre of theearth […] it will follow that there is no horizontal balance with unequal arms and weights inreciprocal proportion of the length of the arms, so that these weights equilibrate.26 (A.6.9)

The demonstration of what is said above, which I do not carry over, uses the ideaof static moment developed by Giovambattista Benedetti that the effect of a forceon a lever is proportional to its intensity and distance of its line of action from thefulcrum. Or better he reproduces the argumentation of Benedetti of § 5.5.2.

Torricelli then deals with the lever with arms of equal length:

Now place B as the centre, and AC a balance with arms of equal length with two equalweights in the extremities A and C, the moments or gravities of which are measured fromthe perpendiculars DF and DE, as Giov. Battista de Benedetti declares in his book of mathe-matical speculation, chapter III or IV [29]. It follows that the moment of the weight in A, to

24 pp. 9–10.25 p. 15.26 E. Torricelli, Galilean collection, manuscript n. 150, c. 112.


A D C

EF

B

Fig. 6.1. The equilibrium of a lever with equal arms

the moment of weight C, is reciprocally as the line BC is to the line AB, that is reciprocallyas the distance of the weights from the centre of the Earth. And here we have not only thatthe weight closer to the centre, while it is in the balance, weighs more than the farther, butwe yet know to what proportion it weighs more.27 (A.6.10)

The argument still uses the law of static moments – for which the ratio betweenweights A and C is as the ratio between the distances DE and DF also equal to thatbetween AB and CB – is conclusive if it is accepted that the absolute gravity ofbodies does not vary with distance.

6.4 Torricelli’s principle

Torricelli applies his ideas on the centres of gravity in the De motu gravium [229],published as a book of Opera geometrica in 1644 but almost certainly based on amanuscript dating at least to 1641. The goal was ambitious: to rebuild the Galileandynamics on a more solid foundation than the one Galileo had given in the first edi-tion of theDiscorsi e dimostrazioni matematiche of 1638. To do this it was necessaryto clarify certain ‘assumptions’ of Galileo not sufficiently convincing, in particularwhat Torricelli called ‘theorem of Galileo’, which states:

Proposition IIMoments of equal heavy bodies over unequally inclined planes, having the same height, arein the reciprocal ratio with the length of planes [229].28 (A.6.11)

the demonstration of which was not reported in the first edition of the Discorsi edimostrazioni matematiche. Torricelli had participated with Vincenzo Viviani in thelast months of Galileo’s life, a period when the master, nearly blind, tried to shedlight on the basic principles of his mechanics, and between them the propositioncited above. Correspondence between Torricelli, Nardi and Galileo bear witness tothis work.

In 1641 Galileo wrote to Torricelli that he has sent him the proof of his theoremby means of Nardi [276].29 When Torricelli wrote the Opera geometrica in 1644 hewas therefore aware of the demonstration of Galileo. But because this was not yet

27 E. Torricelli, Galileian collection, manuscript n. 150, c. 112.28 p. 100.29 p. 16.

6.4 Torricelli’s principle 145

printed,30 and because it was based on principles other than his own, he considers ituseful to provide his own version:

I know that Galileo during the last years of his life tried to prove this proposition. But be-cause his argumentation was not published with his book on Motion, we assumed to placebefore our booklet these few things on the moments of heavy bodies. So it appears Galileo’ssupposition can immediately be proved from the theorem he himself assumed for granted inhis Mechanics [Discorsi e dimostrazioni matematiche] in the second part of the sixth propo-sition of accelerate motion, that is moments of equal bodies over unequally inclined planesare as the perpendiculars of equal parts of the planes [229].31 (A.6.12)

This quotation raises some doubts. Torricelli claims that Galileo gave as proved histheorem, and not that he proved it. This may mean either that Torricelli did not knowexactly or did not approve Galileo’s demonstration. The two hypotheses are not mu-tually exclusive. It is possible that Torricelli did not know Le mecaniche of Galileo,published at that time only in French in 1634 [117]. Besides, it is also possible thatTorricelli did not approve the proof of the law of the inclined plane that Galileohad sent to him and that will be published in the second edition of the Discorsi edimostrazioni matematiche, because it was based on the questionable principle ofvirtual displacement [276].32

If Torricelli did not know Le mecaniche he could not have known that Galileo’sdemonstration of the law of the inclined plane was based on the law of the lever,considered an unquestionable principle by all.33 And then one can justify his claim,at first sight unreasonable, that Galileo in the Le mecaniche only supposed that law.

To prove Galileo’s theorem Torricelli puts a premise now known as the principleof Torricelli. The premise is followed by an explanation and a justification if not ademonstration, so somehow it retains the logic status of a principle.

PremiseTwo equal bodies joined together cannot move by themselves if their common centre ofgravity does not descend. When indeed two heavy bodies are joined together such that tothe motion of one it follows the motion of the other, the two heavy bodies will be as onesingle body compound of the two, be it a balance, a pulley, or whatever mechanical devices.Such a heavy body will never move if its centre of gravity does not descend. When all isdisposed so that the common centre of gravity cannot descend in any way, the heavy bodywill remain at rest, because otherwise it will move in vain, i.e. with a horizontal motion,which no way tends downward [229].34 (A.6.13)

30 It will appear in the second edition of theDiscorsi e dimostrazioni matematiche [120]. It is worthnoting that the proof of the law of the inclined plane shown in the Dialogo [116], pp. 215–218, isdifferent from that reported in the Le mecaniche [119], p. 181. The first is based on the principleof virtual work, the second on the law of the lever.31 p. 98.32 p. 18.33 It is interesting to note that Torricelli in De motu gravium refers also a demonstration of the lawof the inclined plane, alternative to that developed with his principle, very similar to that of Galileoin the Le mecaniche. This, if not to commit the sin of plagiarism to Torricelli, could prove that heactually did not know Le mecaniche.34 p. 99.


The premise, made explicit with modern terms, states that two heavy bodies con-nected to each other in any way, cannot move by themselves from the configurationin which they are if [nisi] their overall centre of gravity did not sink for a genericvirtual displacement of the two bodies compatible with constraints. Using a compactlanguage:

Common centre of gravity cannot sink→ equilibrium

It is then only a sufficient condition for equilibrium because it is not stated explicitlythat there is not equilibrium when the common centre of gravity sinks.

Torricelli begins the justification of his premise by stating that the two heavybodies, as they are joined, have to be treated as a single body and returns as examplecases of joined bodies of mechanics. Among them it is particularly significant thepulley where one has two weights connected with a wire inextensible but flexible.

The assimilation of the two bodies to a single body is not at all obvious, in fact,the common centre of gravity of two bodies is a purely geometric point, with nosubstance, on which one can hardly think that gravity is applied, such as the Me-dieval scholars assumed for the individual body or for an aggregate of contiguousbodies. Torricelli has the ‘ingenius’ idea to extend analogically and unequivocallythe reasoning valid for a body/centre of gravity to the aggregate. This analogicalextension is made possible, perhaps even for the persistence of the idea of a globalaction of gravity, not divided in different parts of a body. I have however shown thatin the analysis of the equilibrium of the lever for actions of gravity that converged,which the mature Torricelli developed (see above), there was clearly the idea thatany portion of the body retains its individuality. Perhaps the mature Torricelli couldnot have conceived his principle.

Once accepted that the mathematical centre of gravity of an aggregate behaveslike a physical centre of gravity, the justification for the premise can be referred to themotion of centre of gravity of an ordinary body (graveautem). The locution “does notlower” (nisi descendant) referred to the centre of gravity is separated into two parts:(a) it rises, (b) it remains horizontal. The case (a) where the heavy body moves withthe centre of gravity that rises is clearly impossible for the very definition of centreof gravity. Case (b) where the heavy body moves and the centre of gravity is kepthorizontal is also impossible. According to the prevailing views on the gravity ofTorricelli’s times, bodies moved down because they have the goal to reach the centreof the world. When a body is in a plane and cannot sink its movement is impossible,because it is without a goal. Since the goal is, according to the Aristotelian doctrine,a formal cause it can be said that the body moves without cause, which Torricelliconsiders absurd.

6.4.1 Analysis of the aggregate of two bodies

Torricelli’s concept of aggregate differs in two fundamental aspects from Albert ofSaxony’s vague concept mentioned at the beginning of this chapter, and also fromGalileo’s slightly more defined concept. First, Torricelli presents a concrete case:the two spheres as heavy bodies. The centre of gravity of the aggregate is deter-


G′GA

C

D

E

B

Fig. 6.2. Equilibrium on the inclined plane of two joined bodies

mined by the Archimedean rules. In this way, Torricelli can check the condition ofthe aggregate equilibrium on the basis of the constraint imposed to lowering the cen-tre of gravity. Second, the aggregate does not maintain the same shape as a singlebody does, but it is constrained in order to take different shapes. Note that Torricellireinforces the concept of conjunction also with explicitly operational terms:

Be connected even with an imaginary rope conducted through ABC, so that from the motionof one if follows the motion of the other [229].35 (A.6.14)

Torricelli suggests not only that a mechanism is needed because the motion of amoving body follows themotion of the other, but also explicitly states the instrument,the (imaginary) rope, with which it is connected and its motion. The implementationin a precise mathematical language puts Torricelli in a position to use his premise toprove the law of the inclined plane, the proposition I of the De motu gravium, whichis preliminary to the proof of Galileo’s theorem.

Proposition IIf in two planes unequally inclined but with the same elevation, two heavy bodies are con-sidered which are in the same ratio as the length of the planes, the heavy bodies will havethe same moment [229].36 (A.6.15)

The mathematical core of the proof, which is interesting because it involved theconcept of centre of gravity of heavy bodies, is in fact very simple. It proves that ifthe weights of two heavy bodies A and B are in the same proportion of the lengths oftwo inclined planes, their common centre of gravity G moves horizontally as shownin Fig. 6.2.

The whole demonstration of proposition I, is divided into three steps, first Tor-ricelli denies it (step 1) “They do not have, if possible, the same moment, but pre-vailing one, they move, and the heavy body A rises toward C, while heavy body Bdescends in D [229].37 Then he shows that the centre of gravity of the two bodiesdo not sink (step 2) so the two bodies have acted without cause, which is absurd, so(step 3) proposition I cannot be denied. In the words of Torricelli:

35 p. 99.36 p. 99.37 p. 100.


G1

GG2

O

Fig. 6.3. Lack of equilibrium for converging lines of gravity

Two equal bodies joined together were moved and their common centre of gravity did notdescend. This is against the premised law of equilibrium [229].38 (A.6.16)

Note also that the proof of proposition I, about the truth of which Torricelli has nodoubt, could be seen as an attempt to validate the premise of equilibrium. Proposi-tion I so could be regarded as a methodological principle of the theory and not as atheorem to prove.

The proof would not have succeeded if Torricelli had considered the ‘real’ situa-tion in which the directions of the lines of gravity are converging toward the centreof the world. In fact, when the common centre G of two heavy bodies – assumingit exists – moves on a horizontal plane, it varies its distance from the centre of theworld O (to not change it, the common centre of two heavy bodies should move on asphere) and can even come close to it, as clear from Fig. 6.3 where one can see thata movement on the plane from G1 or G2 towards G leads to an approach toward thecentre of the world.

Having established Proposition I, Torricelli goes to show Proposition II, i.e. Gali-leo’s theorem, thus ending his project of re-foundation of the Galilean dynamics. Itis outside my purpose to express my thoughts on the fact that the objective has beenachieved.

6.4.2 Torricelli’s principle as a criterion of equilibrium

I have shown how Torricelli introduced his principle for a very ambitious reason,to re-establish the mechanics of Galileo. He realizes that his principle furnishes alsoa criterion of equilibrium, but he does not take full advantage of the fact and limitshimself to applying it on an occasional basis.

Torricelli uses his principle in statics at least on two occasions. The first, alreadycommented in another respect, in the Dimensione parabolae, where he intends to

38 p. 100.


H

L

I C

B E

DM

F

A

G

L

Fig. 6.4. A lever with suspended weights

show that a balance with weights inversely proportional to the arms is equilibratedregardless of its inclination. This case study is the balance of Fig. 6.4 from the endsof which weights with inextensible wires are hung (the body CBF is suspended in Ifrom its centre of gravity L).

Considering such a balance in any configuration, different from the horizontal,Torricelli determines the centre of gravity and establishes that it is below the fulcrum,on the vertical line from it, and concludes:

From this reason the magnitudes suspended from the balance AC will equilibrate. Indeedif they moved, their common centre of gravity, which has been proved to be in the verticalDF, would rise. Which is impossible [228].39 (A.6.17)

i.e. for Torricelli the balance is in equilibrium according to his principle becausein any possible motion the centre of gravity of the weight-rod system rises. Theconclusion is true, but the argument is not perfectly developed. It seems wrong,because in reality the centre of gravity of the system of the two weights instead ofrising remains at the same level as it is easy to see with simple calculations (in factthe centre of gravity does not change at all its position).40 Torricelli’s ‘mistake’,perhaps not just an oversight, suggests, that it was not so unnatural to him to thinkof two weights connected to each other as one body, and then apply to their centreof gravity, the same rules for the centres of gravity of monolithic bodies.

Most important and challenging, is the second use of the principle, in a situationwhere one might think that Torricelli would not have thought possible the applica-tion. In fact, Torricelli’s principle, as it provides only a sufficient condition for theequilibrium, allows to recognize the equilibrium in cases now classified as stable(for example a ball placed on the lowest point of a gutter), but not in those classifiedas unstable (for example, a ball placed over a hump) where the centre of gravity canbe lowered and yet the system is in equilibrium. In Torricelli’s times there did notexist a theory of stability – for this one must wait for at least Lagrange with his Mé-canique analytique of 1788 – but for certain situations, such as the ball over a hump,

39 pp. 14–15.40 It seems strange that Torricelli could have made a such trivial error; there is the possibility thathis text is simply imprecise. Indeed it is a mystery why Torricelli uses the body BFF with a sideparallel to the beam AC, as if it were fixed to it. If this were actually fixed the centre of gravity ofthe whole could rise.


BS

L

Rr

M

A

C

G OF

ZQ

fE

b mn

o s

Fig. 6.5. Unstable equilibrium of an inclined shaft

common sense was enough to feel the problematic nature of equilibrium, that alsoseemed to subsist for the principle of sufficient reason.

The case of equilibrium studied by Torricelli, reported in an undated manuscript[233],41 regards the very unstable equilibrium. An inclined shaft is supported with-out friction by two orthogonal walls, one vertical and the other horizontal, as shownin Fig. 6.5. The horizontal force should be determined which must push F towardE in order to maintain equilibrium in the shaft, heavy by itself or by a weight sus-pended from a point C. This issue was raised a century earlier by Leonardo da Vinci[284]42 who attempted a solution. Vincenzo Viviani and other mathematicians alsowrestled with the job, not always with satisfactory results [284].43 Torricelli had al-ready addressed this problem in a few letters from/to Michelangelo Ricci [233]44

and knew that there was a position of equilibrium and what it was. In these letters heprovided a solution based on a principle of virtual work, limited to the case whereto the ends A and C of Fig. 6.6 two forces concurring in B are applied:

The power at A to the power at C, is as the line CB to the line CA. The proof [...] dependson the speed because moving the beam AC, along the two lines of the right angle ABC, thespeed of the point A to the speed of the point B, is as BC to BA [233].45 (A.6.18)

In Torricelli’s letters anyway there is no mention of the fact that he considered prob-lematic the equilibrium, or that he recognized what we now call its instability. Sincethe topic seems particularly interesting I carry out nearly in full Torricelli’s consid-erations [276]:

41 vol. 3, pp. 243–245.42 vol. 4, pp. 65–67.43 vol. 4, pp. 65–67.44 vol. 3, pp. 91–92; pp. 96–99; pp. 99–100.45 vol. 3, p. 100. In [284], vol. 4, p. 64, there is the reference to a Torricelli’s manuscript whichfaces the same argument with the law of static moments, while maintaining a tone that suggestssome doubt on the solution found.


A

B C

Fig. 6.6. The equilibrium of an inclined bar

Among the observed effects of mechanics, for what I know, there is one not yet felt by somepeople, even though from it notions of some interest and curiosity may derive. Assumed avertical wall AE in the horizontal plane EF at which the line EF is normal, and also assumedthe straight beam BCF the centre of gravity of which is C, that with the extremity B supportsthe wall above mentioned, and with the extremity F can slide freely on the floor EF, we lookfor the proportion of the weight of the beam to that force, which directly applied in F pushingin direction FE can balance the moment of the beam and to flow under its own weight inthe direction EF. Suppose the required force be given by a weight attached at the point Zof a rope of given length FEZ, which we call λ, and that passes through the point E. Fromthe centre of gravity C trace the normal CG over FE, and the ratio of BF to FC be the sameas the ratio of 1 to x, we will have for the similar triangles BE : CG = BF : FC = 1 : xand consequently, assumed P the weight of the beam, the distance from the horizontal EFfrom the mentioned weight, which can be understood collected in the centre of gravity ofthe beam, will be equal to P · CG, or really writing x · BE instead of CG, the distance ofthe line EF to the weight P will be x · P · BE [233].46 (A.6.19)

In this first part the system to be analyzed is defined. In an intermediate position C ofthe shaft BF there is a weight P. Its position is defined by x so that xBF represents thedistance from P to F. The shaft is held in equilibrium by a weight Q, connected by ahorizontal string of length λ to the end F, which provides the inward thrust. Torricellidetermines the distances of the weights P andQ from the horizontal wall, meaning inreality, the ‘weighted distance’, i.e. the distances multiplied by the weight associatedwith them.

Let the searched weight be hung at point Z of the rope FEZ equal to Q, the distance of thenamed weight from the horizon will be equal to Q · ZE, and the distance from the commoncentre of gravity of the two weights P and Q from the horizon is equal to Q · ZE – P · CG =Q · ZE – x · P · BE. Consider the centre L and draw the quart of circle SMs with radius LMequal to BF and draw the ordinate OM, which necessarily will be equal to EF, and from pointM trace the tangent to the circumference nM parallel to the straight line LrR and extend theordinate MO until it meets the straight line LrR in R; the distance of the common centre ofgravity of the two weights P and Q from the straight line EF, which is proved to be equal toQ · ZE - x ·P · BE, still equal to Q ·λ−Q · EF - x · P · BE, i.e. equal to Q ·λ−Q · OM -x · P · LO [233].47 (A.6.20)

The centre of gravity of the twoweights P andQ is given by the algebraic summationof the weighted distances dP (= x ·P · BE) of P and dQ (= (λ – EF) · Q) of Q, andthen as the difference dP− dQ, then dividing for the summation P+Q. Accordingto Torricelli’s terminology the difference dP− dQ is the ‘distance’ of the centre ofgravity from the horizontal line EF.

46 vol. 3, pp. 243–244.47 vol. 3, pp. 243–244.


Torricelli then assumes that P,Q and x are related by the expression xP/Q =EB / EF. This choice, which Torricelli knows is associated with equilibrium – thisis the result reported by Torricelli to Ricci, commented on the previous pages – isnot currently justified. With a simple but elegant reasoning, he shows that for thegiven values for P, Q and x, when P and Q move congruently with the constraints,the centre of gravity of the shaft moves along an arc around the point of maximumdefined by xP/Q = EB / EF. Then he concludes: “So by virtue of the lemma abovethe centre of gravity of the twoweights will remain motionless in the proposed case”.

Here Torricelli makes one nontrivial step and not easily explainable. Basicallyhe says that under the “previous lemma” the centre of gravity in our case in factdoes not move downward, that is the system is in equilibrium. Unfortunately it isnot clear what is the invoked lemma. Vassura believes it is a lemma that Torricelliwould have written, but then it went missing [233].48 It seems to me that the lemmacould be simply Torricelli’s principle, which is introduced by him as a premise ofequilibrium, and therefore in some way as a lemma. In fact, if a differential approachof the modern type is adopted with the use of infinitesimal displacements it would beobtained that, for a very small change in the configuration of the system, the centre ofgravity would not be lowered as if it moved in accordance with the horizontal tangentof an arc of circle and then one can invoke Torricelli’s principle for the equilibrium.

The hypothesis that Torricelli is workingwith infinitesimals, of course intuitively,does not necessarily represent an undue modernization of his thought. Apart fromhis studies on the indivisibles where the concept of infinitesimal is touched, this pos-sibility is also corroborated by another passage contained in the undated manuscript.At one point, Torricelli argues that as a result of the infinitesimal (the term is mine)shift of the point F for which the shaft (Fig. 6.5) moves from the configuration BF tothe configuration b f , the centre of gravity sinks of the quantity mn. This reasoningwould be valid only if he neglects the change of slope of Lr with respect to LR. Andthis process will be typical for those who will manipulate the infinitesimals in theXVIII century.

But Torricelli’s reasoning can be explained even without explicit reference to in-finitesimals. For example he might refer to the work of Descartes that a few yearsearlier had reached a conclusion similar to that of Torricelli, by applying the princi-ple of virtual work to the equilibrium of a heavy material point which moves alongan inclined plane of curved profile, also subject to an external force [96].49 Accord-ing to Descartes (see Chapter 7) it is necessary to consider the displacement alongthe tangent and not the actual displacement. Torricelli could have known the ideasof Descartes by Mersenne with whom he was in contact. Considering the motionsin their birth, however, was implicit in the modus operandi of many mathemati-cians of the XVI century. For example, Galileo and Roberval often replace the realconstraints with other constraints, simpler, giving rise to the same infinitesimal dis-placements (see Chapters 5 and 7).

48 p. 234.49 vol. 1, p. 233.

6.5 Evolution of Torricelli’s principle. Its role in virtual work laws 153

AF

B

DG

E

C

A FB

D

C

Fig. 6.7. Application of Torricelli’s principle to the pulley

6.5 Evolution of Torricelli’s principle. Its role in virtual worklaws

Vincenzo Viviani was the first to highlight the potential of Torricelli’s principle, asthe foundation of statics [284].50 Then the principle was extended and generalizedand Torricelli’s principle was going to indicate a fundamental principle of staticsand the fundamental role in dynamics assigned to it by Torricelli was completelyforgotten.

Viviani applied Torricelli’s principle to study the behaviour of weight suspendedfrom a pulley and showed that only if the two weights are equal can their commoncentre of gravity not be lowered, and then they are in equilibrium; this result more-over is trivially known.

If the two equal weights A and B (Fig. 6.7) are attached to a wire passed over a pulley orother support which can run, these will stay in balance, wherever they are located. (A.6.21)

Because if they moved the one that descends as much acquires as the other which rises loses,since their ways are equal and for perpendicular lines. And if it were possible they can movefrom sites A and B to sites C and D, it is clear that because their centres of gravity move in astraight line, the common centre of A and B will be in the middle, that is in F, and the centreof gravity of C and D will be in the middle, that is in F. Then because AC and BD are equaland parallel to each other, CD and AB joined meet in the same point, i.e. in the middle, sothe common centre will not be moved, and will not have acquired anything, so that heavybodies A and B will not move from the site where they were situated. (A.6.22)

But if the weight B that will sink will be greater than the weight A, because their com-mon centre is not in the middle of BA, as in E, but more close to B it can sink along theperpendicular EG [284].51 (A.6.23)

To point out the use by Alfonfo Borelli (1608–1679) of Torricelli’s principle, orbetter, the idea of evaluating the mechanical behavior of an assembly of bodies withreference to the common centre of gravity [46].52

50 vol. 5, pp. 21–22.51 vol. 5, p. 22.52 pp. 311–312.


The subsequent evolution of Torricelli’s principle, occurring for insensitive steps,consisted in replacing the language of proportions with the algebraic one. If p1 andp2 indicate two weights connected to each other and with Δh1 and Δh2 their heightvariations for a possible motion allowed by the constraints, Torricelli’s principlerequires that for balance it should be satisfied that:

p1Δh1 + p2Δh2 ≥ 0, (6.1)

that expresses in formulas the fact that the centre cannot fall and which for infinites-imal displacements assumes the expression:

p1Δh1 + p2Δh2 = 0. (6.2)

The above formula can be given the mechanical meaning that if, in a system ofconstrained heavy bodies, oneweight descends anothermust rise and the relationshipbetween the descent and ascent is inversely proportional to the ratio of the weights.The principle of Torricelli is then a virtual work law, equivalent to Jordanus’ theoremfrom a mathematical point of view. Note that the mechanical nature of Torricelli’sprinciple is hidden by the Archimedean approach which seeks to reduce the physicsas much as possible.

For sure Torricelli would not have had problems to extend his principle to a sys-tem of n weights. If he had done this and had written the results with an equation hewould have obtained the relation:

n

∑i=1

piΔhi = 0. (6.3)

To the reader the opinion on the level of anachronism is introduced with these con-siderations.

6.5.1 A restricted form of Torricelli’s principle

In some textbooks of physics Torricelli’s principle is evoked also for a single body,according to which it is impossible for the centre of gravity of a body in equilibriumto sink from any possible movement. It must be said however that this statement isalmost a logical consequence of the definition of centre of gravity and it does notrequire the degree of abstraction necessary to formulate a statement similar in formbut valid for a set of bodies, as does Torricelli.

Despite its poverty, the restricted Torricelli’s principle has some interesting appli-cations; among these is the proof of the containment polygon theorem, which showsthat a solid based on a horizontal plane is in equilibrium if and only if the vertical linefrom its centre of gravity falls within the (convex) perimeter of the base. This crite-rion has been exposed in a formal manner by Juan Bautista Villalpando (1552–1608)[241] in 1604 and Bernardino Baldi in his Aristotelis problemata exercitationes. ElioNenci believes that Baldi knew the work of Villalpando and then he was inspired byhim [20]. More dramatically Duhem believed that both Villalpando and Baldi wereinspired by Leonardo [305].53

53 vol. 2, p. 119; p. 133.

6.5 Evolution of Torricelli’s principle. Its role in virtual work laws 155

A D

E

KCMB

HL

I

FG

Fig. 6.8. The containment polygon

A

KE

F CB

H

Fig. 6.9. A measure of stability against tilting

In the following I refer to what Baldi wrote in the Aristotelis problemata exerci-tationes in problem XXX, where he studied the equilibrium of the body shown inFig. 6.9:

Assume it moves, and from the semidiameter BE with centre in B the arc of circle EH is de-scribed which cuts BG in H and BF in I. And because EH perpendicular to the semidiameterBK does not pass through the centre B, EM is shorter than BK, i.e. of BI. Cut LB equal toEM from BI. Point L will be then below point I, i.e. closest to the centre of the world of I.Because the wall would collapse it is necessary that the centre of gravity E, after the rotationaround B, reaches I before arriving eventually in H. But I is farther from the centre of theworld than E and L, so the heavy body will rise against its nature, but this is impossible; thisis what to prove [18].54 (A.6.24)

The body is in equilibrium. Indeed assume that it is not, and perform a rotationaround B (the same applies to C); the centre of gravity describes the arc EIH andthus moves away from the centre of the world. But this is absurd, then the body isin equilibrium.

In problem VIII Baldi takes as a measure of stability in the degree of raising ofthe centre of gravity in the rotation around a corner:

Let the triangles ABF and ACF [Fig. 6.9] be equal and of the same weight, with AFB a rightangle. Join F and C with EC greater than EF. Rotate the triangle around the point C and letEC become perpendicular to the horizontal plane as CH, and from E draw the parallel EKto the horizontal plane. Moving the triangle, the centre of gravity E will displace in H, butKC is equal to EF, less than CH, then the centre of gravity will be raised from E to H, i.e.above K for the whole space KH. This raising makes the motion difficult [18].55 (A.6.25)

54 p. 176.55 p. 2.

7

European statics during the XVI andXVII centuries

Abstract. This chapter presents Dutch and French contributions to mechanics, witha nod to the English in the XVI and XVII centuries. The first part describes thecontribution of Gille Personne de Roberval who proved the rule of composition offorces with Jordanus’ displacement VWL. The central part describes the contribu-tion of René Descartes who was among the first to base statics on a VWL accordingto the idea of Jordanus de Nemore. Descartes also introduces the idea of virtual dis-placements as the birth of virtual motion (draft of infinitesimal displacements). Animportant part is devoted to Simon Stevin. Given his role as the founder of modernstatics I did not limit myself to presenting his contribution to the VWLs, which iscontroversial, but I also present some of his other contributions that are less docu-mented in the manuals of the history of mechanics. The final part shows that IsaacNewton does not avoid important considerations on VWLs based on velocities, de-spite the fact that his mechanics is normally considered an alternative to them.

7.1 French statics

In the early XVII century, when Italy was still the leading nation in Europe, theonly text in French about mechanics was a translation of Cardano’s De subtilitateby Richard le Blanc [55]. In 1615 Salomon de Caus (1576–1626) who worked as ahydraulic engineer and architect under Louis XIII, published Les raisons des forcesmouvantes avec diverses machines, a book having as subject the functioning of ma-chines more than their equilibrium and which concentrated on a steam-driven pumpsimilar to one developed by Giovanni Battista della Porta (c. 1538–1615) fourteenyears earlier [90]. The text of de Caus is quoted by Pierre Duhem [305]1 and RenéDugas [308]2 giving significance to the fact that he used (for the first time?) the wordwork (travail) to indicate precisely what today is called work. There is not however

1 vol. 1, pp. 290–292.2 p. 124.


158 7 European statics during the XVI and XVII centuries

any historical evidence that de Caus’ use influenced Coriolis who was for sure theperson who spread the use of the term work (see Chapter 16).

In the following I refer to a passage in which the word work is used, which alsoshows how de Caus takes a law of virtual work to explain the operation of machinery,at least the lever and pulley:

Vitruvius mentions this kind of machine, called by Greeks trochlea [pulley, block andtackle], in which the motion has its way by means of pulleys […] one end will be attachedto the pulley and the other will serve to raise the burden. As it may be seen from the figure ifone pulls the said piece of rope marked G one foot down, the burden attached to the pulleyE simultaneously will rise half a foot and then, since the rope is doubled in the pulley, ifone pulls 20 feet of rope, the burden will move 10 feet. So a man will raise a heavy bodywith this machine, as he would be two, if the machine were simple but the two men togetherwill draw twice the height i.e. 20 feet, before the other had pulled 10, and if in the trochleathere were two pulleys, as shown in figure M [Fig. 7.1], the force would be quadruple, butthe burden would rise only 5 feet by pulling 20 feet of rope.The toothed wheels have still the same ratios as the previous ones, because the force in-creases proportionally as the time increases […] so that a single man, will use equal forcepulling a load in this machine as eight […] but as the eight men took one hour to lift theirweight, a man will take eight hours to lift his [90].3 (A.7.1)

Fig. 7.1. Work of a machine (reproduced with permission of Master and Fellows of St. John’sCollege, Cambridge)

In 1634, two translations and a new text, quite important, were published. AlbertGirard (1595–1632) translated the Tomus quartummathematicorum hypomnematumde statica by Stevin [217], Martin Mersenne (1582–1648) translated Le mecanicheby Galileo [117], Pierre Herigone wrote Cursus mathematici tomus tertius, a courseof mechanics with text both in Latin and French [130].

Pierre Herigone was born in France in 1580 and died probably in Paris about1643. Mathematician and astronomer, he taught in Paris for a long period [290]. His

3 p. 7.

7.1 French statics 159

book was a success and helped to spread Stevin’s ideas on mechanics. Important in-fluences though not made explicit are those of Guidobaldo dal Monte and Jordanusde Nemore. Herigone did not use a single principle for his demonstrations. For ex-ample he demonstrated the law of the lever with an Archimedean approach; the lawof the inclined plane was studied as in Stevin but also as in Jordanus. Accordingto Duhem, Descartes received from Herigone Jordanus’ ideas on the law of virtualwork. Of some interest is the use of a compact mathematical notation to indicate re-lations and operations; among these the sign ⊥ for orthogonality, the sign π for theratio; 2/2 to indicate the equality, 2/3 to indicate less than, 3/2 greater than, < angle.

To illustrate the use of a law of virtual work in Herigone I limit myself to demon-stration of the laws of the lever and the inclined plane:

A

D E

F B

G

C

Fig. 7.2. The inclined plane of Herigone

Herigone proves the law of the lever with an Archimedean approach, close to thatused by Stevin, then he proves as a theorem the following law of virtual work:

For weights in equilibrium, the space of the lighter is to the space of the heavier as theheavier is to lighter; the same holds for the motion in vertical direction of the lighter to thevertical motion of the heavier [130].4 (A.7.2)

In the demonstration of the law of the inclined plane Herigone uses this theorem asa principle of general validity:

For the same time the weight G descends from point C to point B, the weight D rises frompoint A to point E and BC will consequently be the perpendicular of weights G and EF thatof weight D. But since D is to G, the perpendicular BC to the perpendicular EF, the weightsD and G are balanced [130] .5 (A.7.3)

Note, however, that Herigone’s use of the law of virtual work is different from thatof Jordanus in at least two respects. The first because instead of assuming the equiv-alence of rising p to h and p/n to hn, he assumes the equivalence of rising p to hand lowering p/n to hn. The second because he derives from this result the equilib-rium and not just an equivalence. This allows Herigone to arrive directly at its resultwithout recourse to an indirect (absurd) reasoning.

4 p. 290.5 pp. 301–302.


7.1.1 Gille Personne de Roberval

Gille Personne de Roberval was born in Roberval, near Sen-lis, probably in 1602 and died in Paris in 1675. His name wasoriginally Gilles Personne, that of Roberval, by which he isknown, comes from the place of his birth. Roberval was oneof those mathematicians who occupied their attention withproblems only soluble by methods of infinitesimals. Sinceonly few writings were published by Roberval during his lifehe was for long eclipsed by Fermat, Pascal, and, above all,by Descartes his irreconcilable adversary. Serious research

on Roberval dates from approximately the end of the XIX century, and many of hiswritings still remain unpublished. In 1666 Roberval was one of the charter membersof the Académie des Sciences in Paris [290].

The best-known contribution of Roberval to statics is his Traité de méchanique of1636 [209]. The treatise was entered by Mersenne in hisHarmonie universelle [170]and Duhem believes there existed an even more extensive edition in Latin [305].6

Roberval at the beginning of his treatise cites only Archimedes, dal Monte andLuca Valerio. Nevertheless, he knew for sure Stevin’s work very well, and his trea-tise is almost a completion of Stevin’s, needed to remove some imperfections, suchas that in the proof of the parallelogram of forces law. And of course there is theinfluence of Galileo, at least in the demonstration of the law of the inclined plane.One thing quite interesting and perhaps new is the assimilation of the weights withthe muscle forces (powers), which is more evident than in previous authors, Stevinincluded.

Roberval’s treatise is very formal in some places, for the author will not wantto leave any doubt on the validity of the proof. If this pedantry can be criticized, itshould be said that Roberval reaches his goal.

7.1.1.1 The inclined plane law

The proof of the law of the inclined plane follows the same line of thought pursuedby Galileo, and is substantially equivalent to it when a force parallel to the plane isconsidered.

The key point of the demonstration of Roberval is the substitution of the constraintoffered by the inclined plane with the arm ac of a lever with fulcrum c, as shown inFig. 7.3. Unlike Galileo, the substitution is made by Roberval not so much becausefor small displacements the two types of constraint (plane and lever) are equivalent,but rather because the inclined plane is able to provide support along the direction caequivalent to that of the lever. The motivation of this equivalence, which culminatedin his axiom IV [209],7 is perhaps the least clear part of Roberval’s treatise.

The determination of the force acting in directions not parallel to the inclinedplane, which was given without proof by Stevin, begins with a similar argument.

6 vol. 1, pp. 322–323.7 p. 6.


b e

cgh

k

zd

ul m

ny

2a

fx

f 1

q

po

r

s

Fig. 7.3. Inclined plane with force parallel to it

Fig. 7.4. A multiface figure

The inclined plane is still being replaced by a lever, only now the powers are notboth orthogonal to the arms. Fig. 7.4 illustrates the situation; note that Robervalconcentrated in this illustration more figures, including the case of a body lifted bytwo ropes presented in the subsequent section.

7.1.1.2 The rule of the parallelogram

Roberval presents two proofs of the rule of the parallelogram of forces. The firstexploits the law of the inclined plane, or at least brings everything back to the lever,the second uses a law of virtual work. The problem is reduced to determine the forces


E

C F

K

Q

A

B

L

N

2

Fig. 7.5. The rules of the parallelogram and the lever

of two ropes inclined at any given angle supporting a weight A. Fig. 7.5, extractedfrom Fig. 7.4, illustrates the situation for the first proof.

The reasoning is developed (not completely without error) by imagining one ofthe two ropes that support the weight, for example the AC, as fixed at one end.According to Roberval it may be replaced either by the rigid arm AC of a lever orthe inclined plane LN2:

Now for the second proposition [that of the inclined plane] we have seen that if CA is thearm of a balance on which the weight A is retained by the rope AC so it does not slide alongthe arm AC, and as CB is to CF, the weight A is to the power Q or E pulling with the ropeQA, the power Q and E will hold the balance CA in equilibrium, and the rope QA beingattached to the center of the weight A, the balance remains unloaded. The weight A will besupported partly by the power Q or E, partly by the plane LN2 perpendicular to the balanceAC, or by the rope CA, by the fourth axiom of the Scholium [209].8 (A.7.4)

In this way the determination of the force of the rope AQ is reduced to a known case,the lifting of a weight on a plane with a force of given direction. The same can bemade for the rope AC.

The secondRoberval’s proof requires the use of a law of virtual work. It resemblesthat of the angled lever of Jordanus de Nemore being based on the impossibility thatthe sum of the product of the weights that go up by the values of the ascent is differentfrom the sum of the weights that go down by the values of the descent.

Scholium VIII. [...] the weight is located in A on the ropes CA and QA sustained by thepowers C and Q or K and E; with the weight that is to the powers as the perpendiculars QGand CB are to the lines CF and QD [i.e. P : K = QG : CF; P : E = CB : CQ].[…]If below the weight A, in its line of action, one considers the line AP, if the weightA descendsto P, dragging the ropes and making the powers K and E to rise, the ratio of the path thepowers will make in raising to that made by the weight in descending will be greater than

8 p. 22.


B

P

AVG

FC

K

E

QD

Fig. 7.6. The rule of the parallelogram and virtual work

the ratio of the weight to the two powers considered together. So the powers will raise morein proportion than the weight will descend, what is against the common order. If above theweight A, in its line of action, one considers the line AV, and the weight rises until V, theropes rise because of the powers K and E that descend. It will be a greater ratio of the paththe weight will make in raising to the path that the powers will make in descending thanthe ratio of the two powers considered together to the weight. So the weight will rise morethan the powers will descend; and this is also against the common order. Now that the ratioof the weight A and the powers in rising or descending are such as we said, and against thecommon order, is proved in our Mechaniques,9 because it is too long to be reported here.Concluding as the weight A will remain in its place, for the reasons of the 3rd proposition,all goes according to the natural order. What is wanted to remark [209].10 (A.7.5)

In essence Roberval considers a weight A balanced through two ropes by the twoweights K and E, inversely proportional respectively to GQ and CB of Fig. 7.6. Heproves the equilibrium by a reduction to the absurd. Suppose there is not equilibrium,for example the weight P falls and K and E rise. Roberval said, without reportingthe details, that the sum of the products – a modern interpretation – of the weightsE and K for their ascent is greater than the product of the weight A for its descent.But this is impossible, then there cannot be motion. Hence the absurd.

In previous demonstrations Roberval gave two different criteria for determiningthe tension in the ropes, but he did not explicitly set out the rule of the parallelogramof forces. He does this explicitly in a Scholium, by affirming: “If for any point madein the line of the direction of the weight, the line parallel to one of the strings and tothe other are drawn, the side of the triangle thus formed will be homologous to theweight and the two powers” [209].11

9 It seems there exists another treatise where the geometrical conclusions on the variation of ropesby Roberval are proved in detail.10 pp. 35–36.11 p. 28.


7.1.2 René Descartes

René Descartes was born in La Haye en Touraine (nowDescartes) in 1596 and died in Stockholm in 1650. Muchmodern Western philosophy is a response to his writings.Descartes was the greatest mathematicians of the first halfof the XVII century and one of the founders of analytic ge-ometry, the bridge between algebra and geometry. A mostimportant writing for science is his Principia philosophiae[96]. Here he depicted his view of the world, where matterhas only a geometrical nature and the whole amount of mo-

tion, the force, is conserved forever. I.e. he considered the philosophy of nature ascoinciding with mechanics.

The ideas of Descartes on statics are contained in three letters, two of which arequite long and with titles. The first is a letter dated October 5th 1637 to ConstantinHuygens (father of Christiaan), entitled Explication des engins par l’ayde desquelson peut, avec une petite force, lever un fardeau fort pesant [96], the two other lettersare written toMersenne in July and September 1638 [96], the first, July 13th, entitledExamen si un corps pese plus ou moins, estant proche du centre de la terre qu’enestant esloigné, is the most interesting.

7.1.2.1 The concept of force

The contribution of Descartes to the development of laws of virtual work consistsmainly of a framing of the problem; still important are even some of his more strictlytechnical considerations. He was the first to give a mechanical sense to the product ofthe weight for the vertical displacement. This coincides essentially with the modernwork, which he calls action, or more frequently still force, with a little unhappy termbecause Descartes also calls force the muscular effort, the power and in dynamics,the absolute value of the quantity of motion. He repeatedly says that it takes the same‘force’ to lift a weight to a certain height, that to raise a double weight to half height.For instance he writes to Mersenne in July 1638:

It needs neither more nor less force to lift a weight to a certain height than to raise a lowerweight to a height as greater as the weight is less heavy, or to lift a heavier weight to a lessheight.[…]This will be given easily, if it is considered that the effect must always be proportionate tothe action that is necessary to produce it, and thus if it is necessary to use the force by whichone can raise a weight of 100 pounds to the height of two feet, to lift one at a height of onefoot only, it means that it weighs 200 pounds [96].12 (A.7.6)

And also:

Above all it must be noted that I have spoken of the force that is used to lift a weight at someheight, force that always has two dimensions, and not the force used to hold the weight atany point, which always has only one dimension. These two forces differ from one another

12 vol. 2, pp. 228–229.


ds

p

f

ds

Fig. 7.7. Different ways to see the force of Descartes

as a surface differs from a line. In fact the same force a nail needs to support 100 pounds fora moment of time is sufficient, when it does not lessen, to support them for a whole year.But the same quantity of that force used to lift that weight up to a foot is not enough to liftit to the height of two feet, which is not less obvious than two plus two makes four, be it isclear it would need a double force [96].13 (A.7.7)

Descartes does not define his force in an algebraic way, explicitly as the productof weight for shifting, but rather in a geometrical way, as the area of a rectangle.The force which Descartes talks about concerns weight; with a modern language itis the work made to raise a weight and its value is measured by the product of theweight and the space covered. It is probably not far from the ideas of Descartes – notexplicit in this regard – to represent the ‘force’ as the product of a muscle force bythe motion of its point of application, which can be in any direction. For example,with reference to Fig. 7.7, the force is given by the rectangle p · ds but also fromf ·ds.

Descartes clarifies his concept of force by adding that the equality of the workof the ‘forces’ can only be accomplished with the use of machines that transformrectangles of equal area in different forms:

Because I did not simply say that if the force can lift a weight of 50 pounds a height of 4feet, it shall be able to raise 200 pounds a height of one foot, but I said that it could bewhen it is applied. Now it is impossible to apply [this force], but by some other machineor invention that makes this weight [200 pounds] to rise one foot, while the force will actalong the length of four feet, and transforms the rectangle by which the force required to liftthat weight of 200 pounds to the height of one foot into another that is equal and similar tothe one that represents the force required to lift a weight of 50 pounds to a height of 4 feet[96].14 (A.7.8)

In substance Descartes says a man cannot raise indifferently a weight of 200 poundsand one of 50 pounds, because probably he cannot exercise the muscular force nec-essary to raise the greater one, he can however choose opportune machines that canperform this operation. The above passage is followed by an application to the caseof the inclined plane.

13 vol. 2, p. 353.14 vol. 2, p. 357.


AE

HP

GFONA

C

D

D

L C B

LD

Fig. 7.8. The conservation of force on an inclined plane

Consider the inclined plane ABC of Fig. 7.8 with AB = 2 AC. The two-dimensionalforce to lift D along AB is represented by the rectangle FGH, with GH = AB. Con-sider then the weight L required to lift D along AB. The force in two dimensions toraise L to the height of AC is equal to that FGH required to lift D along AB. The forceneeded to lift the weight D without the intermediary of the plane AD is still equal toFGH but one of its dimensions, AC, is half of AB then the other dimension, whichrepresents the force required to lift the weight will be double. Hence the weight L ishalf of the weight D.

Descartes believes that the rule laid down by him, namely that the force neededto raise p to h is the same as that required to raise p/2 to 2h should be consideredthe only principle of statics. Principle because it can explain the operation of allmachines. Principle because evident since one cannot challenge the simple consid-eration according to which:

It is the same to lift 100 pounds to the height of one foot, and again another 100 to the heightof one foot, as to raise 200 to the height of one foot, and the same also to raise one hundredto two feet [96].15 (A.7.9)

In fact, the justification above, coinciding with that of Galileo referred to in § 5.6.2,though ingenious, does not withstand critical analysis as noted by Mach [355].16

Indeed, the admission that to lift 100 pounds in two stages is equivalent to 200 in one,although intuitive, is not logically deductible and it is not contradictory to imaginethat it is not true.

Descartes believes that his principle gives a causal explanation, i.e. that it allowsone to understand the why. It is natural to ask whether, given the importance of thisprinciple, Descartes does not count it among the laws of nature found in thePrincipiaphilosophiae of 1644 written after the letters to Huygens and Mersenne. Accordingto Sophie Roux [211] this is because the law is formulated by involving the weightand not the categories of matter and motion, the only ones that can give rise to clearand distinct concepts.

15 vol. 2, pp. 228–229.16 p. 84.


It must be said that Descartes is not always consistent with his statements. Inthe applications presented, in particular the law of the pulley, but also the law of thelever, he does not use his principle as such, but rather recognizes equilibrium by othermeans. From this point of view, that of Galileo who reduced actually and clearly allthe machines to a single principle, the lever, was a clearer approach, which also hadcausal value. The unifying approach of Descartes is considered attractive by modernhistorians of science because, being of algebraic nature, one has to apply always thesame formula, and it is easier to use in intricate situations than that of geometricnature of Galileo, which can take a significant technical skill and imagination.

7.1.2.2 The application to simple machines

The letter of 1637 to Constantin Huygens looks like a small treatise on mechanicsin which all the simple machines are analyzed. Here Descartes refers to forces ofgravity – or lines of descent – parallel to each other, though he admits that this isonly an approximation. The treatise opens with a passage similar to that reported inthe letter to Mersenne:

The invention of all those engines is based on one single principle, that if the same forcethat can lift a weight, for example of 100 pounds to a height of two feet, it can also raise aweight of 200 pounds to one foot, or a weight of 400 pounds to the height of 1/2 foot, andothers.[…]Now the engines used to make this application of a force acting on a large space to a weightthat it raises with a minor space, are the pulley, the inclined plane, the wedge, the wheelwith the shaft, the screw and some more [96].17 (A.7.10)

The explanation of simple machines follows:

The pulley. Let ABC be a rope passing around the pulley D, in which the weight E is appliedand suppose first that two men support or raise equally each of the two ends of the rope, it isclear that if the weight weighs 200 pounds, each of the two men take, to support or lift, theforce required to support 100 pounds, because each holds only one half. Let then that A, oneend of the rope, being attached to a nail, the other C is still supported by a man, it is clear

CC

A

B

D D

E E

B

HK

A

Fig. 7.9. The pulley

17 vol. 2, pp. 435–436.


that this man in C would need, as before, only the force required to support 100 pounds,because the support is to do the same service of the man who was supposed before. Finally,suppose that this man in C pulls the rope to lift the weight E; it is clear that if he uses theforce required to raise 100 pounds to the height of two feet, he will raise this weight whichweighs 200 pounds to the height of one foot. This because the rope ABC being wrapped asit is, it must be pulled by two feet from the head C to raise the weight E as two men pulledit, one from the head A the other from the head B, each the length of a foot only. There ishowever something that prevents this calculation being correct, such as the weight of thepulley and the difficulty to make the rope to slide and hold it. But, this will be negligiblecompared to what it gets [96].18 (A.7.11)

It is worth noting that Descartes proves the law of the pulley directly with simpleconsiderations of equilibrium, the same of Hero, and then verifies that it complieswith the law of virtual work, contrary to the declared intentions of wanting to take asingle law of statics. In his letter to Mersenne of September 1638, there is an inter-esting comment on the result for the pulley, which clarifies even further Descartes’sconcept of ‘force’:

So to not deny that the nail A [Fig. 7.10a] supports half the weight of B, one can onlyconclude that by this application [of the pulley], one of the dimensions of the force, thatmust be in C to lift that weight, is one half, the other therefore double. Thus, if the line FG[Fig. 7.10b] represents the force required to hold the weight B at some point without the helpof any machine, and the rectangle GHwhich is enough to lift the height of a foot, the supportof the nail A reduces to one half the dimension represented by the line FG, and doublingthe rope BDC it doubles the other dimension represented by the line FH, so the force whichmust be in C to lift the weight E to the height of a foot, is represented by the rectangle IK[see Fig. 7.10b]. And since it is known in geometry that a line added or removed to an areaneither enlarges nor diminishes, here you will observe that the force with which the nail inA supports the weight B, having only a single dimension cannot ensure that the force in C,seen in its two dimensions, should be less greater to lift the weight in this manner than tolift it without the pulley [96].19 (A.7.12)

H

K

F I G

E

A

B

D

C

a) b)

Fig. 7.10. Two dimensional force

18 vol. 2, pp. 437–348.19 vol. 2, p. 356.


In the following I will report only the demonstration of the law of the lever. Thatof the inclined plane is the same as that already shown in the letter to Mersenne butmore concise and the demonstrations of the laws of the wedge, the screw and the axisand the wheel do not present elements of particular interest. Different is the situationof the lever for which Descartes says, not without surprise to a modern reader, thatit is the case more complicated to prove.

The lever considered by Descartes is that of Fig. 7.11. The weight is applied at theend H, the force at the end C. It is OC = 3 OH. When the lever is in the position GB,Descartes admits with Galileo, that the constraint of the lever is equivalent in G withthe inclined plane GM tangent in G to the circle KHF. With the law of the inclinedplane Descartes can determine the apparent gravity or relative gravity, as opposedto the absolute gravity of a body free from constraints, which acts perpendicular tothe lever and then parallel to power in B. At this point, Descartes takes for grantedthe law of the lever with powers perpendicular to it and determines the power in Bsaying that it is equal to one third of the relative gravity of the weight. From thearguments of Descartes it is clear that the difficulty lies not in the law of the leverin itself, which is presupposed, but in the fact that at one end of the lever the poweracts perpendicularly to it while at the other end the weight acts vertically, and thatthe efficacy of this weight depends on the inclination of the lever.

Lever. And to accurately measure this force which must be at each point of the curved lineABCDE, it is known that it works the same way as if the weight moved on an inclinedcircular plane, and that the slope in each of the points of this circular plane is to be measuredby that of the straight line that touches the circle at this point. Such as when the force is atpoint B, to find the proportion that the heaviness of the weight that is at that moment in Gmust have, it must draw the tangent GM and consider that the heaviness of this weight isproportional to the force required to drag it on this plane, and thus to rise it according to thecircular arc FGH, as the line GM is to the line MS. Then, since BO is three times OG, it issufficient that the force in B is to this weight in G as the third part of the line SM is to thewhole GM. The same is true when the force is at point D [96].20 (A.7.13)

A

B

C O

D

E

K I

H

GFM

S

PN

Fig. 7.11. Descartes’ lever with a force and a weight

20 vol. 2, p. 445.


Descartes, therefore, studies, or at least considers, the lever starting from the in-clined plane, with a procedure completely opposite from that taken by Galileo. Thisis largely due to the fact that because of his concept of work, the inclined plane,which consider changes in height, naturally lends itself to play a paradigmatic role.

7.1.2.3 The refusal of virtual velocities

In the above analysis on the simple machines it seems to have more to do with themotion than with the equilibrium. Descartes’ argument, however, is contrary to theuse of virtual velocities, and it is not suitable for him to establish a principle ofstatics, because the velocity (real) depends on many factors, such as the resistanceof the medium, the velocity of application of the force and so on:

Many people regularly confused the consideration of the space with that of time, or speed,so that, for example, in the lever, or equally, in the balance ABCD [see Fig. 7.12a], havingassumed that the arm AB is twice as long as BC, and that the weight in C is twice the weightin A, and that they are in equilibrium, instead of saying that the cause of this equilibrium isthat because the weight C lifts or is lifted by the weight A, it will not go for half the space ofit [the weight A] they say that it moves half more slowly, and this is a mistake, among themost insidious to be recognized, because it is not the difference in speed that determines theweight to be in equilibrium, but the difference of displacements [96].21 (A.7.14)

B

D

a) b)

C H

G

FA

Fig. 7.12. The effect of velocity

For Descartes it is not the difference of velocity which determines that one of thetwo weights must be double the other, but the difference of displacement:

As it is shown, for example, that to lift the weight F with your hand to G [Fig. 7.12b], it isnot necessary, if one wants to lift it with twice the speed, to use a force that is exactly twicethat otherwise required. It is required instead a force that is more or less double, accordingto the variable ratio that can have the speed to the factors that will resist, but to raise it at thesame speed of twice the height, up to H, a force is needed that is exactly twice, I say that isexactly twice, just as one plus one makes two: in fact a certain amount of that force must beused to lift the weight from F to G and then again the same amount to raise it up from G toH [96].22 (A.7.15)

And to the contrary, take a fan in your hands, you can lift it with the same speed with whichit could descend by itself in the air when you leave it to fall, without any effort except that

21 vol. 2, pp. 353–354.22 vol. 2, p. 354.


necessary to sustain it. But for lifting and lowering two times more fast it will be necessarythat you employ a force greater than double the other, unless it be zero [96].23 (A.7.16)

He continues to justify his position, without being very convincing, saying that heknows very well that the ratio of velocities can be equal to that of displacements, butthis is not enough to accredit a principle of statics:

Now the reason because I am criticizing those who use speed to explain the force of thelever and the like, is not that I deny that the same proportion of speed does always occur,but because the speed does not explain why the force increases or decreases, as does theamount of space [motion], and that there are many other things to consider for the speedsthat have not been explained [96].24 (A.7.17)

The criticism of Descartes, with considerations similar to those of Stevin, is directedalso against the traditional formulations of the laws of virtual work, in which it isconceptually irrelevant to consider virtual velocities or virtual displacements, be-cause both are hypothetical and not real. This fact was perfectly clear to Galileo,who for his ideas was the subject of an attack by Descartes, shown below, ungen-erous toward a man who for sure had understood the problem of equilibrium betterthan him:

What Galileo wrote about the balance and lever [in Le mecaniche], he explains how but notwhy, as I do with my principle. And for those who say that we must consider the speed,as Galileo, instead of the space to explain the machines, I believe, between ourselves, theyare people who speak only by fantasy, without knowing anything about this subject [96].25

(A.7.18)

7.1.2.4 Displacements at the very beginning of motion

An important contribution, of technical character, offered by Descartes to laws ofvirtual work is surely to have guessed, but not fully developed, the concept of itsinfinitesimal character in the case of bodies constrained to move on a curved path.And paradoxically, this character makes it easier to talk of virtual velocities, theuse of which Descartes opposed, than of virtual displacements. Indeed velocities ina constrained motion can always be really possible, while displacements generallyonly approximate the real motion, the smaller they are.

In the letter to Mersenne of July 1638, Descartes modifies in part the concept ofgravity accepted in the letter to Huygens. There the lines of action of weights wereconsidered as parallel, here converging. This choice complicates things unnecessar-ily – at least in the eyes of a modern scholar – however, it leads Descartes to say thatthe relative gravity of a body is measured by reference to the motion in its birth andit can vary along the same inclined plane.

The relative weight of each body, or that is the same thing, the force that must be used tosupport it and prevent it from descending when it is in a certain position, shall be measuredfrom the beginning of the motion that the force that sustains [the weight] should make,

23 vol. 3, p. 614.24 vol. 3, p. 614.25 vol. 2, p. 433.


either to raise or to follow if it sinks. The proportion that exists between the straight line[the tangent] describing the motion and that which defines the approach of the body to thecentre of the earth is the same as that which exists between the absolute and relative weight[96].26 (A.7.19)

In his analysis of the descent of a body on an inclined plane Descartes first argues thatthe relative gravity varies along an inclined plane as the angle between the directionsof force of gravity and motion. In the generic point D of Fig. 7.13 the ratio betweenthe relative and absolute gravity is given by the ratio between the sides FN and NP.

F

BG C

M

K

E D

PN

AH

Fig. 7.13. Motion over a curve path

Let AC be an inclined plane on the horizon BC and AB tend directly to the centre of theearth. Those who write about mechanics shall ensure that the heaviness of the weight F ,when it is on the plane AC, has the same proportion with its absolute weight as the line ABto the line AC.[…]Which is not entirely true, however, except when it is assumed that the heavy bodies tenddownward along parallel lines, an assumption that is commonly made when Mechanics isconsidered to be useful, since the little difference that can cause the inclination of theselines, that tend toward the centre of the earth, is not sensitive. […] And to know how muchit weighs in every one of the other points of the plane with regard to this power, for exampleat point D, we must draw a straight line, as DN, toward the centre of the earth, and from thepoint N, arbitrarily assumed on this line, draw NP perpendicular to DN, which meets ACin P. As DN is to DP, so the relative gravity of the weight F in D is at its absolute gravity[96].27 (A.7.20)

Descartes then considers what would happen if one admitted that the weight wasfalling not on an inclined plane but on the curved surface EDG:

Note that I say, begin to descend, not just descend, because it is at the beginning of thedescent to which it is necessary to refer. So if, for example, the weight F is not supported at

26 vol. 2, p. 229.27 vol. 2, pp. 232–233.


the point D on an inclined plane, as ADC is supposed, but on a spherical surface, or curvein any other manner, such as EGD, provided that the flat surface, which is imagined tangentat point D, it is the same as ADC, it will not weigh any more or less, for the power H, whichmust be applied to the plane AC. Because, although the motion that would make this bodygo up or down from point D to E or to G on the surface EDG, would be completely differentthan it would be on the flat surface ADC, however, being in the point D of EDG, it will beforced to move as if were on ADC, toward A or C. It is evident that the change of positionresulting in the motion, when the body has ceased to touch the point D, cannot change theweight that it has when it touches it [96].28 (A.7.21)

What is reported above very clearly by Descartes can be repeated with a bit moremodern language. With reference to Fig. 7.13 it can be seen that if the weight F isin equilibrium on the inclined plane AK, this equilibrium will not be upset if theplane turns, in the points where it is not in immediate contact with the weight, withthe surface EDG, or any other surface. To check the equilibrium it is sufficient toconsider infinitely small displacements which the smaller they are the more theyare parallel to the tangent to the curved surface (in technical language, infinitesimaldisplacements). If one considers finite displacements the weight would be movingon a surface of different slope than the inclined plane in which he had found theequilibrium and this would no longer exist.

7.1.2.5 A possible precursor

It has been said beforehand that Descartes could have borrowed his ideas on staticsfrom Herigone. But it is also possible that there was no direct personal influence andthat Descartes drew from formulations, more or less defined, of the laws of virtualwork which were part of the background knowledge of the period. In this respect itseems interesting to refer to the little known work of Genevan Michel Varro.

Not much is known of this author who is considered a minor scientist and thereare few specific studies on him [363, 266]. Of his writings a treatise on mechan-ics (tractatulum) in Latin is known, entitled De motu tractatus [239], which studiesequilibrium, making reference to a virtual work law based on speed. Varro’s treatiseis very slim, less then fifty pages, and considers just general aspects. From this pointof view it is quite different from the treatises of the period which almost all concen-trated on explanation of the operation of simple machines, an approach that to someextent did not escape even Galileo and Stevin (to a lesser extent for the latter) withwritings substantially contemporary to Varro.

Varro says he was inspired by Archimedes in his mathematical approach to stat-ics, not so much in the principles he uses but rather for the method. Like Archimedes,Varro argues that a treatise on mechanics should not have to deal with special casesbut should report the general theory:

It is for this that I think it is necessary to insist first in the theory, because what is appliedcould be consideredwithout any difficulty. In some respects there is the danger that if we stopto deal with special cases, people are satisfied with these, and so it happens that the universalknowledge is neglected and the search for causes and science end [239].29 (A.7.22)

28 vol. 2, pp. 233–234.29 Preface.


In a series of definitions concerning the nature of forces – not just weight and mo-tion – Varro, despite his thinking being in part still inside Scholasticism with theuse of concepts like natural motion etc., sets the stage for a quantitative study of thevirtual work laws of equilibrium and reaches out to formulate his virtual work lawsas ‘theorems’ of equilibrium, as the following for instance:

Theorem ITwo forces connected so that their motions will be inversely proportional do not move butare in equilibrium. [239].30 (A.7.23)

This theorem is enough for Varro to formulate a correct law of the inclined plane,although he gives the result no particular emphasis [239].31

Historians of mechanics [363],32 credit Varro with having given some contribu-tion to the rule of composition of forces [239],33 however, the text of Varro is notso clear on the subject and I will not comment on the fact. As far as this chapteris concerned, it is interesting to note the following comment that Varro adds at theend of his treatise, which recalls the letters of Descartes to Constantin Huygens andMersenne:

To therefore conclude this small treatise, or to close in a summary, to produce the motion,three things must be considered: the force by which we want to make the motion, the forcethat we want to move, and the motion with which we want to move. Any two of themdetermine the third. Indeed if we want to move a large force with a small one, we can moveit by a small motion, if on the contrary we want to move some force by a large motion, itrequires a large driving force. For example, if we want to move 100 pounds with the aid of1 pound, the motion must be by 100 times. If we want to use 1 pound to move another forceso that it is driven 100 times faster than the weight of 1 pound, that force must be 100 timessmaller. If we want to move 1 pound so that it is moved 100 times faster than the force thatmoves, it will need a force 100 times greater. Nature does not allow that in all these casesa new force arises. Indeed, if the proportion were by any means violated, there would beperpetual motion, or as it is named, perpetual motion in perpetual matter [239].34 (A.7.24)

Certainly there is a substantial difference in the use of speed, contrasted, instead ofdisplacement, accepted, by Descartes, but the tone is the same and also the numericalvalues of weights referred to in the quotation are the same, so it would not be difficultfor Descartes, supposing he had read Varro’s text, to translate it in his metaphysics.Among other things it is probable that Descartes had read the De motu tractatus. Itwas known in France as it is mentioned in Varignon Nouvelle mécanique ou statique[238]35 where some of Varro’s statements are flanked to Descartes’. It should alsobe noted that Descartes’ claim to use the law of virtual work as the only law of staticsemerges also from the reading of Varro’s text, even if it is not specifically stated andthere are no applications to the various cases. I will not insist on this reconstruction,

30 p. 19.31 pp. 35–37.32 p. 121.33 pp. 37–38.34 pp. 42–43.35 p. 321.


which is certainly not fully founded, but certainly it offers matter to reflect on howthe laws of virtual work were entrenched in mechanics of the late XVI century.

Finally I would like to point out the introduction of the concept in an embryonicform of potential energy, which I think is one of the first in the history of mechanics.Towards the end of his work Varro says “that a lot of things can be raised beyondtheir natural place to use them when it is needed to produce motion”[239].36

7.1.3 Blaise Pascal

Blaise Pascal was born in Clermont-Ferrand in 1623 and diedin Paris in 1662. He was a mathematician, physicist, writerand Catholic philosopher. Pascal’s earliest work was in thenatural and applied sciences where hemade important contri-butions; he wrote a significant treatise on projective geome-try at the age of sixteen, and later correspondedwith Pierre deFermat on probability theory, strongly influencing the devel-opment of modern economics and social science. Pascal alsowrote in defense of the scientific method. About mechanics

his most significant contribution is on hydrostatics where he made important studieson fluids and clarified the concepts of pressure and vacuum. In 1654, he had a ‘seri-ous’ conversion, abandoned his scientific work, and devoted himself to philosophyand theology. Of this period are his most famous writings, the Lettres provincialesand the Pensées [354].

At the beginning of his Traité de l’equilibre de liqueurs [185]37 Pascal presenteda new type of simple machine to add to the lever, the inclined plane and so on. Toexplain its functioning he assumed the virtual law according to which “the motionis increased with the same proportion as the force”:

From which it appears that a reservoir of water is a new principle of mechanics and a newmachine to multiply the forces to the degree one wants, so that a man by means of it couldraise any weight he would.And it should be appreciated that in this new machine is found the constant order found inthe ancient machines, i.e. the lever, the wheel with the shaft, the screw and so on, which isthat the motion is increased with the same proportion of the force [emphasis added]. Becauseit can be seen when one of these holes [Fig. 7.14] is one hundredth the other, if the manwho pushes the small piston by one inch, the other will move only one hundredth: because itdepends on the incompressibility of the water which is commonwith the two pistons [185].38

(A.7.25)

To explain in detail the operation of the new machine, known today as the hydraulicpress, Pascal curiously assumed another principle, that of Torricelli. He, however,did not mention Torricelli of whom almost certainly he knew the work, perhapsbecause he followed the fashion of the period to not give much evidence of the

36 p. 45.37 It seems however that Pascal’s treatise was composed in 1653.38 p. 183–184.


sources. Perhaps because he considered now Torricelli’s principle as backgroundknowledge.

I take for granted that never does a body move for its weight if its centre of gravity does notdescend. From it I prove the two pistons of the figure [Fig. 7.14] are in equilibrium and theircommon centre of gravity is a point which divides the line joining their individual centresof gravity, in the inverse ratio of their weights. Assume for absurdity that they move. Theirmotions will be inversely proportional to their weights as we assumed. Now if one takesthe common centre of gravity in this second situation, it will be exactly at the same point asbefore, because it will be always at the point which divides the line joining the individualcentres of gravity in the inverse ratio of their weights. So, because of the parallelism ofthe line of their displacements, it will be in intersection of the two lines which join theirindividual centres of gravity in the two situations. Then the two pistons considered as aunique body are moved without their common centre of gravity having moved down. Thisis against the principle; then they cannot move, they will be at rest, that is in equilibrium.As it was to prove [185].39 (A.7.26)

Fig. 7.14. The hydraulic press (reproduced with permission of Biblioteca e Archivio AccademiaNazionale delle Scienze, Torino)

Pascal above has considered a parallelepiped reservoir on which there are two cir-cular holes, one great on the right and one smaller one on the left (see Fig. 7.14).After filling the container with fluid, a weight proportional to the area of the holes isapplied to the two pistons. The proof of equilibrium is for reduction to the absurd.Suppose there is not equilibrium and a piston moves up and the other down. For thegeometry of the pistons, for their weights and for the incompressibility of the fluid itis easy to infer that their common centre of gravity is not lowered. Then the motioncannot occur, hence the absurd.

7.1.4 Post Cartesians

After Pascal and Descartes in France there were no longer written texts of valuefor statics. The only works of some importance were those of de Challes, Pardies,Rohault, Lamy. French statics emerged again with Pierre Varignon (1654–1722), towhich I will refer in Chapter 8.

39 pp. 186–187.

7.2 Nederland statics 177

Claude François Milliet de Challes (1621?-1678) combined the talents of mathe-matician, teacher, and writer. HisCursus seumundusmathematicus [91] is a remark-able and well-written course onmathematics and subjects such as optics, magnetism,mechanics, navigation, pyrotechnics, astronomy andmusic. De Challes was to incor-porate the works of previous mathematicians into a coherent system and to explainthe intricacies of the mathematical sciences with ease and accuracy. He assumedsome form of virtual work law in his argumentations.

Jacques Rohault (1620–1673) was a mechanistic Cartesian and experimentalphysicist. His Traité de physique [210] was a standard text for nearly fifty years.John Clarke and Samuel Clarke, rather than writing a Newtonian physics, translatedRohault’s work into Latin and English [70] and added Newtonian footnotes to cor-rect Rohault’s mistakes.

Ignaces Gaston Pardies (1636–1673) was a Jesuit. His collected mathematicalworks were published in French and in Latin. Of particular interest is his La statiqueou la science de forces mouvantes [184], where there are contributions also to thestrength of materials.

Bernard Lamy (1608–1679) was professor of classics at the Jesuit Collège deCésar in Vendome. His major publication in statics was his Traité de mécanique, del’équilibre des solides et des liqueurs inwhich the parallelogramof forces law is given[155]. Pierre Varignon discovered the parallelogram of forces law independently, atabout the same time, and he saw more consequences of it than Lamy did.

7.2 Nederland statics

In 1581, seven of the seventeen Low Countries refused to recognize Philip II as theirking and originated the so-called Republic of the Seven Provinces, partially coincid-ing with the modern Nederland. Thus began a period of great political and religiouschanges and a large cultural and economic development; it usually is referred to asthe Golden Century of Nederland. Great stimulus to the development of sciencesin general and of mathematics and mechanics in particular came from commercialneeds of the new state. The republic promoted the dissemination of scientific knowl-edge with the creation of new schools at the local level. Also higher studies wereenhanced and the university of Leiden, founded in 1575, became a very importantschool.

A special role for the development of mathematics was played by surveyors, whofaced complex problems for the preparation of reliable nautical and land charts re-quired for the trade policies of the new state. This fervor of scientific activity wasrooted in an important cultural tradition. Just remember that, in the city of Deventer,Nicholas Cusanus (1401–1468) and Erasmus of Rotterdam (1466–1536) appearedon the scholary scene. Certainly a notable influence in the development of Dutchscience was also due to the long stay of Descartes, started in 1617, with the fruitfulcollaboration of Isaac Beeckman (1588–1637). There is therefore no wonder that inthis land, florid and tolerant, then as now, people with genes as unique as Simon


Stevin and Christiaan Huygens were born, however, separated by a vast amount oftime. In this section I illustrate with some detail only the contribution these twoDutch scientists made to the development of laws of virtual work; the contributionthat albeit has involved only a small part of their production is nevertheless impor-tant [384].

7.2.1 Simon Stevin

Simon Stevin was born in Bruges in 1548 and died in Lei-den (or maybe in Den Haag) in 1620. He was for some yearsbook-keeper in a business house at Antwerp; later he securedemployment in the administration of the Franc of Bruges. In1583 he entered the university of Leiden. From 1604 Stevinwas an outstanding engineer who advised on building wind-mills, locks and ports. Author of many books, hemade signif-icant contributions to trigonometry, mechanics, architecture,musical theory, geography, fortification, and navigation. He

introduced the use of decimals in mathematics in Europe [384].Inspired by Archimedes, Stevin wrote important works on mechanics. His books

De Beghinselen der Weegconst (Principles of the Art of Weighing) and De Begh-inselen des Waterwichts (Principles of the Weight of Water), published in 1586,deal mainly with equilibrium. Although he undertook his mathematical work earlierin his life, Stevin collected together some of his mathematical writings and editedand published them during the years 1605 to 1608 inWiskonstighe Ghedachtenissen(Mathematical Memoirs, in LatinHypomnematamathematica) [215, 216, 217, 218].As a custom of the times he did not quote his predecessors, with the exception ofArchimedes, Commandino and Cardano but in the last case only to criticize his(wrong) result for the inclined plane; for some comment on the matter see [346].40

Assessing Stevin’s contribution to the history of mechanics is not simple becausehis ideas were originally written in Dutch and then read by few. When they weretranslated into Latin (1605) and French (1636) the state of mechanics was alreadychanged. He is indeed, in any case, the founder of statics in the modern sense. Thename statics is in the title of his major work in mechanics Tomus quartus mathe-maticorum hypomnematum de statica, at least in the Latin version. And although hedefines statics as the science of weights:

Definition I. Statics41 its the science of the reasons, proportions, qualities and heaviness ofheavy bodies [215].42 (A.7.27)

in fact he often introduces forces applied by ropes that can be tightened by weightsor by human hands (muscle force).

The Tomus quartus is divided into five books, plus an Appendix and some Addi-tions to the Dutch edition of 1586. The approach is of Euclidean type, in the sense

40 p. 94.41 In the Dutch text instead of ‘statica’ there is written ‘art of weighing’ [218], p. 97.42 p. 5.


that for every book there is a different topic; first there are definitions, then postulatesand finally theorems, that are linked together.

In the first part of the first book Stevin demonstrates the law of the lever, with anargument similar to that used by Galileo in Lemecaniche. Starting from a continuousprismatic body with geometric considerations in the wake of Archimedes he findsthe law of inverse proportionality between weight and arm length. In the second partof the same book Stevin gives his famous demonstration of the law of the inclinedplane, determining the value of the force parallel to the slope enough to maintaina heavy body in balance. Stevin extends his result to the case where the upliftingforce is not parallel to the inclined plane. Gilles Personne de Roberval (see previoussections) found Stevin’s proof not satisfactory and gave a much more convincingproof; in a subsequent section I will discuss the legality of Stevin’s extension.

Based on the law of the inclined plane generalized to a force of any direction, witha rather complex argument that is developed with many theorems and corollaries,Stevin puts the groundwork for the proof of the rule of the parallelogram of forceswhich is satisfactory if the generalized law for the inclined plane is accepted.

The second book of the Tomus quartus regards the evaluation of the centres ofgravity of plane and solid figures, and it is definitely less interesting. The third bookis on practical statics in which lifting of bodies more complex than those treated inthe first two books is considered. The fourth and fifth books are dedicated to hydro-statics. They are fundamental texts on the subject that however I do not commenton because they are not related to the subject of my work. The Appendix containsvarious comments, including perhaps the most interesting about the criticism of theprinciple of virtual velocities to be discussed below.

In the Additions Stevin considers and devises demonstrations for pulleys, andtreats with some generality the case of forces applied by means of ropes in a sectioncalled spartostatica. In this section statics has already became the science of equi-librium of force and no longer of weights. It contains the wording of the rule of theparallelogram which is a rule of composition of forces, even though it is presentedas a way to determine the tension of two ropes which sustain a weight [215]. Thischange of attitude is a fundamental Stevin’s contribution to modern statics, and itdoes not matter if the rule of composition of forces is given an imperfect proof; it ishowever a rule which works. In the final part of the spartostatica Stevin considers forthe first time fundamental arguments that can be conceived only in the new frameof reference, i.e. the funicular polygon of forces, the weight sustained by more thantwo ropes in the plane, and the non-coplanar ropes.

The reading of Stevin’s mechanical work offers a much more modern view thanthat of Guidobaldo dal Monte (1577) [86] and Galileo (1594) [119]. The approachof Archimedean kind is equally rigorous, but less verbose. Unlike Galileo, Stevindoes not bother to set up statics on a single principle, that of the lever. He usesthe Archimedean geometric proof for the lever, but then he relies on the law of theinclined plane using an empirical principle, then in part still controversial, the im-possibility of perpetual motion.

Stevin among other things, is among the first to realize that the centre of gravityof a heavy body is not unique if one admits that the lines of descent of bodies are


converging toward the centre of the world. He first shows that since the actions ofgravity converge toward the centre of the world, the centre of gravity in the sense ofPappus and Commandino cannot exist for a body other than the sphere:

From this it follows there is no body in Nature, speaking mathematically, other than thesphere which can be suspended from its centre of gravity and maintains any position. Orsuch that the plane passing from it splits the body in equally weighting parts. But for the var-ious and infinite configurations there will be various and infinite centres of gravity [215].43

(A.7.28)

On the other hand, however, he argues that this is only a theoretic conclusion andin practice, because the actions of gravity differ by a very small angle, the centre ofgravity determined with the Archimedean rules meets the demand of Pappus to bethe point of suspension of neutral equilibrium.

But this difference is not observable for the practice of men and the beam should be somemiles long because it can be detected. So we postulate that the verticals be parallel eachother [emphasis added] [215].44 (A.7.29)

A CE

B

D

G

M

L

H

O

F

K

N

Fig. 7.15. The centre of gravity of a body

7.2.1.1 The rule of the parallelogram of forces

Demonstration of the rule of the parallelogram for composition of forces was car-ried out by Stevin with a long series of theorems and corollaries (about twenty) thatleave the modern reader a little upset . Also because the demonstration of each the-orem and corollary is carried out with rather slender mathematical reasonings, veryclose to the modern sensibility, it is difficult to understand the reason for Stevin’sprolixity. A part of this difficulty might be overcome by assuming that Stevin’s ob-jective originally was not to formulate the rule of composition of forces, of which

43 p. 11.44 p. 11.


Plate 3. A Latin and a Dutch edition of Stevin’s books on mechanics (reproduced with permission,respectively, of Biblioteca Nazionale Centrale, Rome, and of Max Planck Institute for the Historyof Science, Berlin)

perhaps he did not understand the full extent, but only to make a series of commentson the way weights can be lifted. In fact, the explicit formulation of the rule of theparallelogram is in the section of the Additions named spartostatica.

Below I refer with some detail to Stevin’s demonstration, although it in no wayaffects the laws of virtual work. This for two reasons: the first to illustrate the difficul-ties of the proof of the composition of the forces, a rule that was and still is alternativeto laws of virtual work. The second reason is that normally Stevin’s demonstrationis not reported so faithfully in the textbooks of history of science, perhaps becauseit is too complex.

The starting point is the law of the inclined plane. For reasons that will appearclear later he refers to a prism that is moved along an inclined plane as shown inFig. 7.16. In corollary V to the law of the inclined plane reported in the second halfof the first book [216]45, it is easy for Stevin to show that the ratio between theweightM of the prism, i.e. the force to lift it, called the direct uplifting force, and theforce E needed to move it on the inclined plane, called the oblique uplifting force,is equal to the ratio of the segments LD and DI identified by the intersection of theropes with the prism (because M : E = AB : BC = LD : DI).

45 p. 36.


LH

M D

A C N

P

E

F

BO

Q

I

Fig. 7.16. Equilibrium of a prism over an inclined plane

In corollary VI Stevin considers a horizontal uplifting force measured by weightP in Fig. 7.16. Imagining a rotation of ninety degrees, the horizontal uplifting forcebecomes vertical and the plane ABC turns into a tilted plane whose slope is as NBof the triangle NCB. Following this rotation the ratio between direct and obliqueuplifting forces is equal to that between the segments DO and DI. Stevin believesthat this relationship is maintained even when the rope carrying the load P is effec-tively horizontal. At this point, he can say that in the vertical, in the inclined and inthe horizontal directions, the values of the forces necessary to keep the prism in bal-ance are proportional to the length of the segments DL, DI, DO, intercepted by theropes on the prism, to conclude (improperly) that this fact applies to all directions.Stevin’s argument is interesting only for its strong rhetorical value, at least for thegeneralization to the case of any direction. The belief of the reader is made possibleby the choice of a prism as the body to be lifted. It should be stressed however, thateven if the reasoning cannot convince the result is correct.

Below Stevin’s proof of corollary VI follows, to allow the reader to judge thelawfulness of the reasoning:

Let BN be conducted cutting AC and extended to N, and the same DO cutting in O theextension of LI, so that the angle IDO is equal to the angle CBN, and then let the upliftingforce P be applied along DO, taking the column in its position (with weights M and Ebalanced); then as LD is homologous to BA in the triangle BAC and DI with BC, it followsthat BA is to BC as the weight on BA is to the weight on BC, by the second corollary. Andalso DL is to DI as the weight belonging to DL is to that to DI, i.e.M to E. Similarly the threelines LD, DI, DO being homologous to the three segments AB, BC, BN, then BA is to BNas the weights that belong to them, and also LD to DO will be like the weights that belongto them, i.e. M to P. Because this proportion is invalid not only at that elevation where DIis perpendicular to the axis, but for all sorts of angles [216].46 (A.7.30)

Stevin continues his argument with corollary VIII, which states that the relationsfound for the prism that moves on the inclined plane remain valid if the constraint

46 pp. 36–37.


LH

M D

G

F

P

O

Fig. 7.17. Prism supported by a fixed point

of the inclined plane is replaced with that provided by a fixed point, as shown inFig. 7.17.

Even in this case the relationship between the segments intercepted by the variousropes that support the cylinder is proportional to the forces necessary to balance thecylinder. In particular in the case of Fig. 7.17 the ratio between DL and DO is equalto the ratio of the direct and horizontal uplifting. Stevin does not pause to justify thelawfulness of the replacement of the inclined plane with the fixed point G. Readingbetween the lines it can be understood that, because for every inclination of the ropethe intercept with the side of the prism provides the force necessary to maintain theequilibrium whichever is the inclination of the inclined plane, the inclined planecan be replaced with a constraint that performs its essential function, i.e. to offera support to the prism. The result of Stevin, namely the determination of the forcenecessary to support the prism constrained to a fixed point, could have been extendedquite easily to the case of a body of any shape to get a rule of equilibrium as efficientas the vanishing of the static moments. But Stevin does not do it.

The next step, basically the definitive one, consists in the analysis of the situationof Fig. 7.18 for which Stevin states the following theorem XVIII:

If a column is maintained in equilibrium by two oblique uplifting forces as the line of theoblique uplifting force is to the line of the direct uplifting force, so each oblique upliftingforce is to its direct uplifting force [216].47 (A.7.31)

Notice that if points E and F have the same distance from the centre of gravity ofthe prism the vertical uplifting force I and K will be the same, so LE and FM havethe ratio of G and H. From this theorem, of which I do not give a demonstration, itis very easy to arrive at the parallelogram rule. Stevin does this in the Additions.

To get the rule of the parallelogram from theorem XVIII it suffices to considerthe case where the two points E and F of Fig. 17.18 coincide with each other andwith the centroid as shown in Fig. 7.19a. In this case it can be affirmed that theproportion between segments CI, DC, CE is the same as the direct, and inclinedforces (corollary III of the Additions); but this is the rule of the parallelogram. The

47 p. 48.


E

F

NL I K

O

M

AC

BD

H

G

Fig. 7.18. The prism sustained by two ropes

E

I

C

DD

A

a) b)B

EI

C

G

HA

B

Fig. 7.19. The law of the parallelogram

proof is perfected by translating downward the prism as in Fig. 7.19b, which doesnot change the value of uplifting inclined forces (corollary 4) and finally replacedwith a weight of any shape (corollary 5).

7.2.1.2 The law of virtual work

On Stevin and the law of virtual work different opinions have been reported, somelike those of Mach and Duhem make him one of the most modern supporters, otherslike Dijksterhuis deny this claim.

Stevin’s considerations on the law of virtual work are listed in the Appendix andAddition of his Tomus quartus de mathematicorum hypomnematum de statica. Inchapter I of the appendix, he writes:

The cause of the equilibrium of the lever, as the chapter title says, does not lie in the arcs of acircle described by its ends. Common sense is enough to prove that equal weights suspendedat equal distances are in equilibrium with a lever. But to say that different weights suspendedat different distances are in equilibrium when these weights are in inverse proportion to thedistances from which they are suspended, does not seem so obvious. The ancients thoughtthat the reason for this was in the arcs described by the end of the lever. This view can be


found in theMechanica of Aristotle and the work of his followers.Wewill try the inaccuracyof this view as follows:A That what is in equilibrium does not describe a circle,E two weights in equilibrium are motionless,A so two weights in equilibrium do not describe a circle, So there is no circle.Once one deletes the circle, the cause that should reside in it also disappears. So the causeof the equilibrium cannot be found in the circle [215].48 (A.7.32)

Stevin uses, probably ironically, the syllogistic notation of medieval treatises onlogic, directly derived from Aristotle. The letter A indicates the universal positiveproposition (it is the first vowel of the latin verb adfirmo), the letter E indicates theuniversal negative proposition (it is the first vowel of the verb nego). Stevin’s reason-ing is very natural, because the concept of virtual motion in a situation of equilibriumis far from intuitive. Essentially then Stevin denies that the laws of virtual velocitieshave explanatory value in mechanics. Indeed in his mechanical theory he does notmake any use of them: a large part of Stevin’s mechanics is based on the theory ofcentres of gravity, another part on the theory of forces.

Although Stevin declares his opposition to the principle of virtual velocities forwhich the equilibrium of a body depends on its possible motion, in at least one im-portant situation he seems to contradict himself.

In the proof of the law of the inclined plane Stevin considered a chain that wrapsaround it, as shown in Fig. 7.20. Stevin claims that the chain must be in balance in agiven configuration otherwise, because the relative configuration of the chain cannotchange, if it is not equilibrated in one configuration it is not equilibrated in any otherconfiguration, then would occur perpetual motion, which is impossible:

It is not possible that a given motion has not end [215].49 (A.7.33)

The law of the inclined plane was immediately followed by a comparison of weightsof the chain that rely on the two opposing inclined planes (see Fig. 7.20).

Fig. 7.20. The chain of spheres on an inclined plane (reproduced with permission of BibliotecaNazionale Centrale, Rome)

48 p. 151.49 p. 35.


Notice that Stevin considers as unproblematic negating the perpetual motion, anddoes not assume it explicitly as a principle of statics though it is as fundamental forhis mechanics at least as the law of the lever. The simple justification is that probablyStevin did not want his book to appear too new by introducing at the beginning anon-standard statement. Note also that the perpetual motion of which Stevin speaks,although he is not precise in this regard, is not the inertial motion, ideally possible,the physical perpetual motion of Leibniz [158],50 but a perpetual motion where onecan always get work, the mechanical perpetual motion [158],51 actually impossible.So the reasoning of Stevin seems safe from such criticisms made against him byDijksterhuis [292].52

Stevin returns to the subject in the Addition, where he speaks of Trochleostatica,and shows that for a simple system of pulleys as in Fig. 7.21 the power F is half theweight B because B is carried by two ropes:

Proposition. To search the quality of lifting of weights with the pulley.Before starting to speak of the subject, we will say in general that when we speak of a givenweight, we will assume a weight suspended from the lower pulley; regarding the weight ofthe rope we will neglect it. Examen of weight raised along a straight line: Let A in the firstfigure [Fig. 7.21] be a pulley, from which the weight B is suspended, and the rope CDEF,the part FE and CF of which are parallel and vertical. This posited, the weight B will besustained equally by the two parts EF and CD, because the pulley acts equally on both. Soif one would sustain the weight B with his hand in F , by keeping the weight in this position,he will sustain one half of B. From this it results that it is easier to lift a weight with a pulleythan without it [215].53 (A.7.34)

Fig. 7.21. The pulley (reproduced with permission of Max Planck Institute for the History of Sci-ence, Berlin)

50 p. 472.51 p. 472.52 IV, p. 65.53 p. 171.


And he comments:

The following axiom of statics is valid: as the space of the agent to the space of the patient,so the power of patient to the power of agent [215].54 (A.7.35)

Here Stevin states unambiguously a law of virtual work. His remarks, according toMach, are more mature than these purely geometric statements of Guidobaldo dalMonte, reported in the previous chapter [355].55 So to say that Stevin had a role inthe development of the law of virtual work or to deny it, it’s just a matter of wording.The law of virtual work for Stevin is not a foundational principle of mechanics, buta theorem, surely relevant.

Stevin with his consistency in considering only motion and not speed, will con-tribute to the development of the principle of virtual displacements, operationallyequivalent but conceptually not to that of virtual velocities. In this way it is eas-ier to separate the law of virtual work from Aristotelian dynamics, destined to beabandoned. Descartes and Wallis are the natural successors of Stevin in wanting todeny the dignity of an equilibrium law based on virtual velocity and to develop theapproach of virtual displacements.

7.2.2 Christiaan Huygens

Christiaan Huygens was born in Den Haag in 1629, anddied in Den Haag in 1695. He generally wrote his name asHugens, but I follow the usual custom in spelling it as Huy-gens. He was probably, with Newton, the greatest scientistof the XVII century. The most important of Huygens’s workwas his Horologium oscillatorium published in 1673. Theincreasing intolerance of the Catholics led him to remainin Nederland where he devoted himself to the constructionof lenses of enormous focal length: three of these of focal

lengths were 123 feet, 180 feet, and 210 feet. In 1690 Huygens published his trea-tise on light in which the undulatory theory was expounded and explained [384]. Itmust be added that almost all his demonstrations, like those of Newton, are rigidlygeometrical, and he would seem to have made no use of the differential or fluxionalcalculus, though he admitted the validity of the methods used therein.

Huygens in his vast production also used some form of law of virtual work instatics. The originality and the importance of his contribution lies in the fact he in-troduced infinitesimal virtual displacements and forces. He gave his thoughts onsome papers joined together today in his Oeuvres, in the chapter called Spartista-tique [135].56 Is doubtful that the writings of Huygens were known by contempo-raries, and then it may not have had any impact on developments of the law of virtualwork, especially on Bernoulli. His considerations are in any case important, as they

54 pp. 171–172.55 p. 49.56 vol. 6.


Fig. 7.22. Equilibrium of ropes (reproduced with permission of Biblioteca Guido Castelnuovo,Università La Sapienza, Rome)

witness the level of maturation of the ideas on the law of virtual work at the end ofthe XVII century.

The first Huygens’s application refers to three ropes converging together into aknot, illustrated in Fig. 7.22 on the left. He is looking for a relationship between theforces in the ropes in the state of equilibrium. The text of Huygens is laconic, just anote, in which Latin and French are mixed. To notice the adoption of an algebraiclanguage see [135].57

Let the node be in E. And PE be shortened by DEQE by AESE elongated by BE

Then it is necessary DE in p + AE in q - BE in t = 0. (A.7.36)

In the final equation ‘in’ means multiplication, p,q, and t are the forces that pullthe ropes P, Q and S, while DE, AE, BE are the respective variations of lengths. Inmodern terms, the equation expresses the vanishing of virtual work.

The second application, more complex, refers to the equilibrium of the four ropesPRST shown in Fig. 7.22 on the right. Huygens had to determine the forces prst thatpull the ropes, so that they are balanced. He considered separately four infinitesimalvirtual displacements, each perpendicular to a rope. He then writes the virtual workdepending on the other three ropes. In modern terms, his calculations are equivalentto the projection of three forces, one at a time on the straight lines orthogonal to thefourth [135].58

CA in P - CD in R -CO in S = 0

sin TEP in P – sin TER in R - sin TES in S = 0sin PER in R – sin PES in S - sin PET in T = 0sin RES in S – sin RET in T - sin REP in P = 0sin SET in T + sin SEP in P - sin SER in R = 0

57 vol. 19, p. 51.58 vol. 19, p. 52.

7.3 British statics 189

on

p =f r− ct

er =

es+atd

s =ap−br

et =

dp− f ttb

.

(A.7.37)

In the above equations ‘in’ again means multiplication, ‘sin’ is for sinus, the vir-tual displacement is simplified. Note that the forces p, r, s, t issued by the last fourequations are not exactly univocally determined, but only the one over the others. Inmodern terms it can be said that the system is statically indeterminate and admits asimple infinity of solutions.

Fig. 7.23 shows, for clarity, the case of virtual displacements orthogonal to thedirections T and P, corresponding to the first two equations of virtual work, that withthe symbols of the figure become:

psinα− r sinβ− ssinγ= 0

r sinδ− ssinε− t sinα= 0.

R

SE

T

a) b)

αγ

β

P

pt

s s

r r

α

R

S

E

T

− δπ2

εP

A

Fig. 7.23. Equilibrium of four ropes

7.3 British statics

Also in England, as in France, the XVII century saw a revival of the sciences ingeneral and the exact ones in particular. Not that there had been no great British sci-entists – William Harvey (1578–1657) and William Gilbert (1544–1603), to namejust two – but these were sporadic cases. The real flowering of British science startedfrom 1640, with the beginning of the Puritan Revolution, until the restoration of 1658[395, 361, 338]. For Webster [395] reference should be made at least to 1626, theyear in which the rise of the Puritan movement started, among other things, coin-ciding with the year of the death of Francis Bacon. Not all historians however agreein attributing a close connection between the Puritan movement and scientific de-velopment; some argue that the fact that British science developed during and after


the Puritan Revolution is just a coincidence. But in fact, this coincidence exists andthe conception of science and mathematics of the Puritan movement was certainlyfavourable to its development.

The following list of scientists, to which should be added Isaac Newton, suf-ficiently show the impressive way British science grew in the second half of theXVII century: John Wallis (1616–1703), Robert Boyle (1627–1691), Isaac Barrow(1630–1677), Christopher Wren (1632–1723), Robert Hooke (1635–1703). With re-spect to statics the only direct contribution came from John Wallis who, from a cer-tain point of view, can be seen as the last important representative of the ancientstatics, where the lifting of weights was still an important aspect. Newton made acontribution mainly from a methodological, or philosophical, point of view by sub-ordinating statics to dynamics.

7.3.1 John Wallis

John Wallis was born in Ashford in 1616 and died in Oxfordin 1703. He learned Latin, Greek, Hebrew, logic, and arith-metic during his early school years. In 1610 he received thedegree of master of art and was ordained a priest; shortly af-terward he exhibited his skill in mathematics by decipheringa number of cryptic messages from Royalist partisans thathad fallen into the hands of the Parliamentarians. In 1645Wallis moved to London. His appointment in 1649 as Savil-ian professor of geometry at the university of Oxford marked

the beginning of intense mathematical activity that lasted almost uninterruptedly tohis death. He also discovered methods of solving equations of degree four. Wal-lis contributed substantially to the origins of Calculus and was the most influentialEnglish mathematician before Newton. He studied the works of Kepler, Cavalieri,Cardano, Roberval, Torricelli and Descartes, and then introduced ideas of differ-ential analysis going beyond these authors [290]. Wallis’s most famous work wasArithmetica infinitorum which he published in 1656.

In 1670 he wrote an important treatise on mechanics, Mechanica sive de motu[245] where he made important contributions to mechanics in general and staticsin particular. Wallis’ exposition is axiomatic deductive, based on definitions (verydetailed), principles (not clearly marked) and theorems, as is typical of most mathe-maticians; the influence of Descartes is evident. The book is not easy to read, how-ever it had a good success among scientists of the XVII century.

Wallis considers gravity to be a force as the others, directed toward the centreof the earth. But he makes no comment on its causes, for which he proposes a fewhypotheses:

Gravity is the motive force, i.e. toward the centre of the Earth.Here we do not discuss what is the principle of the Gravity from a physical point of view,or which quality it has, or passion of the body, or with whatever name it could be called.Either it is innate in the body, or comes from the common tendency toward the centre of theEarth, or from an electric exhalation which attracts like chains, of from something else (of


what it is not here the case to speak about). It is enough that with Gravity we intend whatwe know from senses. The force which moves downward either for the heavy body itself orfor the less constraint toward the centre of the Earth.With weight I mean the measure of Gravity [245].59 (A.7.38)

One of the founding points of Wallis’ statics is the idea that the relationships ofequilibrium occurring between weights also apply to ordinary forces, with appropri-ate adjustments. Every time he establishes a theorem or a definition for weights, herepeats it for ordinary forces:

Prop. IHeavy bodies gravitate according to their weight. And in general, motive forces act accord-ing to the law of forces [245].60 (A.7.39)

Prop. IIHeavy bodies, unless constrained, descend, or get closer to the centre of the Earth. Andin general any motive force, [moves] according to its direction, if there are not constraints[245].61 (A.7.40)

Wallis calls Descensus et Ascensus the virtual descent and ascent of heavy bodies;for forces he introduces the terms Progressus for displacement in the direction of theforce and Regressus for displacement in the opposite direction:

Prop. IIIFor Heavy bodies Descensus is greater when [the body] becomes closer to the centre ofEarth, Ascensus when it becomes farther. And in general, Progressus of the motive force isgreater if [the body] moves according to its direction, and inversely for Regressus [245].62

(A.7.41)

But Descensus and Ascensus are not measured only by motion but also by weight,in the sense that they are proportional to them. And this holds also for Progressusand Regressus:

Prop. VDescensus of Heavy bodies, compared among them, is proportional to the ratio of theirweights and the value of descent. The same for the Ascensus. This is so if the weights areequal, [are proportional to] the ratio of the values of displacement, and if the displacementsare equal to the ratio of weights. If weights and displacements are equal, or are in inverse pro-portion, [Ascensus or Descensus] are equivalent. And in general for the motive forces. Pro-gressus and Regressus are proportional to the ratio of forces and to the regress and progressaccording to the line of action [245].63 (A.7.42)

With these definitions Wallis is able to introduce his law of virtual work, where therole of work is played by Ascensus, Descensus, Progressus and Regressus: a bodyis equilibrated if in a virtual motion the Ascensus (Regressus) and the Descensus(Progressus) are equal.

59 Chapter 1, pp. 3–4.60 Chapter 2, p. 33.61 Chapter 2, p. 33.62 Chapter 2, p. 34.63 Chapter 2, p. 37.


In an aggregate of bodies Wallis’ virtual law says that there is equilibrium if thesum of Ascensus equals that of Descensus.

Prop. VIDescensus and Ascensus of aggregates: if the Descensus is prevalent, simply it occurs adownward motion; otherwise if the Ascensus prevails an upward motion [occurs]. If theyare equal, there is no motion.When many ascending or descending bodies are joined it is their summation which is rele-vant [245].64 (A.7.43)

The above is exemplified in some cases in which Wallis introduces the algebraicnotation of Ascensus and Descensus, as the product of weight and displacement.With his symbols:

For example […], compare the Descensus of a weight 2P for the displacement 3D, with theDescensus of a weight 3P for the displacement 2D: they are equivalent (because 2× 3 =3× 2), in a virtual motion, so they are balanced. But the Descensus of a weight 2P for adisplacement 4D is prevalent aver the Descensus of a weight 3P for a displacement 2D(because 2×4 > 3×2) it in a virtual motion; then it prevails [245].65 (A.7.44)

Basically Wallis generalizes the law of Descartes, or if one wants Torricelli’s prin-ciple, from the case of two weights to the case of n weights or n forces. Using analgebraic language, as Wallis does although in an embryonic form, for n forces fiand n motions ui, his results are summarized in the following relation:

n

∑i=1

fiui = 0 (7.1)

where the signs are positive in the case of Progressus, negative in the case of Re-gressus.

The above is valid for constant forces that move always parallel to themselves.For displacements along curved paths, however,Wallis suggests the solution alreadyproposed by Descartes. One should consider motions in the direction of the tangentto the curve on which the heavy body moves.

Prop. XVThe slope of the descent of a curve line in a point is given by the tangent, and for a surfaceby the tangent plane [245].66 (A.7.45)

64 Chapter 2, p. 38.65 Chapter 2, p. 39.66 Chapter 2, p. 47.


7.3.2 Isaac Nevton

Isaac Newton was born in Woolsthorpe, near Grantham in1642 and died in London in 1727. Mathematician and physi-cist, one of the foremost scientific intellects of all time. Lu-casian professor of mathematics in 1669 at the university ofCambridge [290]. His most famous book is the Philosophiaenaturalis principia mathematica of 1687 [175]. Newton hasbeen regarded for almost three hundred years as the founderof modern physical science, his achievements in experimen-tal investigation being as innovative as those in mathemati-

cal research. With equal, if not greater, energy and originality he also plunged intochemistry, the early history of Western civilization, and theology.

For Newton, statics is a special case of dynamics. And it is a trivial theoremof mathematical analysis to prove that a material point is in equilibrium when theresultant of its forces is zero. To check the balance it is therefore enough to disposeof a rule of composition of forces. This is provided by the rule of the parallelogram,which is a corollary to the second law of motion:

Corollary IA body by two forces conjoined will describe the diagonal of a parallelogram, in the sametime thai it would describe the sides, by those forces apart [176].67

The study of equilibrium of a constrained material point is more complicated. Notcovered in the Principia, it is customary to solve it with the assumption that the con-straints exert reactive forces. One thus has the criteria of balance set out in Chapter 2of this text

It is clear even from these observations that Newtonian statics is quite differentfrom that based on laws of virtual work. The differences are epistemological, on-tological, mathematical. But there is a contact point, the idea that the equilibriumis a dynamic concept, a balance of tendencies contrary to the motion. To Newtonthe force expresses a tendency to motion; this trend, however, is not evaluated byobserving motion, it is estimated before. To the contrary the virtual velocity, whichis required in the application of the laws of virtual work, cannot be measured beforethe motion is imagined.

Newton however was never disconnected from the idea that the tendency to mo-tion and thus the force is measured by velocity, virtual or real. In the introductorypart of the scholium to the Principia, Newton refers to the law of virtual veloci-ties, presenting it as a special case of his third law of motion, I must say somewhatdisconcerting to a modern reader:

And as those bodies are equipollent in the congress and reflexion, whose velocities are recip-rocally as their innate force, so in the use of mechanic instruments those agents are equipol-lent, andmutually sustain each the contrary pressure of the other, whose velocities, estimatedaccording to the determination of the forces, are reciprocally as the forces [176].68

67 p. 84.68 p. 93.


A long passage follows in which Newton says more or less explicitly that forcesand weights are more or less effective depending on their velocity. So there is adifference in the effectiveness and the force that Newton introduced in technicalsense, as magnitude measured by the acceleration it imposes. To this efficacy onewould be tempted to give the name of work, or better power (product of force forspeed), but Newton does not it.

So those weights are of equal force to move the arms of a balance; which during the playof the balance are reciprocally as their velocities upwards and downwards; that is, if theascent or descent is direct, those weights are of equal force, which are reciprocally as thedistances of the points at which they are suspended from the axis of the balance; but if theyare turned aside by the interposition of oblique planes, or other obstacles, andmade to ascendor descend obliquely, those bodies will be equipollent, which are reciprocally as the heightsof their ascent and descent taken according to the perpendicular; and that on account of thedetermination of gravity downwards. And in like manner in the pulley, or in a combinationof pulleys, the force of a hand drawing the rope directly, which is to the weight, whetherascending directly or obliquely, as the velocity of the perpendicular ascent of the weight tothe velocity of the hand that draws the rope, will sustain the weight.The force of the screw to press a body […]. The form by which the wedge presses or drivesthe two parts […]. The power and use of mechanics consist only in this, that by diminishingthe velocity we may augment the force, and the contrary: from whence in all sorts of propermachines, we have the solution of this problem; Tomove a given weight with a given power,or with a given force to overcome any other given resistance. For if machines are so contrivedthat the velocities of the agent and resistant are reciprocally as their forces, the agent willjust sustain the resistant, but with a greater disparity of velocity will overcome it […] But totreat of mechanics is not my present business. I was only willing to show by those examplesthe great extent and certainty of the third Law of motion [emphasis added]. For if we estimatethe action of the agent from its force and velocity conjunctly, and likewise the reaction ofthe impediment conjunctly from the velocities of its several parts, and from the forces ofresistance arising from the attrition, cohesion, weight, and acceleration of those parts, theaction and reaction in the use of all sorts of machines will be found always equal to oneanother. And so far as the action is propagated by the intervening instruments, and at lastimpressed upon the resisting body, the ultimate determination of the action will be alwayscontrary to the determination of the reaction [176].69

69 p. 94.

8

The principle of virtual velocities

Abstract.This chapter is almost entirely devoted to Johann Bernoulli, who considersthe equilibrium for a set of infinitesimal displacements and forces. In the first partthe different conceptions of the force of the XVIII century, including those of deadand living forces, are summarized. In the central part Bernoulli’s VWL is presentedwhich enforces equality of positive and negative energies, the energy being definedas the scalar product of the force by the displacement of its application point, namedvirtual velocity. He offers a number of applications to the various cases includingthe fluid but does not provide any demonstration. In the final part a comparison ofVarignon’s mechanics, based on composition of forces and Bernoulli’s mechanicsbased on his VWL is considered.

8.1 The concept of force in the XVIII century

At the beginning of the XVIII century the concept of force had not yet a shared status.There was the static force measured by weight, there was the confusion concept ofNewtonian force, there was the Cartesian concept associated to bodies in motionand the Leibnizian concept of living and dead forces [340, 304]. Before consideringwith some detail Bernoulli’s concept of force, the Newtonian and Leibnizian onesare presented.

8.1.1 Newtonian concept of force

Newton assumed the following principles of mechanics which he referred to as laws,probably to emphasize that he considered them of experimental nature:

Law I. Every body perseveres on its state of rest, or of uniform motion in a right line, unlessit is compelled to change that state by forces impressed thereon.Law II. The alteration of motion is ever proportional to the motive force impressed; and ismade in the direction of the right line in which that force is impressed.Law III. To every action there is always opposed an equal reaction: or the mutual actions oftwo bodies upon each other are always equal, and directed to contrary parts [176].1

1 p. 83.


196 8 The principle of virtual velocities

These laws are quite familiar to a modern reader even though some particularity bothformal and substantial does not escape, mainly for Law II. First there is nowhere thefamous formula f = ma, commonly known as the second law of Newton, or betteryet, no formula is referred to.Mass is not named explicitly but it is absorbed in [quan-tity of] motion; in the end no reference is made to acceleration. A scrutiny shows thatalso the impressed force, apparently the only familiar element in the second law, can-not be identified with the modern concept of force. Indeed, the integration of the lawof motion, considered in modern sense as f = ma, over a finite interval T of timegives: ∫ T

0f dt = m

∫ T

0adt = mΔv, (8.1)

where the second part is the variation of the quantity of motion, or according toNewton’s terminology, the “alteration of motion”. Comparison of the analytical ex-pression just obtained with the Law II of motion, shows that what Newton calls forcemust be equal to

∫ T0 f dt.

Newton chose the use of the word force to indicate a founding quantity of dynam-ics, but he did not reconnect it to any of the concepts today named in the same way.Newton’s force is likened to the whole force introduced by previous scientists suchas Descartes or Torricelli. This concept today is scarcely used and anyway is notreferred to with this name; the most common name for it is the impulse of the forcef . In the scholium which follows the three laws of motion, Newton said verbatimabout the force of gravity considered as an example of a force acting continuously:

When a body is falling, the uniform force of its gravity acting equally, impresses, in equallyparticle of time, equal forces upon that body, and therefore generates equal velocity; and inthe whole time impresses a whole velocity proportional to the time [176].2

That is the whole variation of velocity is proportional to the whole force, which isproportional to time. In the Principia the whole force can also represent the intensityof a pulse, and the action of continuum force, as the gravity for instance, is describedusually as a sequence of pulses, divided by a constant time step Δt, which in the limitturns to zero.

Regarding the ontology of force, Newton was quite ambiguous. He introducedforce as a dominating concept together with that of absolute space and time, at thebeginning of the Principia:

Definition IVAn impressed force is the action exerted upon a body to change its state, either of rest, or ofmoving uniformly forward in a right lineThis force consists in the action only; and remains no longer in the body, when the ac-tion is over. For a body maintains every new state it acquires by its vis inertiae only. Im-pressed force are of different origins as from percussion, from pressure, from centripetalforce [176].3

2 p. 89.3 p. 74.

8.1 The concept of force in the XVIII century 197

So the impressed force is what different kinds of ‘forces’ have in common from amechanical point of view. Here it seems that Newton wanted to say that a force canbe measured only by its effect.

8.1.2 Leibnizian concept of force

Leibniz considered two types of force, the dead force and the living force, whichare somehow related to our concept of force and kinetic energy respectively. Theseconcepts cannot be understood without the comprehension of that of conatus whichin Leibniz derived from Hobbes. The Hobbesian conatus is defined as the motionmade in the shortest possible time and space; its use is associated not always to mo-tion but also to the efficient cause of any change. Leibniz did not maintain a choerentposition; at the beginning he assumed conatus, according to the theory of Cavalieri’sindivisibles, as the distance traveled in an indivisible element of time; thereafter theindivisible became an infinitesimal. In the Specimen dynamicum, he gave the fol-lowing definitions: “Velocity taken together with direction is called conatus, whileimpetus is the product of the mass (moles) of a body and its velocity” [159].4 Thesentence is consistent with previous formulations only if ‘velocity’ is the infinitesi-mal velocity dv.

Already in a letter dated 1673, to Edme Mariotte, Leibniz used the terms forcemort and force violent ou aimeé. In the Essay de dynamique he used the term visviva, as opposed to vis mortua. The following passage of the Specimen dynamicumprovides explanations in terms of non-quantitative relationship between dead andliving forces:

Hence force is twofold: the one elementary, which I call also dead, because motion (motus)does not yet exist in it, but only a solicitation to motion (solicitatio ad motum), such as thatof the ball in the tube, or of the stone in the sling, even while it is held still by that chain;the other however, is the ordinary force, united with actual motion, which I call living. Andan example of dead force indeed is the centrifugal force itself, and likewise the force ofgravity or centripetal force, the force also by which the tense elastic body (elastrum) beginsto restore itself. But in percussion, which arises from a heavy body falling already for sometime, or from a similar cause, the force is living force, which has arisen from an infinitenumber of continued impressions of dead force [159].5 (A.8.1)

The relation between dead and living forces is commented on also in an importantletter of Leibniz to Burchard de Volder (1643–1709) of 1699:

Consequently, in the case of a heavy body receiving an increase of speed equal and infinitelysmall at every moment of its fall, the dead and the living force can be calculated at thesame time. The speed increases uniformly with time but the absolute force as the square ofthe time, that is, as the effect. So according to the geometric analogy or our analysis, thesolicitations are as dx, the speed is as x, the forces as xx, or

∫xdx [160].6 (A.8.2)

In this letter, in the definition of dead and living forces, the mass of the body wasleft in the shadow and according to a use of the times was regarded as a constant of4 p. 237.5 p. 238. Translation in [161], p. 674.6 p. 156.


proportionality. By making explicit mass, the previous quotation says that the deadforce – actually Leibniz says solicitation – is proportional to the mass (m) multipliedby the infinitesimal speed (dv) and the living force is proportional to the mass mul-tiplied by the square of speed (v), i.e. mv2. Speed and force are linked by a simpleintegration

∫xdx, where time does not appear.

There is some disagreement among historians about the relationship betweendead and living force, simply because while living force is defined by Leibniz with amathematical expression (mv2), dead force is not. Some argue that the living force isthe integral of the dead force over an infinitesimal distance, for example René Dugas[308]7 believes that the relationship dead-living force expresses the theorem of liv-ing forces. A similar position was held by Ernst Cassirer. For them, where Leibnizseems to explicitly refer to integration in time, it would be an inaccurate languageand, for example when talking about a heavy body which has fallen for some time, tosimply report a qualitative description of the phenomenon. Other authors, includingWestfall [396], argue that Leibniz has not grasped the true link between dead andliving force. According to them, the natural Leibniz’s concept of variation would bewith respect to time (the monads evolve over time) and then he should integrate thedead force overtime and this would simply give speed and not its square.

What is certain is that Leibniz says in several places that the dead and the liv-ing force are in the same ratio as points to straight lines and then the dead force isinfinitesimal and the living force finite. They are related by a simple integration ordifferentiation:

The equilibrium consists of a simple effort (conatus) before the motion, and that is what Icall a dead force that has the same relationship as respects the living force (which consistsin the simple motion) as the point to the line. Now at the beginning of the descent, whenthe motion is small, the motion, the velocity or rather the elements of velocity are like thedescents, instead after the integration, when the force has become alive, the descents are asthe square of the velocity [158].8 (A.8.3)

Note that Leibniz does not say that the descent are as the speed, but that the elementsof speeds are as the descents and then, at the rising of motion, the elementary dis-placements are proportional to the elementary speeds. I know of no other passagesin which Leibniz presents the concept of dead and living force in a different way.In particular, there are no passages in which Leibniz ‘calculates’, or puts in rela-tion with an explicit formula dead and living forces, or simply gave an analyticalexpression of the dead force.

The concept of dead force should allow a direct link from statics to dynamics;of course the price is the acceptance of the metaphysical hypothesis of conservationof forces, according to which the dead force becomes the living force without anyloss, a hypothesis that is not very different from that adopted by Newton that a causeproduces its effect, by conserving in some way.

7 p. 211.8 p. 480.

8.2 Johann Bernoulli mechanics 199

8.2 Johann Bernoulli mechanics

Johann Bernoulli was born in Basel in 1667 and died in Baselin 1748. At Basel university Johann took courses in medicinebut he also studiedmathematics with his brother Jakob. Jakobwas lecturing on experimental physics at the university whenJohann entered the university; after two years of studyingtogether, Johann became the equal of his brother in mathe-matical skill. Johann Bernoulli’s first publication was on theprocess of fermentation in 1690, apparently not a mathemat-ical work, but it was really on an application of mathematics

to medicine, being on muscular movement. In 1692 Bernoulli met Pierre Varignon,who later became his disciple and close friend. This tie also resulted in a voluminouscorrespondence. In 1693 Bernoulli began his exchange of letters with Leibniz, whichwas to grow into the most extensive correspondence ever conducted by the latter. In1713 Bernoulli became involved in the Newton-Leibniz controversy on Calculus.He strongly supported Leibniz and added weight to the argument by showing thepower of his calculus in solving certain problems which Newton had failed to solvewith his methods. Although Bernoulli was essentially correct in his support of thesuperior calculus methods of Leibniz, he also supported Descartes’ vortex theoryover Newton’s theory of gravitation. This in fact delayed acceptance of Newton’sphysics on the Continent. Bernoulli also made important contributions to mechanicswith his work on living forces, which, not surprisingly, was another topic on whichmathematicians argued over for many years.

Johann Bernoulli attained great fame in his lifetime. He was elected a fellow ofthe academies of Paris, Berlin, London, St. Petersburg and Bologna. He was knownas the ‘Archimedes of his age’ and this is indeed inscribed on his tombstone [290].

8.2.1 Dead and living forces according to Bernoulli

Johann Bernoulli in the Discours sur les loix de la communication du mouvementdeclared to have adhered to the Leibnizian concepts since 1714 [35],9 but it is forsure that his ideas had not matured at the time. As it will be clear from the followingsections, in that period he reformulated the concept of dead force, replacing it withthat of energy. Dead force identifies the force in the usual meaning, while the energyof a force f is an infinitesimal pulse defined explicitly by the relation f dx, wheredx is the infinitesimal virtual displacement of the point of application of f in itsdirection. In static situations there is equilibrium when the energies of various forcesare balanced; in dynamic situations the energies add up to give the living force.

Bernoulli was the first to introduce an analytic relationship between living anddead forces (actually his energy). The following passage illustrates quite well theideas of Bernoulli:

9 p. 40. More precisely he says twenty eight years after the publication of Leibniz’s famous Brevisdemonstratio erroris memorabilis Cartesii [157].


B D H

NT

OK

MV

D L

P J A C G E F

Fig. 8.1. Dead and living forces

1. I suppose two whatever given straight lines AC, BD, which I assume to represent two setsof small equal springs and equally compressed. I suppose also that two equal balls start tomove from the points C and D toward F and J, when the springs begin to dilate. Let CML,DNK two curve lines, the ordinates GM and HN of them express the speed acquired at pointsG and H. I name BD = a , the abscissa DH = x, its differential HP, ou NT, = dx, the ordinateHN = v, its differential dv. I assume the abscissas CG and CE of the curve CML, such theyare to the abscissas of the curve DNK as AC is to BD, or, which is the same, I make BD :AC = DH : CG = DP : CE. Supposing AC = na, it will be CG = nx, GE = ndx; let also GM= z. All this supposed I reason in this way.2. When the balls arrive at points G and H, each spring, both that contained in the intervalAC and the interval BD, will be equally extended, because AC : CG = BD : DH. Each ofthese springs will have lost, on both sides, an equal portion of elasticity and each spring willmaintain the same elasticity. So the pressure or dead force [emphasis added] the balls havereceived are equal to each other. I name this pressure as p. But the elementary increasingof the speed in H, i.e. the differential TO or dv, is for the known law of acceleration, in acompound ratio of the motive force, or the pressure p, and of the little time the mobile takesto pass the differential HP, or dx, that can be expressed by HP : HN = dx : v. It will be thendv = [pdt =]pdx : v, or vdv = pdx, which by integration gives 1

2 vv =∫pdx. For the same

reason it is dz = p× GE: GM = p×ndx : z, or zdz = npdx and by integration 12 zz = n

∫pdx,

from which it follows vv : zz =∫pdx : n

∫pdx = 1 : n = a : na = BD : AC. But BD is to AC

as the living force acquired in H is to the living force acquired in G. Then the two forcesare to each other as vv to zz; so the living forces of bodies with equal mass are as the squareof their speeds, and the speeds themselves are as the square root of the living forces [35].10

(A.8.4)

Bernoulli starts from the metaphysical assumption: as in the cause so in the effect,translating it in the statement that the action of the pulse of the dead force-energybecomes living force. Each pulse of dead force-energy has the expression pdx; theirsummation is the integral

∫pdx which is shown to furnish the living force:∫

pdx ∝ mv2. (8.2)

It should be noted that after Johann Bernoulli the previous relation was generallytreated only as amathematical theorem, and not as a principle of conservation. This isthe case of Euler, d’Alembert and even perhaps of Lagrange. They, while somehowcould give a physical meaning to the second member, because the idea of livingforce was certainly familiar to them, did not seem to know how to deal with the firstmember, to which neither a name nor a mechanical meaning was given. Especially

10 pp. 46–47.


for the scientists of the XVIII century it had not the meaning of mechanical work,understood as a physical quantity that can be converted into various forms of energy.Bernoulli’s idea of conservation of mechanical energy will be resumed with successonly in the XIX century.

8.2.2 The rule of energies

It is more or less universally acknowledged that the wording now used for the lawsof virtual work has its core in the regle d’energie of Johann Bernoulli, better knownafter Lagrange as the principle of virtual velocities. Until a few years ago little wasknown about the origin of this principle. Even the date of its first statement wasreported incorrectly. In fact, in theNouvelle mécanique ou statique of 1725 by PierreVarignon a letter dated 1717 is reported which sets out Bernoulli’s famous principle.There’s actually a misprint and the correct date of the letter is February 26th, 1715[238].11

With the occasion of the new edition of the works of the Bernoullis [38] the corre-spondence of JohannBernoulli, of which only a fraction of the letters were published,has been reconsidered. From it Patricia Radelet de Grave, one of the curators, made amore complete reconstruction. In what follows, I draw inspiration from the study ofDe Grave to reconstruct the development of the concepts of Bernoulli, not so muchfrom the chronological point of view, but rather as an evolution of contents.12

All started with the publication of De la theorie de la manoeuvre des vaisseauxin 1689, by the naval engineer Bernard Renau d’Elizagaray [205] and from the criti-cisms about it by Christiaan Huygens. Johann Bernoulli joined in the discussion, ini-tially taking the side of Renau, then that of Huygens. The debate between Bernoulliand Renau is embodied in some letters written in 1713, published in 1714 as ad-dendum of the booklet Essay d’une nouvelle theorie de la manoeuvre des vaisseaux[33], in other letters to Renau and especially in some letters to Pierre Varignon in1715, including the one above cited.

In the absence of a shared concept of force, the debate between Bernoulli andRenau was heated and difficult to disentangle. In showing that Renau’s solution con-tradicts the fundamental principles of statics, especially the composition of forces,Bernoulli put the debate on a methodological level. Renau distinguished forces forwhich this principle is valid – for weights for instance – and forces for which it isnot – the forces of wind for example. Bernoulli did not accept such a distinction; ina letter to Renau, November 9th 1713, he began to reflect on the nature of the forceexerted by the wind, and concluded that it had nothing special compared to otherforces acting in a continuous way, for example he mentions the magnetic force, butalso the force of gravity. In this way the various problems involved in the theory ofthe vessels can be reconnected to statics:

11 vol. 2, pp. 175–176.12 Prof. Radelet De Grave sent me some typewritten Bernoulli’s letters; part of them are reproducedbelow [39].


The distinction you make between the forces of weight and of wind gives no reasons toadmit the principle of statics for those [forces of weight] and to reject it for these [forces ofwind], because the distinction concerns only the causes of the two forces. But it does notmatter how the forces are produced, it is enough they exist, by any cause they derive theywill have always the same action, and consequently the same effect, when the forces areapplied the same way [33].13 (A.8.5)

The assimilation of the force of wind to a generic force can be a consequence orcause of Leibniz’s idea of dead and living force that he is discussing and elaboratingjust in this period. In the case of wind the existence of pulses is evident, in the caseof gravity or magnetic forces there still are pulses, even if they are not so evident.

Among the topics under discussion between Bernoulli and Renau, two had a spe-cial role in the formulation of the principle of virtual velocities. The first, associatedto Fig. 8.2, refers to the determination of the speed of a vessel bound by the stringBZ of infinite length, which would move if it were not constrained in the directionBQ of the wind speed. The second associated to Fig. 8.3, brings to statics the caseof a vessel called by the wind in two directions.

C E B

Q

D

Z

Fig. 8.2. A vessel constrained by a rope

B

R

γ

ε

PC

R

L

G

N

O

FM

Fig. 8.3. A vessel constrained by a rope. Static model

13 p. 212.


Bernoulli introduces the word puissance perhaps because he feels force to beimprecise, a term he would likely refer to the living force.

I use the word power here instead of force to make myself more intelligible by showing thatthe force of winds has no prerogative to another kind of power continuously and uniformlyapplied [33].14 (A.8.6)

In a letter to Varignon of June 1714 [39], Bernoulli discusses the problem of Fig. 8.2in the special case where the water’s resistance to motion is zero. Bernoulli arguedthat the ship, without the rope, would be ready to move in the direction of the windwith its speed. In fact if the speed of the ship were less than that of the wind thewind would push the ship, conversely, if more it would restrain it. To Varignon whoholds that the velocity should be as BE, as it would be dealing with the rule of theparallelogram, Bernoulli replies that in this case of velocity composition the rule ofthe parallelogram cannot be applied:

I may be one of the most zealous defenders of the composition of forces, as you have seenin my book and in other occasions, but let me tell you that here you are abusing of thisgreat principle of Mechanics. You do not make a good application to our subject. To showit to you, let us see what this principle says. There are mainly two cases: the first is whentwo dead forces acting together, but in different directions, originate a third medium force,the second of such cases is when two living forces are to apply immediately and in a shorttime following different directions on a moving body, which each separately would generatecertain velocities. These forces would produce in the mobile if they act together, an averagevelocity, which will be as in the case of dead forces the diagonal of the parallelogram. […]To get to our subject, the first of our two cases does not apply, because we are not concernedwith dead forces, the second there cannot be applied either, because the ship is not pushedby the wind like a ball by a single instant shock, but by a force applied continuously [39].(A.8.7)

The speech is not entirely clear as Bernoulli seems to limit the validity of the rule ofthe parallelogram, but in fact it is not so. Bernoulli simply says that the rule of theparallelogram applies to forces and not necessarily to velocities. Very interesting isthe following passage, which comes closely after the previous one:

However, as the wind acts very differently on the sail by its continuation, we can considerits action as repeated bursts at any time, each of which adds a new level of speed infinitelysmall to the vessel until the overall speed of the vessel is so large that the wind can addnothing more to it. This happens when the ship, as I said, flees across the wind with thewhole speed of the wind [39]. (A.8.8)

Above Bernoulli seems to apply the Leibnizian language to the transformation ofdead force into living force, by means of subsequent pulses.

The first time Bernoulli refers to a law of virtual work is however in a letter toRenau of August 12th, 1714, after the publication of Essay d’une nouvelle theoriede la manoeuvre des vaisseaux. The reference is to the diagram of Fig. 8.2, whichnow has lost any reference to navigation and is reduced to an ordinary problem ofstatics.

14 pp. 217–218.


Bernoulli had already studied this case in another letter to Renau of July 12th,1713 [33],15 by requiring that the centre of gravity of the three weights was as low aspossible, taking up a similar analysis of Huygens [135].16 Now Bernoulli introducesthe terms ernegie and vitesse virtuelle. Energie is not given a precise mechanicalmeaning. However it is not the dead force (Leibniz meaning) but the dead force(Bernoulli meaning) multiplied by the infinitesimal velocity and as such it has littleto do with the modern concept of energy:

In the demonstration you make about the equilibrium of weights you say the powers or theforces are like themassmultiplied by the velocity and this is very true in a sense, but considerin the application you made in the equilibrium of the three sails whether you confuse forceor power with the energy of the power or the force [emphasis added] and you confuse thecurrent velocity of the wind, which multiplied by the mass produces the absolute force, withthe virtual velocity, which multiplied with the absolute force produces the momentum or theenergy of this force [376]. 17 (A.8.9)

Immediately after, Bernoulli specifies that virtual velocity is identified with the in-finitesimal displacement, energy with the product of the power or force multipliedby the virtual velocity. Note that at this stage forces and virtual velocities have thesame direction and Bernoulli makes no distinction between force and power.

I mean with virtual velocity the only tendency to move the forces have in a perfect equi-librium, where they do not move actually. So in your figure [Fig. 8.3], which is here thesecond, if the weight B inseparable from the line MB is in equilibrium with the weights Nand O, the virtual velocity is the small line BP, and the virtual velocity of N and O are CPand RP, and then the product of the weight B by BP, which is the energy of the weight B,is equal to the products of weight N multiplied by PC, and the weight O multiplied by RP,which are their energies. Wherefore to avoid ambiguity, instead of saying that their powersor forces are as the products of the masses by their velocity you might have done better,to express yourself well, to say that the energies of powers or forces are as the products ofthese powers or forces by the virtual velocity [376]. 18 (A.8.10)

Bernoulli will come back on this in a letter to Varignon of November 12th, 1714.

The essential point [of the divergence with Renau] can be put on half a page, but this isprecisely where Mr. Renau grossly errs in that it merges the forces of winds with the energyof forces, forgetting that to have energy that the Latins called momentum [emphasis added]of the wind, it is not enough to take, as he does, the square of the wind speed, which wouldgive the sheer force of the wind, but it is necessary to multiply the square of the velocitymultiplied by its virtual velocity, i.e. by the distance from the centre of support, about whichthe applied force tends to move [39]. (A.8.11)

The last step is the famous letter to Varignon of February 26th, 1715. Here Bernoullispecifies his principle and affirms its generality, in the sense that he sees the princi-ple of virtual velocities as the possible and only foundation of all statics, includinghydrostatics. He is argumentative with Varignon who proposes to establish statics

15 p. 164.16 vol. 3.17 p. 18.18 p. 18.


on the law of composition of forces, an intention declared in the Project d’une nou-velle mechanique [237] and completed with publication of the Nouvelle mécaniqueou statique [238], after his death (see last section of this chapter):

Conceive several different forces acting along different trends or directions to balance apoint, line, surface, or body; conceive also to impress on the whole system of these forcesa small motion either parallel to itself in any direction, or around a fixed point whatsoever:you will be glad to understand that with this motion each of these forces will advance orretire in its direction, unless someone or more forces had their trends perpendicular to thedirection of the small movement, in which case this force or these forces, neither advance norretire anything. These advancements or retirements, which are what I call virtual velocities,are nothing but what each direction increases or decreases by the small movement. Theseincreases or decreases are found by drawing a perpendicular to the end of the line of actionof any force. This perpendicular will cut in the same line of action, displaced in a closeposition by the small motion, a small part that will measure the virtual velocity of this force.Take, for example, any point P in the system of forces that is in equilibrium, F one of thoseforces which push or pull the point P in the direction FP or PF; Pp a small straight line thatthe point P describes because of the small motion, for which the trend FP takes the directionf p, which will be exactly parallel to FP if the small motion is made in all parts of the systemalong a given line, or will have, being prolonged, an infinitely small angle with FP if thesmall motion of the system is around a fixed point. So draw the perpendicular PC to f p, andyou will have Cp for the virtual velocity of the force F , so that Cp×F is what I call energy.Note that Cp is negative or positive relative with respect to the others: it is positive if thepoint P is pushed by the force F , and the angle FPp is obtuse and is negative if the angle FPpis acute, but otherwise, if the point P is pulled, Cp will be negative when the angle FPp isobtuse, and positive when acute. All this being understood, I form this general proposition:In any equilibrium of any forces in any way they are applied and following any directions,either they interact with each other indirectly or directly, the sum of the positive energieswill be equal to the sum of the negative energies taken positively [39] [238].19 (A.8.12)

F

CP

f

p

Fig. 8.4. Definition of the virtual velocity

In the passage above some things should be underlined; the first one is that the “gen-eral proposition” expresses a necessary but not sufficient condition for equilibrium.Secondly, the small movement is a rigid motion of all points and forces of the sys-tem – in particular, though not explicitly, a plane rigid motion, because it reducesto a translation or a rotation, to which it is natural to associate a system of planeforces – but these forces are not necessarily applied to a single rigid body. Thirdly,

19 vol. 2, pp. 175–176.


the motions are supposed to be small so not to affect, or to affect in an infinitesimalway, the position of the forces participating in the motion, as clear from Fig. 8.4where FP and f p are the forces before and after the displacement Pp. The idea thatthe forces are involved in the virtual motion is typical of the whole literature of theXVIII century.

While highlighting the infinitesimal character of virtual velocities, Bernoulli doesnot stress the need that such motion is according to internal and external constraintsof the system of bodies. Bernoulli then has never commented on the importance ofthe principle to eliminate the constraint forces from the equations of equilibrium.And in fact sometimes he will consider also virtual velocities incompatible withconstraints and the work of the reactions.

The letter to Varignon continues with applications of the energy rule to all casesof simple machines and also to fluids. In each case the results obtained are comparedwith known ones. The applications refer only to single degree of freedom systems;in this case the vanishing of the sum of the energies is also a sufficient condition forequilibrium. Riccati, Angiulli and Lagrange (Chapters 9 and 10) will clarify that theequation of energy becomes a sufficient condition for equilibrium if the validity forall possible virtual velocities is imposed; Servois (Chapter 12) will add interestingcomments on the difference between necessary and sufficient conditions.

In the following I will present only a few applications to show the difference ofBernoulli’s formulation with Descartes’ or Wallis’.

In the case of the inlined plane of Fig. 8.5; the two weights A and B are connectedby an inextensible wire. The force which equilibrates the weight B laying on theinclined plane is furnished by the weight A. Supposing a virtual motion of the twobodies with A that moves into a and B into b , the virtual velocity of weight A isAa, that of weight B is the line BC (i.e. the component of Bb in the direction of theweight force in B). Assuming equal energies leads to

A×Aa = B×BC = B× LNLM

, (8.3)

a well-known result. The procedure is more or less the same of that of Descartes.However now an algebraic equation is written down, and no recourse is made to anabsurd reasoning for the sufficiency of equilibrium.

a b

A

P

B

L

N M

C

Fig. 8.5. Equilibrium on the inclined plane


P

A a BD

C c

b

pn

m

Fig. 8.6. The composition of forces

More or less the same is true for the pulley, where the kinematical analysis of thepoints of application of power and resistance is enough. Some more attention shouldbe devoted to two particular cases. The first is the proof of the composition of theforces, the other the evaluation of the pressure a body exerts on the support.

For the first case, Bernoulli considers the situation of Fig. 8.6, where there arethree forces A, B, C converging into P. Imagine a horizontal translation of P andthese forces – notice that also the forces are moved. The virtual velocity of forces Aand B are pm and pn respectively, the virtual velocity of the force C is zero:

So we will have A× pm = B×Ppn+C× 0 = B× pn, i.e. A : B = pn : pm = sinus of theangle pPn : sinus of the angle pPm [39]. (A.8.13)

Bernoulli says that similar relationships are obtained by imagining motions of thesystem of forces in the directions Pm perpendicular to A and Pn perpendicular toB. The relations obtained are the same as that obtained by applying algebraicallyVarignon’s rule of force composition.

The second Bernoulli’s case refers to Fig. 8.7. The goal is to find the impressionthat each of the two inclined planes CA and CD receives from the ball of weight P.Bernoulli determines an impression at a time, thinking of replacing one of two planes

e

d

c n

A

Ca

fR

b

DB

Fig. 8.7. The impression on a support


with a force. For example the plane CD with a force R orthogonal to it. This is theclassic approach in statics to replace a constraint with forces and apply the rulesof equilibrium to the resulting system as if the body were not constrained. Bernoulliimagines the displacement Cc along the plane AC. The virtual velocity of the weightB which moves to b is represented by Cn (the projection of Cc along the vertical),the virtual velocity of the force R that replaces the constraint of the plane Cd isrepresented by Ce (the projection of Cc along R). It is not hard to find the relationshipof these two virtual velocities as a function of the angles ACD – the angle formedby the two planes – and Ccn, which is the slope of the plane CD:

P : R = sinus ACD : sinus Ccn. (8.4)

For fluids, assumed as incompressible, Bernoulli considers the case of the siphonand the hydraulic paradox. For simplicity, I will refer only to the latter. Bernoulliconsiders the tube SNns of Fig. 8.8, that extends into the cylinder SDABEs. The baseAB can move inside the cylinder without allowing the fluid to drain. The weight Pis at the end of a scale, to the other end of which there is the base AB. The systemis filled with a fluid until F f , in order to equilibrate the weight P. By neglecting theweight of the base, it is found that the weight P necessary for the equilibrium equalsthe weight of the column of water with base AB and height JF, and not only theweight of the portion in gray of Fig. 8.8, which seems paradoxical.

The mystery is explained by applying the rule of energies. Bernoulli imagines todivide the fluid into n layers of the same height, p in the cylinder and n− p in thetube, the weight P is also imagined divided into n equal parts. The virtual velocityof the weight P is equal to Aa as that of the base AB. Then each of the p layers ofthe cylinder has the same energy, Aa× AB, as each element of the weight P. Thevirtual velocity of a layer of the tube is given by f nwhich for the preservation of the

N n

F f

G g

Ca b

D

A J

J M

P

B

S s E

Fig. 8.8. The hydraulic paradox


volume of incompressible fluids is related to Aa by the relation f n = Aa×AB/Gg,then the energy of each of the n− p layers of the cylinder being proportional to theproduct f n×Gg = Aa×AB is it the same as one of the remaining n− p elementsof the weight P still not considered in the equilibrium. Thus the energy of the loadP and the water column is the same and so the paradox is explained.

After this exchange of letters with Varignon in the years 1714–1715, Bernoullireturned to his principle just once, a few years later, in 1728 in his Discours sur lesloix de la communication du mouvement [35]. At the beginning of chapter III hedefines virtual velocity:

I call virtual velocities, those that two or more forces brought into equilibrium acquire whena small movement is impressed to them, or if these forces are already in motion. The virtualvelocity is the element of velocity, that every body gains or loses, of a velocity alreadyacquired during an infinitesimal interval of time, according to his direction [35].20 (A.8.14)

The above definition is not equivalent to that contained in the letter to Varignon. Atrue velocity is considered rather than a displacement, moreover, it is in general thevariation dv of a motion. This new point of view is justified by the fact that Bernoulliis now considering the motion of bodies and not just their equilibrium. No reference,or comment is made to his earlier definition of the virtual velocity, as if he had neverwritten anything about it. Slightly further down Bernoulli continues, with the title ofHypothesis I:

Two agents are in equilibrium, or have the same moments, when their absolute forces are inthe mutual relationship of their virtual velocities, either the forces acting on each other arein motion or at rest.This is a normal principle of Statics and Mechanics, I do not stop to prove it, I prefer ratherto show how the motion is produced by the force of a pressure which acts continuously, andwithout further resistance in addition to those resulting from the inertia of the mobile [35].21

(A.8.15)

One must wonder about the very little weight Bernoulli now attaches to his prin-ciple, which in 1715 he had seen as a key of statics, and to the few references hemade to it, declaring it, inter alia, to be a principle of ordinary mechanics and there-fore well known. At the same time time he should have known that not all scientistsaccepted it. Moreover, Varignon’s Nouvelle mécanique, reporting his famous letter,was not released before 1725.

20 p. 23.21 p. 23.


8.3 Varignon: the rule of energies and the law of compositionof forces

Pierre Varignon was born in Caen in 1654 and died in Parisin 1722. Educated at the Jesuit college and the universityin Caen, he received his master’s in 1682 and holy ordersthe following year. He became professor of mathematics atthe Collège Mazarin in Paris in 1688 and was elected to theAcadémie des sciences in Paris in the same year. He waselected to the Berlin academy in 1713 and to the Royal so-ciety in 1718 [354] Varignon was in touch with Newton,Leibniz, and the Bernoulli family. His principal contributions

were to mechanics. With l’Hôpital, Varignon was the earliest and strongest Frenchadvocate of differential calculus. He simplified the proofs of many propositions inmechanics that were based on the composition of forces. An interesting publicationof his concerned the application of differential calculus to fluid flow and to waterclocks.

8.3.1 Elements of Varignon’s mechanics

In 1687 Varignon published the Project d’une nouvelle mechanique [237], whichgave rise to the Nouvelle mécanique ou statique of 1725 [238], after Varignon’sdeath (1722). In the premise of the Project Varignon explained the reasons that ledhim to undertake his work. He declared to have been very impressed by Descartes’claims for whom there was no sense in reducing the pulley to the lever, as dal Monteand Galileo did. This led him to conclude that it made not much more sense to re-duce the inclined plane to the lever, or to reduce one machine to another machine.To Varignon it was better to find a single simple principle which explained the op-eration of all machines. For him Descartes’ approach with the virtual displacementwas interesting but it had the inconvenience of considering more the necessity thanthe sufficiency for equilibrium and of not furnishing a causal explanation.

For Varignon the examination of all cases of equilibrium studied made it clearthe need for a causal principle that serves to explain the reasons of equilibrium:

I remain in the opinion that to understand equilibrium it is necessary to know how it isestablished and to see in it all the proprieties that all the other principles prove at most as anecessary condition [237].22 (A.8.16)

He found this causal mechanism in the law of composition of forces by means of therule of the parallelogram which he assumed to be the only principle of statics.

The law of composition of forces according to the rule of the parallelogram,known in the XVIII century as Stevin’s theorem, was reformulated by Varignonand demonstrated on a dynamic basis as Newton did in the same year in his Prin-cipia, with the difference that Varignon assumes proportionality between forces and

22 Preface.

8.3 Varignon: the rule of energies and the law of composition of forces 211

velocities instead of forces and accelerations. I do not want to comment here on thelegality of this transaction, I will only refer to the wording of the law of compositionof forces as proposed by Varignon:

To prepare the imagination to compound motions, conceive [Fig. 8.9] the point A with noweight moving uniformly toward B along the straight line AB, while this line moves uni-formly toward CD, along AC by remaining always parallel to itself, i.e. by making alwaysthe same angle with this fixed line AC. Of two motions started at the same time let the ve-locity of the first to the velocity of the second be as the sides AB of the parallelogram ABCDalong which they [the motions] occur. Whatever the parallelogram ABCD be, I say that forthe effect of the two forces producing these two motions in the mobile A, this point will passthe diagonal AD of this parallelogram, during the time that each of these [forces] would havemake to pass along each of the correspondent sides AB and AC [238].23 (A.8.17)

Fig. 8.9. The composition of forces according to Varignon (reproduced with permission of Bib-lioteca Guido Castelnuovo, Università La Sapienza, Rome)

The Nouvelle mécanique had a great influence on statics for nearly a century. Thisinfluence did not so much derive from the theoretical content of the text, but ratherfrom the large number of applications. Results of such applications obtained with ge-ometric considerations on various parallelograms of forces were expressed bymeansof formulas, mainly proportions, which gave Varignon’s treatment a partially alge-braic aspect that made it easier to solve static problems than the rigidly geometricapproach of the lever. Varignon applied the composition of force rule also to con-strained systems, replacing the constraints with equivalent forces. Among the alge-braic relations that he established between the forces is an important one that we nowcall the Varignon theorem, which in modern terms says that the static moment of theresultant with respect to a pole is the sum of the static moments of the componentswith respect to the same pole [238].24

23 vol. 1, p. 13.24 pp. 84–85.


Varignon in his book, probably for the first time, introduced the modern Frenchterm moment from the latin word momentum, with the meaning of static moment,i.e. the product of a force by its distance from a reference point (the arm):25

Definition XXIIThe product of a weight or absolute power by their distances from the fulcrum of the lever towhich it is applied is called in latin,Momentum […] So we will continue to call itmoment toremain close with the ordinary use. The reason of this name stems without any doubt fromthe fact that these products are equal or different as the actions of two powers in a lever[238].26 (A.8.18)

At the end of the first volume of the Nouvelle mécanique ou statique, for the leversubjected to various forces at various points of application and with different direc-tions, Varignon presents as a theorem (really a corollary to a more general theorem),proved with the composition of forces, the rule of equilibrium based on the vanishingof static moments:

The contrary expressions of moments will always be equal to each other, i.e. the sum of themoments conspiring to turn the lever in a sense about its support will always be equal tothe sum of moments conspiring to rotate in the opposite sense on this support, as we havealready seen in Corol. 9 of Th 21[238].27 (A.8.19)

The semi-algebraic approach of Varignon evolved toward a purely algebraic ap-proach, for which the balance of forces results in forcing to zero the sum of thecomponents of the forces and static moments, which today are called cardinal equa-tions of statics. There is still no precise reconstruction in the literature of the way inwhich the modern form of cardinal equations of statics was obtained. D’Alembertis commonly credited as the first to give these equations, in the Recherches sur laprécession des equinoxes in 1749 [82] followed by Euler [101, 106]. To the best ofmy knowledge they were Fossobroni with his Memoria sul principio delle velocitàvirtuali in 1794 [109],28 Prony with his Sur le principe des vitesses virtuelles in1797 [202]29 and Lagrange, and only in the second edition of the Mécanique ana-lytique of 1811 [148],30 which collectively gave the first modern expression to thecardinal equations of statics. But it was only with the Mémoire sur la compositiondes moments en mécanique by Poinsot in 1804 [193] that they were fully under-stood.

25 In specialised treatises of mechanics, static moment is not a term of mechanics but rather ofgeometry, like area or moment of inertia. The product of a force by its arm is simply called moment(of a force). Historians of mechanics however are used to speak about static moments to distinguishthem from the Galilean moment. In the book I will follow this use.26 p. 304.27 pp. 385–386.28 pp. 86–87.29 p. 194.30 pp. 46–58.


8.3.2 The rule of the parallelogram versus the rule of energies

In his letter of February 1715, Bernoulli declared the superiority of his principle:

Your project of a new Mechanics is filled with a great number of examples, some of which,to judge from the figures, seem very complex. But I challenge you to propose one at yourchoice, and I will solve it on the field and as for a joke with my rule [39]. (A.8.20)

After having asserted that with his rule it is possible to solve all the problems ofstatics, Bernoulli added:

The principle that you pretend to substitute to mine, and which is based on the compositionof forces, is nothing but a little corollary of the energy rule. I have so the right to consideras the first and great principle of statics that on which I based my rule: in any equilibriumthere is an equality between the energies of the absolute forces; i.e. between the product ofthe forces multiplied by their virtual velocities [39]. (A.8.21)

and suggested that Varignon replace the rule of the composition of forces with therules of energies:

I beg you to think, you will find in it an inexhaustible fund to enrich mechanics and to makethe study incomparably more comfortable and simple than it was in the past. The completetreatise of this science, that you promise so long, could appear much more estimable, if itwill be founded over a principle so universal, so simple, so clear and so certain, like that itis concerning and of which I showed so many advantages [39]. (A.8.22)

In March 1715 Varignon replied that, yes it is true that the rule of energy is interest-ing, but that the rule of composition of forces is easier and more fruitful:

But mechanics, from this proposition and from the general one you added to your last letter,far from being the great and first principle of statics, is in my opinion only a corollary ofcompound motions, or of another principle, which proves this proposition, i.e. your equalityof the sum of energies, by deducing with its aid or by supposition, the incipient motions ofMr. Descartes, which you call virtual velocities, that with the powers elsewhere evaluated,with the assumption of their equilibrium, it is all needed for the equality of the sum of theenergies, of which one could ever have the right to think that it could be derived from oneof these principles [39]. (A.8.23)

He saw the law of energy rather as a corollary of the composition of forces. In orderto take the rule of energy as a principle of statics, for him it must be proved with thelaw of composition of forces or with some other principle – presumably Varignonthought the law of the lever.

Varignon, rightly, traced the law of energy to Descartes:

Cartesians, according to the letter I cited of their Master,31 had already deduced from hisprinciple the same equality of Moments or energies, or the quantity of motion, that you use,for two powers in equilibrium on simple machines, and in fluids, from the incipient motionthat Mr. Descartes prescribes in this letter. But you are the only one, for what I know, whoextended the equality of energies to as many powers as you like, acting in any directionand in equilibrium with themselves. This point is very nice, but (as I have already said) itsupposes the equilibrium among them and does not prove it [39]. (A.8.24)

31 [94], letter 73, vol. 1, pp. 327–346.


and asserted that Bernoulli stated his principle only as a necessary condition, thatis if there is equilibrium then the energies are the same for virtual motions, as theCartesians that demonstrated the sufficiency of the balance with an ad absurdumargument:

The equilibrium from non-equilibrium, they make only a demonstration ad absurdum [39].(A.8.25)

Bernoulli replied, arguing the logic superiority of the rule of energies, stating that itapplies equally well to solids and fluids, while this is not true for the compositionof forces. Diplomatically he then ended by asserting that what counts for him is thathis rule is correct and works very well:

I am afraid of falling into a long verbosity if I try to discuss all you are saying regardingmy rule of energy, that I pretend to be general for the whole of mechanics, both for fluidsand solids […] let us avoid that verbosity. It is only sufficient to establish the truth and theuniversality of my rule of energies against your objections. That this rule be a principle or atheorem of another rule, it does not matter; it is enough that it is true, general and comfort-able, without any exceptions, uniform and simple to use. Advantages that the compositionof forces does not possess [39]. (A.8.26)

To the dispute that Descartes preceded him, Bernoulli added, with a touch of con-troversy, that Varignon too was not very original:

You cite Mr. Descartes’s letter to prove that this author has already had the idea to explainthe equilibrium of powers by means of the equality of energies by considering their incipientmotion, that I call virtual velocities. I reply that I am not proud to be the first inventor of thisidea; no more should you be proud to be the first to explain the equilibrium by means of thecomposition of forces [39]. (A.8.27)

To Varignon, who asked permission to report Bernoulli’s principle in the book hewas writing – i.e. the Nouvelle mecanique –, Bernoulli had no difficulty in grantingthe permission in a letter of July 1715, provided he did not present it as subordinateto the rule of composition of forces:

You can make what you like of my rule of energies, adding or not adding it to your me-chanics. I allow both of them. But to pretend that it is a corollary of the principle of thecomposition of motions or forces, I may still hold the reasons given in my previous letters,to prove the contrary, if you want to engage me in a challenge that will cost us time andtroubles. So I will prefer to leave to you the pleasure to believe that the principle of the com-position of forces should precede that of energies, to try a long and lengthy contestation. Itis enough that the second could be applied both to fluids and solids, it is more general thanthe first that is useful only for solids, moreover it will need one more principle from whichit could be deduced, because the composition of forces is not so clear as to be assumed asan axiom. Then it looks to me more reasonable that the principle of energies, as the moregeneral and at least as clear as the composition of motions, contains the last as less general[39]. (A.8.28)

Varignon however did not respect the desire of Bernoulli. Indeed in the second vol-ume of the Nouvelle mécanique ou statique, he wrote a chapter titled Corollairegénéral de la théorie précédente, where he claims that the rule of energies is nothing


but a corollary of the rule of composition of forces, qualified as TheoremXL [238].32

Actually Varignon was not successful in proving the rule of energies in general, butonly for various cases: a weight supported by many strings, the pulley, the wheel,the inclined plane with the weight pulled by a force with any direction, the lever, thescrew and the wedge, by checking that values of forces and displacements, evalu-ated respectively with the rule of composition of forces and with simple kinematicalanalysis, respect the equation of energies. The applications considered by Varignon,probably to avoid being accused of plagiarism, did not coincide with the applica-tions suggested by Bernoulli in his letter of February 1715, in particular there are noapplications to fluids.

32 pp. 174–176.

9

The Jesuit school of the XVIII century

Abstract. This chapter is devoted to the principle of action by Vincenzo Riccati andVincenzo Angiulli, whose VWL is similar to that of Bernoulli. In the first part thecontribution of Angiulli and his demonstration of VWL is presented in the founda-tional route. This is perhaps the first convincing demonstration of a VWL. In thesecond part the contribution of Vincenzo Riccati is presented.

Unlike France which, as reported in the previous chapter, saw a revival of interestin mechanics, Italy in the second half of the XVII century started a slow decline,except for some recovery in the second half of the XVIII century, that will stop onlyafter the unification of the nation in the late XIX century. This decline is particularlyevident in the so-called exact sciences, including mathematics and mechanics. Thelast great Italian scientist in this field was Giovanni Alfonso Borelli (1608–1679).Not that clever and educated people were missing, except that the lines of researchpursued in Italy were no longer included in those being conducted in Europe by Huy-gens, Newton, Leibniz, the Bernoullis, for example. The reasons for this delay werenumerous and probably outside the nature of science itself, largely due to economicbackwardness and independent political status compared to the national states thatwere consolidating in Europe, where scientists were asked to solve pressing practicalproblems, related for example to navigation and military activity.

In this climate, Italian scientists found themselves almost compelled to thinkabout old problems using old categories. To give an idea of this type of studies it isuseful to cite Girolamo Saccheri (1667–1733) who dedicated his efforts to the studyof classical geometry, providing interesting contributions to non-Euclidean geome-try, that however will be resumed in Europe only toward the end of the XVIII cen-tury. One of the mathematicians who fitted well in the mainstream of EuropeanCalculus was Jacopo Francesco Riccati (1676–1754), who studied the equation thatbears his name. The schools of Bologna and Naples [367] must also be referencedwith regard to studies ofmechanical theory, and yet another Riccati (Vincenzo)madea very important contribution, influenced in part by his father’s studies.


218 9 The Jesuit school of the XVIII century

Vincenzo Riccati supported the ideas of Leibniz, following the Italian attitudewhere Newton’s ideas were not yet widespread. This was not actually much differentfrom what was happening in Europe where, besides Leonhard Euler (1707–1783),who developed Newton’s mechanics to make it an effective instrument operatingthrough the use of differential equations, there was still Johann Bernoulli who, withhis principles of virtual velocities in statics and the theorem of living forces in dy-namics, had proposed an alternative approach to mechanics, close to that desired byLeibniz. In Italy the tradition of Galilean mechanics was still alive and in it the lawsof virtual work were very important. For sure the continuity with the approach ofGalileo also influenced Vincenzo Angiulli, an intelligent Riccati’s pupil, to assumea law of virtual work as the basis of statics.

But the greatest ‘Italian’ mathematician and mechanician of the middle of theXVIII century was Ruggiero Giuseppe Boscovich (1711–1787). Boscovich left nowritings of statics, but in some of his works he made use of a law of virtual work,applied to a new area, the mechanics of structures, which will provide fertile groundfor modern laws of virtual work. Probably it is not a coincidence that Boscovichand Riccati were both Jesuit and the two greatest mathematicians of the order. It islikely that Boscovich had knowledge of the research of the ‘brother’ Riccati, eventhough the publication of Riccati’s studies on laws of virtual work [207] was sixyears subsequent to Boscovich’s first applications [138].

9.1 Vincenzo Angiulli and Vincenzo Riccati

The work of Vincenzo Riccati and Vincenzo Angiulli presented in this chaptershould be seen as an interesting attempt to defend the approach of mechanics basedon virtual work laws, which in Italy had its roots in the works of Galileo and Tor-ricelli. The former showed his ideas in the Dialogo di Vincenzo Riccati della com-pagnia di Gesù dove ne’ congressi di più giornate delle forze vive e dell’azioni delleforze morte si tien discorso of 1749 [207] and the De’ principi della meccanica of1772 [208], the latter in theDiscorso intorno agli equilibri in 1770 [4]. Both took onas a foundation of mechanics, with various reasons, the principle of action, which isa possible version of virtual work laws.

In the following I will set out a summary of the thought of the two Italian schol-ars, spending more time on Angiulli, who will be considered first. This is because,although Riccati’s contribution is more original, he first proposed the principle of ac-tion, the production of Angiulli and his attempts at justification are more effective.

9.1.1 The principle of actions of Vincenzo Angiulli

Vincenzo Angiulli was born in Ascoli Satriano (Foggia) in 1747 and died in Naplesin 1819. Very young he got a degree in law in Naples where he attended the enlight-enment circles of the city. Only twenty three he became member of the AccademiaClementina, of the Istituto delle science of Bologna and professor of mathematics atthe Real accademia of Nunziatella in Naples. On the occasion of the death of Charles

9.1 Vincenzo Angiulli and Vincenzo Riccati 219

III of Borbone he wrote and read in Ascoli Satriano an interesting funeral oration.In 1800 he was jailed for the role he had during the French occupation. He knewRiccati in Bologna where he followed his courses on mathematics. The Discorso in-torno agli equilibri was the only scientific work of Angiulli written on the occasionof his professorship at the Nunziatella when he was a young man. After this experi-ence Angiulli dedicated himself to administrating his properties and to a consistentpolitical activity influenced by the Enlightenment [5].

9.1.1.1 The action of a force

The basic concept of Angiulli is that of action of a force, as developed by Riccati.To introduce this concept, Angiulli begins to point out that although the presenceof a ‘force’ is a necessary condition for changing the state of equilibrium, it is noteven sufficient. For example, in a heavy body suspended by a thread, although it issubject to gravity, there is no change in state. Therefore onemust distinguish betweenthe force and its action, only the latter can produce the change in state. From theforegoing it is evident that in Angiulli’s mechanics, like that of most scholars of theXVIII century, there is no room for constraint reactions and the ‘forces’ are onlythose now classified as active forces.

After stating that Galileo was the first to distinguish between force and action,when he introduced the idea of the force of blow, Angiulli presents, by means ofexamples, his concept of force, which he calls power:

For power, therefore, we do not mean anything but the pure and simple pressure, or thateffort, which the gravity or other force makes against some invincible obstacle, as preciselyit is what a ball of lead makes against a fixed table, or against the hand that sustains it [4].1

(A.9.1)

In this quotation the power does not appear as a purely static force, able to balancea weight, as it was mainly up to Galileo, it always makes an effort against an ob-stacle and therefore it plays an active role. In the next quotation Angiulli recalls thedefinition of Leibniz’s living force and its relationship with the dead force which isidentified with the power:

So if a ball, for example of lead, will be located above a fixed table, the gravity, whichresides in it, will be the only pushing force, and therefore dead force. But if the obstacle isremoved, that is the table, in the ball will soon be a change of state, […] the mechaniciansimagined the force to give the body a boost, which, however, just born, was destroyed bythe invincible obstacle, and so according to their method mathematicians represented thedead force with the idea of an infinite small impulse […]. But because the mechanicianscould form a clearer idea of the action of the force, as they represented it under the idea ofa pulse, which in the process of its birth is extinct and destroyed by the invincible obstacle,thus removing the invincible obstacle, conceived that all pulses […] were conserved in thesame body, and then thought the action of the power not to be but the sum of all the pulsesaccumulated, and stored in the body. Then the amount of energy generated in the body forthe action of the force [...] is called living force [4].2 (A.9.2)

1 p. 5.2 pp. 5–7.


Power or dead force is thus presented as an infinitesimal pulse that is constantly beingrenewed, by gravity or other causes, and continuously destroyed by constraints.Withconstraints removed, pulses can accumulate and the action of power lies preciselyin the effect of cumulative pulses that are not destroyed. The action of the powergenerated then the living force. In the following quotation it is possible to see howAngiulli conceives the relationship between power, action, living force and changeof state:

But if it is as real as it is said, that the power should be considered as a pulse less than anyother, and that the action of the power is the sum of all the pulses communicated to the bodyand stored in the body, there will certainly be the same proportion of the power to the actionof the power, as that passing between an infinitesimal quantity and a finite one [4].3

From above it appears, that the power acting in the body to which it is applied generates in itthe living force, and this produces the state change. So the living force must be considered asan effect of the power and as a cause of the change of state that is induced in the body. Andas in this case we speak of entire and total causes, the Ontological axiom will take place forwhich the causes must be proportional to the effects and the effect to the causes. So these arethe two ways of measuring the living force, i.e. either with their effect, which is the changeof state, or with the extent of their cause, which is the action of the force [4].4 (A.9.3)

Note that the action is identified as the cause not of motion but only of the livingforce that is the true cause of the motion.

The analysis of the text clearly shows that Angiulli is linked to the school of Leib-niz and Bernoulli, but it shows also the call to the Italian school; Angiulli strives tolink the results of Galileo to those of Leibniz. For example, in his way of explainingthe concepts of power and its action it is possible to see Torricelli’s language of theLezioni accademiche, published only in 1715 [276].

After the presentation of the power and its action in the book of Angiulli, therefollows some definitions connected to power, useful for the introduction of the prin-ciple of action, reported in the next paragraph. A fundamental concept is that of thecentre of the power P. This is conceived as that point, considered fixed, where thesource of the power itself is located. For example in the case of gravity the centreof power is the centre of the earth, in the case of a force due to an elastic spring thecentre is the fixed extremity of the spring. The line joining the centre of the power toits point of application is the direction of the power. In addition, the space of accessand space of recess are the amount by which the point of application of the powerapproaches or moves away from the centre P. In this nomenclature the influence ofJohn Wallis is clear (see Chapter 7).

In Fig. 9.1 there are given by way of example, three powers, AD, CD and DB,of centres A, B and C, respectively, applied at point D with the direction indicated.Assuming that Dmoves in a, the distance Da is the space of recess of power in A, thedistance Db is the space of access of power in B and the distance Dc is the space ofrecess of power in C. For small motions, the access or recess spaces coincide with theprojections of the displacement in the direction of power, both in the starting positionor in that varied, because they differ by infinitesimal quantities from each other.

3 p. 7.4 p. 22.


A

D

a

B

C

b c

Fig. 9.1. Access and recess spaces

9.1.1.2 The principle of actions

In his definition of the action of the power, Angiulli has left as indefinite one impor-tant aspect: he does not specify how the pulses should be counted. Notice that at thispoint Angiulli, as Bernoulli, because of the need to give a mathematical expressionto the dead force, is obliged to redefine it. This is made by distinguishing power,dead force and infinitesimal action, i.e the action associated to a pulse. Up to now,on the one hand dead force and power were considered as synonymous, on the otherhand infinitesimal actions and dead force are assumed to be both pulses, so there isno difference between them. From now on, dead force will be considered to be apower in the traditional sense, i.e. not a pulse, while the infinitesimal action will beconsidered as a pulse. If f denotes the measure of the power, x the measure of spaces or time t, the measure of the infinitesimal action is, according to Angiulli, givenby f dx.

He argues also that it cannot be said in advance if pulses of action replicate inspace or time. The indecision between space and time is not only due to a rhetori-cal need for objectivity, but it also has another origin. The definition by Angiulli ofthe action of a power, by his own admission, is not original but goes back to thatof Vincenzo Riccati of 1749 which contains the same problem of choice. VincenzoRiccati introduces the power and the action of the power in a way formally similarto Angiulli’s, but he attributes to them a different ontological status. Riccati’s poweris the Newtonian force, acting to produce any change of motion of a body. The ac-tion of the power is still an aggregate of pulses, but the aggregate must be regardedsimply as an associated mathematical quantity. In this purely mathematical sense itis reasonable to consider both f ds and f dt as equally representative of the infinitesi-mal. The choice of either option is left to their more or less usefulness in establishingthe laws of equilibrium. For Angiulli instead the power is the Leibnizian dead force.The pulses associated with it are hypostatized: they are created and destroyed andaccumulate only when they are not destroyed.With their accumulation, which repre-sents the action of the power, they generate the living force (and it is not the action ofthe Newtonian force, as for Riccati) which is responsible for the change of motion.


Prior to leaving the reservation whether to measure the replicas of the pulse ofpower in space or time, Angiulli tackles the problem of defining a criterion for equi-librium, reaching the conclusion that it is provided by the equality of the infinitesimalactions. He begins by stating that a policy of equilibrium, to be metaphysically wellfounded, must be based on some equality, without which motion emerges. In con-sidering the various possibilities, Angiulli immediately discards the criterion thatassumes the equality of powers. Although desirable, it is generally not empiricallyverified; for example the equilibrium of the lever with different arms is accomplishedwith two different powers between them. Then he criticizes the approach of the‘ancient mechanicians’, which had placed equality of static moments – defined byhim in the modern sense of force multiplied by arm – as the basis of equilibrium. Itdoes not provide a causal explanation, since the moments have no physical reality,they are not beings – as powers are – able to act causally:

So by saying with the Ancients, that the cause of the equilibrium is in equality of the mo-ments, they seem having said, nothing but the equilibrium depends on the equality of thosequantities from the equality of which the equilibrium depends [4].5 (A.9.4)

and then one encounters in a petition of principle. Angiulli concludes that sinceequality must relate in some way to powers, there is nothing but to consider the ac-tions and define the equilibrium as a result of equality of actions of the powers thatact one way and another. Then, if the actions which are the causes are prevented, theeffects, i.e. motions, also are prevented, and there is equilibrium:

The equilibrium comes from the fact that the actions of the powers which must be equili-brated, if born, would be equal and opposite, and therefore the equality, and opposition, ofthe actions of the powers is the actual cause of equilibrium.[…]The equilibrium is nothing but that the impediment of the motions, that is of the effects ofthe powers, and it is not surprising if it matches the prevention of the causes, i.e. of theactions themselves [4].6 (A.9.5)

To explain better the contents of this quotation, Angiulli states that the balance ofthe actions of powers should not be considered as a cause in the strict sense of equi-librium. Each action by itself (through the intervention of the living force) is thecause of a motion; if there is equality between contrary actions, there is equality be-tween the possible motions, and then there is no motion, and then balance. If at thebeginning of a virtual motion the actions are equal to each other, then the motionis impossible, but if they are different, nothing will prevent the greater to make itseffect and there will be motion. Therefore, Angiulli concludes, the general criterionof equilibrium is contained in the following theorem:

Then we establish a principle, that is a general criterion to know when it will happen thatbetween the forces there is balance, and it is what is contained in the following theorem: Theforces will be in balance if they are in such circumstances that if an infinitesimal motion wasborn, their infinitesimal actions would be the same. And this principle must take place in allequilibria [4].7 (A.9.6)

5 p. 15.6 pp. 16–17.7 p. 18.


Note the presence in the last passage of the term infinitesimal motion, which under-lines the idea of motion in the process of being born, whose introduction is essentialto arrive at a correct formulation of the criterion of equilibrium.

Angiulli has qualified as a theorem his statement because he believes to havedemonstrated it with metaphysical considerations. He followed trying to bring otherarguments in favor of the principle of action, which for the moment is still quitegeneral, not specified in its magnitude. The principle does not “Only administersa general method to examine the equilibrium, but it is also the real way in whichthe equilibria will go to establish themselves”. In fact, he says, suppose to put aball between two opposing inclined planes, it just will not be placed in equilibrium,because there are no absolutely rigid bodies in nature, and the inclined planes willyield to pressure of the ball, and provide an elastic power, whose action is opposedto the gravity of the ball. The elastic powers give rise to actions, which being equaland opposite to that of weight ensure that the ball remains in equilibrium [4].8

However these considerations confuse rather than clarify. Indeed Angiulli intro-duces the elastic forces, trying to give physical sense to constraint reactions, thatotherwise may remain obscure, a mere fiction. The concept of elastic constraint re-action, however, is incompatible with the concept of hard body, i.e. a body withonly passive function, capable of absorbing all dynamic actions; a dominant con-cept in the mechanics of the XVIII century, which Angiulli accepts on a number ofoccasions.

9.1.1.3 The measure of actions

Finally Angiulli switches to solve the problem of the concrete measure of the action,i.e. to decide if the pulses of powers are replicated in space or in time. He recog-nizes at this point that this choice is the object of the dispute between Cartesian andLeibnizian, which in Italy was still alive in his time:

Because the famous dispute of the living forces, which is to establish whether these aremeasured by the mass multiplied by the speed, or by the mass multiplied by the square ofthe speed, reduces to this other question, namely whether the action of the force should beproportional to time rather than space [4].9 (A.9.7)

That the choice of the measure of the actions conditions the selection of the measureof living forces, is justified by claiming that there are two ways of measuring theliving force: or by measuring the cause, which is the action, or by measuring theeffect, that is the change in velocity. Since causes and effects should be proportional,so themeasure will be reflected on each other. If it is proved that the action of a powershould be measured with the space it would also be proved that the living force ismeasured by the square of the velocity.

Angiulli refers to a law of motion established by Galileo [118],10 to state that thepower multiplied by the space is proportional to the mass multiplied by the square of

8 p. 19.9 p. 22.10 pp. 287–288. Galileo says simply that the space of descent is proportional to the square of ve-locity.


velocity, but this subject was largely explored by other scholars. Of some interest areBoscovich’s considerations on his De virus vivis dissertation of 1745 [47],11 wherehe considers the possibility to integrate the equation of motion in time, to obtain thevelocity, or the expression of force in space, to obtain the square of velocity [358].Boscovich also names Vincenzo Riccati among Leibniz supporters.

The arguments with which Angiulli arrives soon after to what he considers a fairmeasure of the power, are only partly convincing, as I shall explain later. He saysthat the action of powers cannot be obtained by combining powers with time becauseif the action was based on the time, and is the same time for all actions, the actionswould be proportional to the power and the criterion of equality of action coincideswith that of the equality of powers. This is not acceptable, because the equality ofpowers is not a universally valid criterion of balance, as argued above, so one mustcombine the power with the space:

Not being able therefore to measure the action of a power by the power multiplied by time,is necessary to turn to space. In all known balances it is true, as discussed in the followingchapters, that making an infinitesimal motion, the powers are in the inverse ratio of theirrespective space of access, or recess from the centre of the same powers […]. So whetherthe action of the force will be measured by the power multiplied by the distance, to whichthe force acting carries the body, making it closer to the centre, or making it away from thecentre, it will be saved in the equilibria the equality between the actions of powers […].So the action of the power has truly to be measured by the power multiplied by the spaceaccording to the method of Leibniz [4].12 (A.9.8)

Angiulli claims to have shown in this passage that the measure of the action as thepower multiplied by the space of access or recess is a consequence metaphysicallycertain, calculated with exact reasoning from the premises. In fact, even acceptingthe evidence that the extent of the actions should be based on space instead of ontime, it cannot be seen why the powers should be multiplied simply for displacement(and not their square, for example) and because these displacements should coincidewith the previously introduced spaces of access and recess.

The operational statement of the criterion of balance is eventually provided bythe following theorem (the term is Angiulli’s):

The powers are in balance, if they are in the situation that making an infinitesimal motion,some powers become as close to their centres, some others move away from their centres, thesum of products of positive powers multiplied by the respective spaces of access or recessis equal to the sum of similar negative products [4].13 (A.9.9)

With a choice that is questionable to a modern reader, he will then call the above‘proved’ theorem by the name of the principle of action. The criterion of equality ofthe action is then considered by Angiulli as a theorem, when viewed from a meta-physical point of view, as a principle, when viewed from a purely mechanical pointof view, as non-deductible by other laws of mechanics. The principle of the actions

11 p. 4; pp. 13–14; pp. 28–29.12 pp. 25–26.13 p. 28.


stated above can essentially be considered as equivalent to Bernoulli’s rule of en-ergies. Angiulli is also conscious of this and according to him “it does not seem todiffer” from Bernoulli’s principle of which he only refers to the brief mentions inthe definition of 1728 (see § 8.2).

According to Angiulli the coincidence between Bernoulli’s rule and his principleappears immediately as soon as it is recognized that virtual velocity and access orrecess space are the same thing. Angiulli however does not declare that he referred toBernoulli, but rather to Galileo, Descartes, Borelli and other sublime mechanicians.His position should not be seen as a disavowal of the merits and priority of Bernoulli,it should be seen rather as the realization that his principle of action is one of thepossible expressions of the principle of virtual work, to which formulation the Italianschool has greatly contributed. Since Angiulli deals with fundamentals, it is right thathe calls more on Galileo than on Bernoulli, because Galileo in Le mecaniche andthe Discorsi sulle cose che stanno in su l’acqua had dealt with the basic conceptsof virtual work laws, while Bernoulli contributed mainly from a technical point ofview, recognizing the infinitesimal nature of the virtual displacements.

9.1.1.4 The principle of action and the principles of statics

Angiulli had already said that the criterion of balance of moments, on which the lawof the lever is often based, had no clear metaphysical evidence and therefore the lawof the lever, referred to as the principle of ancients, is not ‘obvious’. Angiulli, follow-ing Riccati’s steps, believes that the law of the lever in the past was not achieved withmetaphysical certitude, or only by recourse to a priori categories, and it was a ‘sim-ply’ experimental principle, i.e. endowed with only physical evidence. He claimsthat the attempts carried out by Aristotle, Archimedes, Galileo, Stevin, Huygens, togive the law of the lever the metaphysical certitude have failed. Analogous criticismsare expressed about the demonstrations concerning the wedge and the screw.

Even when passing over the difficulty of its proof, the law of the lever is notsufficiently general, it cannot be used for the equilibrium of fluids and even in simplemachines such as the pulley, the inclined plane and then the wedge and the screw.For the pulley, for example, the law of the lever cannot be applied strictly becauseit is not possible to identify a priori the centre of rotation of the pulley and thus thefulcrum of the lever itself; this difficulty had already been removed by Descartes

GE

I

F

L

M

B

XP

K

Fig. 9.2. The inclined plane reduced to the lever


F

Z

BE

b

a

M

N

GA C

X

Fig. 9.3. The law of the lever

[4].14 But according to Angiulli the law of the lever cannot be applied always to theinclined plane but only in the case of a rolling such body as a sphere of Fig. 9.2,which rolls having L as a fixed point. In such a case it is possible to assume a leverwith fulcrum in L, the weight concentrated in E and a power P applied to any pointB of the sphere. When instead the body slides on the inclined plane, for Angiulli,it is not possible to see any lever. This position could indicate that Angiulli did notknow Galileo’s Le mecaniche, where the lever is applied also – probably only – tosliding bodies.

According to Angiulli the principle of equality of the actions, instead, is generallyvalid and true in all cases. In the following, with reference to Fig. 9.3, I report thepreliminaries of the proof of the law of the lever.

The thesis:

I say, from the principle of actions it can be deduced that when in the rod ABC there isequilibrium, the power Z is to the power X as CN : CM, i.e. that the equation Z · CM = X ·CM is valid [4].15 (A.9.10)

And the proof:

Let the points Z and X be the centres of the powers Z and X. Conceive now an infinitesimalmotion be born in the rod ACB, so that the points A and B describing the arches Aa, Bbcome in a and b. From point b to point X draw the line bX, and from point A to point Z theline aZ, then with the centre Z, and the interval aZ describe the arc that matches AZ in F,and similarly with the centre X […]. This makes it evident that AF is the minute space ofaccess of the power of Z, and bG the minute space of recess from the centre of the power X.The principle of action requires, that to have equilibrium in the rod ABC, the power Z is tothe power of X as bG : AF [4].16 (A.9.11)

The text allows comparison of the proof as given by Angiulli with that reportedin Chapter 2. In Angiulli the virtual displacement is in real time with the points A

14 pp. 35–37.15 p. 42.16 pp. 42–43.


and B that describe the arc of a circle; the directions of the powers depend on themotions of A and B. In a modern discussion the virtual displacements of A and Bare considered to occur along the tangent, i.e. along the perpendicular to the lineAB; also it is not assumed that such motions really act on the points A and B andtherefore the direction of the forces shall be deemed unchanged. The two approachesare equivalent if one assumes, as Angiulli does, infinitesimal displacements. Thenthe change of direction of the forces is negligible, the arc can be confused with thetangent and the virtual displacements projected on the forces are indistinguishablefrom the spaces of access and recess, so the virtual work, calculated as the sum ofthe products of virtual displacements multiplied by the components of forces alongthem, coincides with the action, measured as the sum of the products of forces bythe access or recess spaces.

The fifth chapter of the Discorso intorno agli equilibri is dedicated to the prin-ciple of equivalence, i.e. the composition and decomposition of forces by the rule ofthe parallelogram, a principle which had been assumed ‘recently’ by Pierre Varignonin hisNouvelle mécanique ou statique, as the foundation of statics. Angiulli first crit-icizes as not metaphysically obvious the demonstrations of the principle as given byNewton and Varignon, because they assume that two forces produce the same mo-tion both when they are applied together and when they are applied separately. Heenhances instead that of Daniel Bernoulli, appreciated by Riccati [339].

Also the principle of equivalence of course can be proved with the principle ofaction. The proof refers to Fig. 9.4 and considers three powers AD, AC and ABforming the sides and diagonal of a parallelogram respectively.

To demonstrate the equivalence between the powers AD and AC with AB,Angiulli demonstrates the equilibrium, assuming that the power AB is directed to-wards A, while the others start from A. He considers a general virtual displacementof the point of application of the three powers from A to R. For this motion there

F

B

C

F

RA

M

N

po q

D

Fig. 9.4. The rule of the parallelogram


are the small spaces of access oR, qR, and recess pR (notice pR is opposite to -AB).The principle of action ensures the balance provided it is CA×oR + AD×qR = AB×pR, but for a purely geometrical lemma, shown previously [4],17 this conditioncan be satisfied whatever is the displacement of A, if and only if AC, AD and ABare the sides and the diagonal of a parallelogram. The demonstration follows a fairlyfaithful proof by Bernoulli and Varignon [267] and is less general than that reportedby Riccati in his De’ principi della meccanica, which considers the equality of theactions accounted for two types of virtual displacements, one along AB and the otheralong the perpendicular to AB, reaching two equations of equilibrium [208].18

After showing the fertility of the principle of action, however, Angiulli concludessomewhat surprisingly:

Note secondly, that making comparisons between the principle of equivalence and that ofactions, both of them must be estimated to be equally fruitful and extended, with that dif-ference, for which in some cases the principle of equivalence can be used with more skilland elegance, in other cases, it is more convenient and appropriate to use the principle ofactions.And finally it is to be noted that the method of composition, and resolution of the forces isnot the true method of nature, but it is a method Geometers have developed for the easiestand quickest solution of their problems. Nature in its work never composes or resolves theforces, but always uses actions, that being equal and opposite, causing the equilibrium to beproduced [4].19 (A.9.12)

Therefore in part he gives up the claim to make of the analytical principle of actionsthe cornerstone of statics, holding that in practice the approach based on the law ofcomposition of forces is often convenient. The principle of actions remains the honorof being the general principle from which all methods of solving static problems canbe derived. This position of Angiulli, that after Lagrange’s Mécanique analytiqueseemed unjustified given the high fertility of both theoretical and applicative shownby the principle of virtual velocities, is today supported by most scholars of appliedstatics, who while recognizing the theoretical importance of the modern virtual workprinciple prefer, in applications, to introduce directly the cardinal equations of staticswhich are the analytical counterpart of the principle of composition of forces, inwhich constraint forces are introduced as unknown quantities.

9.1.1.5 The applications to simple machines

In the final part of his book devoted to applications, Angiulli considers the equi-librium of the simple machines: the lever, the shaft with the wheel, the pulley, thewedge, the screw and the inclined plane. The principle of action is used as a neces-sary criterion and provide the results of equilibrium in a very simple way, showingits great advantage in dealing with constraints. I refer as an example to the analysisof the equilibrium of the simple pulley:

In the fixed pulley, to have equilibrium, it is asked the equality between power and weight.Let [Fig. 9.5] AB be a fixed pulley, which has around it the rope EABD, to the end D of

17 pp. 56–57.18 pp. 26–29.19 pp. 63–64.


which the weight P is attached, to the other end E the power is applied, which supports theweight. I say that for there to be equilibrium it is necessary, that the power applied in E isequal to the weight P [4].20 (A.9.13)

The proof is very simple and is developed linearly by Angiulli:

Be an infinitesimal motion in the direction of the power applied in E, so that the end of therope E comes in G, while the end D comes in H. It is too obvious, that EG is the minute spaceof access to the centre of the power, and DH the minute space of recess from the centre ofthe weight. So because there is equilibrium between power and weight, it is necessary thatthe first is to the second as DH : EG. But DH = EG, for supposing that the rope does notpractice any strain, but remains always of the same length, the length DAE will be equal tothe length HAG; then, DH and EG will remain equal between themselves. So because therebe equilibrium in the pulley it is asked the power be equal to the weight. What is needed todemonstrate [4].21 (A.9.14)

C

A

E

G

P

B

H

D

Fig. 9.5. The equilibrium of the pulley

In the statics of fluids, treated at the end of the applications, there is the influenceof Galileo [115] and of Riccati [208]. The principle of equality of actions is some-times used as the principle of equality of Galileian moments, as when Angiulli showsthat the free surface of a fluid is horizontal. More articulated is the discourse onthe calculation of the pressures on the bottom of a container of any shape and thedemonstration of equality in the level of communicating vessels that I refer to as anexample.

Let [Fig. 9.6] GHPQ be a whatever trap, if an arm of it GH is filled with homogeneous fluid[...]. Given that the fluid poured in the trap will be in equilibrium, it will raised to the sameheight in one arm and another of the trap [4].22 (A.9.15)

The demonstration takes an infinitesimal motion for whichGHwill drop up to IK andat the same time, in the other arm, the surface PQ is brought into RS; in the abovethere is implied an admissible motion of the fluid congruent with the constraints

20 p. 89.21 pp. 89–90.22 p. 126.


V

V

Vdx

G H

KI

R

P

x

dz

z

S

Q

V

T U

Fig. 9.6. The equilibrium of fluids

imposed by the walls and its incompressibility. The infinitesimal motion of all thefluid occurs by translation, along the communicating vessels, of a fluid volume equalto the elementary volume indicated below with ΔV. Named x the height of the singlefluid layer of the vessel GH and z that corresponding to the vessel PQ, given that theweight of the volume ΔV is proportional to the same ΔV, the total actions in the twovessels are proportional respectively to

∫ΔV dx and

∫ΔV dz, where dx and dz are

changes in the level of the elementary volumes ΔV into which the fluid is supposedsplit and the integrals are extended along the two vessels. Imposing the equality ofthe actions and in view of the constancy of volume ΔV, one has

∫dx =

∫dz. That is

to say that the fluid being in equilibrium, their perpendicular GT and QU have to bethe same and that is what was to demonstrate [4].23

The demonstration of Angiulli is a generalization of that reported byGalileo [115]and Bernoulli [39] in which cylindrical communicating vessels were considered withuniform sections. The generalization is made easy by the use of Calculus.

9.1.2 The principle of actions of Vincenzo Riccati

Vincenzo Riccati was born in Castelfranco Veneto in 1707and died in Treviso in 1775. Riccati was the fourth sonof Jacopo Riccati. He began his studies at the College ofSt. Francesco Saverio in Bologna, run by the Society of Jesusunder the guidance of the mathematician Luigi Marchenti.In 1734 he went to teach Latin and Italian literature at theCollege of St. Caterina of Parma and in 1735 began thestudy of theology, before in the Educandato of San Roccoin Parma, then (1736–1739) in the Institute of St. Ignazio in

Rome. From 1739 he taught mathematics in the College of St. Francesco Saverioin Bologna, succeeding Marchenti. In February 1741 he took his vows. Vincenzo

23 pp. 126–128.


Riccati remained in Bologna until 1773 when, due to the suppression of the Societyof Jesus, he returned to Treviso, host of his brothers Montino and Giordano and inthe same year refused the chairs of mathematics at the universities of Bologna andPisa. After less than two years he died in Treviso. He was, with Ruggiero GiuseppeBoscovich, one of the greatest mathematicians of the Society of Jesus [252].

In what follows I will refer mainly to the work the De’ principi della meccanicadel 1772 [208], which incorporates the ideas of the previous Dialogo di VincenzoRiccati della Compagnia di Gesù dove ne’ congressi di più giornate delle forze vivee dell’azioni delle forze morte si tien discorso [207], but exposes them more clearlyand in more mature form. Riccati argues that although many ‘writers’ who dealt withthe method of statics make use of Bernoulli’s rule of energies, the principle of actionis no different from it, and is clearer and tested more solidly. The other scholars,according to Riccati their conclusion based only on the success of the method ofenergies, in some cases to extend it into more general situations. Here is a commentby Riccati on Johann Bernoulli’s version of the principle of virtual velocities, limitedto what was reported by Varignon; Riccati for sure did not know in full the letter ofBernoulli to Varignon of 1715. To notice that Riccati does not cite Lagrange who in1763 introduced the principle of virtual velocity in the study of the libration of thelune (see Chapter 10):

I only warn that the famous theorem of the incomparable Johann Bernoulli, who was shownin all themachines by themost learnedMr. Varignon, is simply a consequence of the equalityof contrary actions, which is necessary in any equilibrium. Bernoulli’s theorem is as follows:In any equilibrium of how many and various powers they want, in any way applied, andagents for any direction, the sum of positive energies is equal to the sum of negative energies,as long as you take them as affirmative. By name of energy Mr. Bernoulli does not mean butthe product of the power and the virtual velocity of the same power, which will be positiveif it follows the direction of the power, negative if it follows the opposite direction. Andwho does not see that the virtual velocity of the power is proportional to the space, of whichthe body, or the power, approach the centre of the forces, or whether the powers are elasticropes, to the contraction or relaxation of the ropes. So Bernoulli’s energy is the same, or atleast proportional, to what is called by us action of the power [208].24 (A.9.16)

Riccati begins his enunciation of the principle of action, in a way that was taken al-most exactly by Angiulli. He notes that the nature of the equilibrium requires equal-ity between quantities dependent on forces which he, as Angiulli, indicates with thename of power, but the equality cannot be directly between the powers themselves.The equilibrium depends on how the powers act, i.e. on their action. To explain thedifference between the powers and their actions Riccati refers to a weight suspendedby a thread, even as presumed by Angiulli and writes:

To declare as the powers and their actions are distinct, I conceive a heavy body suspendedby a wire, which prevents him to descend, and to approach the earth. So far I do not mean butthe power of gravity applied to the body, which is contrary to the elasticity of the wire, whichcontrasts it, and does not leave any chance to have effect. I cut the wire, and the elasticitycontrary to gravity is removed. Now the power subsequently and continuously replicatesits impulses or stress to the body, which is obliged to change its state. The sum and theaggregate of pulses is named the action of this power, and the effect, i.e. the mutation of

24 p. 237.


state is proportional not to the power but to the aggregate of its pulses. Therefore it can bedistinguished three quantities, namely the power considered in itself, that is usually calledpressure, the action, which is the aggregate of its impulses, with which the power pushesthe body, composed of the power and the number of pulses, and the effect, i.e. the muta-tion of state of the body, effect that has not proportion to power, but to its action [208].25

(A.9.17)

He takes from Leibniz the concept of dead force, the power that generates a sequenceof infinitesimal impulses continuously replicated. If the pulses of this type are notdestroyed, they can be added and should become the action of the power, which isresponsible for the change in state. The constraint does not exert a force then, as itis accepted in modern statics, but has merely the role to destroy the power of pulses.Note that Riccati says that the action of the power is directly responsible for thechange of state, and the language is here closer to Newton than to Leibniz, as wasfor Angiulli. According to Leibniz, the action of the dead force produces the livingforce which – and not the action of the force – is responsible for the change in state.Also the definition of the living force is separated from that of Leibniz and assumesa Newtonian notation; for Riccati the living force is merely the inertia of a body inmotion which requires a force to be stopped:

So that, therefore, although the centrifugal force is not really anything else than the inertiaof the body in some considered circumstances, it is neither useless to introduce it in thereasonings, nor ought it be excluded from physics: in fact it will be profitable to fix itslaws, thus the theorems elaborated around such a force by the learned and deep ChristiaanHuygens, will be recognized to be true and beautiful.I will answer similarly around the living force. It is by no means so distinct from the forceof inertia, rather it is the same force of inertia with some special conditions changed: andit will be useful to consider it with this name, and to fix the laws, which in many problemsand research can be of great benefit [208].26 (A.9.18)

I pass over the treatment of Riccati which is substantially the same as that reportedby Angiulli to comment on an important observation that the latter will not refer towhat Riccati writes:

To put in good view our method, and the use of the principle, I must not omit an observation,that appears important to me. When only one motion is possible, as happens to bodies thatrotate about any axis, then if the slightest motion is conceived, the spontaneous actionsmeasured by the space of access and recess are found equal, and without hesitation theequilibrium can be deduced. But when more motions in different directions are possible,whether devising some arbitrary motion, I still find the equality of the above actions, but Icannot claim a full equilibrium, but only say that that motion is impossible, and that in thatdirection the powers are balanced [208].27 (A.9.19)

Riccati then realizes that, for the equilibrium, when one has a system of bodies withmore than one degree of freedom, it is not sufficient to impose the vanishing of theaction in a single degree of freedom, because then equilibrium would be assumed

25 p. 13.26 p. 26.27 pp. 23–24.

9.2 Ruggiero Giuseppe Boscovich 233

only with respect to the motions allowed by this degree of freedom, but it is neces-sary to impose the vanishing of the action in all motions permitted by the degreesof freedom of the system. And he makes the verification by considering the equilib-rium of three parallel forces applied perpendicularly to a straight line, consideringthe cancellation of the actions for displacements resulting from rotation about twoseparate points. If this condition is met, Riccati shows that the actions are canceledfor the rotation of the system of forces around a generic point of the plan.

More interesting is the application to the equilibrium of three forces applied toa point. This demonstrates that the rule of the parallelogram has some significancebecause it is part of the discussion on the validity of the proof of Daniel Bernoulli,‘based only’ on a priori considerations. Riccati in 1746 had written a work specifi-cally on the rule of the parallelogram entitledCausa physica compositionis ac resolu-tionis viribus [339], in which hewas already using the principle of action. Proving therule of the parallelogram, he proved to have validated the principle of actions, since,as Bernoulli, Riccati believed that the rule of the parallelogram could be proved apriori.

Even Riccati, as Angiulli, uses the ‘demonstration’ of the validity of his principleof action, to join the controversy over the living forces and support the thesis ofLeibniz. After presenting the principle of action Riccati passes then to the applicationof his principle to fluids and to all the large chapter on dynamics, which he studiesbased on the law f ds = mvdv, instead of f dt = mdv.

9.2 Ruggiero Giuseppe Boscovich

Ruggiero Giuseppe Boscovich was born in Dubrovnik (for-mer Ragusa) in 1711 and died in Milan in 1787, his motherwas Italian, and his culture was Italian too; for this reason heis often considered as an Italian scientist. At fourteen he wassent to continue his studies in Rome at the Collegio Romanoof the Society of Jesus, where in less than thirty years he be-came one of the most distinguished teachers in the chair ofmathematics and geometry, dealing with a broad spectrumof disciplines, from natural philosophy – with the develop-

ment of a new theory that unifies the physical and chemical forces in a single law– algebraic and geometric calculus problems posed by the type of application inconstruction engineering, optics, geodesy, meteorology, hydraulics. He was also theauthor of verse, both in the traditional scientific-didactic poem of Lucretius and inthe context of the Arcadian academy, of which he was a member. As a Jesuit and aneminent scientist he was asked to perform delicate diplomatic tasks in a time whenscientific and technical expertise were considered important in resolving conflictsover political boundaries, geodetic measurements, possession of watercourses, andso on. His main contribution to theoretical mechanics is the Theoria philosophiaenaturalis [48]; note also some more technical works including studies on the motionof solid bodies.


In late 1742 and early 1743, with Thomas Le Seur, François Jacquier, he gave tworeports to the press on their views on the static conditions of the dome of St. Peter.They are the Parere di tre mattematici sopra i danni, che si trovano nella cupoladi S. Pietro [137], referred in the following as the Parere, and the Riflessioni sopraalcune difficoltà spettanti i danni, e risarcimenti della Cupola di S. Pietro [138].The three mathematicians were commissioned by Benedetto XIV, a pope learnedand sympathetic to the new science, which would shed light on the health of his‘residence’, worried by alarming rumors on the spread of cracks.

The opinion of the three mathematicians, even for the drama of the conclusionsrelated to a possible collapse, instead of putting an end to rumors, provoked a livelydiscussion among architects, mathematicians, and ‘gentlemen’, which ended withthe appointment of Giovanni Poleni. His suggestions, in fact coinciding with theremedies proposed by the three mathematicians and consisting of the introductionof new metal rims, were finally accepted by the pope and put into practice by thearchitect Luigi Vanvitelli (1700–1773) in 1748.

9.2.1 A virtual work law for Saint Peter’s dome

Of the two reports of the three mathematicians the most interesting is the Parere. It isimportant in the history of architecture because it represents one of the first attemptsto set up on a mechanical basis the testing and design of a structure as complex as adome. From my point of view, the report is important because it represents the firstapplication of a law of virtual work for the study of a complex system, by introducingsome innovation.

While in the Parere the division of scientific expertise is not made explicit andno one knows who did what, a few years later Boscovich will consider himself theauthor of the particular form of the virtual work law used for the static analysis ofthe dome of St. Peter:

I even used some of the research that I had already made twenty years ago on the greatdome of St. Peter’s in Rome, and especially the theory which led me to know the force withwhich an iron ring pulled out by force applied perpendicularly to all points, resist, finding itgreater a little more than six times than it would be for the same iron bar pulled directly intothe direction of its length, i.e. in proportion of the radius to the circumference of the circle,whence then the Marquis Polini28 conceived of the idea of that experience where a wire ofsilk, octagon in shape, pulled out from all the angles to be broken, needs a force about sixtimes greater than when another companion wire was pulled directly [49].29 (A.9.20)

The reasons for which Boscovich has used a law of virtual work for the static analysisof the dome of St. Peter are not known. It is possible that he has read the works on theprinciple of actions of Vincent Riccati, his contemporary and brother of the Societyof Jesus.

28 Boscovich perhaps deliberately misspells the surname of Giovanni Poleni.29 p. 54.

9.2 Ruggiero Giuseppe Boscovich 235

9.2.1.1 The mechanism of failure and the forces

After an accurate description of the ‘sufferings’ of the dome in the first part of theParere, the three mathematicians go to the second part to verify the degree of resis-tance of the dome. And this in two stages, first identifying the failure mechanism,namely the virtual motions that can support the system of the bodies of the structure,after they have identified the actual and potential failure points, then considering theresistant forces and those which tend to cause the failure of the dome.

The failure mechanism, i.e. the ‘system’ is shown in Fig. 9.7. The dome is set ona drum called attico which in turn rests on another drum of much greater thickness,it is reinforced by sixteen great ribs which are constrained by sixteen buttresses anda number of metal chains. To analyze the failure mechanism, the whole structureis divided into sixteen slices, each corresponding to a rib, and the possible motionof each slice is analysed. The base of the drum, cracked vertically moves like asingle rigid body with the buttresses of the drum. This complex is represented by theexternal rectangle ABF in Fig. 9.7b, which shows a section corresponding to the ribsand buttresses. The wall of the drum, the inside of the base and the attic behave asanother rigid body, is identified by the internal rectangle BDHI. Each rib identified inFig. 9.7b by IHMN acts like a single rigid body, bringing with it a piece of the dome.

Note that the three mathematicians consider the sixteen ribs of the dome as theonly structural elements, neglecting the resistance of the remainder of the dome.

Fig. 9.7. Mechanism of failure of the dome (a). Layouts of the threemathematicians (b, c) (modifiedfrom [279])


Also note that the failure mechanism considered, although not stated explicitly, isthree-dimensional, consisting of sixteen radial elements similar to that of Fig. 9.7aconnected by round chains, and it is mainly for this link that it is not reducible to aplane system.

Once the failure mechanism was determined, Boscovich applied its principle ofvirtual work, ensuring that the sum of the positive work, due to resisting action, isequal to the sum of the work of negative actions that lead to failure. Notice thatBoscovich uses the Galilean term moment for virtual work.

Two forces push out on the spring gi [HI in Fig. 9.7], i.e. the weight of the lantern and theweight of the ribs with the portions of the dome, and likewise the two forces, which resistthe impulse, i.e. the circular chains, or circles, and the support […], reduced to two distinctcomponents, the first of which is the drum HI, with the interior part of the base CDF and thesecond the buttresses with the outside part ABF of the detachment of the base itself. […] It isnot possible to make a precise estimate of the detachment of the parts and their resistance. Itdepends largely on the quality of the concrete and the diligence of the work. To evaluate theforces and whether these are in equilibrium, it is convenient first to determine the absolutequantity of them, and then that which by mechanicians is called moment. To get the absoluteamount of force, with which on one hand the lantern and the vault of the dome act, and onthe other hand the base, the drum, the buttresses act to contrast the pressure, it is convenientto know their weights [138].30 (A.9.21)

For the evaluation of the moments of the weights Boscovich has no difficulty; it isenough to multiply the weight of each masonry mass (ribs, buttresses, etc..) for thevertical displacement of its centre of gravity. More complex is the calculation of thevirtual work of the chains that connect the ribs. Boscovich does it elegantly, with areasoning by analogy, by assimilating the chain to a straight bar of length equal tothe circumference of the chain, subject to the same stress. In this way he finds thatthe moment of each chain is obtained by multiplying the length change of the radius,given by the horizontal displacement of point H in Fig. 9.7, by 6π.

Assuming this principle, first it seemed to us, that the energy of a chain of iron, bent in acircle must grow above the absolute force, which would have if it were lying on the straightposition, in the same proportion which has the circumference of the circle to the radius,that is a little more than six. Conceive a force distributed throughout the circumference ofa circle which is forced to relax and dilate in the act of breaking up, and an iron rod of thesame length pulled by another force, such as would do a weight hung vertically. In the lattercase, the descent of the weight in tending the fibers would be equal to the sum of all of theextensions of the fibers arranged along the same rod, but in the first by expanding the circle,and growing so its circumference, the force that compels it advances as much as the radius ofthe circle grows, while the sum of the extensions of the same fibers arranged around wouldbe equal to an increment of the entire circumference [138].31 (A.9.22)

The conclusion of Boscovich’s work is that the chains are not strong enough to en-sure the resistance of the dome, and therefore they should be replaced with morerobust chains. For more details, see [279, 280]. Here I will only add that Boscovicha few years later applied the yet to be born law of virtual work to study the resistanceof the cathedral of Milan [279, 281, 257].

30 pp. 23–24.31 pp. 26–27.

10

Lagrange’s contribution

Abstract. This chapter is entirely devoted to Lagrange’s VWL. In the first part thefirst introduction of the law of Lagrange is reported, which has a wording similarto that of Bernoulli. Lagrange calls his VWL and Bernoulli’s the principle of vir-tual velocities. In the central part the wordings of VWL in the two editions of theMécanique analytique in terms of virtual displacements (following a foundationalroute) and the Théorie des fonctions analytique in terms of virtual velocity (follow-ing a reductionist route) are presented. In the final part an overview of D’Alembert’smechanics is presented aimed at an understanding of the extensibility of VWLs todynamics.

A hundred years after the first edition in 1687 of Isaac Newton’s Philosophiae nat-uralis principia mathematica [175], in 1788 the first edition of Joseph Louis La-grange’s Mécanique analytique [145] saw the light. Between the two publishingevents that symbolize, respectively, adolescence and maturity of classical mechan-ics, there was an intense and fruitful process of understanding, systematization andgeneralization of the possible approaches to mechanics. During this period, the ageof the Enlightenment, a huge effort was made: of systematisation and developmentof concepts elaborated in the previous century, especially in mathematics and inphysics; of development of the Baconian sciences; of development of technology.

In recent past historians of science assumed Isaac Newton’s contribution as ex-pounded in his masterpiece as the climax of classical mechanics and that scholarsof the Enlightenment added little to it. Today historians realise that this was mis-leading and that period far from being a dark century was filled with fundamentalcontributions and most concepts of mechanics were laid down then [356, 389, 388].Newton’s mechanics was for sure incomplete; it allowed only the study of the equi-librium and motion of material points free in space, with a mathematical apparatusnot completely developed, based on an uncertain Calculus. Problems related to sys-tems of constrained points remained unapproachable, as did the study of continuumbodies either rigid or deformable. Moreover Newton had to face, mainly on the Con-tinent, people scarcely disposed to follow the religious metaphysics behind his work.


238 10 Lagrange’s contribution

Thus he was not well considered and Cartesianism was still dominant. This situationpromoted a profound innovation in the mechanics as formulated by Newton. New-ton’s main concepts, that of force included, remained dominant among scientists buta different approach based on work and energy became a serious contender. Thecompetition was not only on the basis of a more or less appealing ontology but alsoof either a more simple or a more complex mathematical formulation.

At the beginning of the XVIII century Newton was surely seen by his contem-poraries as one of the most prominent mathematicians and physicists but not as theone who carried mechanics to its final form. This appears clear from the literatureof the time. Newton’s mechanics was considered unsatisfactory by many scholarsfrom both epistemological and ontological points of view, also because of the intro-duction of forces acting at a distance, which were considered occult entities. Morefundamentally for scientists, Newton’s mechanics was considered to be incomplete,because it was limited essentially to material points free in space, and were unsuit-able to solve problems raised by the technology of the times. As an example of theopinions of the period, some comments by Daniel Bernoulli and Leonhard Euler arereported below:

Theories for the oscillations of solid bodies that up to now authors furnished presuppose thatinto the bodies the single points relative positions remain unchanged, so that they are movedby the same angular motion. But bodies suspended at flexible threads call for another theory.Nor it seems that to this purpose the principles commonly used in mechanics are sufficient,because clearly the mutual dispositions of points is continuously changing [30].1 (A.10.1)

But as with all writings composed without analysis, and that mainly falls to be the lot ofMechanics, for the reader to be convinced of the very truth of these propositions offered, anexamination of these propositions cannot be followed with sufficient clarity and distinction:thus as the same questions, if changed a little, cannot be resolved from what is given, unlessone enquires using analysis, and these same propositions are explained by the analyticalmethod. Thus, I always have the same trouble, when I might chance to glance through New-ton’s Principia or Hermann’s Phoronomiam, that comes about in using these, that wheneverthe solutions of problems seem to be sufficiently well understood by me, that yet by makingonly a small change, I might not be able to solve the new problem using this method [100].2

(A.10.2)

So Newtonian principles alone did not seem enough; it was necessary to look forsome other more fundamental principles. Problems faced byXVIII century scientistswere less demanding from a philosophical point of view than those faced by Newton,nothing less than the search for the universal laws, but this notwithstanding theywere not simpler. They concerned for example the search for the oscillation centreof a rigid body and the study of vibrations of a chain. The search for the centre ofoscillation was quite a relevant and difficult problem; it is equivalent to finding thelength of a simple pendulum with the same period. The problem was substantiallysolved by Christiaan Huygens in his Horologium oscillatorium of 1646 [135], bymeans of a first formulation of the theorem of living forces. Jakob Bernoulli cameback to the subject in 1713, with a completely different and promising approach, in

1 p. 108.2 Preface, translation by I. Bruce.

10 Lagrange’s contribution 239

the paperDémonstration générale du centre du balancement a toutes sortes de figuretirée de la nature du levier [32] whose redaction preceded Newton’s Principia. In itone can find roots of both D’Alembert’s principle and the angular moment equation.Johann Bernoulli also faced these problems at the end of the XVII century; a relevantexample is published in hisOpera Omnia of 1742 [37], where he first introduced theconcept of angular acceleration. The problem of a vibrating chain was studied byscientists such as Euler, D’Alembert, Johann and Daniel Bernoulli.

Among the other problems that occupied the minds of scientists of the XVIII cen-tury it must be remembered the study of the equilibrium of elastic rods, the motionof bodies on mobile surfaces such as, for example, the motion of a heavy body uponan inclined plane were without friction. Johann Bernoulli studied the motion of amaterial point with a Newtonian approach by introducing among the external forcesalso the constraint reactions, calling them immaterial forces, as they were outsidethe bodies in touch [37]. Notice that the assimilation of constraint reactions to ordi-nary forces were quite common in statics, but in dynamics the problem was muchmore complex conceptually, because reactions should be endowed with activity. Eu-ler himself, who developed principles of mechanics which made it easy to introduceconstraint reactions, tried as much as possible to avoid their explicit use. To con-clude, in the solution of the various problems no reference was made to a uniqueprinciple and were sought as analogies with already solved problems.

This way of thinking is sufficiently documented by the introduction to part IIof Lagrange’sMécanique analytique, which sets forth the history of dynamics fromGalileo up to the end of the XVIII century. The description of Newton’s contributionis relegated to the first part and occupies a small space, but in this regard Lagrange’sposition is certainly not objective.Most of the considerations are devoted to the studyof Jakob, Johann and Daniel Bernoulli, Euler, D’Alembert and Hermann. In the lat-ter part of the introduction Lagrange presents the ‘principles’ most frequently used:the conservation of living forces, the conservation of the motion of the centre ofgravity, the principle of the areas and the principle of least action. And it was on theprinciple of least action that had focused the attention of Lagrange as a young manwho, in 1762 published the Application containing the results of his studies on theprinciple of the minimum action applied to dynamics, making a fundamental contri-bution to the development of this principle, then seen as the most promising way tosolve complex mechanical problems. Despite earlier formulation by Maupertuis andEuler, Lagrange claims for himself the credit for having moved the principle frommetaphysics to science.


10.1 First introduction of the virtual velocity principle

Giuseppe Lodovico Lagrangia, better known as Joseph LouisLagrange, was born in Turin in 1736 and died in Paris in1813. In 1755 he was appointed assistant professor at theRegie scuole di artiglieria of Turin. In 1757 with GiuseppeAngelo Saluzzo (1734–1810) and Giovanni Francesco Cigno(1734–1790) he founded the Società privata torinesewith theaim of promoting research in mathematics and in sciences,which later became the Reale accademia delle scienze ofTurin. In 1766 D’Alembert, who will become a great friend

of Lagrange, knew that Euler was returning to St. Petersburg and wrote to Lagrangeto encourage him to accept a post in Berlin. Lagrange finally accepted. He succeededEuler as director of mathematics at the Berlin academy in 1766. For twenty yearsLagrange worked at Berlin, producing a steady stream of top quality papers and reg-ularly winning the prize from the Académie des sciences de Paris. During his years inBerlin his health was rather poor on many occasions, and that of his wife was evenworse. She died in 1783 after years of illness and Lagrange was very depressed.Three years later Frederick II died and Lagrange’s position in Berlin became a lesshappy one. In 1787 he left Berlin to become a member of the Académie des sciencesde Paris, where he remained for the rest of his career. Lagrange was made a memberof the committee of the Académie des sciences to standardise weights and measuresin 1790. He married for a second time in 1792; his wife, Renée-Françoise-AdélaideLe Monnier was the daughter of one of his astronomer colleagues at the Académiedes sciences. In 1794 the École polytechnique was founded with Lagrange as itsfirst professor of analysis. In 1795 the École normale was founded with the aim oftraining school teachers. Lagrange taught courses on elementary mathematics there.Lagrange was, with Euler, one of the greatest mathematicians of the second half ofthe XVIII century [265, 353, 289].

10.1.1 The first ideas about a new principle of mechanics

Even before the publication of tome II of the Miscellanea philosophica mathemat-ica societatis privatae Taurinensis, where he presented his version of the minimumaction principle [140], there were indications that Lagrange thought of a more fun-damental and general principle than that of least action. In a letter to Euler dated 24November 1759 [318],3 Lagrange wrote that he had composed elements of differ-ential calculus and mechanics and had developed the ‘true metaphysics of its prin-ciples’. This can be considered as a symptom of Lagrange’s comprehension of thefundamentality of the principle of virtual work [262].4 Today a virtual work princi-ple is considered more general than that of minimum action, because it allows to takeaccount also the non-conservative forces. But to Lagrange a virtual work principle

3 p. 107.4 pp. 216–218.

10.1 First introduction of the virtual velocity principle 241

would appear to be more fundamental, mainly because it could be assumed alone asthe foundation for all of mechanics. The minimum action principle instead requiredsupport of the living force principle; a principle assumed as true but not completelyevident

To confirm Lagrange’s change of mind in his 1759 letter, I refer to a memoirpublished in the same tome II of the 1762Miscellanea by one of his students, Davidde Foncenex (1734–1799), who was actually older than Lagrange. In this paper, aswell as an attempt to rationalize andmake precise D’Alembert’s mechanics, a simplevision of the principle of virtual velocities is presented, inspired by Lagrange, whohad since 1760 understood the generality of his principle:

The composition of forces just as it was made can be used to demonstrate the equilibrium ofthe lever, and conversely this last proposition once proved, can easily give the compositionof forces. It also gives us a highly simple demonstration of the principle of virtual velocities,which can rightly be considered the most fertile and most universal of mechanics: indeedany other principles easily reduce to it, the principle of conservation of living forces, andgenerally any principle imagined by a fewMathematicians to facilitate the solution of severalproblems, are nothing but a purely geometrical consequence or better, are this same principlereduced to formulas [107].5 (A.10.3)

These are the same words Lagrange will use in the Mécanique analytique. For thisreason it is generally felt that the Foncenex memoir was influenced by Lagrange andcan be seen as a witness of the turn of Lagrange regarding the principle of mechanics.

An interesting and fairly comprehensive analysis of times and ways of the transi-tion from one to another principle is that of Galletto, who reports the named letter toEuler dated November 24th 1759. Here and elsewhere [314, 260], it is assumed thatLagrange had already developed his method during the preparation of his courses forthe Regie scuole di artiglieria in Turin, where the approach to mechanics with theprinciple of virtual work would have been better understood by students, comparedto that based on the principle of least action which required knowledge of analy-sis too complex for that time. Unfortunately there is no evidence that this is true,and maybe there will never be because the manuscripts of Lagrange’s lessons onmechanics are unavailable.

This chapter aims to highlight and critically analyze the principle of virtual worklaw which is at the foundation of Lagrange’s masterpiece, the Mécanique analy-tique. The study is conducted with numerous references also to the first publicationavailable on this subject, the Recherches sur la libration de la Lune; it will be consid-ered in some detail also a few years later in a work, dealing with the same subject, theTheorie de la libration de la Lune. For a historical reconstruction of the events thataccompanied the publication of these works, I have examined mainly the secondaryliterature [318, 260].

5 p. 319.


10.1.2 Recherches sur la libration de la Lune

The Recherches sur la libration de la Lune [142] is the first of many memoirs ofastronomy written by Lagrange and the work he presented, winning, for the com-petition held by the Académie des sciences de Paris in 1764 on the topic: “If it canbe explained by any physical reason why the moon always presents the same facetoward us, and how it may be determined by observations and by theory, if the axisof the satellite is subject to some motion of its own, similar to what is known for theearth’s axis, which produces the precession of the equinoxes”. The work belongs tothe Italian period of Lagrange.

In [260] there are reported elements of some interest for the reconstruction of thescientific training of Lagrange; analyzing the works produced from 1755 to 1764,one can understand how the main ideas, that made him famous, were already broadlydefined. In 1759 the first volume of the Miscellanea philosophica mathematica so-cietatis privatae Taurinensis (Melanges de Turin) was published [139] with threeLagrange memoirs, of which the most important, Recherches sur la nature et la prop-agation du son, is a long work on the solution of the equation of vibrating stringswhich already showed signs of the greatness of Lagrange, the same work that Eulerexpressed a flattering opinion of it [260].

In 1762 the second volume of the Miscellany was published, in which there isanother memoir about the nature of sound propagation, but mainly there are twomemoirs: Essai d’une nouvelle méthode pour determiner les maxima et les formulesdes integrales minimum indéfinies [140] and Application de la méthode exposée dansle mémoire précédente à la solution des problèmes de dynamique differents [141].These two memoirs are not minor works, but they should rather be regarded amongthe most important contributions of Lagrange to analysis and dynamics [255]. Thefirst memoir lays the foundations for the calculus of variations and the second is acoherent treatise on mechanics, based on the principle of least action.

The following year, 1763, Lagrange ended the Recherches sur la libration de laLune and, in 1765, before finally leaving Turin for Berlin, he wrote a vital work onastronomy, Recherches sur le inégalités des satellites de Jupiter causées par leurattractions mutuelles [142, 143].

In the Recherches sur la libration de la Lune (hereinafter referred to as theRecherches), for the first time, Lagrange formulated the dynamic equations of mo-tion using a “new principle of mechanics” alternative to the least action: a variantof Bernoulli’s rule of energies, named by him and after him the principle of vir-tual velocity. The originality of the presentation of the virtual velocity principle inthe Recherches is not always fully understood [314, 254] . It is widely acceptedamong historians of science that the ingredients that allowed Lagrange the use of avirtual work law for the study of dynamic problems were already available at thetime. In reality it is not. The principle of virtual velocity was not yet an operationaltool.

Theoretical difficulties remained open to the epistemology of time, to present itas a principle. It is likely that Lagrange until the time of the publication of the Mé-canique analytique in 1788 was convinced of the substantial evidence of the virtual


velocity principle. Most likely, until after that date, as a result of heated discussionsthat followed in which the positions that deny the evidence of the principle wereprevalent, Lagrange changed his mind (see also § 10.3.2) and in the second editionof the Mécanique analytique, immediately after the extremely positive opinion ofthe first edition, he added the remarks that it is not obvious enough in itself to beelected as a founding principle.

There were also technical difficulties such as those related to the types of admissi-ble displacements to be considered, such as whether or not they should be compatiblewith the constraints. Moreover, the issue of the Recherches is dynamic and the useof the virtual velocity principle requires its generalization to dynamics. It shouldbe “combined with the principle of dynamics due to D’Alembert, it is a species ofgeneral formula containing the solution of all problems concerning the motion ofbodies” [145].6 The interpretation of the principle of D’Alembert by Lagrange, nowclassic, although not corresponding to the original formulation, is that the accelerat-ing forces with reversed sign ‘balance’ the applied forces: “This method reduces allthe laws of motion of bodies in those of their equilibrium, so reduces Dynamics toStatics” [145].7

Before entering the merits of the use of the virtual velocity principle in theRecherches it is useful to give a short account of Lagrange’s conception of force.The discussions among post-Newtonian scientists on the ontological and epistemo-logical status of force were not yet over, comparing on the one hand the positions ofNewton and Euler, who considered it as a primitive quantity, and for whom f = mais a law, on the other hand the positions of D’Alembert and Lazare Carnot who seesforce as a derived quantity, and for which f = ma is essentially a definition. La-grange assumed a pragmatic and instrumentalist position, which will be followed bymost scientists of the XIX century. To them force is a useful concept for mechanicsand if one is not too ‘exacting’ it does not create any problem in the development oftheories. It is symptomatic in this regard the comment of Poinsot, a few years afterthe publication of the Mécanique analytique:

The force is therefore any cause of motion. Without considering the force in itself, howeverwe conceive very clearly that it is acting in accordance with a direction and with a certainintensity [195].8 (A.10.4)

Lagrange explicitly addresses the idea of force essentially with two brief commentsboth contained in the introduction to the first part of theMécanique analytique. Rightat the beginning of the 1788 edition, Lagrange will simply say:

In general, force or power is defined as the cause, whatever it is, that impresses or tends togive motion to the body to which it is supposed applied, and it is by the amount of motionimpressed or about to be impressed, that the force should be estimated. In the state of equi-librium, the force does not have an active role, it produces only a mere tendency to motion,but it must always be measured by the effect it would have if it were not blocked. Taking any

6 p. 12.7 p. 179.8 p. 2.


force, or its effect, as a unit, the expression of all other forces is only a ratio, a mathematicalquantity that can be represented by numbers or lines. It is from this point of view that inmechanics the forces should be considered [145].9 (A.10.5)

As can be seen on the one hand there is the definition of the ontological status offorce, conceived as the cause, to which, however, Lagrange does not appear to givespecial weight. On the other hand, it is shown how to measure it, and how it entersinto mechanics as a physical quantity. Even D’Alembert (and to some extent LazareCarnot), did not consider unlawful the use of the concept of force in mechanics, forthe explanation of qualitative character. But for the quantitative determination ofthe forces it is necessary to refer only to the effects that – according to him – arethe only ones able to be measured. In this sense, for D’Alembert, ma is more than adefinition of force, it represents the only possible measure of the force-cause basedon the effects, which for simplicity is referred to as force.

The other comment on force by Lagrange is given in the second edition of theMécanique analytique, where while comparing the principle of the composition offorces and the law of the lever, he wrote:

It can however cannot but recognize that only the law of the lever has the advantage ofbeing based on the nature of equilibrium considered in itself, [that is regarded] as a stateindependent of the motion, so there is a fundamental difference in the way the powers thatare in equilibrium are considered in these two principles, so that, if they were not linkedbecause of the results, one could reasonably doubt that it would be allowed to replace thefundamental principle of the lever to that resulting from independent considerations on thecompound motions [148].10 (A.10.6)

This problematic position reflects Lagrange’s embarrassment that is always presentto address the problems of statics when power (force) is defined as ma and viceversa when addressing dynamic problems starting from the static concept of power.D’Alembert and Carnot strive to bring their concepts of force to the pre-Newtonianone; Euler conversely tries to base his dynamics on the concept of power, identifiedas a force capable of producing acceleration. Lagrange instead does not analyze indepth the problem and makes a substantial dichotomy between dynamic force andstatic force without any concern for the inconsistencies that can be determined at atheoretical level.

It is worth noting that in the historical introduction to statics in the Mécaniqueanalytique, Lagrange, to indicate force, uses either the words power (puisssance) andforce (in 8500 words or so, one has 78 times power and 87 force). In the historicalintroduction to dynamics instead he uses almost only the term force (in 8300words orso, one has 4 times power and 127 force). The same applies basically for the properlyscientific parts of the Mécanique analytique, where when dealing with statics theuse of power is frequent, and when dealing with dynamics the use of force is almostexclusive.

This asymmetry reflects an asymmetry of concepts. In statics, force has alwaysbeen considered a primitive concept, lacking of very challenging metaphysical con-notations, it is the muscle force, that can always be replaced by a weight and the9 pp. 1–2.10 p. 17.


word power is consolidated by a long tradition, previous to Galileo. In dynamics,things are different. Here, force has an ontological connotation not yet completelyclear. In the ‘scientific’ parts on dynamics of the Mécanique analytique, Lagrangeuses a language similar to that of D’Alembert, and then it would seem that for himthe dynamical force is for definition equal to ma. It is classified as both accelerat-ing force or driving force. The accelerating force is simply equal to the acceleration:“one can always determine the value of the force acting on bodies in each instant, bycomparing the velocity gained in this time with the length of this time” [148].11 Thedriving force is instead equal to the product ma. Lagrange does not appear to confera different ontological status to the two terms, although the former is preferred whendealing with a single particle.

In the following passage from the second edition of the Mécanique analytique,Lagrange returns to his conception of force:

As the product of the mass by velocity expresses the force over a body in motion, so theproduct of the mass by accelerating force, that we have seen to be represented by the velocitydivided by the element of time, expresses the elemental or rising force, and this amount,when taken as a measure of the effort that the body can make because of the elementaryvelocity that has acquired or tends to acquire, is what is called pressure. But if it is consideredas a measure of force or power to give the same velocity, then it is called motive force[148].12 (A.10.7)

The accelerating force, considered as elemental or rising force, is compared withfinite force of Cartesian conception, that is the product of mass and velocity. In thefinal part of the passage it also seems to be a reference to Leibniz’s idea of dead force,according to which pressure is generated by the destruction of elementary impulses.

The failure to merge concepts of static and dynamic forces becomes really em-barrassing when the way Lagrange treats constraint forces, which are assimilated to‘real’ forces that constraints exercises ‘really’, is considered. These forces cannot beframed neither in D’Alembert dynamics nor in Euler’s. In the first case because con-straints do not exert forces and in the latter because they are placed in an impreciseway.

10.1.2.1 Setting of the astronomical problem

In the study of the motion of the moon Lagrange assumes a coordinate system,X ,Y,Z, centred in the lunar centre of gravity as shown in Fig. 10.1.

As the first coordinated plane he chooses the plane τ parallel to the ecliptic, i.e. theorbit of the earth around the sun. The X axis is directed toward the first point of Ariesϒ, the Y axis is perpendicular to X and contained in τ, the Z axis is perpendicularto τ. The moon is considered as a rigid body, not necessarily spherical in shape. Ageneric element of it dm is subject to the forces of gravity of the earth and the sunwhich have the expression:

TR2 dm,

SR′2

dm, (10.1)

11 p. 240.12 pp. 245–246.


S

R

R

R′

R′E

dm M

M

X

Y

Z

X

Y

Z

M

E

ϒEcliptic plane

Aries point

Moon detail

Fig. 10.1. Moon, earth and sun configurations for two different instants

where T and S are the masses and R and R′ are the distances of the moon from theearth and sun respectively – Lagrange, as a custom of the times, avoids exhibitingthe gravitational constant. In addition to these forces there should also be consideredthose quantities that, with a terminology borrowed from D’Alembert, are called ac-celerating forces, given by:

d2Xdt2

dm,d2Ydt2

dm,d2Zdt2

dm, (10.2)

with X ,Y and Z that define the position of dm.Immediately Lagrange introduces the ‘principle of D’Alembert’, by quoting:

“These accelerating forces taken in the opposite direction and combined with theforces T/R2dm and S/R′2dm, keep balanced the system of all points dm, i.e. the en-tire mass of the moon, in equilibrium around its centre of gravity, supposed fixed”.The analysis of the reference system and of the forces is completed in the first twoparagraphs.

Note that the reference system chosen by Lagrange is not fully inertial and then inaddition to the forces he listed there should be considered also the dragging forces.Since the reference system has axes that do not rotate with respect to the fixed stars,drag forces are reduced to a homogeneous field defined by aLdm, where aL is theacceleration of the centre of the moon in its motion around the earth and the sun. Ifone considers the moon as a rigid body, these forces have no effect on its motionwith respect to the X ,Y,Z system as they are equivalent to a single force applied inthe origin M (the centre of the moon). It is unclear whether Lagrange, in ignoring thedrag forces, was conscious of the above considerations, or he has simply let himselfbe guided by instinct.


10.1.2.2 The symbolic equation of dynamics

Paragraph III of the Recherches is certainly one of the most important of the work.It begins with the enunciation of the principle that lays at the basis of Lagrange’smechanics:

There is a generally true principle in statics, that if any system of bodies or how many pointsyou wish, each solicited from arbitrary powers, is in equilibrium and if someone gives thesystem a little motion, arbitrary, because of which each point moves along an infinitelysmall space, the sum of each power multiplied by the distance traveled by the point whereit is applied, in the direction of this power will always be zero [142].13 (A.10.8)

Expressed in modern language this principle states that if a system of particles isin equilibrium, the active forces fi to which it is subject have to satisfy the relation∑ fi · dui = 0, being dui the generic infinitesimal displacement of the applicationpoint of fi and dot the scalar product, i.e. if a system is in equilibrium the virtualwork of active forces fi shall be zero for any virtual displacement.

Lagrange here does not specify the nature of the infinitesimal displacements, itwill be understood from the context that they are compatible with constraints. Withno other comment Lagrange begins to ‘calculate’ as follows:

Imagine that for an infinitesimal variation of the position of the Moon about its centre, thelines X ,Y,Z,R,R′, assume the values:

X +δX , Y +δY, Z+δZ, R+δR, R′+δR′

it is easy to see that the differences:

δX , δY, δZ, δR, δR′

express the distances passed at the same time by point dm in the opposite direction to thatof the powers:14

d2Xdt2

dm,d2Ydt2

dm,d2Zdt2

dm,TR2 dm,

SR′2

dm

acting on that point. It will then hold, for the condition [necessary] of equilibrium, the generalequation:15∫

L

[d2Xdt2

dm(−δX)+d2Ydt2

dm(−δY )+d2Zdt2

dm(−δZ)+TR2 dm(−δR)+

SR′2

dm(−δR′)]

13 pp. 8–9.14 In the transcription I made a slight typographical variation of Lagrange’s formulas, from thebeginning and then even in relation (A), instead to indicate the element of mass with α, I used thesymbol dm, putting it after the force per unit mass rather than before. The same notation was usedin Lagrange’s Théorie de la libration de la Lune.15 The negative signs are due for the first three terms to the principle of D’Alembert, for which theaccelerating forces must be treated with sign changed, for the last two terms, to the convention onsolar and terrestrial gravity forces which are considered positive if attractive, while the change ofdistance is positive if there is an increment of distance.


i.e., changing sign:∫L

[d2Xdt2δX +

d2Ydt2δY +

d2Zdt2δZ

]+T

∫L

δRR2 +S

∫L

δR′

R′2dm (A)

[142].16 (A.10.9)

Lagrange’s comments on how he arrives at equation (A) are extremely scanty. Hewill remedy, but only in part, his terseness in the next paragraph. The language offormulas is however sufficiently clear. According to the principle “true in general”one has to sum (integrate) the products of the elementary gravity and acceleratingforces multiplied by the motions of their points of application and then impose thissum equal to zero.

It is not entirely clear whether Lagrange considers the balance of forces existing inthe portion of space occupied by the moon, or considers the equilibrium of materialpoints dm, rigidly connected to each other, which form the moon. Distinguishingone case from the other, allows one to see if Lagrange is applying the principle ofvirtual velocities in exactly the form given to him by Bernoulli, which affects thebalance of forces alone or, if he enters in the classical tradition of virtual work laws,according to which the equilibrium concerns bodies more than forces.

The analysis of the entire Lagrange’s text seems to lead to the second possibility,and when he speaks of the elements dm it seems implied that he refers to a materialpoint. When he is proving the theorem of living forces he will apply the “princi-ple true in general in statics” explicitly to a system of particles and finally whenhe is concerning the changes δX ,δY,δZ,δR,δR′ he speaks about the motion of theelements dm and not the points of application of forces.

On the other hand, this interpretation is put a bit in crisis by the fact that Lagrangedid not mention at all the internal forces between individual elements dm. Theseinternal forces are certainly compatible with the concept of force of Lagrange, whoadmits the physical reality of the constraint forces (see Section 10.2.1), and thereforecan consider them originated from the interaction of individual particles. Lagrange’ssilence can be explained, however, in two ways. With the first way, which I thinkis the more plausible, it is arguable that the concept of internal forces was not tooclear to Lagrange, which can perform as Euler when, in his study on the motion ofrigid bodies in 1750, decided not to consider the influence of internal forces based onthe heuristic principle that they could not give a global contribution. With the secondway it can be assumed that the internal forces are associated with constraint reactionsand, in view of the application of the virtual velocity principle they can be ignoredbecause their virtual work is zero. It should be noted that, for a comprehensive andsatisfactory analysis of the internal forces, it is necessary to wait for the work ofCauchy in 1822 [63], in which the concept of tension in continua is presented, as theinternal forces of contact.

The above equilibrium equation (A) is now under the name of symbolic equationof dynamics. In the Mécanique analytique Lagrange will refer to this as the general

16 p. 9.


formula of dynamics. I would like to stress that to get it, without highlighting thething, he used two independent principles: the first, referred to as “true in general inStatics”, is clearly the virtual velocity principle, while the second, which consists inthe use of the accelerating force with sign reversed, is, as will be explained furtherbelow, the version of the principle of D’Alembert given by Lagrange. It is absolutelynot true that the virtual velocity and D’Alembert principle were widely known andshared by ‘geometers’, and indeed it can be argued with good reasons that Lagrangewas the first to enunciate clearly and disseminate them. And Lagrange in the Mé-canique analytique, had a different approach with the two principles, for the virtualvelocity principle he presented a demonstration – at least in the second edition andin the Théorie des fonctions analytiques – and for the principle of D’Alembert heprovided a comment with a rather extensive historical analysis.

The paragraph IV of the Recherches is intended to clarify what stated in the pre-vious paragraph. Given the clarity of Lagrange, it is best to report his comments infull:

The principle of Statics that I outlined, in the end is nothing but a generalization of whatis usually called the principle of virtual velocities [emphasis added], which has long beenknown by Geometers as a fundamental principle of equilibrium. Mr. Jean Bernoulli is thefirst, for what I know, who has seen this principle in general form and applied it to all mattersof statics, as it can be seen in section IX of the newMechanics of Mr. Varignon, where suchskilled Geometer, after referring Bernoulli’s principle, shows, in various applications, itleads to the same conclusion as that of the composition of forces [142].17 (A.10.10)

Note that though Lagrange was the first to name Johann Bernoulli’s rule of energiesprinciple of virtual velocities, to identify the virtual work he used instead of the termenergie the Galilean term moment.

The next part of the Recherches, paragraph V, is dedicated to making explicitthe symbolic equation of motion and the solution of the differential equations thatfollow. Of some interest is the presentation, at the end of section IV, of probably thefirst ‘satisfactory’ proof of the theorem of living forces, obtained as an immediateapplication of the symbolic equation of motion [267]. In the following I will onlyexpose some hints on how to make explicit the symbolic equation of motion, withthe introduction of the Lagrangian coordinates.

The symbolic equation of dynamics contains virtual displacements not yet ana-lyzed; for them the space of admissible values is not defined. Lagrange expressesthe idea of admissibility using the concept of independent variables (known todayas Lagrangian variables or coordinates) already introduced in the Applications [282]and developed, after the Recherches, in the Théorie de la libration de la Lune and intheMécanique analytique, by recognizing that the virtual velocity principle leads tomany balance equations as there are independent variables. In the words of Lagrangea few years later:

In the following, keeping in mind the equations of condition among the coordinates of thedifferent bodies, given the nature of this system, the variation in those coordinates will be

17 p. 10.


reduced to the smallest possible number, so that the resulting variations are completely in-dependent and totally arbitrary. Then if the sum of all terms with each of these variationwill be equated to zero, it will be obtained all the necessary equations for determining themotion of the system [145].18 (A.10.11)

In the three-body problem that Lagrange is studying, consisting of earth, sun – repre-sented by two material points located in their centre of gravity – and moon – treatedas a rigid body – there are three degrees of freedom each for the earth and the sun (thethree motions of their centre of gravity) and three degrees of freedom for the moon(the three rotations). In total there are nine degrees of freedom and the equation ofsymbolic dynamics is expected to deliver nine equilibrium equations. In the simpli-fied treatment of the three bodies that Lagrange is considering, he takes as known –without ever explaining clearly that assumption – the positions of the earth and thesun and, therefore, virtual displacement involves only possible motions of the moon,identified by three angular coordinates, to which, through the symbolic equation ofdynamics, are associated three balance equations.

10.1.2.3 The virtual velocity principle

It is not entirely clear what Lagrange meant by the term ‘generalization’ used in theopening of paragraph IV: “The principle of statics that I come to expose in substanceis nothing but a generalization of what it is usually called the principle of virtualvelocities”. On the one hand, Johann Bernoulli’s principle of virtual velocity thatrefers to concentrated forces, is certainly extended to distributed forces, and its rangeof validity is specifiedmore precisely. On the other hand, the generalization concernsthe extension of the principle to dynamics by assimilating accelerating force, withsign reversed, to ordinary forces.

Bernoulli’s statement does not justify in full the applications that Lagrangemakes. Here it should be noted that Lagrange for sure knew Bernoulli’s rule of ener-gies only from what was reported by Varignon in his Nouvelle mécanique, becausethe letter of Bernoulli to Varignon in 1715 was not published. From what was re-ported by Varignon it can be evinced as follows:

• virtual velocities are not necessarily compatible with constraints. Or better Bernoulligives no particular attention to the problem of constraints, and he considers a gen-eral system of forces in equilibrium, which may also contain constraint reactions;

• the class of virtual velocities is limited to a single degree of freedom;• the wording of energy rules is expressed in the language of geometry and the

displacement and force vectors are represented as oriented segments;• the rule is limited to systems of particles and concentrated forces.

Lagrange will change all these points.

• the virtual velocities, according to the classical formulations of virtual work lawsare always compatible with constraints, implicitly assumed holonomic and inde-pendent of time;

18 p. 197.


• the virtual velocities have a variation defined by the number of degrees of freedomof the system under examination;

• applications are purely analytical, as Lagrange works with components he needsto not consider the projection of forces in the direction of the displacement, sincethe components of the forces and those of the displacements have the same direc-tion;

• forces are not necessarily concentrated, and indeed in the Recherches they areonly distributed. The idea to consider distributed forces other than those due toweight, and in particular the accelerating forces, was not common also after 1750,when Leonhard Euler, in a memoir of the Academy of sciences of Berlin [101]proposed that Newton’s second law ( f = ma) could also apply to entities otherthan the material points, such as an infinitesimal element of a continuum (d f =dma). From this point of view the generalization from concentrated to distributedforces, did not seem trivial.

For the generalization to dynamics see § 10.4.

10.1.3 The Théorie de la libration de la Lune

The Théorie de la libration de la Lune is another of Lagrange’s great work of astron-omy, published in the memoirs of the Academy of sciences in Berlin, for the year1780 [144] and written partly in response to questions about motions of the moonthat remained open after publication of the Recherches. Here Lagrange introducesand uses the virtual velocity principle, initially without any reference to the spe-cific astronomical problem, but referring directly to the general case of an indefinitenumber of bodies. A significant improvement in the analytical aspects should alsobe registered, in particular in the calculation of the virtual work of inertia forces,so that the developments of the Théorie de la libration de la Lune are substantiallysimilar to those of theMécanique analytique and also contain a statement of the nowfamous Lagrange equations.

At the beginning of section I there appears an introduction of the virtual velocityprinciple and the principle of D’Alembert:

1.The principle provided byMr. D’Alembert reduces the laws of dynamics to those of statics,but the search for these laws by the ordinary principles of equilibrium, the lever and/or thecomposition of forces, is often long and painful. Fortunately there is another principle ofStatics, more general, and, above all, that has the advantage that it can be represented byanalytical equations, which alone contains the conditions of equilibrium of any system ofpowers. This is the principle known as the law of virtual velocities. It usually will be setthis way: when two powers are in equilibrium, the velocities of the points to which theyare applied, estimated according to the direction of these powers, are in inverse ratio to thepowers themselves. But this principle can be made more general as follows.2. If any system of bodies, reduced to some points subject to any forces, is in equilibriumand if this system is given any little motion for which each body moves along an infinitelysmall space, the sum of the powers each multiplied for the displacement of the point whereit is applied along the direction of this power is always zero [144].19 (A.10.12)

19 pp. 15–16.


Lagrange presents here, as he will also do in theMécanique analytique, two differentstatements of the virtual work law. The first is indirect, for only two bodies, andresembles the wording of the law of the lever; the second refers to that of JohannBernoulli. Saying that the first ‘principle’ can be made more general by the second,Lagrange wishes to emphasize that the statement of Bernoulli is somewhat impliedby a description of the law of the lever generalized, scaling down in some way theoriginality of Bernoulli.

To make explicit his virtual work law, following shortly after, Lagrange adoptsfrom the outset an approach that foreshadows its application to dynamics. Insteadof powers he speaks of accelerating forces – which this time as with D’Alembertare just accelerations. Although the virtual work law is applied to the case of anynumber of points and not only the earth-moon system, the accelerating forces areconsidered always acting toward a centre.

If P,Q,R, ...,P′,Q′,R′, . . . are the accelerating forces acting on the mass pointsm,m′, . . . toward the centres p,q,r, ..., p′,q′,r′, . . . , and if there is equilibrium, thefollowing equation is obtained – the symbols are Lagrange’s:

m(Pδp+Qδq+Rδr+ . . .)+m′(P′δp′+Q′δq′+R′δr′+ . . .)+ · · ·= 0. (10.3)

To get the values of the changes

δp,δq,δr, . . . ,δp′,δq′,δr′, . . .

the expression of the distances p,q,r, . . . , p′,q′,r′ should be differentiated considering thecentres of forces as fixed [144].20 (A.10.13)

The Théorie de la libration de la Lune continues to consider the virtual work of theinertia forces, providing a general expression, and introducing technical refinementsto the Recherches. I will limit myself here to indicating only the additional evidenceLagrange gives of independent variables:

Furthermore, given the mutual positions of the bodies, there will be many constraint equa-tions between the variables x,y,z, x′,y′,z′, . . . by which it is possible to express all the vari-ations one over the other or rather by other variables in small number and such that theyare entirely independent and correspond to the various motions that the system can receive[144].21 (A.10.14)

10.2 Méchanique analitique and Mécanique analytique

In 1788 the first edition of Lagrange’s masterpiece was published in one volumewith the title Méchanique analitique; it was published in a second edition in twovolumes with a slight change of title, Mécanique analytique, reflecting changes inthe written French language; the first volume released in 1811 with Lagrange stillalive, the second in 1815 [145, 148, 150]. The third edition was published in 1853–1855 by Joseph Bertrand (1822–1900); it differed from the second edition only for

20 p. 16.21 p. 20.

10.2 Méchanique analitique and Mécanique analytique 253

the different typeface and the addition of notes by Bertrand. The fourth edition isthe one shown in the complete works of Lagrange edited by Joseph Alfred Serretand Gaston Darboux. It was printed in 1888–1889 and it is simply a reproduction ofBertrand’s edition. An English edition is also available [153].

In the following, as it was indicated previously, I shall refer to the work of La-grange as to theMécanique analytiquewhen it is not necessary to specify the edition.Otherwise I will talk about the first edition of the Mécanique or the Méchanique, orof the second edition of the Mécanique analytique.

According to the not uncommon use of time, theMécanique analytique containedhistorical ‘hints’, except that these hints were quite substantial and well made, so thatthey are generally regarded as the first example of a history of mechanics, an exam-ple that has influenced modern history for a long time. TheMécanique analytique isdivided into two parts, one dealing with statics and the second with dynamics; eachof the two parts is preceded by a history. The history of the first part, the only one Iwill examine below, presents the virtual velocity principle as the most recent of theprinciples used in mechanics. It was preceded by the principle of the lever and thelaw of composition of forces. Lagrange’s task is precisely to implement the appli-cation of the virtual velocity principle to all problems of mechanics. Only after thepublication of his work, prompted by criticism from his colleagues, will Lagrangecommit himself to a demonstration of the principle.

Those who have hitherto written on the principle of virtual velocities have dedicated them-selves to prove the truth of this principle, by the conformity of its results with those of theordinary principles of statics, rather than to show the uses that can be made to directly solveproblems of this science. We have proposed to dedicate ourselves to that subject with allthe generality of which it is liable and to deduce from the principle at issue the analyticalformulas, which contain the solution to all problems of equilibrium of bodies, in the sameway the formulas of subtangents, the osculating radius, etc. contain the determinations ofthese lines among all the curves [145].22 (A.10.15)

10.2.1 Méchanique analitique

In the historical part of the first edition of theMécanique analytique, after a brief his-torical summary, citing Galileo, Torricelli, Descartes, and Wallis, Lagrange sets outthe virtual velocity principle in substantially the same form in which it was exposedin the Recherches, attributing its formulation to Johann Bernoulli:

If any system of as many bodies or points one wishes, each solicited from any powers, is inequilibrium, and if this system is given an arbitrary small motion, under which each pointpasses along an infinitely small space, which will be its virtual velocity, the sum of powers,multiplied each by the space that the point where it is applied passes in the direction of thatpower, will always be zero, considering as positive the small spaces in the direction of powerand as negative the spaces in the opposite direction [145].23 (A.10.16)

22 pp. 44–45.23 pp. 10–11.


Plate 4. Front page of two editions of Lagranges’sMécanique analytique (reproduced with permis-sion, respectively, of Accademia Nazionale San Luca, Rome, and of Biblioteca Guido Castelnuovo,Università La Sapienza, Rome)

The historical introduction of statics ends with the following passage:

I think I can say in general that all the general principles that can still be discovered in thescience of equilibrium, will not be but the same as the principle of virtual velocities, givenin a different way, and from which they differ only in form. But this principle is not onlyitself very simple and general, it has, in addition, the valuable and unique benefit to result ina general relation that contains all problems that can be posed on the equilibrium of bodies.We will expose this relation in all its extensions, we will also try to present it in an evenmore general way than what has been made to date, and provide new applications [145].24

(A.10.17)

Here the confidence of Lagrange, in both the capacity of the virtual velocity principleto solve whatever mechanical problem and its simplicity of use, is clear.

Theoretical-technical aspects begin to be addressed after the historical part. La-grange cannot bring himself to apply directly Bernoulli’s virtual velocity principle,and tries to justify it with a simpler and more traditional statement, as he made inthe Théorie de la libration de la Lune:

The general law of equilibrium in machines is that the forces or powers, are among themin inverse proportion to the velocities of the points where they are applied, estimated in thedirection of these powers [145].25 (A.10.18)

This virtual work law, less general than Bernoulli’s, as it refers only to two forces,is more simple and intuitive. Before presenting Lagrange’s proof is worth noting

24 pp. 11–12.25 p. 12.


that the two forces, to which the above statement refers, must not be thought of asapplied to the same point; they may be and generally are applied to different pointsof a machine, which transfers forces from one point to another. The two points aremoving in the direction set by the kinematics of the machine itself; the statementcontains the law of the lever as a special case.

Lagrange begins his demonstration from three forces, P,Q and R in equilibriumwith each other, generally applied to three distinct points p,q and r, somehow con-nected. For the principle of solidification (see also § 14.2.1), the equilibrium is notdisturbed if any of the points of application of the forces is supposed fixed. Assum-ing, for example, r as the fixed point, the forces P and Q are still balanced betweenthem. Denote by dp and dq the virtual velocities, i.e. the infinitesimal displacements,of the points p and q estimated in the direction of P and Q, in the case of any act ofmotion. The above virtual work law gives:

PQ

=− dqdp

(10.4)

or Pdp+Qdr = 0.For the three powers, considered two by two, then it is:

Pdp+Qdq = 0; Pdp+Rdr = 0; Qdq+Rdr = 0. (10.5)

To obtain an equation of virtual work valid for the three forces, Lagrange uses anargument, not too sharp, which will resume and improve in the second edition of theMécanique analytique [274].26 In essence he argues that because the points p,q andr influence each other, among their displacement dp,dq and dr there must exist arelationship, linear because infinitesimal displacements are concerned. For example,it can be written:

dp = mdq+ndr, (10.6)

where m and n are numerical values that depend on the constraints and the type ofvirtual displacements considered. If dr = 0 is assumed, as in the first of (10.5), it isdp = mdq; dq = 0, as in the second, it is dp = ndr. Substituting these values in thefirst two of (10.5), one obtains:

Pmdq+Qdq = 0Pndr+Rdr = 0,

(10.7)

which adding member to member give:

P(mdq+ndr)+Qdq+Rdr = 0 (10.8)

or, for (10.6):

Pdp+Qdq+Rdr = 0. (10.9)

26 p. 137.


A similar result is obtained by combining (10.5) according to the other two possiblecombinations (the first with the third and second with the first).

The proof outlined above lends itself to a process of mathematical induction, inthe sense that the virtual work law for n+1 powers is valid knowing it is valid for npower. For any number of them it is then:

Pdp+Qdq+Rdr+ etc. = 0, (10.10)

which is precisely the principle of virtual velocities according to Bernoulli. At thispoint Lagrange introduces the definition of moment: “We will call each term of thisformula, the moment of force, using the word moment in the sense Galileo has givento it, i.e., as the product of force for the virtual velocity. So that the general formula ofequilibrium will consist in the equality of the moments of all forces to zero” [145].27

The term moment will continue to be adopted for a long time until it is graduallyreplaced by virtual work introduced by Coriolis (see Chapter 16).

Lagrange continues: ‘With the use of this formula, the difficulty is reduced todetermine the values of the differentials dp,dq,dr, in accordance with the nature ofthe given system” [145].28 The following is a brief but precise explanation how toget the expressions of differentials:

We replace the expressions of dp,dq,dr, &c. in the proposed equation, and because theequilibrium of the system holds in general and in every sense, this equation must be satisfied,regardless of all the indeterminate quantities. It will be equated to zero separately the sumof the terms affected by the same indeterminate quantity [emphasis added]. There will be asmany particular equations as many indeterminate quantities. Now it is not hard to believethat their number must always be equal to that of the unknown quantities of the configurationof the system, then this method will give as many equations are those needed to determinethe equilibrium state of the system [145].29 (A.10.19)

The indeterminate quantities are those today called Lagrangian coordinates, whichLagrange, as reported in the previous pages, had already introduced in the Additionof 1762. This brief explanation of how to obtain the equilibrium equations will becompleted in the second edition of the Mécanique analytique where Lagrange alsointroduces the concept of generalized forces.

In the third section of the Méchanique analitique, Lagrange starts applying thevirtual velocity principle to determine the equations that are necessary – but that maynot be sufficient – for the equilibrium of a system of material points p, p′, p′′, etc.subject to the forces P,P′,P′′, etc., constrained together in some way (for the rigidbody the equations that are obtained are also sufficient for the equilibrium). Lagrangegets first the equations of equilibrium to translation, then to rotation. In both cases,he divides the generic virtual displacement of the system of material points into twoparts: a global displacement of the system and relative displacements between thepoints. He chooses as global motion that of an arbitrary point p, while the relativemotions are assumed equal to the difference of their total motion and that of p. In

27 pp. 15–16.28 p. 16.29 p.16.


the following I report only the proof of the equilibrium to translation, because it is abit simpler and equally explanatory of the approach used by Lagrange.

Naming x,y,z the coordinates of the point p, x′,y′,z′, those of the point p′,x′′,y′′,z′′, those of point p′′, he poses:

dx′ = dx+dξ, dy′ = dy+dη, dz′ = dz+dζ;dx′′ = dx′+dξ′, dy′′ = dy′+dη′, dz′′ = dz′+dζ′; (10.11)

where dξ,dη,dζ represent the relative motions and dx,dy,dz the global motions.Applying the equation of moments he obtains:

0 = (Pcosα+P′ cosα′+P′′ cosα′′+ etc.)dx+ (Pcosβ+P′ cosβ′+P′′ cosβ′′+ etc.)dy+ (Pcosγ+P′ cosγ′+P′′ cosγ′′+ etc.)dz+ P′(cosα′dξ+ cosβ′dη+ cosγ′dζ)+ P′′(cosα′′dξ′+ cosβ′′dη′+ cosγ′′dζ′)+ etc.,

(10.12)

where α,β,γ, α′,β′,γ′, etc. are the direction cosines of the powers P, P′, etc.Assuming that the system is isolated, it is clear that the constraints only result from

the mutual relations between the material points, and then from ξ,η,ζ, ξ′,η′,ζ′, etc.and not by x,y,z, etc. which may vary arbitrarily. So because the above equation issatisfied for all possible virtual displacements, it is necessary that the coefficients ofdx,dy,dz vanish. Lagrange has therefore come to the following, already well knownas the cardinal equations of equilibrium to translation [145]:30

Pcosα+P′ cosα′+P′′ cosα′′+ etc. = 0Pcosβ+P′ cosβ′+P′′ cosβ′′+ etc. = 0Pcosγ+P′ cosγ′+P′′ cosγ′′+ etc. = 0.

(10.13)

The method of Lagrange to obtain the cardinal equations, and in particular the mean-ing of the terms where relative displacements appear, attracted the curiosity of Fos-sombroni which provided an alternative derivation also based on the virtual velocityprinciple [109].31

In section IV Lagrange introduces his method of multipliers. Let L =const.,M =const., N = const., etc. be the constraint equations that govern a system of ma-terial points with coordinates x′,y′,z′, x′′,y′′,z′′, etc. which differentiated lead tothe conditions dL = 0, dM = 0, dN = 0, etc.; then the following considerationsapply:

Now since these equations must be used to remove an equal number of differentials in theequation of virtual velocities, after which the remaining coefficients of the differentials mustall be matched to zero, it is not difficult to prove through the elimination of linear equations,that it will be the same if to the equation of virtual velocities the different constraint equationsdLN = 0, dM = 0, dN = 0, & c. are added, each multiplied by an indeterminate coefficient,and then the sum of all terms that are multiplied by the same differential is equated to zero.

30 p. 28. The equations of moments are deduced on pp. 28–29.31 pp. 101–115.


This will provide as many particular equations as the number of differential equations. Fi-nally, the indeterminate coefficients by which the constraint equations are multiplied, areeliminated from these equations [145].32 (A.10.20)

Applying the above ‘instructions’ Lagrange gets the equation of the type:

Pdp+Qdq+Rdr+ etc.+λdL+μdM+νdN = 0, (10.14)

that he names general equation of equilibrium. Note that in this equation dp, dq, dretc. can vary freely, as if there were no constraints attached.

In the next sections of the Mécanique, Lagrange addresses problems of someinterest in mechanics, including Maupertuis’s law of rest; they do not cover basicaspects of virtual work laws and therefore will be ignored.

10.2.1.1 Constraint reactions

In mechanics, the idea of constraint reactions has evolved along with those of forceand constraint. For Aristotle, a constraint was essentially an impediment for a bodyto reach its natural place. Removing the constraint leaves the body free to move.Even among the ancient Greeks, especially among engineers, it was clear that theeffect of a constraint could be obtained with a power, a muscular force for instance.If a heavy body was fixed on a hook by means of a rope, it was clear that the roleof the hook could be played by a muscular force, appropriate to support the weight.Therefore, the possibility of interchangeability between constraints and powers isseen from the beginning of mechanics, although they remain distinct concepts, theconstraint does not exert a force on a body but it seems as if it does. One can talkabout constraint reaction as the force that, for equilibrium, has the same effect of theconstraint.

By accepting the rule of the parallelogram as the primary tool for addressing thestudy of static problems, which occurred due to Varignon’s Nouvelle mécaniqueou statique of 1725, the reaction forces in this sense began to appear explicitly asgeometric or algebraic variables in the calculations. As the equilibrium reduces to theannulment of the sum of the forces, if also a constraint contributes to the equilibrium,there is nothing more natural than in the equations of equilibrium symbols appearingto represent the forces equivalent to constraints, e.g the reactive forces. Only afterNewton, with the introduction of forces at a distance as physical magnitudes, theprinciple of action and reaction and the emergence of the corpuscular concept ofmatter and the theory of elasticity, did the ontological status of the reactive forcesbegin to change. The constraints are no longer, in general, impediments to motionbut they become bodies composed of particles that are centres of forces and theconstraint reactions are ‘real’ forces that the constraint-body exert over other bodieswhich are to interact with them.

In the XVIII century, constraints are still generally modeled as hard bodies, thatis, as being capable of absorbing motions and impulses acting at right angles to them,but toward the end of the century with the emergence of the Eulerian and Newtonianconcepts of force, constraints are entities treated as dispensers of forces and thus

32 pp. 45–46.


no longer merely passive. By the XIX century, this second point of view becomesprevalent, especially among the French scientists.

Lagrange is situated in an intermediate position, on the one hand he considersconstraint forces as the forces required to perform the functions of the constraints,on the other hand he gives them the ontological status of active forces, or powers. Heintroduces the reactive forces from the general balance equation given above, notingthat to terms λdL,μdM,νdN one can give, by analogy, the mechanical meaning ofmoment and that “each equation of constraint is equivalent to one or more forcesapplied to the system in accordance with given directions, so that the equilibriumstate of the system will be the same – either one uses the account of these forces,or one refers to the equation of constraint [145].33 The direction of the forces isorthogonal to the surfaces L = 0, M = 0, N = 0.

Regarding the interpretation of the multipliers λ,μ,ν, etc. as forces, Lagrangestates:

Reciprocally, these forces can replace the constraint equations resulting from the nature ofthe given system, so that with the use of these forces the system may be considered as com-pletely free and without any constraint. And the metaphysical reason [emphasis added] canbe seen, because the introduction of the terms λdL+μdM + &c. in the general equilibriumequation, makes that this equation can then be treated as if all the bodies of the system werecompletely free. This is the spirit of the method of this section.Properly speaking, the forces in question shall take account of the resistance that bodieshave to bear because of the mutual constraints, or by the obstacles which, by the nature ofthe system, may oppose to the motion, or rather those forces are not but the same forcesof resistance, which must be equal and opposite to the pressure exerted by the bodies. Ourmethod provides, as we see, the means to evaluate this resistance. This is not one of theminor benefits under this method [145].34 (A.10.21)

As it can be seen, he explicitly introduces the constraint reactions as ordinary forces:“those forces are nothing but the very forces of this resistance, which must be equaland opposite to the pressure exerted by the bodies”, and also says that it is importantto determine the constraint forces. This position puts him at odds with the mechanicsof D’Alembert and Lazare Carnot, where constraints destroy the motions but do notexert forces.

10.2.2 Mécanique analytique

The second edition of the Mécanique analytique registers a significant number ofchanges. With regard to the basic aspects of the principle of virtual velocities, withsome refinements and additions which I have already mentioned, the most notableaddition is a demonstration. Apart from the fact itself, it is important to point outthe awareness by Lagrange of the problematic nature of the principle, which is wellexpressed by the considerations he makes following the assessments reported in thefirst edition (see previous section):

33 p. 49.34 p. 49.


As to the nature of the principle of virtual velocities, it is not so self-evident that it can be as-sumed as a primitive principle, but it can be considered as the expression of the general lawof equilibrium, deduced by the two principles that we set out [that of the lever and composi-tion of forces]. So in the proofs that are given of this principle it is always considered due toone of these, more or less directly. But, in Statics there is another general principle indepen-dent of the lever and the composition of forces, although the mechanicians will commonlyrefer it to them, which would seem to be the natural foundation of the principle of virtualvelocities: you can call it the principle of the pulleys [emphasis added] [148].35 (A.10.22)

According to Lagrange the principle of virtual velocities has its foundation in theprinciple of the pulley. This principle says that if one considers a system of two pul-leys, consisting of a fixed and a movable one, and wraps around them an inextensiblerope, the relationship between power P and resistance R is 1/n, with n the number ofcords. The ‘cords’ are the whipping situated on either side of the pulley, which maybe either an even number or an odd number, not to be confused with the number oflaps. For example, with reference to Fig. 10.2, there are 4 cords on the left and threeon the right, for a total number of 7 cords; the laps are only 3.

Lagrange argued that the principle of the pulley is absolutely self-evident becauseit is clear that all the cords of the rope have the same tension – supposing the absenceof friction – and it is also clear, with reference to Fig. 10.2, that the lower pulleyis sustained by a force equal to the tension of the rope multiplied by the number ofcords. Notice that Lagrange does not attribute to the principle of the pulley the statusof a virtual work law because at the moment he avoids any kinematical analysis.

In the following I briefly summarize Lagrange’s proof, trying to interpret it. Re-ferring to Fig. 10.3, consider three couples of pulleys set out at points A, B, C – thenumber of pulleys is limited only for ease of exposition, the following argumentsapply equally to any number of them. The movable pulleys are placed at A′, B′, C′.An inextensible rope is wrapped P times around AA′, Q times around BB′, R timesaround CC′, or rather P, Q, R, are equal to the number of the cords of the pulleys.After being wrapped around CC′, to the rope is hung a weight Γ.

This system can be taken to represent a system of three forces P,Q,R commen-surable with each other, applied to points A′, B′, C′ of a generic body, or a systemof bodies, linked together and directed as AA′ , BB′, CC′. To show this assume the

34

Fig. 10.2. A simple system of two pulleys

35 p. 23.


C

Γ

β

γ

α

R

R AP

P

B

C′B′

A

Q uB′uA′

uC′

Q′

Fig. 10.3. The system of pulleys equivalent to the system of forces36

weight Γ as the maximum common divisor of P,Q,R and a unit of measure so thatit is unitary. The unitary weight Γ applied to the last system of pulleys will be ableto move a resistance and then to exert a force equal to PΓ = P ×1 = P in the firstsystem of pulleys, to Q = Q in the second and to R = R in the third (notice that anassembly of such pulleys may represent only the central forces of centres A, B andC; for forces with constant directions it is sufficient to take AA′, BB′ and CC′ verylarge).

Points A′, B′, C′move in directions that are allowed by the constraints. Letα,β,γbe the components of the infinitesimal virtual displacements uA′ ,uB′ ,uC′ of thesepoints in the directions of the forces P,Q,R, that is their virtual velocities coincidingwith the variation of distance between fixed and mobile pulleys. Because the rope isinextensible, the virtual vertical displacement Δl of the unit weight Γ is given by:

Δl = Pα+Qβ+Rγ, (10.15)

which expresses the variation of length of the rope wrapped around the three couplesof pulleys.

According to Lagrange, for the system of bodies to be in equilibrium it is nec-essary that in the virtual motion the weight does not sink, otherwise it will actuallysink and then there will not be a state of rest. So the relation should be valid:

Pα+Qβ+Rγ≤ 0. (10.16)

However, reversing the direction of the virtual displacements, which Lagrange con-siders always possible, and repeating the argument, one must also have:

P(−α)+Q(−β)+R(−γ)≥ 0. (10.17)

36 The figure is a variant of Fig. 54 in [355], p. 66.


The only way to to satisfy both relations (10.16) and (10.17) is to accept the sign ofequality, that is to have:

Pα+Qβ+Rγ= 0. (10.18)

But Pα, Qβ and Rγ are the moments of the forces P,Q and R because α,β and γ arethe virtual velocities of their points of application while the number of wrappingsof P, Q, R, can be interpreted as P,Q,R, and then the expression (10.18) says that anecessary condition for the equilibrium of a system of bodies is that the sum of themoments of all powers vanish. Notice that the reversibility of the virtual velocitiesconsidered as not problematic by Lagrange is possible only for ‘regular’ constraints.In such a case if, for example, f (u,v,w,etc.) = 0 is a constraint equation, fudu+fvdv+ fwdw = 0 is the constraint equations for infinitesimal displacements whichare linear, and if it is satisfied by du,dv,dw, it is satisfied also by −du,−dv,−dw[152], [274].37

The relation (10.18), if satisfied by any possible value of α,β and γ is also asufficient condition for the equilibrium. Suppose that indeed (10.18) is satisfied fora set of virtual velocitiesα,β andγ. Given the linearity, it is also satisfied by−α,−βand −γ. According to Lagrange, since there is no way to prefer one or the other oftwo possible motions of the system they must both be zero and so the system is inequilibrium.

The proof can be easily extended to the case of powersP,Q,R,etc. not commensu-rable with each other, using a limit process because it is known that all propositionsproved for commensurable quantities can be proved equally when these quantitiesare incommensurable by means the reductio ad absurdum [148].38

It is clear that the demonstration of the virtual velocity principle referred to abovedoes not require only the principle of the pulley. For the proof of the necessary partof the principle, Lagrange assumes that the weight Γ naturally tends to sink if theconstraints, as determined by ropes and pulleys, allow it to do so, and this certainlyupsets the equilibrium, then the weight Γ cannot sink. This principle of natural de-scent of heavy bodies is probably the most natural of all of mechanics, but it is notdeductible a priori and it is true because of everyday experience. It is more intuitivealso of the principle of the impossibility of perpetual motion. For the sufficient part ofthe demonstration Lagrange then uses a principle that is generally accepted as valida priori, i.e. the principle of sufficient reason, which states that if there is no reasonthat a motion should be done in one way or another, the motion is not realized at all.The application of this principle, however, does not leave one completely satisfiedbecause it is not so obvious that there is no reason to prefer one or the other of thetwo motions. Lagrange’s proof of the second edition of the Mécanique analytiquereferred to above, reconnects to a demonstration of 1798, published in the Journal ofthe École polytechnique along with the three demonstrations by Fourier [147] (seeChapter 12). It differs only for formal aspects, such as, for example, to consider aweight of 1/ 2 instead of a unit weight and for a more extensive discussion.

37 p. 137.38 p. 26.


10.2.2.1 Criticisms of Lagrange’s proof

Lagrange’s proof for the virtual velocity principle as reported in theMécanique ana-lytique was the object of a series of indirect and direct criticisms, during Lagrange’slife and after. Indirect criticisms are certified by the series of attempts to furnish anew proof of the principle; see for example the argumentations of Prony, Fossom-broni and Servois in subsequent chapters.

The following comment by Bertrand, the editor of the third edition of the Méca-nique analytique is instead an example of direct criticism:

It is opposed, with reason, to this affirmation of Lagrange, the example of a heavy materialpoint equilibrated on the highest point of a hill. It is clear that an infinitely small displacementwill make it to descend; this notwithstanding this displacement does not occur at all. Thefirst rigorous proof of the principle of virtual velocities is due to Fourier (Journal of the Écolepolytechnique, Volume III, Year VII). The same volume of the Journal contains Lagrange’sproof reported here [148].39 (A.10.23)

According to Bertrand, the demonstration is imperfect and the criterion of equilib-rium based on the assumption that a weight must necessarily go down if it is allowedby constraints, has exceptions. The same criticism of Bertrand was reported in moredetail by Jacobi in his lectures on mechanics of 1847–1848 [375].

Perhaps the criticism of Bertrand and Jacobi applies to the reasoning of Lagrange,because it is likely that Lagrange’s conception of infinitesimal displacements is thatassigned by them, and the situation of Fig. 10.4a would reveal the weakness of La-grange’s reasoning because here there is a material point which is in equilibriumnotwithstanding the possibility of a downward motion. In a few places, however,Lagrange seems to adopt the infinitesimal displacement as in Fig. 10.4b – as a mod-ern mathematician would do – when he says that the displacements of points inthe system are reversible (as already mentioned just above) and in the Théorie desfonctions analytiques where he interprets the virtual displacement as velocity, and avelocity in the case of Fig. 10.4 is directed horizontally and no downward motion isallowed.

An apparently more relevant criticism is that by Mach, who sustains the claimthat Lagrange’s proof is circular because it assumes the principle of the pulley, i.e.a simplified version of virtual work law – that is the law which is to be proved –

dudu

P

pp

P

b)a)

Fig. 10.4. Various kinds of infinitesimal displacements

39 p. 24. Note by Bertrand.


[355].40 Mach’s criticism is however not convincing from two points of view, firstlybecause Lagrange uses the principle of the pulley not as a law of virtual work, butsimply as a principle of equilibrium of parallel forces, secondly because in any casepassing from a simple proposition to a complex one should be considered as a formof proof.

The criticism that a modern reader – myself, for example – can turn to Lagrangerefers to various aspects. The first criticism is formal of logical character; Lagrangedoes not specify that he is in fact assuming smooth constraints. The issue is not eventouched. A second criticism is methodological and concerns the possibility of replac-ing the real system of forces with a series of pulleys and wires completely ideal, i.e.infinitely flexible, with no mass or weight and without frictions. One more criticismconcerns the admission of linear constraints, at least for infinitesimal displacements.Lagrange’s proof seems quite convincing, but it leaves some points of dissatisfac-tion, on the other hand it is not possible to dismiss it in an easy way. In my opinionLagrange’s proof is the most convincing of all attempted up to now, after that ofPoinsot (see Chapter 14) which however has a completely different nature, becauseLagrange’s proof has a foundational character, while Poinsot’s a reductionist char-acter.

10.3 The Théorie des fonctions analytiques

The Théorie des fonctions analytiques had two editions issued by Lagrange, thefirst in 1797, the second in 1813 shortly before his death. In the section devotedto mechanics of the first edition, Lagrange addresses the characterization of re-action forces. Lagrange assumes without any criticism the postulate P3 of Chap-ter 2 of the present book and tries to determine the reactions that come to be es-tablished as a result of internal constraints. He considers a constraint of the typef (x,y,z,ξ,ν,ζ, . . .) = 0, between the coordinates of the material points p, p′, ….According to the axiom P3, the components of constraint forces R at the points p areorthogonal to the surface f (x,y,z,−, . . .) = 0, where ξ, ν, ζ are assumed as fixed,i.e.:

Π∂ f∂x

, Π∂ f∂y

, Π∂ f∂ z

(10.19)

and that R′ in p′ are orthogonal to the surface f (−,ξ,ν,ζ, . . .) = 0, with x, y, y fixed,i.e.:

Ψ∂ f∂ξ

, Ψ∂ f∂ν

, Ψ∂ f∂ζ

(10.20)

with Π and Ψ arbitrary constants.Lagrange showswith the use of the virtual velocity principle thatΠ = Ψ, or which

is the same, R = R′:

40 p. 67.

10.3 The Théorie des fonctions analytiques 265

If one impresses to each body forces equal and contraries to those [R and R′], the effect ofthese forces will be destroyed by the resistance we referred above; and consequently thesystem should remain in equilibrium. […] But for the principle of virtual velocities, thesum of the forces multiplied by the velocities of their points of application, according to thedirection of the force, should vanish in the case of equilibrium, […] for the equilibrium ofthe concerned forces it will be satisfied the equation:

−Π f ′(x)x′ −Π f ′(y)y′ −Π f ′(z)z′ −Ψ f ′(ξ)ξ′ −Ψ f ′(ν)ν′ −Ψ f ′(ζ)ζ′ −&c. = 0

which should be valid with the equation of constraint f (x,y,z,ξ,ν,ζ) = 0 […]. But if it istaken the derivative of this equation, with respect to time t, on which the variables x,y,z,ξ,&c., depend, it holds:

x′ f ′(x)+ y′ f ′(y)+ z′ f ′(z)+ξ′ f ′(ξ)+ν′ f ′(ν)+ζ′ f ′(ζ)+&c. = 0

and it is clear that this relation is satisfied with the previous one for all values of x′,y′,&c.only if Π = Ψ = &c. [146].41 (A.10.24)

In the second edition of the Théorie des fonctions analytiques Lagrange changescompletely the structure of the paragraph concerning the constraint reactions, takinginto account Poinsot’s comments [197] who believed neither useful nor necessarythe use of the virtual velocity principle to show that Π = Ψ. Lagrange virtually re-verses the setting of the first edition; instead of using the virtual velocity principleto characterize constraints, he uses the characterization of constraints in order todemonstrate the principle. The demonstration takes on two fundamental principles:the law of the pulley and the rule of composition of forces.

For the sake of brevity I will change the order of presentation of Lagrange’s ar-guments, keeping the same logic. He considers first the case of only two materialpoints, identified by the coordinates x,y,z,ξ,η,ζwhich are subject to a quite generalcondition of constraint represented in the form:

F(x,y,z,ξ,η,ζ) = 0. (10.21)

Actually, the generality is limited by the fact that taking an expression of the type(10.21), where F is an ordinary function, is the same as considering holonomic andindependent of time constraints.

To the above mathematical expression of constraints Lagrange associates a geo-metric-mechanical model consisting of two fixed and two mobile pulleys connectedby a taut rope of fixed length, as shown in Fig. 10.5.

The geometric-mechanical constraint can be made locally equivalent – in a waythat gives rise to the same infinitesimal virtual displacements – to the analytical con-straint defined by equation (10.21) provided the position of the pulleys and the num-ber of rope turns are chosen in an appropriate way. In particular, line RM joiningthe centres of the first two pulleys, fixed and mobile, should be orthogonal to thesurface defined by F = 0 with ξ,η,ζ assumed as fixed, while line SN of the secondtwo pulleys should be orthogonal to the surface F = 0 with x,y,z assumed as fixed.Now it is easy for Lagrange to define the direction of the reactions, by introducingin a non-problematic way the assumption of smooth constraints. For the system of

41 pp. 255–256.


M

N

m

R

F(x, y, z, ) = 0

F( , x, h, z) = 0

n

S

T

Fig. 10.5. Reaction of a constraint

pulleys the constraint reaction R in M is clearly directed along the line MR joiningthe centres, while the reaction R′ in N is directed along the line NS. For the equiva-lence it can thus be said that the reaction forces R and R′ associated with (10.21) areorthogonal respectively to the surfaces F(x,y,z,−) = 0 and F(−,ξ,η,ζ) = 0 andtherefore defined by relations like these:

ΠF ′(x), ΠF ′(y), ΠF ′(z)ΠF ′(ξ), ΠF ′(η), ΠF ′(ζ), (10.22)

where Π is an arbitrary constant of proportionality.Lagrange thus obtains the same result which will be obtained in 1805–1806 by

Poinsot [194], starting not by the law of the pulley but by the ordinary principles ofstatics (see Chapter 14):

The derivatives of the same function [the constraint conditions] considered with respectto the different coordinates are always proportional to the forces that act according to thesecoordinates [in the points having these coordinates] and depend on the constraints expressedby this function [149].42 (A.10.25)

The extension of the above considerations to any number of points does not presentany difficulty. A condition of constraint, for example with three points, like this:

Φ(x,y,z,ξ,η,ζ,x, y, z), (10.23)

admits a geometric-mechanical model with three fixed and three mobile pulleys.Similarly in the case with four points and so on.

For more than one constraint condition Lagrange assumes, without any justifica-tion, and then without taking into account Ampère’s comments to the first edition ofthe Theorie des fonctions analytiques (see Chapter 13), that constraints do not affecteach other. Therefore, two boundary conditions for a single material point:

F(x,y,z) = 0, Φ(x,y,z) = 0, (10.24)

42 p. 384.

10.3 The Théorie des fonctions analytiques 267

may be replaced by two forces of components, using Lagrange’s symbols:

ΠF ′(x)+ΨΦ′(x), ΠF ′(y)+ΨΦ′(y), ΠF ′(z)+ΨΦ′(z), (10.25)

where Π and Ψ are arbitrary constants of proportionality.Now Lagrange may formulate the equations of equilibrium for any number of

material points subject to any number of constraints and forces. In his own words:

Let X ,Y,Z be the forces applied to one of the bodies in the directions of coordinates x,y,z,Ξ,ϒ,Σ the forces applied to another body in the directions of coordinates ξ,η,ζ, and X, Y, Zthe forces applied to a third body according to the direction of the coordinates x, y, z; fromwhat said it results:

X = ΠF ′(x)+ΨΦ′(x), Y = ΠF ′(y)+ΨΦ′(y), Z = ΠF ′(z)+ΨΦ′(z)Ξ = ΠF ′(ξ)+ΨΦ′(xi), ϒ = ΠF ′(η)+ΨΦ′(η), Σ = ΠF ′(ζ)+ΨΦ′(ζ)X = ΠF ′(x)+ΨΦ′(x), Y = ΠF ′(y)+ΨΦ′(y), Z = ΠF ′(z)+ΨΦ′(z)

(a)

and from the equilibrium equation it will result:

Xx′+Yy′+Zz+Ξξ′+ϒη′+Σζ′+Xx′+Yy′+Zz′ =ΠF(x,y,z,ξ,η,ζ,x, y, z)′+ΨΦ(x,y,z,ξ,η,ζ,x, y, z)′. (b)

The second member of this equation is clearly zero as a consequence of the constraint equa-tions, because the indeterminate quantities Π, Ψ are multiplied by the derivatives of theseequations; it will be then:

Xx′+Yy′+Zz+Ξξ′+ϒη′+Σζ′+Xx′+Yy′+Zz′ = 0 (c)

the general equation of the principle of virtual velocities for the balance of the forces X ,Y,Z,Ξ,ϒ,Σ, X, Y, Z, where the derivatives x′,y′,z′, ξ′, …express the virtual velocities of thepoints to which the forces X ,Y,Z, Ξ …, estimated according to the directions of these forcesare applied (see the first part of the Mécanique analytique).After all, one should not be at all surprised to see that the principle of virtual velocitiesbecomes a natural consequence of the formulas which express the forces resulting from theconstraint conditions, because the consideration of a thread that acts on all bodies throughits uniform tension and induces forces assigned, is sufficient to lead to a general and directproof of this principle, as I showed in the second edition of the work cited [149].43 (A.10.26)

The text of Lagrange is sufficiently perspicuous though perhaps the transition from(a) to (b) is a little fast. To obtain (b) one has to multiply the first of (a) for x′, thesecond for y′, the third for z′, and so on and then add and to recognize in the secondmember the total derivative of two composed functions. It is worth noting that in hisproof Lagrange uses virtual velocities and not infinitesimal virtual displacements,without commenting on the fact and in this he followsmore or less the same approachof Poinsot in his proof of virtual velocity principle [194], without declaring the fact.

Joseph Louis François Bertrand (1822–1900), who edited the third edition of theMéchanique analytique, wrote a comment in the second volume of it, where he de-nounced that absence of any reference to Poinsot:

One might wonder that the illustrious author, usually so careful to know the origin of theideas he presents, do not quote here anything. The passage we just read is, in effect, sevenyears later of the publication of the famous paper on the equilibrium and motion of systems,

43 pp. 384–385.


in which Mr. Poinsot proposes and solves precisely the same question of freeing mechanicsfrom the principle of virtual velocities seeking forces that correspond directly to a givenequation. This memoir much struck Lagrange, as evidenced by numerous autograph notesplaced by him on the margins of a copy that I was able to consult. I reproduce here one ofthese notes, which can leave no doubt on the question of priority [151].44 (A.10.27)

Towards the end of the Théorie des fonctions analytiques Lagrange stresses the com-monality of the proof above with that of theMécanique analytique (second edition),but he believes that this is more direct and general. One can agree with him that itis more direct, as in the proof of the Théorie des fonctions analytiques he uses thelaw of the pulley and the rule of composition of forces while in the Mécanique an-alytique he uses almost only the law of the pulley. It is difficult to agree with theopinion about the generality. Indeed, the proof of the Théorie des fonctions analy-tiques seems more comprehensive because with it the virtual velocity principle canbe extended to the dynamic case without any difficulty. Indeed the characterizationof smooth constraints as made by Lagrange allows the extension of the virtual veloc-ity principle from the static case to the dynamic case without having to go throughthe principle of D’Alembert. It is enough that instead of the equation of statics (a),the equation of dynamics is considered and then to follow the same procedure topass from (a) to (b) and to (c). This possibility of the extension was not, however,ever remarked by Lagrange.

10.4 Generalizations of the virtual velocity principle to dynamics

Although virtual work laws have historically always been considered as principlesof statics, speaking of Lagrange it is impossible not to consider their extension todynamics. As seen in § 10.1.1, he introduced the virtual velocity principle by be-ginning to study a dynamic problem with a generalization made possible thanks toD’Alembert’s ideas. With his usual laconic style he wrote:

The principle of statics that I come to expose, combined with the principle of dynamicsdue to D’Alembert, is a kind of general formula containing the solution of all problemsconcerning the motion of bodies [142].45 (A.10.28)

The principle of dynamics due to D’Alembert is that of paragraph II of the Recher-ches de la libration de la Lune according to which accelerating forces taken in theopposite direction can be treated as ordinary forces.

The meager statements of Lagrange raise at least two problems: what is the prin-ciple of D’Alembert? Is the interpretation of Lagrange permitted? A first answer tothese questions comes by reading the Traité de dynamique [84] byD’Alembert. Fromhere it would seem that the ‘principle of D’Alembert’ reported by Lagrange in theRecherches does not have much to do with the original principle, and therefore his

44 p. 366.45 p. 12.

10.4 Generalizations of the virtual velocity principle to dynamics 269

interpretation is not permissible. But this conclusion is puzzling, because Lagrangeis familiar with D’Alembert’s work and may not have completely misunderstoodit. To try to overcome the problems associated with the name of the principle ofD’Alembert and with the true ideas of Lagrange it is necessary to refer to workssubsequent to the Recherches and in particular the two editions of the Mécaniqueanalytique and the Théorie de la libration de la Lune.

References and interpretations of the principle of D’Alembert in the Théorie dela libration de la Lune (1780) and in the first edition of the Mécanique analytique(1788) are equivalent, but there is a shift in the second edition of the Mécaniqueanalytique (1811). In the introduction to the second part of the 1788 edition of theMécanique analytique, Lagrange wrote:

If now it is assumed the system in motion, and it is considered the motion of each body atany given infinitesimal interval of time as consisting of two motions, one of which is thatthe body will have immediately after, it is necessary the other is destroyed by the mutualactions of bodies and by the motive forces by which they are animated at the moment. Sothere should be equilibrium among these forces and pressures or resistances that resultsfrom the motions that may be regarded as lost from the body, from one instant to another.It follows that to extend the formulas of the equilibrium to the motion of systems it will beenough to add the terms due to these forces [145].46 (A.10.29)

The passage is quite obscure and it is not much clarified by what Lagrange addedin the following pages where he passes to derivation of the equations of motion.Much clearer exposure of the same concepts can be found instead in the Théorie dela libration de la Lune, written some years before, of which I quote a brief excerpt:

It is clear that the motion or the velocity of the body m during the time dt could be regardedas composed by other three velocities expressed by:

dxdt

,dydt

,dzdt

and parallel to the axes x,y,z. It is then evident, when the bodies are free and no externalforces act on them, each of these three velocities will remain constant; but actually in thesubsequent instant they change and become:

dxdt

+ddxdt

,dydt

+ddydt

,dzdt

+ddzdt

so, if the previous velocities are assumed to be composed of these last and of the velocities:

−ddxdt

, −ddydt

, −ddzdt

or, assuming dt as costant:

−d2xdt2

, −d2ydt2

, −d2zdt2

it follows that they should be destroyed by the action of the force acting on the bodies. Butthese velocities are due to accelerating forces equal to:

d2xdt2

,d2ydt2

,d2zdt2

46 p. 181.


and directed parallel to the axes x,y,z (expressing as usual, the accelerating force as theelement of velocity divided by the element of time) or, which is the same, the forces equal to

d2xdt2

,d2ydt2

,d2zdt2

and direct to the contrary.[…]It follows that there should be equilibrium among these different forces and the others actingon the bodies, and so the laws of motion of the system are reduced to those of his equilib-rium, it is that the substance of the nice principle of dynamics of Mr. D’Alembert [148].47

(A.10.30)

In these passages Lagrange’s attempt to interpret D’Alembert’s principle in termsof forces instead of motions is reflected, and this is partly justified by the fact thatin the Traité de dynamique D’Alembert is ambiguous in his use of the words motionand force, meaning for the latter sometimes ma and sometimes mv. The languageof Lagrange is similar to that of D’Alembert, but the concepts are very different.Lagrange’s destroyed motions have nothing to do with D’Alembert’s lost motionsdue to constraints; they are the actual changes of motion with sign reversed, due toall forces. It is not entirely clear if "equilibrium among these different forces and theothers acting on the bodies”, means equilibrium in the sense of absence of motion,or simply a balance of forces, though not made in accordance with the parallelogramrule but according to the principle of virtual velocities.

The sentence of the Méchanique analytique,“It follows that to extend the formu-las of the system’s equilibrium to its motion, would be enough simply to add theterm due to these forces”, then gives the idea of what Lagrange intends with to “re-duce dynamics to statics”. It does not mean eliminating dynamics as a discipline,which would be absurd, but simply to apply the same algebraic method for solvingequilibrium problems or motion problems. Dynamics is implicit in the definition ofmass, in the use of an inertial reference system and in the assumption of f = ma asa fundamental quantity, wether it is treated as a definition or as a law.

In the second edition of theMécanique analytique there is a different presentationof the principle of D’Alembert that, while still attached to the one of the first editionhowever, is farther from D’Alembert’s original exposure. To clarify the ideas ofLagrange it is enough to report what he says in the introduction to part II of hisbook:

If motions are imposed to so many bodies that they [the motions] are forced to be modifiedby their interaction, it is clear that these motions can be seen as consisting of those thebodies would follow really and other motions that are destroyed by which it follows thatthese motions must be such that bodies on which only them are imposed are in equilibrium[148].48 (A.10.31)

But the difficulty of determining the forces that must be destroyed and even the laws of theequilibrium among these forces often makes the application of this principle embarrassingand difficult.[…]

47 pp. 17–18.48 p. 255.


If one wants to avoid the decomposition of motions, that this principle requires, it will beenough to establish directly the equilibrium between forces and resulting motions, but takenin the opposite direction. For it is imagined that the motion is impressed on each body in theopposite direction to that which must follow, it is clear that the system will be reduced torest [emphasis added]. Therefore it will be necessary thesemovements destroy those that hadreceived the bodies and that they would have followed without mutual interaction; so thereshould be equilibrium among all these movements, or among the forces that can producethem.This way of reducing the laws of dynamics to those of statics is less direct than that resultingfrom D’Alembert’s [original] principle, but it offers greater simplicity in applications. It issimilar to those of Hermann and Euler applied to the solution ofmany problems ofmechanicsand it is sometimes found in the treaties under the name of Principle of D’Alembert [148].49

(A.10.32)

The idea here seems much easier than in the first edition, at least for a modern reader,and perhaps reveals a different conception of force, closer to that of Euler than to thatof D’Alembert. If to a system of bodies in motion appropriate forces are applied (ifmotions are impressed) in the opposite direction to the actual motion of each body,the system remains in equilibrium. The main difference compared to the first issueis that here the motions are not destroyed by the active force and constraints, butby the fictitious forces, −ma. Note that this time the reference is to equilibrium inthe strict sense (“The system will be reduced to rest”), and not just to the balance offorces.

For clarity, I refer for simplicity to a single material point constrained to moveon a surface; if f is the active force and a the acceleration, assumed φ= −ma, theforces f and φ are balanced with each other, not in the sense that f +φ = 0, butrather in the sense that the material point of mass m is at rest under the action ofthe forces f and φ on the surface to which it is constrained (in modern terms, thebalance equation is satisfied with intervention of the reactive forces).

Unfortunately, Lagrange’s attempt to bring balance of forces to equilibrium, as-suming that my interpretation is correct, gives rise to a rule that does not alwayswork, and when the forces acting on a system depend on the velocity, the appli-cation of the forces φ = −ma may be unable to keep the system in equilibrium.In fact, if f (v) indicates the force dependent on the velocity v, the addition of theforces φ=−ma to the system leads to the balance f (v)+φ= 0, but not even thatf (0)+φ = 0, between the force φ and that which would act on the system at rest(v = 0).

The position of the first Lagrange formulation, for which equilibrium and balanceseem identified, which is even more problematic from several points of view, is notsubject to this criticism. If the idea of equilibrium is generalized to the dynamiccase, as a balance of accelerating and active forces, all the arguments of support-ers of the virtual work laws as a criterion of rest (Aristotle, Galileo, Riccati), fall.But Bernoulli’s arguments do not fall because his principle of virtual velocities isset out with reference only to a balance of forces. In this statement there is no diffi-culty in inserting accelerating forces as balancing active forces applied and thereforeLagrange is justified in the use of the virtual velocity principle.

49 p. 256.


S x1

r1

m1

p1

x2

r2

r3

m3

p3a3

a2

a1

m2

p2

x3

Fig. 10.6. Static analysis of an oscillating compound pendulum

Lagrange, in the Mécanique analytique, says that his idea of considering φ =−ma as forces, is similar to the approaches pursued by Hermann and Euler. He cer-tainly refers to Hermann’sPhoronomia of 1716 [134] and to various of Euler’s worksafter 1750, on vibrating strings [102, 105]. He forgets to mention Clairaut who in hiswork Sur quelques principes qui donnent la solution d’un grand nombre de prob-lèmes de dynamique of 1742 [69] introduced reaction forces to study the motion ofa simple point system, requiring that they be self-equilibrated.

In the study of a compound pendulum dynamics, Hermann considered the ele-mentary masses mi subject to the weight of pi, the driving force miai and forceswhich constrain the masses to belong to a rod. Adopting as a criterion of balancethe law of the lever, Hermann ignored the constraint forces and required the equiv-alence of the static moments with respect to the centre of suspension between piand miai:

∑miai× ri = ∑ pi× xi (10.26)

where × is the ordinary product. From this equation it is then easy for Hermannto deduce the law of motion of the compound pendulum. Fig. 10.6, illustrates thesituation.

Euler, in the study of the dynamics of a taut string, considered as a set of ele-mentary masses mi connected by the string, said that the accelerating forces miai ofthe elementary masses considered in the opposite direction must be balanced by therestoring force of the string:

As currently it is the case to determine the movement of the rope due to the force stressing it,i.e. the accelerating force P for which the point M of the rope is accelerated toward the axisAB, it is clear that all those forces by which each element of rope is urged in the directionof AB, taken together, shall be equivalent to the force from which the rope is in fact tight,which we have indicated with AF = F . Well if we conceive forces opposing and equal to P,


L

M

P

D

CAF

GB

l

m

p

Fig. 10.7. Static analysis of a vibrating string

applied according to ML on each point M of the rope, then they must be in balance with theforce that stretches the rope [102].50 (A.10.33)

Even today, thanks to Lagrange’s remarks, the term principle of D’Alembert meansdifferent things: the original principle of the Traité de dynamique, D’Alembert prin-ciple of Lagrange and even the symbolic equation of dynamics. (It should be empha-sized, however, that Lagrange is not responsible for the enunciation of the principleof D’Alembert in the form: the forces of inertia balance the active forces, in whichthe forces of inertia are hypostatized and treated as ‘true’ forces.) I cannot avoidthinking that Lagrange (and Euler and Hermann) though if he has not introduced anew principle has at least had a very good idea.

10.4.1 The calculus of variations

Another aspect of Lagrange’s dynamic generalization is the introduction of the timefactor in defining the pattern of a system of material points. Although at any momentthe dynamic problem can be studied with the same formulas of statics, i.e. the useof the virtual velocity principle, it must be taken into account that the configurationof this system changes with time and consequently also the virtual displacementschange, which, considering the motion in all its duration, become functions of time.The expression of the virtual velocity principle in dynamics, along the trajectory ofthe system, has then the form ∑ f (t) · u(t) = 0, where u(t) is the vector of virtualdisplacements in the configuration at time t. In principle, the vectors u(t) may becompletely unrelated to each other at different times. Here Lagrange introduces thecalculus of variations and treats the virtual displacements u(t) as the variation ofcertain functions of time that represent motion. The virtual displacements, indicatedby δu and thought of as a function of time, are so endowed with some degree ofregularity, in particular Lagrange acknowledges at least the existence of the first-order derivative.

The introduction of the calculus of variations with the consequent possibility of‘regularization’ of motions does not limit the generality of the virtual velocity prin-ciple, because its validity does not depend on the absolute value of the motion of vir-

50 p. 73.


tual velocity in an instant, and makes simpler some fundamental deductions. Amongthem there is the one regarding the theorem of living forces, where real changes aretreated as virtual variations due to the fact that both du and δu are smooth functionsof time (assuming constant intervals dt). This same conception of virtual displace-ments allows us then to discover the general procedure for calculating the virtualwork of accelerating forces from the expression of kinetic energy, used in the Mé-canique analytique, using the possibility of the permutation of δ with d in the ex-pression d2δu, which is conceivable only if δu is a continuous and differentiablefunction.

For continuous systems, such as those included in the Recherches, virtual dis-placements of each point are identified by their coordinates x,y and z, instead ofby a label. Again the use of the symbol δ implies some regularity with respect tovariables x,y and z (beside t).

10.4.2 Elements of D’Alembert’s mechanics

Jean le Rond D’Alembert was born in Paris in 1717, and diedin Paris in 1783. He was the illegitimate child of the cheva-lier Destouches. Being abandoned by his mother on the stepsof the little church of St. Jean le Rond, which then nestledunder the great colonnade of Notre Dame, he was taken tothe parish commissary, who, following the usual practice insuch cases, gave him the Christian name of Jean le Rond. Hewas boarded out by the parish with the wife of a glazier wholived near the cathedral, and here he found a real home. His

father appears to have looked after him, and paid for his going to a school wherehe obtained a fair mathematical education. Nearly all his mathematical works wereproduced during the years 1743 to 1754. During the latter part of his life D’Alembertwas mainly occupied with the great Encyclopédie, with Diderot. For this he wrotethe introduction, and numerous philosophical and mathematical articles; on geom-etry and on probabilities. His style is brilliant and faithfully reflects his character,which was bold, honest, and frank. The most famous of his books is perhaps theTraité de dynamique, published in 1743 [84], in which he proposed laws of mechan-ics other than the Newtonian ones [335].

According to D’Alembert the principles of geometry and algebra, with the ad-dition of the assumption of the impenetrability of bodies, were enough to developmechanics, which appeared to be a completely deductive science marked by the sealof evidence; as geometry and algebra. Motion and its property are the main objectof mechanics.

To be assumed as foundations of mechanics, to be its principles, all the rele-vant concepts must be subject to scrutiny by the philosopher: Only concepts witha sufficient clarity and distinction (in a Cartesian sense) could be accepted. AndD’Alembert identified only two fundamental concepts of the kind: those of spaceand time, which are the only elements and principles of mechanics. The laws ofmechanics are theorems which can be deduced by the two principles. The various


concepts of force, together with that of motive cause, are to be rejected as “darkand metaphysical beings, only capable to spread shadows in a science clear in it-self” [84].51

Scepticism toward force does not originate directly in D’Alembert. Maupertuistoo had this same conception. This scepticism was already present in Descartesand mainly in Malebranche. Descartes thought there were no forces in bodies, eventhough he conceived the concept of cause. According to Descartes, God is the primacausa, but after the creation there is only a cause which is mechanical: the impactof impenetrable bodies. Malebranche did not accept this model and assigned to Goda greater space. The concept of force is avoided by Malebranche not only becauseits assumption reduces God’s power but also because it is not well-defined. Accord-ing to Malebranche force cannot be observed or measured directly, it looks like asimple word, made-up by philosophers to hide their illiteracy. Berkeley wrote somesentences in which D’Alembert will recognize himself, though it is difficult to de-termine if D’Alembert knew Berkeley’s work, published only in England in 1720.Hume too was contrary to the concept of force; but also in this case it seems difficultthere was any influence, notwithstanding that Hume was familiar the Enlightenmentphilosophers [271].

Without force, motion can be described by geometry alone. So D’Alembert leftno space for what today is called Newton’s second law. Here is what he wrote onthe matter:

Why should we recur to this principle of which everybody recurs today, that the force ac-celerative or retarding is proportional to the element of velocity divided by time? […] Weneither will examine in any way whether this principle is a necessary truth […] nor, as someGeometer [Daniel Bernoulli] a purely contingent truth […] we will limit ourselves to ob-serve that, true or false, clear or dark, it is useless in mechanics and consequently has to bebanished [84].52 (A.10.34)

According to D’Alembert there were two species of causes, and then of forces inmechanics: a) causes which derive from the mutual actions of bodies because of theirimpenetrability, which are the “main causes of the effects” we observe in nature; b)causes not immediately reducible to impulsion or pressure. These causes have tobe equally considered as distinct, if one considers as possible their reducibility toimpulse but cannot prove the fact. Causes of the first kind have well-known laws;this is not true for causes of the second kind. They are known only through theireffects; one speaks about a cause because one sees an effect.

Among the causes of the second kind there is gravity, which because it couldnot be reduced to the impact, and then to geometry, must be excluded from thenecessary laws of mechanics and considered as a contingent truth. D’Alembert as-serted that also the causes of the first kind which look evident, are so only im-properly:

51 p. XVII.52 p. XII.


What we call causes, also of the first kind, are such only improperly; they are effects whichdetermine other effects. A body pushes another body, or a body in motion meets anotherbody, one must then have necessarily a change in the state of bodies in this occasion [em-phasis added]. Because of their impenetrability, the laws of these changes are determinedby means of sure principles; and consequently impelling bodies are considered as causes ofthe impelled bodies. But this way to speak is improper. The metaphysical cause, the truecause is not known to us [81]. (A.10.35)

As a matter of fact, D’Alembert notwithstanding the rational framework of his me-chanics, affirmed his empiric faith. One has to do only with effects. In mechanics itis called cause of an effect, another effect; the true causes remain hidden. A chainof explanations cause-effect is nothing but a relation among effects, which howevercan be connected by necessary laws.

Anyway D’Alembert felt the need to introduce a quantity called force that, atleast from a mathematical point of view, played the role of the force as commonlyintended in statics and in dynamics by most physicists. Force is defined simply asthe product of mass by accelerations, where both concepts are previously defined:

So we will intend in general with motive force the product of mass multiplied by the elementof velocity [acceleration], or, which is the same, multiplied by the small space it covers in agiven time because of the cause which accelerates or retardates its motion; with acceleratingforce we will intend the element of velocity only [84].53 (A.10.36)

The definition of force by D’Alembert presupposed that of mass. D’Alembert, how-ever, in his examination of the principles of mechanics passes over this concept,without realizing its problematic character and that the lack of its clarification makesmechanics incomplete. To D’Alembert, and also to Newton, mass is given by thequantity of matter; a concept which could appear clear to anyone who had a concep-tion of matter based upon a crude atomism with all equal atoms.

D’Alembert assumed as fundamental theorems of dynamics the theorem of inertiadivided in two parts (I and II law), the theorem of composition of forces, and thetheorem of equilibrium. Their statements are referred to in Table 10.1 [84].54

Table 10.1. D’Alembert’s mechanics. Laws or theorems

I law A body in rest will remain in rest unless an external cause will force it.

II law A body once put in motion by whichever cause, must persevere uniformly and instraight line, unless a new cause, different from that has caused the motion, will acton it.

Theorem If any two forces act together on a point A to move it, the former uniformly from A toB, during a given time, the latter uniformly from A to C [...] I say that the body A willcover the diagonal AD uniformly, in the same time it will cover AB or BC.

Theorem If two bodies whose velocities are in inverse ratio of their masses, such that one cannotmove without shifting the other, there is equilibrium between these two bodies.

53 p. 26.54 pp. 3, 4, 35, 50–51.


10.4.2.1 D’Alembert principle

From the foregoing it appears that the so called principle of D’Alembert is not a lawof D’Alembert’s mechanics, and it is still the subject of any dispute regarding itsinterpretation and possible logic status [315, 316, 303].

In the following I will present the statement of D’Alembert as formulated by him-self and then I will cite the opinions of some scholars from which one can at leastpartially justify the interpretation given by Lagrange. D’Alembert presents his ‘prin-ciple’ as a procedure for solving a problem where it uses, though not very clearly,the real ‘principles’ of mechanics.

The principle of D’Alembert is reported far enough in the Traité de dynamique,after the presentation of the fundamental theorems.

General ProblemLet a system of bodies be disposed in any way with respect to each other, and suppose wegive each of them a particular motion, that cannot be accomplished due to the interactionwith other bodies. Find in these conditions, the motion that every body should have.SolutionLet A,B,C, &c. be the bodies of the system and suppose they are impressed with the motionsa,b,c, &c. and that they are forced by their interactions to change in the motion a, b, c, &c.It is clear that the motion a impressed to body A can be regarded as composed of the motiona and of another motion α. In the same way it is possible to consider the motions b, c, &c.composed of the motions b, β, c, κ, &c., from which it follows that the relative motions ofthe bodies A,B,C, &c. would be the same if instead to give them the impulse a,b,c it wouldbe given the couples of impulses a, α; b, β; c, κ, &c. Now, because of supposition, bodiesA,B,C, &c. took the motions a, b, c,&c, then the motions α,β,κ, &c. must not to disturb inany way the motions a, b, c, &c. That is if the bodies had received only the motions α,β,κ,&c. they should have destroyed themselves mutually and the system to remain at rest.From this it results the following principle to find the motion of any bodies that interact eachto the other.Decomposed each of the motions a,b,c, &c. impressed to each body in other twomotions a, α; b, β; c, κ, &c., such that if only the motions a, b, c, &c. were impressed to thebodies, the system should have remained at res, it is clear that a, b, will be the motions thesebodies will assume because of their actions. This is what has to be proved [84].55 (A.10.37)

Before the introduction of his principle, D’Alembert had specified the frame of refer-ence: his principle concerns impact among bodies, direct or mediated by rigid rods;the impacts are a result of imposed motions, an expression that D’Alembert uses inplace of imposed velocities. The bodies are to be understood as material points andare ‘hard’, i.e. not deformable, which do not bounce in the collision and in fact be-have as perfectly plastic bodies (apart from the change in shape that does not exist).

In the following I refer to the interpretations of Ernst Mach and Louis Poinsot.They change the principle of D’Alembert from a principle on motions to a principleon forces in a similar manner, although not identical, as made by Lagrange and serveto justify the ‘a bit free’ interpretation by the latter. Mach’s interpretation is impor-tant because it is the most famous, Poinsot’s interpretation, is important because itis substantially contemporaneous (1806) with the drafting of the second edition ofthe Mécanique analytique.

55 pp. 73–75.


According to Mach [355],56 in place of the impressed motions a,b,c,etc. one canconsider the external forcesUi, instead of the actual motions a, b, c, etc., the forcesVi

able to produce these motions and instead of the suppressed motions α,β,κ,etc. theforces or constraint reactionWi. D’Alembert’s principle, according toMach, requiresthat the constraint forces Wi are balanced. If the system is free, constraint forcesare balanced in the sense that their vector sum is zero; if the system is constrainedexternally, the balance is expressed by the annulment of their virtual work. Then,if the virtual work done by the constraint forces Wi is zero, even that of the forcesUi−Vi is zero, becauseWi =Ui−Vi.

In this interpretation of the principle of D’Alembert, the identification of the ac-celerating forces with the actual motion, has a key role and can justify the nameof the principle of D’Alembert to the assimilation of the accelerating forces withchanged sign to ordinary forces. Note that if there were no accelerating forces theabove interpretation would coincide exactly with the virtual velocity principle andD’Alembert would not have said anything new.

Poinsot also provides an interpretation of the principle ofD’Alembert likeMach’s,after establishing the equations of motion in form ma− r = f , where f are the ac-tive forces and r the vectors orthogonal to constraints (which can be interpreted asconstraint reactions). He writes:

It also can be seen that it is useless to refer to the famous principle of D’Alembert, whichreduces dynamics to statics. Under this principle if each impressed motion is decomposedinto two others, one of which is what the body will really take, all the others must balancebetween them. That is, if each impressed movement is decomposed in two others, one ofwhich is that the body loses, the other that the body will take. But it follows immediatelyfrom what was just said, namely, that the actual motion of each point is the result of theimpressed motion and the resisting forces that it receives because of its connection withother [points], and this is self-evident. Thus the principle of D’Alembert is basically thatsimple idea that is barely noticeable in the course of reasoning, and which takes the form ofa principle only for the expression that it is given to it [194].57 (A.10.38)

It is useful to report a comment by Lagrange to this interpretation of Poinsot and itsreply [197] that may serve to clarify the difference in viewpoint between the twoscientists. Lagrange writes:

The advantage of the principle of D’Alembert is to find the law of motion regardless ofresistance or constraint forces [forces constraint] exerted against it [197].58 (A.10.39)

Poinsot replies:

The forces of resistance of which it is discussed are nothing but forces capable of beingbalanced on the system, they are the same ones that employs D’Alembert. It is, if one wants,to shorten that they are called mutual resistance forces [197].59 (A.10.40)

56 pp. 335–337.57 p. 233.58 pp. 72–73, part II.59 p. 73, part II.


Lagrange criticizes the fact that the concept of constraint forces appears; Poinsotagrees and states that he speaks of constraint forces “only to abbreviate” and thatin fact this term means the forces that must be absorbed by the constraints, whichare absolutely comparable to the destroyed motions of D’Alembert, or also boundto maintain balance in the system, and that the difference is only a matter of“words”.

Interpretations of D’Alembert’s principle as that of Mach introducing (a) theforces in place of motion and (b) internal or external constraints with no impactwere subject to many criticisms. I will not enter into the details and I just imaginemyself in the role of Lagrange in responding to these objections. The first claim iscorrect, D’Alembert’s principle was translated from the language of a metaphysicalsystem in which there are no forces, to that of a metaphysical system in which thereare forces and constraint reactions. But I believe that the translation given by Machwould be probably natural for Lagrange who had no particular position on the on-tological status of forces. Note also that applications of D’Alembert’s principle byD’Alembert, in some cases – see for example problem I of the Traité de dynamique[84]60 – which is then taken up by Mach, where D’Alembert studies the motion ofa compound pendulum – the suppressed motions are described as ‘puissances’, thatis, as forces and although D’Alembert for ‘puissances’ means something differentfrom what we consider as force, the fact remains that the translation of his principlein terms of forces is inviting.

The second objection is less serious. To suppose that a collision takes placethrough rigid rods without mass is equivalent to assume a system constrained forinternal constraints, those for which the distances between points does not vary. Theimpressed motions instead of being real as in the case of direct impact are thosewhich would be if there were no constraints and all the reasoning of D’Alembertruns well. Then D’Alembert applied his principle even when there are external con-straints (see again the problem I, when the pendulum is suspended on an externalconstraint).

60 pp. 96–97.

11

Lazare Carnot’s mechanics of collision

Abstract. This chapter is devoted to Lazare Carnot’s mechanics of impact and toa formulation of a WVL generalized to dynamics. In the first part the mechanicsof impact of hard bodies is presented. Through an appropriate definition of virtualvelocity and motion (the geometric motion) Carnot succeeds in formulating a gen-eralization of VWL that allows one to evaluate the velocities of a system of hardbodies each of which the initial velocity is known. In the second part his exten-sion to gradually variable forces is presented with the introduction of virtual or realwork, named moment of activity, as the fundamental magnitudes of applied mech-anics.

Lazare Nicolas Marguerite Carnot was born in Nolay, Côte-d’Or, in 1753 and died in Magdeburg in 1823. He was oneof the very few men of science and of politics whose careerin each domain deserves serious attention on its own mer-its. Nicknamed Organizer of Victory or The Great Carnot, in1771 he entered the Mezieres school of engineering, wherehe had met and studied with the likes of Benjamin Franklin.It was here that he early made a name for himself both in theline of physics and in the field of fortifications. Although in

the army, he continued his study of mathematics. Carnot entered politics in 1791when he was elected a deputy to the Legislative assembly from the Pasde-Calais.In the military disasters in Belgium in the spring of 1793 Carnot had to overridethe demoralized generals and organize first the defence and then the attack to hisown prescription. On August 1793, the Convention appointed Carnot a member ofthe Committee of public safety. As minister of war he reorganized the French army.In 1797 the leftist coup d’état displaced Carnot from government. He took refugein Switzerland and Germany, returning in 1800 soon after Napoleon’s seizure ofpower. Throughout the Napoleonic period he served on numerous commissions ap-pointed by the Institute to examine the merits of many mechanical inventions. Amidthe crumbling of the Napoleonic system, he offered his services when the retreat


282 11 Lazare Carnot’s mechanics of collision

from Moscow reached the Rhine. In those desperate circumstances Napoleon ap-pointed him governor of Antwerp. Carnot commanded the defence. He rallied to theemperor again during the Hundred Days and served as his last minister of the interior(this testifies to the consistency of Carnot yet more decisively than his having voteddeath to Louis XVI some twenty-two years before). In 1816, following Napoleon’sfinal defeat at Waterloo, Carnot, the only general of Napoleon’s army to never be de-feated, went into exile in the German city of Magdeburg, where he occupied himselfwith science [290, 332].

Lazare Carnot wrote four memoirs about mechanics, two of them have the sametitle, Mémoires sur la téorie des machines, written to compete for a prize of theAcadémie des sciences de Paris in the years 1779 and 1781; both are conserved in theArchive de l’Académie des sciences and Institut de France and partially transcriptedby Gillispie [332].1 The other two memoirs are the Essai sur les machines en généralof 17832 and the Principes fondamentaux de l’équilibre et du mouvement of 1803[60].

If it is incorrect to argue that Lazare Carnot was a precursor of Lagrange, becausethe latter had already reported a mature exposition of the principle of virtual veloci-ties in his essay on the libration of the moon in 1764, when Carnot was just elevenyears old, still it is to be noted that the Essai sur les machines en général wherehe exposed and treated in depth a formulation of the virtual work law, was printedbefore the first edition of Mécanique analytique and the first proof of the principleof virtual velocities reported by Lagrange in 1798 in the Journal de l’École poly-technique [147]. For this reason, considering also that the contribution of Carnot isnot widely known, I will report an extensive comment also taking aspects not im-mediately related to virtual work laws from the Essai sur les machines en général(hereafter Essai), a slender writing, slightly more than a hundred pages. Only at afew points will I refer to the Principes fondamentaux de l’équilibre et dumouvement,where he continued and ‘perfected’ the ideas of the Essai; notice that the way to treatvirtual work laws is here influenced by Lagrange’s Méchanique analitique and lessinteresting, at least from my point of view.

Understanding the role Carnot gave to laws of virtual work requires an under-standing of his mechanics and also of his entire epistemology because his point ofview differs from the traditional one. The empiric mind of Carnot is reflected by thetitle and introduction:

It has given to this pamphlet the title of Essai sur les machines en général firstly becauseit has particularly in view machines as the most important part of mechanics and secondlybecause it does not treat any particular machine, but only the properties that are common toall machines [59].3 (A.11.1)

and well documented better below, where he comments on the two ways to approachscience, the rationalist or synthetic and the empiric or analytic approaches:

1 p. 347; pp. 271–296; pp. 299–340.2 Actually the 1783 edition is very rare; reference is usually made to the second edition of 1786[58] and to the version reported in the Oeuvres mathématiques du citizen Carnot [59].3 p. VII.

11 Lazare Carnot’s mechanics of collision 283

Among philosophers interested in the search of the laws of motion, somemake of mechanicsan experimental science, some others make of it a purely rational science. That is, the formercompare phenomena of nature, decompose them to know what they have in common, andso to reduce them to a small number of main facts which serve in the following to explain allthe others, and to anticipate what has to occur in any circumstance. Some others start fromhypotheses, then, by reasoning according to their suppositions, arrive to discover the lawswhich regulate bodies in their motion; if their hypotheses conform to nature, they concludethat their hypotheses were exact; that is bodies actually follow the laws that at the beginningthey had only supposed.The former of these two classes of philosophers, starts then in their researches from primitivenotions which nature has impressed in us, from the experiences that it continuously offers[empiric approach]. The latter starts from definitions and hypotheses [rationalist approach].For the former the names of bodies, of powers, of equilibrium, of motion are considered asprimitive ideas; they cannot and must not define them; the latter, to the contrary, must attainall from themselves and are obliged to define exactly these terms and to explain clearly alltheir hypotheses. But if this method appears more elegant, it is more difficult than the other,because there is noting more embarrassing in most natural science and especially in this[mechanics] than to assume at the beginning definitions deprived of any ambiguity. I wouldthrow myself in metaphysical discussions if I tried to deepen this argument. I will be happyonly to examine the first and simpler.[…]The two fundamental laws from which I started are then purely experimental truths, and Ipropose them as such. A detailed explanation of these principles is out of the spirit of thiswork and could serve only but to tangle things: sciences are as a beautiful river whose courseis easy to follow, when it has acquired a certain regularity; but if one wants to sail to thesource one cannot find it anywhere, because it is far and near; it is diffuse somehow in thewhole earth surface. The same if one wants to sail to the origin of science, one finds nothingbut darkness and vague ideas, vicious circles; and one loses himself in the primitive ideas[59].4 (A.11.2)

In the first part of the above quotation Carnot declared his preference toward theempiric approach; in the second part he declared the two principles assumed in theEssai (the equality of action and reaction and the conservation of momenta in thecollision) as empirical laws.

In the introduction of the Principes fondamentaux de l’équilibre et du mouvementCarnot was a little bit more vague. Here he reasserted his empiric faith:

Ancients established as an axiom that all our ideas come from senses; and this is no longerobject of dispute [60].5 (A.11.3)

Nonetheless, he also expressed the opinion that, notwithstanding the laws of me-chanics drawing much from experience, they seem so evident and clear that there isthe impression they could be derived from reasoning only:

Yet sciences do not all draw equally the basis from experience. The pure mathematics willtake less than all the others, then the mathematical physical sciences, then the physical sci-ences. It would no doubt be satisfactory in every science, to decide the point where it ceasesto be experimental to become rational, that is to be able to reduce to the smallest possiblenumber of truths which we are forced to draw from observation and that, once accepted, to-gether are sufficient for the sole reasoning to embrace all branches of science. But this seems

4 pp. 120–124.5 p. 2.


too difficult. Wanting to go up too high, with the only reasoning, you are exposed to [the risk]to give obscure definitions and vague and lax demonstrations. There are fewer problems inobtaining more information from the experience of what might be strictly necessary.[…]It is therefore from the experience that men have gained the first notions of mechanics. Nev-ertheless, the fundamental laws of equilibrium andmovement that are its basis, are presentedon the one hand so naturally to reason, and on the other hand they are expressed so clearlythrough the known facts that it seems difficult to say that it is of one rather than another thatwe have full conviction of these laws [60].6 (A.11.4)

Results of experimental observation about equilibrium and motion can be expressedby means of laws to which Carnot attributed the name of hypotheses, instead ofprinciples, to underline that they do not posses absolute evidence. He consideredalso the possibility they could be changed where not able to explain the empiricevidence.

Now it has to establish upon given facts, and upon other observations which we still couldhave, some hypotheses [emphasis added] which are constantly in accord with these obser-vations and which we can assume as general laws of nature.[…]We will then compare the consequences resulting from them [the hypotheses], with phe-nomena, and if we find they agree, we will conclude that we can consider these hypothesesas the true laws of nature [60].7 (A.11.5)

and it is not necessary that hypotheses concern phenomena which are unrelated toeach other:

My objective is not to reduce them [the hypotheses] to the smallest number; it is enough forme that they were consistent and clear enough [...] but they are the most suitable to confirmthe principle [the experimental facts], by showing that they are, as to say, nothing but thesame truth which says all the same under different forms [60].8 (A.11.6)

Using the classification of the theories proposed by Antonino Drago [300], that ofCarnot was an approach for problems, and his main problem, at least officially, wasthe study of the behaviour of machines:

Knowing the virtual [initial] motion of any system of bodies (i.e., that each body would takeif it were free) find the real motion that will take place immediately following, due to themutual action between bodies, considering them as they exist in nature, that is, endowedwith inertia, common to all parts of the matter [59].9 (A.11.7)

He did not follow, like Newton, an axiomatic approach – partly because he wasnot sufficiently accurate – giving at once a principle from which to deduce all themechanics, but rather he sought to trace the principles from elements more or lessobvious. The phenomenon Carnot considered as more immediate is that of collisionand from it he built his mechanics, which was seen essentially as the science thatstudies the communication of motion.

6 pp. 3–5.7 pp. 46–49.8 p. 47.9 p. 14.

11.1 Carnot’s laws of mechanics 285

11.1 Carnot’s laws of mechanics

Carnot begins his exposition with an historical comment on two principles hithertomost widely used in mechanics. The first principle claims that because the centreof gravity of the machines tends to move as low as possible, there is equilibriumwhen moving down is prevented. It is a generalization of the principle due to Tor-ricelli, generally known as the Torricelli principle, though not so-called by Carnot.Although Carnot thinks impossible a rigorous proof of this principle without goingback to ‘first principles’ of mechanics, nonetheless he believes that it can be givenan intuitive justification, which is presented as follows. Imagine a machine, subjectonly to weight, in a certain arrangement of its constituent parts. If there is equilib-rium the sum of the resistances of the fixed points or of any obstacles, estimatedin the opposite direction to the weight, equals the total weight of the system. But,Carnot says, if a motion can originate, some of the weight will be used to producethe motion and the fixed points will be loaded only by the remaining part of theweight. The difference between the force of gravity and that of the fixed points willresult in a force that will bring the system from top to bottom as if it were free,then the centre of gravity of the system will fall, then there will not be equilibrium.Such a ‘demonstration’ appears quite confused and not very intuitive to a modernreader. Meanwhile, it is not clear what Carnot means by the word ‘force’; mainlyone does not understand what he means with the power of fixed points, when takinginto account his criticism of the concept of force.

The second principle is “the famous law of Descartes”, according to which twopowers are in equilibrium with each other if they are inversely proportional to thevelocity that arises when a small movement is caused by an infinitesimal prevalenceof a force on the other, estimated in the direction of the force. Descartes certainlywould not have recognized it as his own principle, at least because it refers to theidea of virtual velocities to which he was clearly contrary.

Why Carnot chooses these principles as the most representative of ‘past’ mechan-ics is partly explained by his ideas on mechanics outlined above, which is incompat-ible for example with the Newtonian approach.

The practical reference to machines is helpful to Carnot for the exposition of hisown principles. A machine is defined as the agent used to communicate motion fromone body to another, an intermediary which is always necessary, not recognizingCarnot’s remote actions. To simplify the problem, he admits to having to deal withideal machines, massless and without any friction.

Carnot begins to enunciate what he sees as his new principles, which are alsoreferred to as laws to underline their empirical content. In the Essai there are onlytwo laws:

First law. The reaction is always equal and contrary to the action.Second law. When two hard bodies act each other, because of the collision or pressure, i.e.because of their impenetrability, their relative velocity immediately after the mutual actionis always zero [59].10 (A.11.8)

10 pp. 15–16.


As mentioned earlier, Lazare Carnot’s laws are different from those of Newtonianmechanics. Carnot’s paradigm is collision rather than continued action, which pre-vails in the Newtonian paradigm. He also believes that his laws are intuitive enough:

This Essay on machines is not a treatise of mechanics, my goal is not to explain in detailor demonstrate the basic laws that I have reported, these are truths that everyone feels verygood [59].11 (A.11.9)

The first Carnot’s law of the Essai expresses the fact that all bodies which changetheir state of rest or motion always do it from the action of some other body, to whicha force equal and directly opposite is impressed at the same time. All bodies resistchanges to their state and to refer to this resistance Carnot uses the term inertia forceof which he defends the use, for example against Euler who considered it a confusedconcept, and contributes to its spread. In more precise terms the force of inertia of abody is “The result of its current motion and of a motion equal and contrary to whichit should be in the next instant” [59].12 In the explanation of the first law, Carnot alsoprovides the prevailing meaning he attaches to force: it is the change of quantity ofmotion F = mΔv.

The second law refers to hard bodies, which according to D’Alembert, are per-fectly rigid bodies deprived of any elasticity; to justify the statement Carnot refersto experience. Here he seems not very honest in considering the case of collision ofplastic bodies which, among other things, are anything but hard, as the most repre-sentative. In any case, the hard body model in the sense employed by Carnot, waswidespread in the XVIII century and its use is justified not so much by experiments,as by the need for a simple model of behaviour. However Carnot is aware, he openlydeclares it, that the second principle leaves out the elastic bodies and justifies its ac-ceptance by noting that the case of elastic bodies can be explained by that of hardbodies by assuming the former as consisting of an infinite number of hard bodies sep-arated from each other by elastic springs. It is clear that this explanation of Carnot’sis a forced justification; for example the way to treat the elasticity which is trans-ferred from the bodies to the springs, remains unclear.

From the two laws, Carnot ‘claims’ two other “secondary principles” relative tothe collision of hard bodies, which are commonly used in mechanics, they are:

The intensity of collision or of action between two colliding bodies, does not depend on theirabsolute motions, but only on their relative motion.The force or the quantity of motion they exert on one another, is always perpendicular totheir common surface at the point of tangency [59].13 (A.11.10)

In the Principles fondamentaux de l’équilibre et du mouvement Carnot presents hislaws or principles in a more organized way. They are qualified as hypotheses and areall on the same ground. Besides the four laws of the Essai referred to above, thereare three others, assumed implicitly in the Essai [60].14

11 pp. 17–18.12 p. 64.13 pp. 16–17.14 pp. 49–50.


The fundamental Carnot’s problem, i.e. the study of behaviour of machines, isreduced to assume as a fundamental problem of mechanics the evaluation of themotion of a system of hard bodies as a result of shocks between them. This is thesame problem considered by D’Alembert in the Traité de dynamique (see Chap-ter 10). There is however an important difference. D’Alembert considers it certainlyan important problem, but to be solved on the basis of the general laws of mechanicsalready formulated by him. Carnot considers it instead the fundamental problem ofhis own theory that will be defined in the attempt to solve it. In any case Carnot’s for-mulation of the problem of collision is close to D’Alembert’s. Even the terminologyis similar, as will become clear hereafter; in particular the reference to lost motionsand the decomposition between actual motions, virtual motions and lost motions.

11.1.1 The first fundamental equation of mechanics

By applying his principles to a system of hard bodies or in any way to a systemof bodies separated by inextensible rods, Carnot obtains a first general principle ofmechanics, according to the following reasoning:

For pairs of hard particles define:

m′ and m′′ Masses of two contiguous particles;V ′ and V ′′ their velocity after the collision;F ′ the action ofm′′ overm′, or the force or quantity of motion

the first of the particles exerts on the other;F ′′ the reaction of m′ over m′′;q′ and q′′ the angles between the directions of V ′ and F ′ and be-

tween V ′′ and F ′′.For the second law, after (and during) the collision, the two bodies must have a zerorelative velocity in the direction of the force (which is unique, see Fig. 11.1). It isthen:

V ′ cosq′+V ′′ cosq′′ = 0. (11.1)

Since for the first law F ′ = F ′′, by multiplying (11.1) by F ′ or F,′′ it is:

F ′V ′ cosq′+F ′′V ′′ cosq′′ = 0 (11.2)

q"

F'

V'

V"

F"

q'

m'

m"

Fig. 11.1. Impact of two masses


and considering all the particles:

∑F ′V ′ cosq′+∑F ′′V ′′ cosq′′ = 0. (11.3)

This expression is then rewritten in a more effective way, by introducing new sym-bols and concepts.

Let it be:

• the mass of each particle of the system, m;• its virtual velocity, i.e. the velocity it could take if it were free (the velocity before

collision)W ;• its real velocity (after the collision) V ;• the lost velocityU so thatW will be the resultant of V andU ;• the force or quantity of motion F which each of the adjacent particle impresses

to m, and through which it receives all the motion transmitted by the system;• the angle X between the directions ofW and V ;• the angle Y between the directions ofW andU ;• the angle Z between the directions of V andU ;• the angle q between the directions of V and F .

First, with these new symbols, relation (11.3) assumes the expression:

∑FV cosq = 0. (11.4)

This relationship is not easily interpreted with the modern categories of mechanicssince the term F , that represents the force, maintains a certain ambiguity, or rather,it is not yet reported in one of its ‘classical’ meanings. From a formal point of viewone can say that relation (11.4) has the form of a virtual work law.

Second, to achieve a more convincing recognition, the expression (11.4) is givena different form. The quantityV−W cosX is the velocity ‘gained’ bym for the effectof the constraints, and thenm(V−W cosX) is the component of the forceF , Carnot’smeaning, acting on the particle m in the direction of V , i.e., F cosq (see Fig. 11.2).In place of (11.4) one can then write the expression:

∑mV (V −W cosX) = 0, (11.5)

mU

F

X

mV

mW

Y

q

Z

Fig. 11.2. Impact of two masses


but becauseW is the resultant ofV andU and becauseW cosX =V +U cosZ, Carnotobtains the following relation:

∑mVU cosZ = 0, (11.6)

to which he refers as the first fundamental equation of mechanics.

11.1.2 Geometric motions

At this point Carnot introduced the concept of geometric motion.

If a system of bodies moves from a given position, with arbitrary motions, but such it ispossible the system could have an equal but contrary motion, any of these motions will becalled geometric motion [59].15 (A.11.11)

In the Principes fondamentaux de l’équilibre et du mouvement the definition isslightly different:

Any motion will be called geometric if, when it is impressed upon a system of bodies, it hasno effect on the intensity of the actions that they do or can exert on each other when anyother motion is impressed upon them [60].16 (A.11.12)

The first definition is purely geometric, that is geometric motions are reversible mo-tions congruent with constraints; the second definition seems to refer to mechanicalconcepts, because the word action calls for concepts like force or work. Howeverthis is not the case and also the second is a kinematical definition, because Carnot’smechanics deals with impact of bodies and the impact is characterized kinemati-cally. In any case Carnot thinks the two definitions are equivalent and ‘proves’ atheorem for which the definition of the Principes fondamentaux de l’équilibre et dumouvement implies that of the Essai [60].17

From the examples Carnot gives it appears that geometric motions can also beinfinitesimal [59],18 [60].19 From an operational point of view the finite or infinites-imal nature of geometrical motion makes no difference because what Carnot usesis the velocity u associated to the geometric motion, called geometric velocity andsometimes still simply geometric motion.

In summary, geometric motions are those motions compatible with all constraintsfinite or infinitesimal. For unilateral constraints, not all compatible motions are ge-ometric, but only those that when reversed do not violate the constraints. With ref-erence to the infinitesimal motion of the material point constrained on the concavesurface of Fig. 11.3, that of Fig. 11.3a is a geometric motion while that of Fig. 11.3b,which detaches the material point from the surface, is not, because the contrary mo-tion is not permissible.

15 p. 23.16 p. 108. English translation from [332], p. 43.17 p. 119.18 p. 26.19 p. 130.


a) b)

Fig. 11.3. Geometric and non-geometric motions

In some parts of his writings Carnot reserves the term absolute geometric motionsto motion defined as above and speaks of geometric motions by supposition referringto those motions that could violate the existing constraints but do not actual violatethem in real motion. The use of geometric motions by supposition can be useful tosimplify the mechanical problem under consideration. In substance, Carnot says, as-sume that a unilateral constraint behaves as a bilateral one, then verify whether theassumption made is correct by checking if there are reactive forces that the ‘true’constraint cannot exercise. If this is the case the assumption of geometric motionwas not eligible. For example consider a wire capable only of tension producing theconstraint of Fig. 11.3; then assume a geometric motion (case a). If the actual mo-tion produces a compression in the wire, it means that the assumption of geometricmotion should be disregarded, but it also means that the constraint too can be disre-garded; in both cases, or assuming a geometric motion or disregarding the constraint,the analysis is made simpler. This idea of Carnot anticipates modern iterative calcu-lation procedures to address problems with unilateral constraints. It is not clear if hehad full awareness of this fact, although it probably was just to deal with unilateralconstraints that he introduced geometric motion by supposition.

Carnot gives a great emphasis to geometric motions, considering their introduc-tion as one of his major contribution to mechanics:

The theory of geometric motions is very important; it is, as I have already noted, like a meanscience between ordinary geometry and mechanics. […] This science has never been treatedin details, it is completely to create, and deserves both for its beauty and utility any care bySavants [60].20 (A.11.13)

The property of Carnot’s geometric motions which attracted the attention of Frenchscholars of mechanics, Poinsot and Ampère in particular, is the fact that they arepurely geometric, independent of the forces acting on the system to which they refer.The virtual displacements and velocities, regarded as geometric motions, are purelyimaginary motions taking place in an imaginary time and do not alter the position ofbodies and forces.

In studies by Drago [299] and Drago and Manno [302], some criticisms are ex-pounded about the way Carnot presents geometric motions and the way the first andsecond (see next section) fundamental equations of mechanics are deduced.

20 p. 116.


11.1.3 The second fundamental equation of mechanics

With the use of the concept of geometric motion relation (11.6) can be rewritten inan even more significant way. If u is a generic geometric motion,U the lost velocity,W = u+U is a virtual motion (Carnot’s meaning, i.e. before the impact), of whichUis still the lost motion because u cannot find contrast in the constraints, then Carnotcan write:

∑mUucosz = 0, (11.7)

where now z is the angle between u andU .Carnot calls relation (11.7) the second fundamental equation. This equation

completes the solution to the problem of the impact of hard bodies suggested byD’Alembert in the Traité de dynamique. D’Alembert had come to formulate theprinciple that the solution to the problem of impact is obtained by decomposing themotion before the impact a, b, c into two other motions a, b, c and α, β, γ. The firstmotion if applied alone would not have caused internal or external impact, the sec-ond motion applied alone would have been completely lost in collisions. The motiona, b, c provides the solution of the problem, i.e. the motion after the collision. Thismotion is determined as soon as the lost motion is found, as it holds a = a−α, b =b−β, c = c−γ.

Relation (11.7) when u is varied in the space of all possible geometric motions,can furnish all the equations necessary to derive all unknown quantitiesU and then tosolve the D’Alembert/Carnot collision problem. The role of the second fundamentalequation of mechanics is therefore the same as that of the virtual velocity principlein which, by assuming different virtual velocities, all the equilibrium equations areobtained. In this analogy the lost motions could be compared with a set of balancedforces, because both of them leave a mechanical system unvaried.

Carnot then defines the moment of momentum or the moment of quantity of mo-tion to indicate the scalar product between the momentum and the geometric mo-tion. In particular he introduces the moment of momentum of the current system:∑muV cosy – with u the geometric motion, V the actual velocity and y the anglebetween u and V – the moment of momentum before the collision: ∑muW cosx –withW the velocity before of the collision and x the angle between u andW – andthe moment of momentum lost in the collision: ∑muU cosz.

With simple steps, keeping in mind that for relation (11.7) the moment of quantityof motion lost in the collision is zero, Carnot demonstrates:

∑muV cosy = ∑muW cosx, (11.8)

i.e. the moment of momentum before collision is equal to that after the collision.More precisely he proves the following theorem:

In the collision of hard bodies, either that collision be direct or be made by amachine withoutany flexibility, for any geometric motion, it is invariably:1 – The moment of quantity of motion lost by the whole system is equal to zero.2 – The moment of quantity of motion lost by any part of the system of bodies is equal tothe amount of moment of quantity of motion gained by the other side.


3 – The real moment of quantity of motion of the general system, immediately after thecollision, is equal to the moment of quantity of motion of that system, immediately beforethe collision [59].21 (A.11.14)

This theorem is just the second fundamental equation (11.7) expressed in a differentway. Furthermore it “has a lot to do with what is obtained by considering the staticmoments with respect to different axes, but it is more general”.

The third proposition of the theorem shows that to remain unchanged it is notso much, as Descartes thought, the quantity of motion (understood in the sense ofscalar quantity) and not even the living force, because it always decreases in case ofcollision among particles, but there is a different quantity that neither the obstaclesthat oppose the motion, nor machines that transmit it, can change and this quantityis the moment of quantity of motion

From the above theorem Carnot obtains the first corollary:

Of all the motions of which a system of hard bodies agent on each other, either throughan immediate contact, or through a machine without any flexibility, is capable, the motionwhich will actually take place after the collision among hard bodies, will be that geometricmotion such that the sum of the products of each of the masses by the square of the speedthat it will take is a minimum, i.e. less than the sum of the products of each of these bodiesby the speed that it would have lost if the system had taken a whatever geometric motion[59].22 (A.11.15)

The proof of the corollary, which, for the sake of brevity I do not reproduce here,shows that the differential of the lost quantity of motion is zero when the geometricmotion coincides with the real one. Carnot pointed out that this theorem has manysimilarities with the principle of least action of Maupertuis.

As second corollary he obtains:

In the collision of hard bodies, either that some are fixed or they are all mobile (which is thesame), either the impact is direct or it is made by means of any inelastic machine, the sumof the living forces before the impact is always equal to the sum of the living forces afterthe collision, plus the sum of the living forces that it would have taken place if the residualvelocity of each mobile were equal to that lost in the collision [59].23 (A.11.16)

The proof of the second corollary is the following. PositW the velocity before thecollision, V the velocity after the collision (the actual velocity), U the lost velocityand Z the angle betweenU and V ; from geometry we know that:

W 2 =V 2 +U2 +2UV cosZ, (11.9)

according to what today is known in geometry as Carnot’s theorem.24 Summing upall the hard bodies and bearing in mind (11.6) one has the second corollary:

∑mW 2 = ∑mV 2 +∑mU2, (11.10)

21 p. 42.22 pp. 44–45.23 p. 48.24 Carnot’s theorem is today proved in a few easy steps, based on the property of the inner productbetween vectors:W 2 = (V +U) · (U +V ) =U2 +V 2 +2U ·V =U2 +V 2 +2UV cosZ.

11.2 Gradual changing of motion. A law of virtual work 293

still known inmechanics as Carnot’s theorem, which is clearly valid only for systemsof hard bodies. But according to the Geometers of the XIX century it seemed it couldbe extended to the more general case of a very sudden shock – see for example theworks of Coriolis, Cauchy, and Sturm in the first half of the XIX century.

11.2 Gradual changing of motion. A law of virtual work

A third corollary extends the mechanics of collision to actions and movementschanging gradually, so it is of considerable importance in applied mechanics. It says:

When a system of hard bodies changes its motion for imperceptible degrees [gradually], m isthe mass of each body, V its velocity, p its moving force, R the angle between the direction ofV and p, u the velocity which m would have if the system would take any geometric motion,r the angle formed by u and p, y the angle formed by V and u, dt the element of time, it willhold any of two equations [59].25 (A.11.17)

∑mV pdt cosR−∑mVdV = 0

∑mupdt cosr−∑mud(V cosy) = 0.

In this corollary there appears the concept of moving force; it coincides with forcein the ordinary sense per unit of mass.

By posingVdt = ds, with ds an infinitesimal displacement, in the first of the twoequations, Carnot obtains:

∑mpdscosR−∑mVdV = 0, (11.11)

that clearly represents the differential form of the “principle of conservation of livingforces”.

In the examination of forces that act continuously, in particular those of inertia,Carnot is not completely consistent. Generally, for him, the force is provided bymΔV , with ΔV being the change in velocity of the body; sometimes it is provided bymΔV/dt, dt being the infinitesimal interval in which there is a change in velocity,and this is used as an example in another proof of the theorem of live forces [59].26

Before moving on to deal explicitly with machines, Carnot tries to adapt his lan-guage to that normally used by the ‘practical’ mechanicians, who talk about powerand not of lost quantity of motion. According to him, the power is the effort ex-erted by the agent, i.e. the tension or pressure, which acts on the body. The differentpowers are compared with each other, without regard to agents that produce them,because “the nature of the agents does not change anything about the properties ofpowers, which are required to satisfy the different uses the machines themselves arerequired” [59].27

25 pp. 49–50.26 p. 84.27 p. 64. Carnot also tries to extend his theory to systems with elasticity. First, he points out that(11.7) also applies to bodies that are not hard, while relation (11.6) is no longer valid. He also pro-


But a force or power considered in this sense, according to Carnot, is just a quan-tity of motion lost by the agent that exerts it, whatever that agent be, that it pulls witha rope or pushes with a rod. This power F coincides with the lost quantity of motionmU . So if one denotes by z the angle between the force F and the generic geometricmotion u, the second fundamental equation becomes:

∑Fucosz = 0. (11.12)

“And it is under this form that henceforth we will use this equation” [59].28 Headds that this general principle is the one of Descartes, to which is given a greatergenerality.

Carnot reformulates all the previous theorems and corollaries using the conceptof power F , generally referred to as force or weight. The ‘principle of Descartes’generalized can also be applied to systems in motion if the forces of inertia are addedto other powers. Below the general formulation of the virtual work law according toCarnot is reported, which substantially fits that of Lagrange, D’Alembert’s principleincluded:

Fundamental theoremGeneral principle of equilibrium and motion in machinesXXXIV. Whatever is the state of repose or of motion in which any given system of forcesapplied to a Machine, if it is given any geometric motion, without changing these forcesin any respect, the sum of the products of each of them, by the velocity which the point atwhich they are applied will have in the first instant, estimated in the direction of this force,will be equal to zero [59].29 (A.11.18)

It is interesting also reading what Carnot writes in a footnote which specifies whyand in which sense relation (11.12) still holds in dynamic situations:

It would not be useless to prevent an objection that could be presented to the spirit of thosepeople who have not paid attention to what has been said about the true meaning that must begiven to the word force. Imagine, for example, they would say, a winch to the axle and wheelof which weights are suspended by ropes, either there is equilibrium or uniform motion, theweight attached to the wheel will stay to that attached to the axis as the radius of the axisto that of the wheel, and this is consistent with the proposition [(11.12)]. But it is not thesame thing when the machine takes an accelerated or delayed motion. Thus it would seemthen that the forces are not at all in the inverse ratio of their mutual estimated speed in thedirection of force as it would follow from the proposition [(11.12)]. The answer to this isthat, in the case where this motion is not entirely uniform, the weights in question are not theonly forces applied to the system, because the motion of each body, constantly changing,opposes in each instant, because of its inertia, a resistance to this change of state and onemust therefore take account of this resistance. We have already said how to evaluate thisforce, and we will see later how it has to enter in calculations. It is enough to point out thatthe forces applied to the machine in question, are not the weights but the quantities of motionlost by these weights. Which must be estimated from the tension of the rope with which theyare suspended. That the machine is at rest or in motion, this motion is uniform or not, the

poses an approximate method to deal with the case by introducing a multiplier – it seems invariableover time – of the force of impact F .28 p. 66.29 p. 73.

11.3 The moment of activity 295

tension of the string attached to the wheel, is to that of the string attached to the axis as thebeam axis is to the radius of the wheel. I.e. these tensions are always in inverse relationship tothe speeds of the weights they support, and this is consistent with the proposition [(11.12)].But these tensions are not equal to their weights at all, they are the result of these weightsand their forces of inertia, which are themselves the result of the current motion of bodiesand of the motion equal and directly opposed that it will actually take the next time [59].30

(A.11.19)

So finally equation (11.12) is a classical formulation of a virtual fork law, with geo-metric motions u that play the role of virtual displacements. It is certainly admirablethe way in which Carnot reached his conclusion from phenomena that appear to bevery simple and well defined. According to the categories introduced in Chapter 2,Carnot’s ‘demonstration’ has a foundational approach, which does not require anypreexisting equilibrium criterion. The arguments, however, are not strict, mostly thetransition from the case of collision to that of the continuous variation of motion isnot clear, i.e. the identification of lost motion with power. This creates a prolifera-tion of lost motions. On the one hand there are motions lost by means of constraints,on the other hand those lost by the agents and it is not clear why one should apply(11.12) to only the latter type of motion.

However Carnot’s move produces the ‘miracle’ to transform a law of motion intoa law of equilibrium. With his formulation in terms of power Carnot may eventuallyprove Torricelli’s principle, which he states in relation to machines:

When some weights fitted to a machine are in equilibrium with each other, if you give thismachine any geometric motion, the velocity of the centre of gravity of the system evaluatedalong the vertical should be zero [59].31 (A.11.20)

Carnot’s proof is very simple. Let u be a geometric motion; theorem (11.12) appliedto weights mg gives:

∑mgucosz = 0. (11.13)

But for the geometric properties of the centre of gravity, if u = ucosz denotes thecomponent of u along the vertical, it is:

mgu = MuG = 0, (11.14)

where M is the total mass of the system and uG the velocity in the vertical directionof the centre of gravity, and then ultimately one has uG = 0; end of proof.

11.3 The moment of activity

Lazar Carnot was opposed to introducing force as a founding concept:

There are two ways to deduce mechanics from its principle. The first is to consider it asthe theory of forces, that is the causes which impress motion. The second is to consider

30 pp. 73–75.31 p. 77.


mechanics as the theory of motion in itself. The first way is that generally pursued, as thesimplest; but it has the shortcoming to be founded on a obscure metaphysical concept, thatof force [60].32 (A.11.21)

Carnot preferred the second way. He was not however opposed to the word forcewhich he used often, sometimes with a technical meaning: “it is the product of themass multiplied by the velocity it could take if it were not impeded by bodies havingmotions incompatible with it” [60],33 some other times according to the sense of thecommon language, sometimes even with the meaning of work.

Carnot anyway sustained that, when the motion of a machine is concerned, that offorce was not the most important concept because the effect it produces depends alsoin the way it is applied. To take into account this way Carnot introduced a conceptcoinciding with the modern meaning of work. He was not the first to do this, but hewas the first to give it an emphasis and an operational meaning as a foundation ofmechanics, especially for applied mechanics. The term he used to indicate work ismoment of activity:

If a force P moves with a velocity u and the angle formed by P and u is z, the quantityPcoszudt, where dt is the element of time, is called moment of activity [emphasis added]consumed by force P during dt [59].34 (A.11.22)

The total moment of activity during a finite interval of time T is given by:∫ T

0Pucoszdt. (11.15)

Note that here there is an important shift compared to what had been done so far.Carnot’s laws, the first and second fundamental equations of mechanics (11.6) and(11.7), and the relation (11.12) that is valid for slowly varying forces, are expressedin terms of the geometric motion u; consequently expressions like Fucosz andmUucosz can be given any numerical value because u is not a physical magnitudebut an undefined quantity; they are virtual works. Instead, the expression Pucoszdthas a definite numerical value for it depends on two physical magnitudes, the ‘true’force P and the ‘true’ velocity u, which in any real situation have well-defined nu-merical values, so the moment of activity is a physical magnitude, the real work.

The possibility to replace geometric motion with actual motion allows Carnot toprove some interesting theorems. In particular he can quite easily formulate, as acorollary, a fundamental result of his mechanics, the conservation of work:

Fifth Corollary. In a machine where the motion changes for imperceptible degrees the mo-ment of activity consumed during a given time by the insisting forces is equal to the momentof activity exerted during the same time by the resisting forces [59].35 (A.11.23)

which comes directly from relation (11.12) when the geometrymotion ‘u’ is replacedwith the true motion ‘u’.

32 p XI.33 p. 2.34 p. 69.35 pp. 82–83.

11.3 The moment of activity 297

I sense some relief in a theorem ‘proved’ by Carnot inductively, by showing it isvalid in all cases he knows, which concerns the moments of activity and, using mod-ern language, coincides with the theorem of minimum of potential energy applied tocentral forces.

Let some bodies, subject to an attraction exerted according to any function of distance, eitherby these bodies one on another or by different fixed points, be applied to a machine. If thismachine passes from any assigned position to that of equilibrium, the moment of activityconsumed by the forces of attraction from which bodies are animated, will be a maximumin this passage [59].36 (A.11.24)

The position of Carnot in attributing an important role to the concept of work willhave a considerable influence on later scientists and his son Sadi, who will developa thermodynamic theory based on the concept of work, which he will call movingpuissance. Towards the end of the Essai he addressed the problem of perpetual mo-tion and showed how it is impossible, on the basis of his principles, for the presenceof passive forces.37 But although the impossibility of perpetual motion is easily jus-tifiable, it is not employed by Lazare Carnot as a principle of mechanics, as was doneby his son Sadi Carnot for thermodynamics.

In considering the way in which Lazare Carnot’s ideas of work influenced subse-quent scientists, such as Petit, Coriolis, Navier, Poncelet, Saint Venant and so on, itis interesting to note the following observation of Gillespie, for whom Carnot’s writ-ings were less significant than his interactionwith other scientists and his prominenceas a statesman:

The failure of Lazare’s Essai sur les machines to attract contemporary attention has alreadybeen discussed. Although more often mentioned since, the Principes fondamentaux faredlittle better when it appeared in 1803. It fell into the same obscurity, lasting another fifteenyears. The book was occasionally mentioned prior to 1818, but rather by way of noticing itsexistence than because its point of view affected the treatment of problems.[…]The explanation cannot well be that either father or son was an obscure or neglected per-sonality (except in Lazare’s early years). What seems likely, therefore, is that attention firstto machines and then to heat and power developed in a largely verbal, pedagogical, andpractical way. The subjects constituted a kind of engineering mechanics avant la lettre inwhich problems were posed, principles tacitly selected, and quantities employed becausethat was the way to get results. What further seems likely is that both Carnots participatedin that development personally rather than through their books. Lazare in his latter yearswas a kind of Nestor of engineering busying himself judging inventions for the Institute[332].38

36 p. 114.37 Carnot wondered what passive forces are, what difference there is between them and activeforces. He believed that this is an important issue to which no one has responded, nor even attemptedto answer. The distinctive character of the passive forces, for him, is that they can never becomeactions, while the active forces can act either as active forces or as resistant forces. Those of wallsand fixed points are passive forces because they cannot act as active forces.38 p. 101. The text by Gillispie is presently probably the most exhaustive on a historical analysis ofCarnot’s mechanics.

12

The debate in Italy

Abstract. This chapter is devoted to the debate in Italy on the principle of virtualvelocities as presented in Lagrange’s Méchanique analitique of 1788. Both reduc-tionist and foundational approaches are presented. In the first part those contributionsthat criticize the evidence of the principle of virtual velocities are introduced, whicharrived at slightly different formulations of VWLs. In the second part those moretechnical contributions are presented, aimed to reformulate the principle of virtualvelocity principle without the use of infinitesimals.

Italy was one of the European countries where virtual work laws received the great-est attention, as evidenced by the long list of Italian scholars related to this subject.In previous chapters I have shown that, before Lagrange and after Galileo and Torri-celli, other relevant contributions came from the Italian school in the XVIII century.In 1743 Ruggiero Giuseppe Boscovich, a Dalmatian mathematician deeply rootedin Italian culture, used a virtual work law in his analysis of some damage suffered bySt. Peter’s dome. But perhaps the most interesting contribution was the introductionof the principle of actions by Vincenzo Riccati in 1749. He generalized the princi-ple of virtual velocities presented by Johann Bernoulli in 1715. The same improve-ments were reiterated in 1770 by Vincenzo Angiulli, professor in a military schoolat Naples. Lorenzo Mascheroni published in 1785 a paper on statics of domes [166],where a virtual work law played a meaningful role; the study by Daviet de Foncenex,a Lagrange student, is also remarkable.

The efforts of the Italian scholars of the early XIX century, after the publica-tion of Lagrange’s celebrated book Méchanique analitique in 1788, are surely lessinteresting. They can be divided into two groups:

a) those which are addressed to improve the proof of the virtual work principle,before the second edition of the Mécanique analytique and the proof presentedby Lagrange himself;

b) those which are addressed to improve the mathematical formulation, by dis-cussing in particular the possibility of avoiding the use of infinitesimal displace-ments.


300 12 The debate in Italy

12.1 The criticisms on the evidence of the principle

In the following I will consider the contributions by Vittorio Fossombroni, GirolamoSaladini and François Joseph Servois.

12.1.1 Vittorio Fossombroni

Vittorio Fossombroni was born in Arezzo in 1754 and diedin Florence in 1844. He was a scientist and a statesman ofthe Grand Duchy of Tuscany and was particularly active as ahydraulic engineer. Since 1815 he was the chief of the gov-ernment of the Grand Duchy of Tuscany, achieving impor-tant results in the modernization of the region [294]. Fossom-broni was an important representative of the Italian school ofvirtual work laws. He presented his ideas in theMemoria sulprincipio delle velocità virtuali of 1794 [109], six years after

the publication of Lagrange’sMéchanique analitique. The work of Fossombroni didnot have a great theoretic value, but it contained some interesting contributions: itshowed, probably for the first time, a convincing demonstration of virtual velocityprinciple relatively to rigid bodies, on a purely analytical basis in a mechanics ofreference based on the cardinal equations of statics, with an approach that will betaken up by Prony. Fossombroni also raised the question of verifying the extent towhich the principle of virtual velocities remains valid when considering finite ratherthan infinitesimal displacements.

In Italy, a country then scientifically provincial, Fossombroni’s work was hailedas an event of considerable importance; it is significant that a the note appearedon June 8th 1797 in the Décade philosophique, littérarie et politique, which statedthat “It is the glory of Tuscany, which had the honor of being home to the famousGalileo, who discovered this principle, the repetition by another compatriot, the firstdemonstration”.

In the following I report a long quotation from the preface of Fossombroni’s text,which in addition to the understanding of the character of the man and somehow alsohis cultural level, contains interesting comments on the history of mechanics:

At the rebirth of Sciences, Galileo investigated the Theoretical Foundations of the equilib-rium and motion, subjecting them to geometry, and with the Principle of Virtual Velocitiesspread a new, universal radiation to all simple and compound machines.[…]In fact Mechanics by means of the Principle of virtual velocities, combined with the Ge-ometry shared the same evidence and the privileges to the full extent which this synthesiscould reach. Following the new Geometry (which swiftly fly through the space that the oldmeasured slowly, and reached places that had never penetrated) has met the most flatteringhopes, and Mr. La Grange’s first in his immortal work entitled Analytical Mechanics, notonly showed that the principle of virtual velocities is due to Galileo, but showed also, thatthis principle has the advantage of being translated into algebraic language, i.e. to be ex-pressed into an analytical formula, so all the resources of analysis will apply directly.That principle after Galileo was almost neglected, as a large sword hanging is useless, as

12.1 The criticisms on the evidence of the principle 301

long as their did not arise an arm capable of wielding it. In fact, Mr. La Grange, master ofall the mathematical entity, was able to assess its importance and fruitfulness, creating bymeans of it a new science of Mechanics, that in the universal doctrine of equilibrium, andmotion of solids and fluids, all those difficult problems that had led up to now to the thornyProblems for a thousand different ways, are reduced to regular and uniform procedures. Andto give an idea of how the human mind has progressed, we can say that the motion and theequilibrium of the Heavenly Bodies, the shape and the orbits they describe, do not call inessence, for what belongs to Mechanics, to consider other laws than those which arise incalculating the motion and equilibrium of a lever of the first kind through the difficulties ofpure calculation, and the multitude of objects to contemplate, need a larger and impressiveapparatus.[…]Some were employed to show, that this principle is true, showing the results of its compli-ance with those raised by other methods generally allowed. But really if one could not obtainother genuine proof, we would be far from the purpose for which routinely Geometers strive;in the same manner, that when the followers of Leibniz lacked a convincing demonstrationof calculus, it was weak support for them to observe the uniformity of their results with thoseof the Geometry of the Ancients.[…]That common faculty of primitive intuition, so everyone is easily convinced by a simpleaxiom of geometry, as for example, that the whole is greater than the part, certainly do notneed to agree on the aforementioned mechanical truth, which is much more complicatedthan that of the common axioms, as the genius of the great Men who have admitted theaxiom, exceeds the ordinary measure of human intelligence, and it is therefore necessaryfor those who are not satisfied to obtain a proof resting on foreign theories, such as Riccatilikes (which with somemetaphysical arguments, has considered this particular case in lettersprinted in Venice in 1772), or to rest on the faith of chief men despising the usual reluctanceto introduce the weight of authority in Mathematics. And if indeed this tyranny of reasonwere to appear only once in the Temple of Urania, it could not follow less scandal, that itwas between Galileo and La Grange [109].1 (A.12.1)

12.1.1.1 Invariable distance systems

Themost interesting part of Fossombroni’s work is on the distance invariant systems,i.e. the rigid body, the only ones to which I refer in the following, neglecting, forreasons of space, systems of many rigid bodies and fluids. Probably the most originalparts of the work are the analyses of the validity of the virtual velocity principle inthe case of finite displacement and its proof for a rigid body.

When examining the kinematics of invariant distance systems, necessary for hisdemonstration, Fossombroni introduces a distinction and a notation unnecessarilycomplicated. He indicates with the symbol d the whole motions, infinitesimal orfinite, with the symbol Δ for pure translations, and with the symbols δ, δ′ for themotion associated with rotation (finite or infinitesimal), where the presence or ab-sence of the apex is used to specify which is the axis of rotation. For example δxrepresents a rotation around the y-axis, while δ′x a rotation around the z-axis. Hedoes not give a name to his displacements; for example he does not refer to them asthe virtual velocities. Instead he uses the term virtual velocity in a ‘modern’ sense,meaning the vector and not the component in the direction of the force.

1 pp. 3–27.


First Fossombroni examines the translational motion and the equations of equilib-rium to translation, deriving easily an equation of virtual work. Using his symbols,by indicating the points with p′, p′′, p′′′, the following kinematic relations can bewritten:

Δp′ = Δx′ cosα′+Δy′ cosβ′+Δz′ cosγ′

Δp′′ = Δx′′ cosα′′+Δy′′ cosβ′′+Δz′′ cosγ′′

Δp′′′ = Δx′′′ cosα′′′+Δy′′′ cosβ′′′+Δz′′′ cosγ′′′· · ·

(12.1)

with cosαi,cosβi,cosγi the direction cosines of the forces Pi acting on pi; Δxi,Δyi,Δzi the components of the virtual displacements Δui, finite or infinitesimal, in thedirections of the coordinate axes; Δpi the components of the virtual displacementsin the direction of forces. Note that in the purely translational motion Δxi,Δyi,Δzi areequal for all points, and thus the translational motion is defined simply, for example,by Δx′,Δy′,Δz′.

The equation of equilibrium to translation for the components of the forcesP′,P′′,P′′′ on coordinate axes are:

P′ cosα′+P′′ cosα′′+P′′′ cosα′′′+ · · ·= 0

P′ cosβ′+P′′ cosβ′′+P′′′ cosβ′′′+ · · ·= 0

P′ cosγ′+P′′ cosγ′′+P′′′ cosγ′′′+ · · ·= 0.

(12.2)

By multiplying these equilibrium equations by Δx′,Δy′,Δz′ in the order, adding andconsidering the kinematical relations (12.1), Fossombroni obtains:

P′Δp′+P′′Δp′′+P′′′Δp′′′+ · · ·= 0, (12.3)

which “is the same equation of moments deduced by the Principle of Virtual Ve-locities”. Fossombroni notes that there is no mandatory reason to suppose that themotion be infinitesimal.

12.1.1.2 The equation of forces

In the analysis of the rotational motion and equilibrium of rigid bodies, Fossom-broni wants to see preliminarily if the law of virtual work can be extended to finitevirtual displacements and introduces the distinction between equation of forces andequation of moments to refer to the first when the displacements are finite and to thesecond when they are infinitesimal.

With reference to the various forces Pi applied to points A, B, C of the line AC,as shown in Fig. 12.1, parallel among themselves and perpendicular to the line AC,the equilibrium conditions to translation and rotation, give respectively:

P′+P′′+P′′′+ · · ·= 0

y′P′+ y′′P′′+ y′′′P′′′+ · · ·= 0.(12.4)

The finite virtual displacements in the direction of forces, are defined by a con-stant value Δx′ associated with a translation and values δx′,δx′′,δx′′′,etc., associ-


AB

CA B

X CE F

G

A B Cca bMY

′′′′

′′

′ ′′

Fig. 12.1. Equation of forces. Orthogonality to the line where they are applied

ated with rotation around the point M. Fossombroni expresses the different valuesδx′,δx′′,δx′′′,etc. to δx′ by means of the relation:

δxi =δx′

y′yi. (12.5)

Substituting the value of yi obtained from this equation, in the second of the equationsof equilibrium (12.4), he first obtains:

y′

δx′(P′δx′+P′′δx′′+P′′′δx′′′+ · · ·) = 0 (12.6)

or:

P′δx′+P′′δx′′+P′′′δx′′′+ · · ·= 0. (12.7)

Then, multiplying the first of the equilibrium equations (12.4) by Δx′, consideringthat Δx′ = Δx′′ = Δx′′′,etc., it is:

P′Δx′+P′′Δx′′+P′′′Δx′′′+ · · ·= 0. (12.8)

By adding (12.8) and (12.7), recalling that Δxi + δxi is the total space dpi coveredby the force in its own direction, Fossombroni writes the equation:

P′dp′+P′′dp′′+P′′′dp′′′+ · · ·= 0, (12.9)

that is valid without requiring the displacements be infinitesimal.Fossombroni can then conclude that in the case of forces applied to the points of

a line, perpendicular to it and lying on a plane, the equation of forces is valid forany motion of the plane. He finds the same result when the forces of the previouscase, while remaining parallel to each other are not perpendicular to the line AC, butslanted, as shown in Fig. 12.2 to further generalize with the following theorem:

Theorem. The equation of the forces will also hold as that of the moments, when the bodieswill be established in a straight line, and also though the forces anyway applied that havedirections not parallel to each other, have at least parallel their projections in a plane passingthrough the line of bodies [109].2 (A.12.2)

2 p. 86.


A′′′′

′′

B

CA B

XC

EF

G

A′ B Cca bMY

′′

Fig. 12.2. Equation of forces for parallel forces

This theorem may seem a mere curiosity and it is certainly true that, from a practicalpoint of view, it is not of much use even if it is considered in the most extensiveform, obtained by Poinsot [197]. But it should be seen primarily as a demonstrationof hardship and as a first step towards the elimination of the concept of infinitesimals.This topic will be taken up in Chapter 14, dedicated to Poinsot.

12.1.1.3 The equation of moments

Fossombroni established his theorem after acknowledging that, in general, it is un-lawful to use finite displacements; he then returns to a rigid body subject to anyforces, for which he proceeds to prove Lagrange’s virtual velocity principle assum-ing infinitesimal displacements. This probably is the first proof of the principle for arigid body in the reductionist approach, assuming as a pre-definite criterion of equi-librium the validity of cardinal equations of statics, three equations for translationand three for rotation:

P′ cosα′+P′′ cosα′′+P′′′cosα′′′+ · · ·= 0

P′ cosβ′+P′′ cosβ′′+P′′′cosβ′′′+ · · ·= 0

P′ cosγ′+P′′ cosγ′′+P′′′cosγ′′′+ · · ·= 0

P′(cosα′y′ − cosβ′x′)+P′′(cosα′′y′′ − cosβx′′)+P′′′(cosα′′′y′′′ − cosβx′′′)+ · · ·= 0

P′(cosα′z′ − cosγ′x′)+P′′(cosα′′z′′ − cosγ′′x′′)+P′′′(cosα′′′z′′′ − cosγ′′′x′′′)+ · · ·= 0

P′(cosβ′z′ − cosγ′y′)+P′′(cosβ′′z′′ − cosγ′′y′′)+P′′′(cosβ′′′z′′′ − cosγ′′′y′′′)+ · · ·= 0.

(12.10)

First he proves the necessary part of the virtual velocity principle, i.e. if the cardinalequations are satisfied, the equation of moments holds true. In the following I do notreport Fossombroni’s lengthy passage, also because of his not happy notation for thekinematics. I only signal that he multiplies the equation of equilibrium to translation


by translational motions and the equilibrium to rotation by rotational motions andadd all, obtaining the equation of moments:

P′dp′+P′′dp′′+P′′′dp′′′+ · · ·= 0, (12.11)

which is formally equivalent to the equation of forces (12.9), but here dp′, dp′′, dp′′′,etc. are infinitesimal. Fossombroni concludes by underlining the condition of validityof his result:

It could be concluded that in each system where the equilibrium depends from the equations(1), (2), (3), (4), (5), (6) [12.10], of § LXXI, the property sum of moments = 0 is a propertynecessary and inseparable of the equilibrium [109].3 (A.12.3)

In the following sections from LXXXIII to XCVI Fossombroni also demonstratesthe sufficiency of the vanishing of the sum of moments for equilibrium. He is notcompletely satisfied with the way Lagrange got the equation of equilibrium for rigidbodies [145]4 from the principle of virtual velocities and tries to bring some contri-bution, but he does it so confusingly, though correct, that I omit the analytic passagesof the proof that apart from being very boring do not provide interesting informationto that already given.

Fossombroni poses the question:

It is not possible to deny, that whenever equilibrium takes place, the equation of momentsis necessarily true, but is it certain that whenever there is an equation of the moments thereis always the equilibrium? [109].5 (A.12.4)

After having raised the doubt:

It could be dubious that beside these six equations [the cardinal equations of statics] therecould be some more[109].6 (A.12.5)

He is able to resolve it.The heavy treatment of Fossombroni can be justified, because in his times there

was not available the symbolism of the vector calculation. With it, the proof of nec-essary and sufficient parts of the virtual velocity principle, assuming the criterion ofequilibrium provided by the cardinal equations of statics, would result in very fewsteps.

Fossombroni’s work fell into the hands of Lagrange, who in May 1797 wrotea letter full of praise, but did not discuss its merits. The only point that Lagrangeunderlined was Fossombroni’s idea to consider finite displacements. Here is the textof the letter:

I read your book with pleasure. If there is still something to be desired in mechanics it is thereductio of principles, which serve as its basis, and perhaps even direct and rigorous proof ofthese principles. Your work is a new service for this science. You observe, correctly, thereare cases where the equation of virtual velocities also occurs in relation to finite differences,

3 p. 97.4 pp. 26–30.5 p. 101.6 p. 112.


so the system while changing the configuration still remain at rest. These kinds of equilibriaare midway between the stable equilibria, where the system returns to its first state whenit is disturbed and unstable equilibria, where the system, once disturbed from its state ofequilibrium, tends to move away more and more [206].7 (A.12.6)

Lagrange’s letter is interesting because he shows that he had understood better thanFossombroni what are the cases for which the equation of forces is valid. They arethose today classified as cases of neutral equilibrium.

Lagrange returned to the matter some time later in another letter:

I gave a demonstration of the principle of virtual velocities derived from the equilibrium ofpulleys. An important principle can be proven in many ways. Your work on this subject,besides its own merit, has that of having motivated other works as the memoirs of Pronyand Fourier, whose authors made homage to you [206].8 (A.12.7)

12.1.2 Girolamo Saladini

Girolamo Saladini was born in Lucca in 1731 and died in Bologna in 1813. He was astudent of Vincenzo Riccati and an important Italian mathematician of the end of theXVIII century. In 1808, eighty years old, he wrote a paper on the principle of virtualvelocities [213] which has little relevance from a theoretical point of view, but isinteresting as the evidence of the relative sterility of the mathematics of the time inItaly. Though Saladini was at the top of Italian mathematics, the quality of his mem-oir is not even remotely comparable to that of contemporary French mathematiciansand even to that of Fossombroni.

Saladini aims to prove the principle of virtual velocities from the rule of compo-sition of forces. He considers the case of a free material point subject to three forcesand proceed to show with simple and elegant reasoning that if the system of forcesis in equilibrium then the equations of moments are fulfilled. No account is given ofconstrained material points.

Saladini first proves two geometric theorems about parallelograms. The first the-orem [213]9 relates to Fig. 12.3. Given the parallelogram ABCD and the point X,the relation holds:

AC×OX = DC×MX+BC×NX. (12.12)

The second theorem [213]10 relates to Fig. 12.4. Here Saladini considers the dualparallelogram XVYZ the sides of which are orthogonal to those of the parallelogramABCD (e.g., XY⊥ AC). Theorem I applied to the parallelogram XVYZ, gives:

CT×XY = CP×XZ+CR×XV (12.13)

7 p. 10. See also [122], vol. 13, pp. XXIII–XXIV.8 p. 10.9 pp. 403–405.10 pp. 405–406.


A

X

CB

FD R

M

N

O

P

Fig. 12.3. The parallelogram

CR

TP

ZY

VX

D

B

A

Fig. 12.4. Dual parallelograms

or

AC×CT = DC×CR+BC×CP, (12.14)

which is the second theorem.Interpreting ABCD as the parallelogram of balanced forces such as AC, BC, DC,

and the segment CX as a virtual displacement, CP is the virtual velocity of forceCB, CR that of the force CD and CT that of the force AC. Taking the appropriatesigns, the previous relation provides the law of moments for three balanced forcesconverging in C.

The paper of Saladini appears less interesting if one reflects that theorem I isnothing but the so-called Varignon theorems, and that Poinsot a few years before,gave a very similar proof for the law of moments expressed by theorem II. MaybeSaladini could not know Poinsot’s work, he does not mention him, but this is not avalid excuse, at most it indicates the isolation of Italian mathematics.

For what concerns the proof of the inverse proposition, that is that from the equal-ity of moments there follows the rule of composition of forces, Saladini refers to thepaper by Vincenzo Riccati [339].


In the application of the virtual velocity principle to a system of material points,Saladini is rather hasty. He says that in this case it is clear that one must rely uponthe law of the lever:

If we suppose that there are more points in any way connected, that are moving around anyaxis, who now does not consider that the theory of such motion depends on the principle oflever? [213].11 (A.12.8)

Saladini then takes on two principles underlying the static demonstration of the prin-ciple of virtual velocities. One is the law of the composition of forces, which for himis of “metaphysical and geometric certainty” [213],12 the other is the law of the leverto which Saladini associated a lower level of confidence, seeing it as certainly true,but as a matter of fact and not logically necessary.

Although, as we noted, some are of the opinion that a rigorous proof of the theory of thelever studied by Archimedes and after him by other savant men still leave something to beinvestigated [213].13 (A.12.9)

By evoking two principles with different degrees of evidence, the principle of virtualvelocities may not have more evidence of the less evident, i.e. the law of the lever:

So we still have to be of the sentiment of those who have opined that the principle of reso-lution and composition of forces have a metaphysical infallibility; that of the lever only thepatronage of continuous and constant experience, and finally that of the virtual velocitiesdeduced from the two previous principles, cannot acquire higher degree of certainty of whathas been identified in the principle of the lever [213].14 (A.12.10)

12.1.3 François Joseph Servois

François Joseph Servois was born in Mont-de-Laval in 1767 and died in Mont-de-Laval in 1847. He was ordained a priest at Besançon at the beginning of the Revo-lution, but in 1793 he gave up his ecclesiastical duties in order to join the army. Hisproduction is not abundant, but it is quite original.

Though Servois was a French mathematician, I will present him here becausehis study on the virtual work of 1810 De principio velocitatum virtualium commen-tatis [214] was entered in a prize competition sponsored by the Reale accademiadelle scienze of Turin. Curiously, his memoir was the only entry the academy re-ceived and, because Servois missed the deadline, nobody won the prize. However,the paper was deemed worthy, so the Reale accademia published it and elected hima corresponding member.

The object of the prix was:

Clarify the principle of virtual velocities in its full generality such as it was enunciated byLagrange. Show if this principle should be considered as a truth evident by itself or if itrequires a proof. Give this proof in the case it is felt necessary. (A.12.11)

11 p. 415.12 p. 415.13 p. 416.14 p. 417.


Servois work is pretty interesting, but little known, perhaps because it was written inLatin and in my opinion is worthy of a specific in depth study. In keeping with aimsof the prize, in the first part of his memoir Servois traces the history of the principleof virtual velocities in the XVIII century. He gives some hints too about the state ofthe art in England and Germany, countries where the principle of virtual work wasconsidered less than in Italy and France.

After the historical part follows the theoretical part. Servois distinguishes betweena priori demonstrations, which prove the principle of virtual velocities from simpleand clear, or at least acceptable, considerations of statics without the use of a criterionof equilibrium fixed a priori, and a posteriori demonstrations, in which there is apre-established criterion. Among them, he cites the demonstrations of Varignon,Fossombroni and Poinsot, all depending on the law of composition of forces. In thesedemonstrations, the principle of virtual velocities becomes a theorem, or a corollary.

Servois aims to provide a demonstration a priori. He sets out a series of princi-ples and definitions on statics very easy to accept, with the exception of an ultimateprinciple, the 8th, which he considers his own:

In whichever way two points A and A′ are joined together, if their virtual velocities v andv′ have the same intensity, then the forces P and P′ applied [to A and A′] are equilibrated ifthey are equal [214].15 (A.12.12)

According to Servois the first seven propositions arose from the elements of me-chanics and should be considered as axioms. The last should be assumed as true orat least postulated as such. Moreover it is of little use to claim its evidence. Thisproposition does not refer only to the case where two forces are aligned, but is moregenerally applied to two points connected in some way and that can move in anydirection.

Previous seven propositions got from the elements of mechanics and should be consideredas axioms. The last should be assumed as true or at least postulated as such. Moreover it isof little use to claim its evidence [214].16 (A.12.13)

A′

A

P ′

m ′

P

m

p p

qq

Fig. 12.5. The law of the pulley

15 p. 191.16 p. 191.


On the basis of his principles Servois begins to show a series of theorems of graduallyincreasing complexity. The first theorem of some interest is the law Lagrange calledlaw of the pulley. In the case of the situation in Fig. 12.5 he shows that the law of themoments holds:

Pv = P′v′, (12.15)

with v/v′ = P′/p = 2. To demonstrate this law Servois uses his 8th proposition tostate that the tension p of the rope that wraps a pulley remains constant for all itslength. Then with simple consideration of equilibrium he obtains P′ = 2p and withsimple consideration of kinematics v = 2v′. Based on the law of the pulley Servoisgets to prove the validity of the law of moments in various situations. For examplefor a system of forces concurrent in a point.

Note that though Servois uses the same model considered by Lagrange in hisdemonstration of the principle of virtual work. i.e. the pulley, he uses it in a com-pletely different way. He claims the originality of his argument by stating that besidehim only Lagrange and the British mathematician John Landen (1719 -1790) gaveimportance to the law of the pulley:

I without problem admit that near the ancients the block and tackle and the pulley werecelebrated less than the lever, as it can be known from sources among others the 8-th bookof Pappus’ mathematical collections (the pulley was the third faculty near Hero) and recentlystill less considered, among the principles of the science of equilibrium, and probably onlyby Landen and Lagrange. When the choice of a principle is concerned, one should payattention to the evidence and especially to the fertility of it and nobody will deny that thetheory of the pulley is very useful [214].17 (A.12.14)

Though in all cases considered it has been proved only that given the equilibrium itfollows the sum of moments is zero (necessary part of the virtual velocity principle),Servois claims that it is true also for the reverse, i.e. that by imposing the vanishingof the sum of moments the equilibrium follows (sufficient condition). Indeed the suf-ficient condition results by imposing the necessary conditions for all possible virtualmotions. In this way a certain number of equations between forces are obtained thatdefine completely their relations at equilibrium.

Moreover, somebody will perhaps say that our demonstration is mutilated and incomplete,because it should be discussed not only that from the equilibrium of forces it follows theequation of moments, but also reciprocally, that from the equation of moments it followsthe equilibrium of forces. Let us consider the meaning of the equation of moments, valid forthe system of pulleys: it shows the equilibrium among forces and meantime, because of theconcatenation among the equations, it is obtained the meaning of the moment equations inall possible cases. And it will be clear that this equation should hold because the equilibriumfollows and not only because it follows from the equilibrium [214].18 (A.12.15)

This position seems to me important because it sets clearly a new way of connectingthe necessity and sufficiency of equilibrium in the principle of virtual velocities.Once the necessary conditions corresponding to all eligible virtual displacements

17 p. 220.18 p. 221.

12.2 The criticisms on the use of infinitesimals 311

are imposed, if from them there result relationships among the applied forces suchthat they can be uniquely evaluated, these relationships are also sufficient. Indeedif they were satisfied and there would not be equilibrium, the equilibrium shouldbe found with values of forces that do not meet those relationships, but this is notpossible because they are necessary.

12.2 The criticisms on the use of infinitesimals

In this section I will refer to comments by Giovanni Battista Magistrini, GeminianoRiccardi, and Gabrio Piola. Only to the latter, I will dedicate a quite large space.

Giovanni Battista Magistrini (Maggiora 1777–Bologna 1849) in a paper of 1815[165] in a section titled Del principio delle velocità virtuali, e del modo di evitarnel’uso, criticizes the principle of virtual velocity because it introduces concepts notwell defined, such as that of virtual velocities:

Because here it is necessary the use of the expedient of a mechanical motion, which thoughoriginated wonderful renowned truths, it notwithstanding leaves the desire for a clear, simpleand unique demonstration which proves in a necessary way these properties. Demonstrationof which it can be said it is not yet obtained, if considering the complication, variety anddarkness of the attempts made to find it [165].19 (A.12.16)

Magistrini aims to clean up the virtual velocity principle by redefining the conceptof virtual displacement. With arguments that I have to say are not unexceptionable,he suggests redefining the virtual displacement of a generic point indicated as δqiby Lagrange with the differential as defined in the Théorie des fonçtions analytique,i.e. the ‘aggregate’of first dimension terms resulting from substituting, in the con-straint equation, x+ i,y+ i′,z+ i′′ for the coordinate x,y,z, with i, i′, i′′ arbitrary quan-tities.

Starting from the rule of composition for the forces Q1,Q2,etc., Magistrini re-gains the moment equation in a form which is formally similar to Lagrange’s:

Q1dq1 +Q2dq2 + etc. = 0, (12.16)

but which does not contain the concept of infinitesimal.The work of Magistrini goes no further; he merely limits himself to suggesting a

redefinition of the concept of virtual displacement without bringing in-depth discus-sion of how to do so in the case of a system of constrained material points.

Geminiano Riccardi (Modena 1794–1857), thirty years later, came back to theproblem of virtual velocities, in this case in the defense of Lagrange and with a crit-icism toward the Russian mathematician Viscovatov who in an 1802 paper [242]suggested to substitute virtual displacement with virtual velocities (in the modernsense). The principle of the virtual velocity according to Viscovatov should be mod-ified as follows:

19 p. 450.


If many forces with any direction applied to a system of bodies or points are equilibrated,the sum of these powers, multiplied each for the velocity which it tends to impress [emphasisadded] to the point to which it is applied, is necessarily equal to zero. One can see that thisstatement is included in what was exposed before, but it is purified from the infinitely smallquantities [242].20 (A.12.17)

In his paper of 1842 [206] Riccardi criticizes the way Viscovatov conducts his proof,and in my opinion correctly because the paper by Viscovatov is not very clear.

A part of the discussion on virtual velocities, Riccardi’s paper is interesting forwhat it says about the diffusion of the principle in the didactic of mechanics [206].21

12.2.1 Gabrio Piola

Gabrio Piola, to whom I will return in Chapter 17, refers to his consideration onLagrange’s principle of virtual velocities in a paper which won a prize from theReale Istituto Lombardo delle Scienze in Milan in 1824 and was published in 1825[187]. The object of the prize was to “Explain the application of the main itemsof the Analytic Mechanics by the immortal Lagrange to the principal mechanicaland hydraulic problems, from which it appears the great utility and efficiency of theLagrangian methods”. Piola thinks that the virtual velocity principle as formulatedby Lagrange has two main drawbacks: it is not completely evident and it makes useof the not well-defined concept of infinitesimals.

These reflections persuaded that it would be a poor philosopher who would insist to knowthe truth of the fundamental principle of mechanics [the principle of virtual velocities] asan axiom. So it would lack of the evidence the principle I will assume […] which is thesame assumed by Lagrange in the third part of his theory on functions. But if the funda-mental principle of mechanics cannot be made evident, it should be at least a truth simpleto understand and convincing [187].22 (A.12.18)

12.2.1.1 Piola’s principles of material point mechanics

To study the motion of a material point Piola believes that the only principle reallyevident is that of superposition of motions – displacements, not forces. The principleis empirical, nonetheless it is absolutely clear because it refers to the evidence of alltimes; the same cannot be said of the principle of virtual velocities.

According to the principle of superposition ofmotions, for twomotions due to twodifferent ‘causes’, the resulting motion is the vector sum of the two motions. Piolais well aware that there are cases where this principle does not apply, for examplefor force depending on position:

If a body attracted toward a fixed point passes in a straight line during the time t the spaceφ(t), when another motion is impressed to it […]α(t) […] for the simultaneous action of thetwo motions, it does not cover the space expressed by φ(t)+α(t) but by another functionof time [187].23 (A.12.19)

20 p. 176.21 p. 8.22 p. XVI.23 p. 5.


He criticizes Lagrange for his lack of clarity in the Théorie des fonctions analytique[149] 24 where it seems that the law of the composition of motion is a purely geo-metric theorem. For him this is true for the decomposition of ideal motions, but notfor the real ones.

Although the composition of motions expressly permits non-trivial exceptions,Piola assumes it as a principle. So it seems that his problem is to understand howmuchmechanics can be explained assuming the composition ofmotions; actingmoreas a mathematician than as a physicist.

Due to the uncertainty of the general validity of the principle he used, Piola avoidsgiving an axiomatic structure to his mechanics. The various concepts are introducedwhen they are useful, without attempting to reduce each to other. In the spirit of theThéorie des fonçtions analytique Piola assumes that the general motion of a materialpoint can be developed into series:

x(t) =Vθ+12Xθ2 + etc., (12.17)

where θ= t− t0 is the difference between a reference time t0 and the current time t.The coefficient V of expansion is said to be the velocity or force exerted, the coeffi-cient X is said to be the accelerating force. Note the dynamic characterization of thevelocity which with an apparently ‘Cartesian’ language is treated as a force probablybecause at the beginning of motion the velocity can be considered as proportional tothe action of forces.

With the aid of the principle of composition of motions, Piola not only solves theproblem of motion but also that of equilibrium. A material point is in equilibrium ifand only if the motion components cancel each other, that is, for each θ the followingrelation holds:

(V1 +V2 + etc.)θ+12(X1 +X2 + etc.)θ2 + etc. = 0. (12.18)

From (12.18) it is clear that a necessary and sufficient condition for equilibrium isthat for each instant t0 it is:

V1 +V2 + etc. = 0X1 +X2 + etc. = 0

etc. = 0.(12.19)

The condition becomes less restrictive if the motion is continuous. In this case, forexample, the vanishing at all times of the sum of velocity implies the vanishing ofall terms in the series and then the equilibrium. It would appear that the fundamentallaw of equilibrium for Piola is not that of the cancellation of forces but rather thatof the cancellation of velocity. This approach, interesting and unusual, however, iscomplicated and in fact Piola leaves it, simply checking the vanishing of the sum

24 Part III, Chapter 2.


of the forces. The study of motion of a free material point is thus reduced to theclassical form:

x−X = 0, y−Y = 0, z−Z = 0, (12.20)

where X ,Y,Z are the accelerating forces while x,y,z are the components of motionthat are defined only if the initial values of velocity are provided.

12.2.1.2 System of free material points

The study of systems of material points interacting with each other requires the intro-duction of new principles and concepts, in particular the concept of mass should beintroduced. Piola is aware of the difficulties inherent to the dynamic implication ofhis choice. He solves the problem by admitting the existence of molecules of matterall equal to each other, so that the mass of an aggregate is proportional to the num-ber of atoms. In addition to the concept of mass he must also introduce the principleof action and reaction. According to Piola, this principle, which is never named assuch, may be regarded partly as a principle of reason, partly as an empirical prin-ciple, at least for material points with equal mass. It is an empirical fact that twomaterial points produce motion to each other; it could be a principle of reason thatthey move on the straight line: “it is easy to be convinced that two points, removedany other action […], will move toward each other and this motion will be on theline connecting them” [187].25

The artifice of considering mass points equal to each other also allows extensionof the principle of action and reaction to points with unequal mass. Consider in facttwo material points of mass m1 and m2 to be composed respectively of m1 and m2

single mass points of unit mass. If all points exchange the same force H betweenthem, the two points of mass m1 and m2 exchange with each other an equal forceproportional to m1m2H.

In the end, the equations of motion of a system of material points free from con-straints can be written in the form:

xi−Xi = 0, yi−Yi = 0, zi−Zi = 0. (12.21)

According to Piola it is easy to see that these equations can be deduced from thesingle variational equation:

∑i(xi−Xi)δxi +∑

i(yi−Yi)δyi +∑

i(zi−Zi)δzi = 0, (12.22)

where δxi,δyi,δzi are generic functions, variables with time, independent of eachother and are not infinitesimal virtual displacements, as it was for Lagrange.

25 p. 33.


In the case when the forces Xi,Yi,Zi can be derived from a function Π of x,y,z ,the above equation can be obtained as a variation of the functional:

U = Π+12 ∑(dx2 +dy2 +dz2). (12.23)

In modern terms it can be said that U represents the total mechanical energy of thematerial point system, which today is well known to be constant with time.

12.2.1.3 System of constrained material points

The solution to the constrained motion is obtained by analogy from the techniques ofsolving the problems of constrained minimum. If a stationary problem representedby a function like (12.23) is subject to geometric constraints such as:

L = 0, (12.24)

the solution is obtained by the method of Lagrange multipliers, making stationarythe function:

Π+12 ∑(dx2 +dy2 +dz2)+λL, (12.25)

where λ is an arbitrary coefficient, and this corresponds to Lagrange’s principle ofvirtual velocities. Note that no use is made of infinitesimals.

The reasoning by analogy of Piola is however entirely devoid of any physicalbasis. No one tells us that for a constrained problem the motion is provided by min-imizing the same functional valid for the free motion. Piola is implicitly taking theidea of smooth constraints, assumptions that had shown all its problematic nature,in the attempts to demonstrate the principle of virtual velocities.

Probably in his youthful work, Piola felt the Lagrangian principle of virtual ve-locities as indubitable. Only the need to relate to epistemology of the times led himto attempt the demonstration, then if this proof was valid only at the rhetorical level,it would not matter. Piola surely realized the weakness of his arguments, because inhis subsequent memoirs on mechanics he never attempted to prove the law of virtualvelocity which definitively became for him the indubitable principle of mechanics.

13

The debate at the École polytechnique

Abstract. This chapter is devoted to the debate at the École polytechnique on theprinciple of virtual velocities as presented in Lagrange’s Méchanique analitique of1788, assuming a reductionist approach. In the first part, after a brief mention ofGaspard de Prony’s contribution, three interesting demonstrations by Fourier of theprinciple of virtual velocities are presented. Fourier considers the case of unilateralconstraints also. In the second part the demonstration of Ampère is presented. Notethe use of purely geometric virtual velocities, following the lead of Lazare Carnot.In the third part the probably less interesting demonstration of Pierre Simon Laplaceis reported.

After the excitement of 1789, the young French republic was in difficulty, having tofight against internal and external enemies. In early 1794 the situation became des-perate and the state began to feel a dramatic lack of scientific and technical services.In March 1794 on the initiative of Monge and Lazare Carnot, the Committee of pub-lic safety appointed a commission of public works which formulated the institutionof an École centrale des travaux publics. The École was established by December1794 with headquarters in the old Palais Bourbon. Its teachers were chosen fromamong the greatest names in science and students were recruited with a contest, thenotice of which was spread throughout France. The rules stated that the students ad-mitted to the school were to be salaried in a dignified manner and housed outside thePalais Bourbon, with the general populace.

The first year saw the enrollment of 400 students at different levels. A first roundconsisting of a three-month course allowed division into three groups: those whocould immediately enter the service of the state, those who needed a year of study,and those who needed two years. Since its inception, the school, which will be calledÉcole polytechnique in September 1795, had a well-defined objective. It was to pro-vide its students with a solid scientific training based on mathematics, physics andchemistry. The École polytechnique was preliminary to specialist schools such asthe École du genie, the École de mines, and the École des ponts et chaussées. In tenyears, from 1794 to 1804,many eminent mathematicians were produced by the École


318 13 The debate at the École polytechnique

polytechnique, such as Poisson and Poinsot, physicists such as Malus and Biot, thechemist Gay-Lussac; later Cauchy, Ampère, and with some problems, Saint-Venant.

The excellence of the École polytechnique impelled Napoleon Bonaparte tochoose for his expedition to Egypt some of the most prestigious teachers, Mongeand Berthollet, along with forty two students. To curb the political vocation of stu-dents who urged them to stand up and fight government decisions, Napoleon decidedto give the École a military structure. The headquarters was moved to the Mount St.Genevieve, in the premises of the College of Navarre and the College of Boucurt,and reamined there until 1975. Its motto was:

For the Country, the Science and the Glory. (A.13.1)

In 1814, despite disputes with the empire, with foreign troops on the outskirts ofParis, students who had followed only a few courses of artillery, defended with greatcourage the Barrière du Trone. With the arrival of Louis XVIII they returned to theirhomes.

Unpopular measures, such as the removal of the old Gaspard Monge created seri-ous disturbances, so that, in 1816, the king will suspend the school. Auguste Comtewas one of the students suspended. The courses were resumed only in 1817, withhalf of the students. The École polytechnique was equipped with a new statute, theuniform became civil, the students were in boarding school and discipline, besidesbeing heavy, also imposed religious obligations. However, the objective of train-ing technicians and scientists for the state remained. Throughout the reign of LouisXVIII and still more that of Charles X, the students were in a sharp contrast with thegovernment. Nevertheless they continued to study under the guidance of renownedteachers, mostly former alumni of the École: Arago, Cauchy, Petit and Gay-Lussac.They, however, participated actively in the risings of 1830.

The arrival to power of Louis Philippe brought a little order. The École polytech-nique regained its military status, but students continued to express disagreement,so that they were suspended in 1832, in 1834 and 1844. In 1848 they were still inthe street, but this time as a mediating force between the regime and the insurgents.Even the prince president, who later became Napoleon III, had little sympathy forthe École, whose students did not submit to the central power. However, the coursesfollowed each other regularly on the premises of Mount St. Geneviève providingscientific and technical services. The army absorbed a large part of them and twosoldiers of the École, Faidherbe and Denfert-Rochereau, saved the honor of the armyin the disastrous 1870 war. During the Paris Commune and its bloody repression, theÉcole polytechnique was moved to Bordeaux and Tours, following the advance ofthe Germans. After 1870, despite the recruitment of graduates being mainly depen-dent on the army, sciences were not yet abandoned. As an example we mention oneof the graduates: Henri Becquerelle, Nobel prize winner in physics [256].

The École polytechnique is still active today, although its headquarters since 1975is at Paliseau, in larger premises. The motto still is: For the Country, the Science andthe Glory.

13.1 One of the first professor of mechanics, Gaspard de Prony 319

13.1 One of the first professor of mechanics, Gaspard de Prony

Gaspard Clair François Riche de Pronywas born in Chameletin 1755 and died in Paris in 1839. Educated at the École desponts et chaussées, he was appointed in 1794 a professor ofthe École polytechnique. Pronywas professor to Poinsot whofollowed his courses in mechanics, most likely in 1797 [197]and the study of his writings is useful to better understandthose of Poinsot. In 1821 he invented the Prony brake to mea-sure the performance of machines and engines.

Prony was concerned with the virtual velocity principlewhich was discussed in his best known work, the Mécanique philosophique [203],but the work in which he expresses his ideas more fully on the subject was the Mé-moire sur le principe des vitesses virtuelles, which appeared in the Journal of theÉcole polytechnique of 1797 [202] in which also Fourier and Lagrange publishedtheir contributions to the demonstration of the virtual velocity principle. One aspectof some interest is the reference to the work of Vittorio Fossombroni’s memoir on theprinciple of virtual work, which left some footprints also on Poinsot. Prony writes:

I must also refer the students to a work of which they will find very useful addition to thelessons received at the École on the same matter: it is a memoir published in Italian inFlorence in 1796, and headed by Mr. Fossombroni, Memoria sul principio delle velocitàvirtuali. This treatise will provide a number of exercises especially well suited to those whowant to study the Mécanique analytique [202].1 (A.13.2)

Prony began to put the argument into his own hands. It is true, he says, that Fourierand Lagrange offered excellent demonstrations of the virtual velocity principle, how-ever, these treatments are not appropriate for students and therefore there is the needto develop others that are easier. Prony’s demonstrations, which cover almost allcases of equilibrium of rigid bodies, are actually a little easier.

He immediately shows correctly the necessary part of the virtual velocity princi-ple, i.e. if a system is in equilibrium according to the criteria provided by the cardinalequations of statics, then the sum ofmoments (Lagrange’s meaning) is zero. But thenhe is not satisfied, because:

The previous proofs leave nothing to be desired in rigor, but the equation of virtual velocitiespresented in this way is a consequence [a theorem] rather than a fundamental truth and it isnecessary, because it retains the characteristics of a principle, to deduce it from theorems[principles] of mechanics even more elementary and closer to the truth that derive directlyfrom the definitions than those that I have used. That is what I am going to do by supposingonly the composition of powers applied to the same point and that of parallel powers [202].2

(A.13.3)

So basically he starts over again, only taking for granted the rule of composition offorces. For him, while the cardinal equations of statics could not be regarded as moreimmediate of the principle of virtual velocity, the rule of the parallelogram does.

1 p. 204.2 p. 196.


Prony justifies the principle of virtual velocities by examining many cases of dif-ferent complexity. He limits his analysis to proving the necessary part of the prin-ciple, i.e. if there is equilibrium the equation of moments holds true. After that hereversed the order followed at the beginning and got the cardinal equations of equi-librium for rigid bodies from the equation of moments. His speech on this part is notcompletely rigorous, which is partly justified by the teaching nature of the work. Inwhat follows I will refer only to the proof of the necessary part of the virtual velocityprinciple starting from the rule of composition of forces.

13.1.1 Proof from the composition of forces rule

Prony begins by assuming the rule of composition of forces, expressed in algebraicform, for the forces PI ,PII ,PIII,etc. balanced and all converging toward the point p.

PI cosαI +PII cosαII + etc. = 0

PI cosβI +PII cosβII + etc. = 0

PI cosγI +PII cosγII + etc. = 0.

(13.1)

The component dpi of the generic virtual displacement of p in the direction of theforce Pi is given by:

dpi = e1 cosαi + e2 cosβi + e3 cosγi, i = I, II, etc. (13.2)

where e1,e2 and e3 are the components of the displacement dp along the coordinateaxis X ,Y,Z, and αi,βi,γi are the angles that the forces Pi form with X ,Y,Z.

With a procedure similar to that of Fossombroni, Pronymultiplies the equilibriumequations (13.1) respectively by e1,e2,e3, adds the three equations and in the lightof (13.2) he gets the equation of moments:

PIdpI +PIIdpII +PIIIdpIII + etc. = 0. (13.3)

Then Prony moves on to the case of two sets of forces PI ,PII ,PIII , etc. and PI ,PII ,PIII , etc. applied at the ends of a rod that acts as a link between the two points thatmake up the ends, as shown in Fig. 13.1, so that the overall system is in equilibrium.

P III

P II

P I

-T

PI

PIII

PII

T dt

Fig. 13.1. Equilibrium of forces applied to a rigid rod

13.2 Joseph Fourier 321

This case is quite interesting because it shows how Prony addresses the issue ofconstraints. He does it in a way already traditional for those times, assuming theexistence of ‘real’ forces exerted by constraints. Prony considers the ends of therod as material points, where the balance results from the active forces applied andreaction forces. As for the single material point he has already shown, the equationof moments, he can write the two equations:

PIdpI +PIIdpII + etc.+T1dt1 = 0

PIdpI +PIIdpII + etc.+T2dt2 = 0,(13.4)

where T1,T2 are the forces exerted by the rod at both ends, and dt1,dt2 are the in-finitesimal virtual displacements of both ends of the rod along its direction. At thispoint Prony assumes without any comment a principle of action and reaction forwhich T1 = −T2 = T , which is not completely evident and which will be criticizedby Poinsot. Moreover, because the rigidity of the rod and the infinitesimal displace-ment is dt1 = dt2, the previous relations can be written as:

PIdpI +PIIdpII + etc.+Tdt = 0

PIdpI +PIIdpII + etc.−Tdt = 0.(13.5)

By eliminating the reaction T between the two equations, relation (13.5) gives:

PIdpI +PIIdpII + etc.+PIdpI +PIIdpII + etc. = 0 (13.6)

and then still the equation of moments.Finally, Prony considers a simple not deformable body in the plane, formed by

a triangle, to the vertices of which forces are applied so that the triangle is in equi-librium and finds again the equations of moments. From the triangle to a rigid bodythen the passage is almost immediate. Here however he assumes for granted someother ‘principles’ of statics.

13.2 Joseph Fourier

Jean Baptiste Joseph Fourier was born in Auxerre in 1768and died in Paris in 1830. In 1795 he was appointed adminis-trateur de police, or assistant lecturer, to support the teachingof Lagrange and Monge. In 1798 Monge selected him to joinNapoleon’s Egyptian campaign. He became secretary of thenewly formed Institut d’Egypte, conducted negotiations andheld diplomatic posts as well as pursuing research. After hisreturn to France in 1801, Fourier wished to resume his workat the École polytechnique but Napoleon had spotted his ad-

ministrative genius and appointed him prefect of the department of Isère, centredat Grenoble and extending to what was then the Italian border. In 1808 Napoleonconferred a barony on him. In 1817 he was elected to the Académie des sciences, of


which, in 1822, he became perpetual secretary [290]. He is mostly famous for hiswork on the transmission of heat [111].

Fourier’s studies on the virtual velocity principle are reported in the Mémoiresur la statique contenant la démonstration du principe des vitesses virtuelles, etla théorie des moments of 1797 [110]; they are among the most interesting of hisworks. The memoir opens by claiming the purpose is to prove a virtual work law,in particular Lagrange’s principle of virtual velocities, without any reference to theparticular nature of the system under examination:

I also thought it was not enough to prove in an absolute way, the truth of the proposition,but we must do so regardless of knowledge that we have of conditions of equilibrium indifferent kinds of bodies, since these conditions should be considered as consequences ofthe general proposition. This objective is fulfilled by the demonstrations that I am going torefer. It seems that they leave nothing to be desired both in respect of the scope and accuracy.We will assume as known the principle of the lever, as shown in the books of Archimedes,or Stevin’s theorem on the composition of forces, and some propositions easy to deducefrom the previous [110].3 (A.13.4)

To satisfy his purpose Fourier assumes two principles that were considered indu-bitable, like axioms, at his time: the law of the lever and the rule of compositionof forces, known as Stevin’s theorem. It must be said however that Fourier is notalways completely rigorous, in particular he does not distinguish always the differ-ence between necessity (equilibrium → vanishing of moments) and sufficiency ofequilibrium (vanishing of moments→ equilibrium).

He reports three separate demonstrations. The first is essentially based on the ruleof the composition of forces. The second and third are based instead on the law ofthe lever. In the following I summarize these demonstrations, and although the thirdis probably the most convincing, also the first and the second should be viewed,for a number of interesting observations, including those involving unilateral con-straints.

Here is how Fourier introduces the virtual velocity principle:

If a body is moved by any cause, according to a certain law, each of the quantities that varywith its position, as the distance of one of its points from a fixed point or a fixed plane, isa given function of time, and can be considered as the ordinate of a plane curve in whichtime is the abscissa. The tangent of the angle this curve makes in the origin with the x-axis,or the first reason for the increment of ordinates compared to the x-axis, expresses the rateat which that amount begins to grow, or for the use of a name known, the fluxion of thisamount.Bodies being subjected to the action of several forces, if one takes on the direction of each[force] a fixed point toward which the force tends to carry the point of the system to whichit is applied, the product of this force for the fluxion of the distance between the two pointsis the moment of the force. The body can be moved in countless ways, and each has a corre-sponding value of themoment. If themoment of each force for a given displacement is taken,the sum of all these contemporary moments will be called the total moment, or the momentof the forces for this displacement. We will distinguish the displacement compatible withthe system state, from what one cannot be undertaken without affecting the [constraint] towhich it is subject, and assume these conditions, expressed as far as possible by equations.

3 pp. 21–22.


Now, for the principle of virtual velocities, when the forces which act on a body, of what-ever nature it may be, are supposed in equilibrium, the moment of these forces is zero foreach of the displacements which satisfy the constraint equations. Bernoulli, in place of flux-ions, considers the rising increment [infinitesimal displacements]. So each of the points ofthe system should be considered as describing a small space with rectilinear uniform mo-tion during a time infinitesimally small. This small space, projected perpendicularly to thedirection of force, is the virtual velocity: and if it is multiplied by the force, the product isthe moment. I will adopt this happy abbreviation and all the usual procedures of differentialcalculus [110].4 (A.13.5)

Note the distinction Fourier makes between the method of fluxions – i.e. of thederivatives – which, with modern terminology is called the method of virtual ve-locities, and the method of infinitesimal displacements, which is called the methodof virtual displacements.

Based on this distinction Fourier defines themoment of a force in twowayswhich,though mathematically coincident for infinitesimal displacements, are formally dif-ferent. In a way the moment is the product of the force by the velocity with whichits point of application approaches its centre. In another way, moment is more clas-sically defined as the product of the force by the virtual velocity, with Bernoulli’smeaning, i.e. the projection of infinitesimal displacement in the direction of the force.Fourier declares that he has adopted this second meaning of moment.

For the understanding of the analytical developments it should be noted that forFourier the moment of a force P and a virtual velocity dp is given by −Pdp andnot, as usual, by Pdp. In fact Fourier does not speak explicitly of a negative sign,but he gives an implicit definition – if it is not a mistake – at the end of paragraph4 where he says “If two forces tend to bring close the two points, their momentwill be negative or positive, depending on whether these two points are nearer orfarther” [110].5

13.2.1 First proof

In the first proof Fourier takes as reference the rule of composition of forces, orStevin’s ‘theorem’. From this theorem it can easily be proved that the total momentof n forces P in a general virtual displacement dp of their common point of applica-tion p equals that of the resultant force. According to another theorem, which Fourierproves, it also holds that the moments do not change by moving a force along its lineof application.

Based on these theorems it is easy to show that a rigid body is in equilibriumif and only if the total moment of forces acting on the body is zero for any virtualdisplacement. In fact, if there is equilibrium, it is possible to reduce the system ofcouples of forces to equal and opposite forces. In this situation, the moment is cer-tainly zero, but as the operations carried out over the forces do not change the totalvalue of the moment, it would be zero even for the effective forces. Conversely, ifthe total moment of forces acting on a rigid body is zero for all virtual displace-

4 pp. 22–23.5 p. 25.


ments, it means that the forces can be reduced to equal and opposite forces, so thereis balance.

The system of constrained bodies is discussed in § 13. Fourier considers bodieslinked by inextensible wires and solicited by any force that is in equilibrium, andreasons as follows. The forces that act on each body are not only those applied fromoutside, the active forces, but also those that come from the wires. It therefore can besaid that if every body is in balance then the moment of the total forces – includingconstraint forces – is zero when considering any motion.

Fourier then assumes that constraint forces of each wire are equivalent to twoequal and opposite forces directed along the wire applied at its own ends. With thisassumption, if infinitesimal displacements congruent with constraints are assumed,it is easy to show that the moment for all the constraint reactions due to the wires iszero. So if the system of points is in equilibrium, because the moment of all forces iszero, even the moment of the active forces must be zero. To complete the proof of thevirtual velocity principle the reverse should also be demonstrated, i.e. if the momentof the active forces is zero for any virtual displacement compatible with the con-straints, then even the moment of all forces is zero and the system is in equilibrium.Fourier does not do it.

Fourier’s assumptions are then:

a) a rigid body is in equilibrium if and only if the forces acting on it can be reducedto collinear forces equally and contrary;

b) constraint forces have the same ontological status of the active forces;c) these forces have directions consistent with the direction that defines the con-

straint. This is the case of smooth constraint.

In different parts of his work, Fourier reflects on what happens for virtual motionsthat produce shortening ofwires, which are by definition unilateral constraints. I referbelow to the view expressed in § 6.

If one considers two forces in equilibrium, being applied to the two ends of an inextensiblewire (but not resistant to compression), it is easy to know their moments for a total displace-ment compatible with the nature of the body in equilibrium. Following the previous article,the moment is zero whenever the distance is preserved, i.e. when the equation of constraintis satisfied. For all other possible displacements, the moment is positive [Fourier assumessigns contrary to the usual convention], and the system in equilibrium cannot be disturbedso that the total moment be negative [110].6 (A.13.6)

Although Fourier does not make these considerations on unilateral constraints clear-er, they are worthy of emphasis because they are the first with a certain degree oforganic unity. It therefore seems appropriate that today, under the name of Fourier’svirtual work law, it is meant the statement that says there is equilibrium if and onlyif La ≤ 0, La being the work made by the active forces for all virtual displacementscompatible with constraints – unilateral and bilateral – though Duhem [99]7 ascribesto Gauss a thorough understanding of the meaning of the inequality La ≤ 0 [126].

6 p. 26.7 pp. 44, 195.


The consideration of Fourier on unilateral constraints were considered more indepth by Mikhail Vasilievich Ostrogradsky (1801–1862) in the paper Considera-tions générales sur les moments des forces presented in 1834 at the Academy ofSciences of St. Petersburg [178]. In this paper Ostrogradsky considered also thecase of moving constraints, that he studied more in depth in two subsequent papers[179, 180].

The objections that can be made in this first Fourier’s demonstration are the as-sumption of smooth constraints and the use of different reasonings for different kindsof systems, i.e. rigid, deformable, solid, fluid. The first objection can be removed atalmost all demonstrations of the laws of virtual work, but the recruitment of smoothconstraints seemed so natural at the time that it is difficult to believe that Fourierhas become aware of its problematic nature. The second objection is rather felt byFourier, as he seeks an alternative proof which is based on the lever and no longer onthe rule of composition of forces which, although it has the advantage of encompass-ing a fully algebraic treatment, should seem to Fourier not completely immediate.

13.2.2 Second proof

Fourier begins his second proof in § 17, stating that:

Instead of transforming the forces that urge the system, we replace this system, in whichthey operate, with a more simple body, but capable of being moved in the same way [110].8

(A.13.7)

The problem is set as follows Let P,Q,R,S, etc. be the forces applied at the pointsp,q,r,s, etc., and give an infinitesimal motion that move the points p,q,r,s, etc.along the directions p′,q′,r′,s′, etc. and resulting in the assigned virtual infinitesi-mal displacements dp,dq,dr,ds, etc. along P,Q,R,S, etc. To obtain an equivalentsystem first consider only two points, p and q, as in Fig. 13.2. Let πp be the planeperpendicular to p′ passing through p, πq the plane perpendicular to q′ and passingthrough q. From the point p the perpendicular h draw to the line ρ common to bothplanes πp,πq, and on the plane πq trace a perpendicular h′ to ρ from the intersectionof h and ρ, and finally draw from the point q, h′′ perpendicular to h that meets it ink. The two segments h and h′ can be considered as forming an angled lever whichrotates around the axis ρ. The segment h′′ may be regarded as the mobile arm of alever with fulcrum a point o. If p moves along p′ with virtual displacement dp, theangular lever p−h−k will move the straight lever o−q−k which in turn will moveq in the direction q′. The fulcrum o of the lever oqk can be chosen so that the virtualdisplacement of q is equal to the assigned value dq.

Notice that the point k moves in the direction orthogonal to the plane πq becausethis rotates around the axis p. As a consequence, qmoves in the direction orthogonalto πq and therefore parallel to qq′.

The same operation is made by joining qwith r, r with s and so on. In this way theoriginal system is replaced by an assembly of levers, which will vary by changingthe infinitesimal virtual displacement imposed on the system, but it does not matter.

8 p. 36.


q

pdp

P

p'

h

h'

h''o

ρ

q'

kπq

π p

Fig. 13.2. Reduction of a system to a set of levers

Fourier says that the equilibrium conditions of the original system are equivalent tothose of the assembly of levers and that in the latter case it is clear that the necessaryand sufficient conditions for equilibrium are provided by the annulment of moments.It must be said however that this statement leaves much to be desired because it isnot so obvious. If it is accepted that the equilibrium of a single lever is based on thecancellation of the total moment, it cannot be admitted with equal ease that this istrue for an assembly.

13.2.3 Third proof

Fourier is probably not fully convinced of the second proof and he tries another.Now he assumes that a generic force P acting at a point p, belonging to a system ofparticles, can always be thought of as due to a weight A applied at the end of a lever,the other end of which is attached to a wire which through a pulley is made to beparallel to the line of action of P, as shown in Fig. 13.3. To eliminate the dependence

pp'

P

eOc

A

ab

Fig. 13.3. Reduction of the forces on a system to two weights


of the proof on the infinitesimal nature of virtual displacements, Fourier conceivedthe weight A applied not directly to the end c of the lever, but through a ‘sector’, acurved element, which allows a non-uniform motion of the point p also when theweight A moves uniformly.

Assume now a material system, fluid or solid, constrained in any way subject toforces P,Q,R,S, etc. applied to points p,q,r,s, etc. and assume that the equation ofmoments Pdp+Qdp+Rdr+ etc. = 0 are verified for any value of the virtual dis-placement dp,dq,dr, etc. By repeating the previous reasoning, any weight P,Q,R,S,etc. can be replaced by a lever loaded by appropriate weights A,B,C,D, etc. Becausethe size of the levers and weights are arbitrary, it is possible to choose them so that,while maintaining the assigned forces P,Q,R,S, etc. and the virtual displacementsdp,dq,dr,ds, etc. to the desired values, the weights A,B,C,D all fall with the samevirtual displacement, either they come up or down.

It is possible thus to admit that all the lowered weights can be replaced by a singleweight E and the raised weights and by a single weight F , which through the ringsof reference provide their actions by means of various wires. These, through otherrings of type b, transmit their tension at the ends of type a of the lever, intendedto apply the forces. By the law of the lever – this is a crucial Fourier assumption –the moment associated, for example, with the force P must be equal to the momentof the weight A, in its vertical motion. The same applies to the other forces, so thetotal moment of the forces P,Q,R,S, etc. must be equal to that of the two weightsA,B,C,D, etc.

Because, by assumption, the total moment of forces is zero, the same holds true forthe total moment of weights and since the virtual infinitesimal displacement of E andF are equal and opposite, it can be said that the two weights are the same. To provethat in this condition the system is equilibrated, Fourier imagines to connect with arigid rod fixed in the middle the two equal weightsE and F which is assimilated – notvery convincing I must confess – to the diameter of a pulley as shown in Fig. 13.4.This is allowed because the pulley assures equal and opposite displacements for Eand F .

Fourier spends some words to prove that this system is in equilibrium, with areasoning ad absurdum, probably unnecessary because of the evidence of the fact[110].9

EF

Fig. 13.4. Impossibility of motion for E = F

9 p. 42.


The third demonstration, except for some slight embarrassment in the end, I thinkis the most compelling of those submitted by Fourier. Notice however that he doesnot refer to the demonstration of the necessary part of the virtual velocity principle,that if the system is in equilibrium then the total moment is zero. This demonstrationis however nearly implicit in Fourier’s reasoning.

Of some interest are the considerations at the end of § 23, of historical nature:

If we are content to replace each of the forces with a weight attached to a wire connectedto a fixed pulley, we recognize that for each movement of the system at equilibrium, themoment of the weights that rise is equal to that of the weights that lower, and although thisconsideration cannot be regarded as a proof nevertheless it refers the principle of virtualvelocities to that of Descartes, or the principle used by Torricelli. It is natural to think thatJohann Bernoulli knew some similar construction. There are the same ideas in a work ofCarnot in 1783 printed under this title: Essai sur les machines en général [110].10 (A.13.8)

After the three demonstrations reported in its parts I and II, the Mémoire sur la sta-tique continues with three other parts; in part III there are interesting considerationson the stability of the equilibrium. In part IV Fourier presents his own demonstra-tions of the law of the lever and the composition of forces, in order to make self-referential his work. In part V he makes concluding remarks on the generality of thelaw of virtual work, noting that it is also valid for fluids.

13.3 André Marie Ampère

André Marie Ampère was born in Lyon in 1775 and died inMarseille in 1836. During the French Revolution, Ampere’sfather stayed at Lyon expecting to be safer there. Neverthe-less, after the revolutionaries had taken the city he was cap-tured and executed. This death was a great shock to Ampère.In 1809 Ampère was appointed professor of mathematics atthe École polytechnique. Ampère’s fame mainly rests on hisestablishing the relations between electricity and magnetism,and in developing the science of electromagnetism, or, as he

called it, electrodynamics. Throughout his life, Ampère reflected the double heritageof the Encyclopédie and Catholicism. From this conflict came his concern for meta-physics, which shaped his approach to science; to point out his Essai sur la philo-sophie des scienceswhere among other things he introduced the term kinematics andclassified mechanics in statics, kinematic and dynamics [290].

He treats the principle of virtual velocities in the same issue of the Journal of theÉcole polytechnique of 1806 where the Théorie générale by Poinsot was published,with a memoir entitled Démonstation générale du principe des vitesses virtuelles,dégagée de la consideration des infinitament petite [2]. It counteracts the criticism ofPoinsot and proposes a new demonstration. Just for curiosity, I remember that Pronyand Laplace were the ‘referees’ of Ampère and Lagrange, Laplace and Lacroix those

10 p. 43.

13.3 André Marie Ampère 329

of Poinsot, and that Ampère had a better score than Poinsot. The demonstration ofAmpère is interesting in itself, but perhaps even more interesting are his introductoryremarks, indirectly criticizing the work of Poinsot (see Chapter 14), who probablyhe knew, and that of Lagrange. Ampère declared:

The principle of virtual velocities, which serves as the basis in this admirable work [theMécanique analytique] was considered by its author as a fact of which he then developsall the consequences, then the general proof of this principle has been looked for. Lagrangebrings it in a very simple way to the principle of a system of pulleys, Mr. Carnot to theequilibrium of the lever.11 The proof of this principle has been deduced by Laplace bymeansof a more general way, but too abstract to be made easily understood by beginners. I set outto provide, as far as I can with the same generality, a proof that rests only on the theoryof composition and decomposition of forces applied to the same point, and is free from theconsideration of infinitely small quantities. This is the goal that I set out in the research thatI have the honor to present to the class [2].12 (A.13.9)

He believes that he has provided a simple and convincing discussion, but in fact hiswork is difficult to read and not always well organized. He, like Poinsot, deems itnecessary to avoid using the concept of infinitesimal and to start from first principlesof mechanics, among which the most important is the rule of composition and de-composition of forces. No wonder because, as I have said elsewhere, most scholarsconsidered the composition of forces as the most ‘rigorous’ principle. Moreover itwas easiest to be expressed in formulas, a fact that, with no real valid logical reasons,made it preferable to those scientists who had a sound analytical culture.

Ampère considers the hypothesis of no interaction between the various condi-tions of constraint and concludes that these cannot be justified a priori, but only inretrospect, starting from the equilibrium equations:

The laws of equilibrium are deduced, in the most rigorous way, from considerations verysimple when the forces are applied to one point and become more difficult to prove, es-pecially if one wants to consider [the laws] in all their generality, when the forces act onpoints subject to constraint conditions that contribute to the mutual destruction of forces.The difficulty comes mainly from the need that these conditions of constraint intervene inthe calculation. At first glance, it seems that they can be considered separately, and initiallyassume a single condition, then another and so on. But a little reflection evidences that wehave to show a priori that the effects produced by the union of multiple conditions is the sumof effects arising from each specific condition, without that they have changed from theirunion. Truth that would appear to be rather a consequence of the equations of equilibriumthan a means of obtaining them [2].13 (A.13.10)

Ampère criticizes the principle of solidification too (see §14.2.1).

One more simplification that could be used in the research we are concerned, is to assume asfixed, then, all the points of the system, with the exception of two of them. This is particularlyconvenient because the total derivatives which are necessary are obtained with the union ofthe equations thus obtained, with the partial derivatives for each variable. But a very simpleexample seems sufficient to show that this assumption is not always eligible [2].14 (A.13.11)

11 Here perhaps Ampère confuses Carnot with Fourier.12 pp. 247–248.13 p. 248.14 p. 248.


Ampère’s example is not very adequate and I do not carry-over it, except to say thatthe principle of solidification, at least in the form used by Poinsot (see § 14.2.1),appears not problematic. However, the question raised is worthy of consideration.

In the proof of the virtual principle Ampère uses two basic principles: the com-position of forces and the smooth constraints assumption: “Because it is clear thata force will have no action for or against a motion of its point of application, whenit is perpendicular to the tangent line at precisely the point on the curve which itdescribes” [2].15

In the following passage Ampère exposes what is the essence of the virtual ve-locity principle according to him and what he intends to demonstrate:

The principle known as the principle of virtual velocities, reduces to the fact that if it ismade the sum of the moments of all forces applied to the system, taking with different signthose in which the projections of the forces fall on the same side and those in which forcesand projections fall into opposite sides; in addition, to this sum there are added the equationsdeduced by all the conditions [of constraint] assigned, each multiplied by an arbitrary factor,and reduced so they contain in all terms the derivatives with respect to x,y,z to the first order,if the quantities that multiply each derivative are equated to zero, separately, and all arbitraryfactors are eliminated, the remaining equation or equations express all the conditions forequilibrium [2].16 (A.13.12)

To understand better the above statement of the virtual velocity principle, in partic-ular the meaning ofmoment and projection, the proof of Ampère has to be followed.I will display only a very brief summary that, besides explaining the terms, explainsalso in what sense the proof avoids the concept of infinitesimal. I will not go intoany detail because Ampère’s approach would require a massive use of mathematicalmanipulations, which, if not complex, is at least boring.

Ampère considers in the first instance a system of constrained points so that thereis only one degree of freedom, which he identifies as the parameter s. Under thisassumption each material point moves on a pre-definite curve γ parametrized by s.The tangents to the curves x(s),y(s),z(s) in the various points are defined by thevector t of components x′(s),y′(s),z′(s), where the apex denotes the derivative withrespect to s. The component of the vector t on the force applied at point P is calledby Ampère projection. The product of the force and projection, with the appropriatesign, is the moment of the force. Ampère then eliminated the concept of infinitesimaldisplacement vector and replaced it by the ‘velocity’ t. It is clear why Ampère haschosen s as a parameter of motion instead of time, even if he does not say anythingabout it. He wants to completely eliminate the concept of time by a law of equilib-rium, thus proving to be more demanding of Poinsot himself (see Chapter 14).

The demonstration consists in refering, by means a system of rigid rods, the mo-tion of a point m on the curve γ, subject to a force P, to the motion of another pointμ moving in a straight line and to which another force S is applied, using the prin-ciple of composition and decomposition of the forces and the assumption of smoothconstraints. An examination of Fig. 13.5 partially gives the idea.

15 p. 250.16 p. 253.

13.3 André Marie Ampère 331

mγ

σ

μ M

P

AB

S

Fig. 13.5. Reduction of a motion from a curve to a straight line

The curve γ along which m moves is in general skewed; Ampère can prove thatit is possible to trace back the motion, using a rigid rod mM, first to a curve σ of aplane π, then, by means of another rigid rod Mμ to a straight line AB still belongingto π, so when μ moves on AB under a force S, the point m moves on the curve γ byforce P, preserving the moment.

In the case of many material points m1,m2, . . .mn things can be arranged throughappropriate curves σi, so that there are so many points μ1,μ2, . . . ,μn moving all onthe line AB with the same velocity and therefore they can be assumed as joinedtogether. These points, subject to the forces S1,S2, ...,Sn, shall move the pointsm1,m2, . . .mn of the curves γi, with the forces Pi so that the relation which expressesthe preservation of moments is:

∑Piu′i = ξ

′∑Si, (13.7)

where u′i is the projection of the vector ti associated with the material point thatmoves on γi, along the direction of the force Pi and Piu′i is the moment according toAmpère, ξ′ is the derivative of the common displacements of the points μi. Ampèreassumes that there is equilibrium if and only if the resultant of the forces acting onthe points μi is zero, i.e. ∑Si = 0; the form in which it follows the law of moments:

∑Piu′i = 0. (13.8)

Ampère concludes:

Now, it is a theorem of algebra easy to prove that the equation resulting from the eliminationprocess is the same as they would be obtained by adding the constraint equations, eachmultiplied by an arbitrary factors to the sum of the moments and equating to zero the amountwhich multiply each derivative and eliminating the factors [2].17 (A.13.13)

17 p. 261.


The ‘easy’ theorem of algebra is not really too easy. The speech of Ampère gets a lit-tle vague when it begins to consider systems with more than one degree of freedom.One can therefore conclude that his attempt to address the constraints preventing thesuperposition of constraints, accepted by Lagrange and Poinsot, was not perfectlysuccessful.

13.4 Pierre Simon Laplace

Pierre Simon, marquis de Laplace was born in 1749 inBeaumont-en-Auge and died in Paris in 1827. His career wasimportant for his technical contributions to exact science, forthe philosophical point of view he developed in the presen-tation of his work, and for the part he took in forming themodern scientific disciplines. Laplace’s biography tells of thestory that he, at the age of nineteen, was given two differentdifficult problems which he would solve in one night each.The story may be apocryphal, but there is no doubt that Jean

le Rond D’Alembert was somehow impressed and took Laplace up, securing his newprotégé the appointment of professor of mathematics at the École Militaire. In 1773Laplace became a member of the Académie des sciences de Paris. He was a memberand even chancellor of the Senate, and great officer of the Legion of Honour and ofthe new Order of Reunion. After the downfall of Napoleon he was nominated Peerof France, with the right of a seat in the Chamber, and was raised to the dignity ofmarquis [290].

Laplace’s contribution to the discussion on the virtual velocity principle is citedby Ampère and Poinsot, for the prestige of scientist and of man of power as well asfor a specific important contribution that left its mark in many subsequent demon-strations referred to in the handbooks of mechanics. Laplace is partially outside theÉcole polytechnique and this is reflected in his writings that he processes withouttaking into account the discussions and clarifications already reached.

He deals with the demonstration of the virtual velocity principle at the beginningof his Mécanique celeste of 1799 [156] and with assumptions that are still those ofProny, Poinsot and Ampère, namely the composition and decomposition of forcesand the need for equilibrium of the orthogonality of active forces to the contact sur-face.

The force of pressure of a point on a surface perpendicular to it, could be divided into two,one perpendicular to the surface, which would be destroyed by it, the other parallel to thesurface and under what the point would have no action on this surface, which is against thesupposition [156].18 (A.13.14)

Besides the two assumptions cited above, Laplace adopts the principle of action andreaction for which two material points m and m′ act on each other with two forcesequal and opposite in the direction of the line joining them:

18 Tome I, p. 9.

13.4 Pierre Simon Laplace 333

Two material points with massesm andm′ can act on each other only according the line thatjoins them. Indeed, if the two points are connected by a wire passing through a fixed pulley,their reciprocal actions are never directed according to that line. But the fixed pulley can beconsidered as having, at its core, a mass of infinite density acting on the two bodies, wherethe action of one on the other is only indirect [156].19 (A.13.15)

The last part of the above quotation raises however doubts about the way Laplaceconceived of the principle of action and reaction and the reading of the whole mem-oir does not clarify the matter. One more principle that Laplace considers, withoutexplicitly stating it, is the principle of solidification.

Laplace’s demonstration of the virtual velocity principle starts from a free mate-rial point M, partially reproducing symbols and arguments of Lagrange in the firstedition of the Mécanique [145].20 Consider a point M subject to many forces S ap-plied to it. Denote by s the distance of M from an arbitrary point C on the line ofaction of each force S:

s =√

(x−a)2 +(y−b)2 +(z− c)2, (13.9)

where x,y,z are the coordinates of M and a,b,c the coordinates of C. Let V be theresultant of the various forces S , still applied to M, and u the distance of an arbitrarypoint D on the line of action ofV from M. The components ofV and S, according tothe direction of the x coordinate are given respectively by:

Vδuδx

, Sδsδx

, (13.10)

as it can be seen, for example, for the components of S, considering that:

δsδx

=x−as

, (13.11)

which is the director cosine of the force S and which multiplied by S provides itscomponent along x. The same is true for the directions y and z. From (13.10) andsimilar relations associated with y and z directions, using the rule of composition offorces one obtains:

Vδuδx

= ∑Sδsδx

; Vδuδy

= ∑Sδsδy

; Vδuδz

= ∑Sδsδz

, (13.12)

where the sum is extended to all forces. Multiplying equations (13.12) for δx,δy,δzrespectively, adding and taking into account the total differential expression as afunction of the partial derivatives, one obtains:

Vδu = ∑Sδs, (13.13)

where the meaning of symbols is clear.In order to represent the equilibrium, Laplace says, the resultantV has to be zero,

so there is equilibrium if and only if:

0 = ∑Sδs, (13.14)

19 Tome I, p. 37.20 p. 20.


which is an equation of moments. It is worth noting that, at this stage of his memoir,Laplace makes no mention of Bernoulli’s principle of virtual velocities and givesno name to the product Sδs – the moment – pointing out, too much I think, hisoriginality.

In the case of a material point constrained on a surface, Laplace denotes by R thereactive forces and δr the infinitesimal displacement of their points of application.From (13.14) he obtains:

0 = ∑Sδs+Rδr, (13.15)

but, as the pointm remains on the surface, it is δr = 0 and (13.15) reduces to (13.14).The case of a system of constrained particles m′,m′′, etc. is studied similarly by

writing relation (13.15) for each point. Now, in addition to the forces S andR, internalforces pmust be introduced. To them Laplace applies the principle he has stated forwhich the internal forces act along the line joining the material points. So indicatingwith f the distance between the material points m and m′ and with f ′ that betweenthe pointsm andm′′ and so on, Laplace can write the equation of equilibrium for anymaterial point m:

0 = ∑Sδs+ pδI f + p′δI f ′+ etc.+Rδr+R′δr′, (13.16)

in which δI indicates that only the position ofm changes whilem′,m′′,etc. are treatedas fixed (and then he applies the principle of solidification). In relation (13.16)Laplace has considered also the possibility that m is constrained to two surfacesthat exert the forces R and R′ in the directions r and r′.

For the point m′ it will be similarly:

0 = ∑S′δs′+ pδII f + p′′δII f ′′+ etc.+R′′δr′′+R′′′δr′′′, (13.17)

where now S′ and s′ are respectively the ‘active’ forces of m′ and the displacementsin their direction; δII f represents the variation of f takingm fixed by varyingm′, δ f ′′is the variation of f ′′ which is the distance between m′ and m′′, by varying m′, R′′and R′′′ represent the constraint forces of the surfaces to which m′ is constrained. Byadding (13.16) and (13.17), together with similar relations for points m′′,m′′′,etc.,taking into account that for example δ f = δI f +δII f , represents the total change inlength of f , the relation

0 = ∑Sδs+∑ pδ f +∑Rδr (13.18)

is obtained. Because for every single point it is δr = 0, the last term is reduced tozero. Then if the system is of invariable distance, i.e. a rigid body, it is also δ f = 0.So the equation of moments (13.14) is again obtained.

If the points of the system have not invariable distance, Laplace believes it can bedemonstrated that it is still ∑ pδ f = 0. For brevity I will not discuss his argument,I will only notice that it was not completely correct, as Poinsot pointed out in 1838,after the death of Laplace, in a note of Crelle’s Journal [196].

14

Poinsot’s criticism

Abstract. This chapter is devoted to Louis Poinsot’s criticisms toward Lagrange’sMéchanique analitique of 1788, assuming a reductionist approach. The first partintroduces Poinsot’s mechanics which also takes account of the existence of con-straints. Poinsot believes that VWLs have no interest for mechanics. The final partof the chapter is a report on the demonstration of a law of virtual work similar tothat of Lagrange in which use is made of velocities and not infinitesimal displace-ments. Although Poinsot considers this demonstration only a trivial corollary of hismechanics, it had a remarkable success.

Louis Poinsot was born in Paris in 1777 and died in Parisin 1859. He enrolled at the École polytechnique probably in1794, without much mathematical background, and studiedthere for three years. The influence on him of Prony, whoat that time was a professor at the École, was remarkable.Poinsot is generally considered as a minor figure in the his-tory of mechanics, not comparable to the great ones: Euler,Lagrange, Laplace, Cauchy, etc. This may be true, but it doesnot mean that the attemption to understand the role of virtual

work laws in mechanics, Poinsot’s position is not in the foreground. Among thescholars of some importance in the first half of the XIX century he was the onlyone who rather than enhancing the principle of virtual velocities, tried to prove thatit was neither necessary nor useful for a coherent and efficient foundation of me-chanics. According to him, once this mechanics was established, any demonstrationof a virtual work law reduced to mere geometry. For this reason, and because his‘demonstration’ has influenced most of the treatises of mechanics, I will dedicate alarge space referring both to his work and to Bailhache’s interesting scientific biog-raphy [197].

Poinsot was not a prolific author; his main works reduce to:

• Éléments de statique 1803 (first edition) [195];• Mémoire sur le compositions desmoments et la composition des aires, 1804 [193];


336 14 Poinsot’s criticism

• Mémoire sur la théorie générale de l’équilibre et du mouvements des systèmes,1806 [194];

• Mémoire sur la composition des moments en mécanique, 1804 [193];• Remarque sur un point fondamentale de la Mécanique analytique de Lagrange,

1846 [198].1

The most interesting of Poinsot’s contributions on virtual work principles is perhapsthe Mémoire sur la théorie générale de l’équilibre, which was included in the lat-est editions of the Éléments de statiq ue. This work is derived from a review of amemoir, Sur la théorie générale de la mécanique, of the previous year which wasread by Lagrange. The criticism was not completely in favor and demanded a radicalrevision. Poinsot accepted the request and sent a new draft to Lagrange. Lagrangesent it back to him, with a series of notes, but after it had already been published inthe Journal de l’École polytechnique. Poinsot replied, even orally, point by point.The result of this discussion was that Lagrange realized the value of his interlocutorand had him appointed inspector general of the university. Poinsot was twenty nineyears old.

In the following I will examine first an unpublished work entitled Considerationssur le principe des vitesses virtuelles of 1797, reported in full in [197], then theMémoire sur la théorie générale de l’équilibre and its previous version, along withLagrange’s annotations to it [197].

14.1 Considérations sur le principe des vitesses virtuelles

Poinsot wrote the Considérations sur le principe des vitesses virtuelles when he wastwenty years old, and so when he had not yet fully developed his critical views onLagrange’s virtual velocity principle. And, entering the École polytechnique culturalclimate, he even provided a demonstration, largely following Prony’s approach (seeChapter 13). However, the critical insights that presage the development of Poinsot’sthought can already be seen.

In this regard, an interesting note was reported at the beginning of the work, whenPoinsot introduced the virtual velocities:

Lines aa′, bb′, cc′, &c, are what scholars call the virtual velocities of the points a,b,c, & c.,but if one wants to have only the value of the moment, one multiplies the force estimated forthese lines in the direction of the force, i.e. projected onto them. To shorten, it is thereforeconvenient to call these projections themselves the virtual velocities [197].2 (A.14.1)

This clarification indicates the attention Poinsot put on the virtual velocity concept,understood in the modern sense as a vector quantity.

Poinsot begins his demonstration of his version of the virtual work law taking forgranted, as did Prony, the rule of composition of forces, but he does so in greaterdetail, as follows. Let P,Q and R be three forces in equilibrium on a plane, applied

1 vol. 11, pp. 445–456.2 p. 4, part II.

14.1 Considérations sur le principe des vitesses virtuelles 337

Q

P

a'

R

a

x

x

z z

y

y

q

r

p

a)

Q

Pa'

R

aq

rp

b)


to a point a. Assume that a moves into a′ with an infinitesimal displacement. Asdemonstrated by Varignon, the static moments of the forces P and Q with respect toany point, for example a′, is equal to the static moment of the resulting R – or thatof the balancing force with sign changed – evaluated for the same pole, and withreference to Fig. 14.1a , it is:

Px+Qy+Rz = 0, (14.1)

where x,y and z are the normal to the directions of P,Q and R conducted from a′.Indicating with p,q and r the components of aa′ on P,Q and R respectively, i.e.Bernoulli’s virtual velocities associated with aa′, it may be obtained easily:

Pp+Qq+Rr = 0. (14.2)

In order to prove this, simply extend the lines x,y,z and build on them forces equalto P,Q,R, but rotated by a right angle and with origin in a′ (see Fig. 14.1b), forwhich now the normals are p,q,r. Because the equilibrium of P,Q,R persists also ifthey are rotated, from the balance of statics moments the relation (14.2) is obtained.This expresses the vanishing of the sum of the ‘moments’ – Lagrange and Galileoterminology. The demonstration of Poinsot deserves attention because it shows theclose analogy between Lagrange’s ‘moments’ and static ‘moments’. They derivefrom a different way to observe forces.

Poinsot then argues that the proof of the equation of moments is also valid in thecase of any number of forces applied to a point and even for any number of freepoints, because a system of material points is in equilibrium if and only if all itspoints are in equilibrium

In the case of a system of constrained material points, the equation of momentsis still valid, provided that in addition to the active forces P,Q,R, etc. also the re-action forces H,M,N, etc. and the corresponding virtual velocities h,m,n, etc., areconsidered, so it can be written:

Pp+Qq+Rr+ etc.+Hh+Mm+Nn+ etc. = 0. (14.3)


Note the explicit introduction of the constraint reactions, perhaps according to theteachings of Prony. Poinsot in his later writings, will instead avoid the concept.

To prove the equation of moment in case of a body with ‘immutable distances’ (arigid body) Poinsot bases his argument on the idea that there is equilibrium if (andonly if) all the forces, as a result of translations, compositions and decompositionswith the rule of the parallelogram can be reduced to only three forces – includingconstraint forces – balancing each other. These three forces should satisfy the law(14.2) which requires the vanishing ofmoments. Exploiting the fact that the total mo-ment of forces does not change with the operations of translation and composition,Poinsot can then claim that the criterion of equilibrium of a rigid body is expressedby the annulment of the sum of all moments of forces acting on the body, includingthe constraint forces. But, when considering virtual displacements compatible withconstraints, the moments of individual reactions, and hence their sum, are zero. Themoments of the constraint forces applied to fixed points, because their virtual veloc-ities are zero and “those of the forces normal to the resistance surface, are also zerobecause the projection of the infinitely small arc, described by the root of normalitself is zero”. Note that Poinsot, in his demonstration, in line with the scholars ofthe time, takes for granted the assumption of smooth constraints. Ultimately it canthen be concluded that the criterion of equilibrium of a rigid body can be traced backto the annulment of the sum of the moments of the only active forces.

Instead of proceeding further, Poinsot warns: “Wewill also change the wording ofthe general principle of virtual velocities to avoid the idea of infinitely small motionsand disturbance of the equilibrium, which are ideas foreign to the subject and leavesomething obscure in the spirit”. To clarify his position on the virtual displacement,I quote in full a note written on a separate sheet of the Considerations:

This will exclude the ideas of the infinitely small and disrupting the equilibrium; ideas thatare alien to the subject, and the principle of virtual velocities appear as a simple theorem ofgeometry by ignoring those considerations that always leave something dark in the spirit.But it should be noted that this property of equilibrium that we study was discovered bymeans of these little velocity [motion], because those offer themselves naturally when youperturb a machine in equilibrium. It seems that through these movements the energies of theforces in motion of the machine are estimated. If a system is in equilibrium, you know theabsolute value of each force, but not the effect it exerts on account of its position. Disturbinga bit the system to see what are the simultaneous movements that can take the points whereforces are applied, some of these points are moving in the same direction of the forces, othersare moving in the opposite direction, and the energy is evaluated as the product of forces bythe velocity of the points of application, it is found that the energies that achieve their effectare the same as the energies overcome [197].3 (A.14.2)

Poinsot will succeed fully in order to eliminate the concept of virtual displacementonly in the later works. Now, he limits himself to see under what conditions theequation of moments can be extended to the case of finite displacements. Besidesthe well-known examples of a straight lever and the inclined plane, Poinsot refers toresults found by Fossombroni for parallel forces applied to the points of a line; casesthat he generalizes by showing that the equation of moments is also true for finite

3 p. 7, part II.

14.2 Théorie générale de l’équilibre et du mouvement des systèmes 339

displacements when the forces, parallel to each other, are applied to the points of aplane:

If a free system of invariable form is in equilibrium under all forces that are applied to it,assuming that all the forces acting at the junction of their directions with a plane situated atwill, the equation of the moments will be valid whatever was the displacement of the system[197].4 (A.14.3)

In addition to striving to eliminate the idea of infinitesimal virtual displacements,replacing these with the virtual velocities, Poinsot points to – or rather decides tofollow in – Carnot’s footsteps, contending that the virtual velocities refer to changesof position occurring in a virtual time while the real time is frozen (i.e. that the virtualvelocities and forces are not correlated, even when the forces depend on real motion).The balance is made with forces frozen at the instant in which the equilibrium shouldbe studied:

It must be noted further that the system is supposed to move in any way, without referenceto forces that tend to move it: the motion that you give is a simple change of position wherethe time has nothing to do at all [197].5 (A.14.4)

Toward the end of his text Poinsot writes: “It would therefore be futile to search forthe metaphysics of the principle of virtual velocities and to endeavor to understandwhat they are in themselves the moments of the forces. Everything comes from theparallelogram of forces, where it is seen as the moments combine among them.”

Poinsot is not the only one who uses virtual velocity instead of virtual infinitesi-mal displacement. Fourier seems to put in the same plane the method of ‘fluxions’(i.e. the velocity) and that of infinitesimal displacements. Poinsot, however, is thefirst to emphasize the need to use only virtual velocities, finally abandoning the in-finitesimal displacements. In this he will be followed later by Ampère and Lagrange(in the second edition of the Théorie des fonctions analytique).

14.2 Théorie générale de l’équilibre et du mouvement des systèmes

The Théorie générale de l’équilibre et dumouvement des systèmes is much more ma-ture than the previous text; it begins with historical considerations on virtual worklaws, then develops a mechanical theory completely independent of it to finish byreducing the virtual velocity principle itself to a trivial theorem of ‘Geometry’. I willrefer mainly to the edition of 1806, published in the XIII Chaier of the l’École poly-technique, but when it will be necessary I refer also to the text of 1805 and to the noteson the text reproduced in [197]. For the first part of the present section, which fromcertain points of view is the most interesting for what concerns the virtual velocityprinciple, I refer instead to the version of 1834 published in the Éléments de statique[195]. In it, historical references and comments to Lagrange’s work are much moreextensive and interesting; the wide passage below clearly expresses Poinsot’s ideaswith no need of comment:4 p. 12, part II.5 p. 13, part II.


The principle of virtual velocities was known for a long time as well as the majority of theother principles of mechanics. Galileo first noticed in the machines, the famous property ofvirtual velocities, that is the known relationship that exists between the applied forces andspeeds that their points of application would take if the equilibrium of the machine shouldupset by an infinitely little amount. Johann Bernoulli saw in the full extent this principlethat he enunciated with the great generality it has today. Varignon and the majority of theGeometers were careful to check it in virtually all matters of statics. And although there wasno general proof, it was universally regarded as a fundamental law of the equilibrium ofsystems.But up to Lagrange, the Geometers were oriented more to prove or to extend the generalprinciples of science, than to obtain a general rule for problem solving, or rather they hadnot yet put this great problem, which alone represents all the mechanics. It was then a happyidea by relying on the principle of virtual velocities as an axiom and, without stopping toconsider it in itself, to be concerned only to get a uniform method of calculation to derivethe equations of motion and balance in all possible systems. Thus overcoming all the diffi-culties of mechanics, avoiding, so to speak, to address the Science itself, one transforms itinto a matter of calculation, and this transformation, the objective of Mécanique analytique[Méchanique analitique], appeared as a striking example of the power of analysis.Nevertheless, since, in this work one was at first careful only to consider this beautiful de-velopment of Mechanics, which seemed to derive everything from a single formula, it wasbelieved that natural science was made and one just has to try to prove the principle of vir-tual velocities. But this research has highlighted the difficulties of the principle itself. Thislaw is so general, where vague and strange ideas on infinitely small movements and thedisturbance of the balance mix, does nothing but to become dark in his examination andLagrange’s book is not giving anything more clear than the course of calculations, one seeswell that the mists were not avoided in the way of mechanics, because they were, so to speaktogether, at the very origin of this science.A general proof of the principle of virtual velocities has basically to put the entire mechanicson another basis, because the demonstration of a law that encompasses a science cannot beother than the reduction of this science to another law so general, but obvious, or at leasteasier than before, thus making it useless. So for the reason that the principle of virtual ve-locities contains all the mechanics, and that needs a thorough demonstration, it cannot serveas a primary basis. Trying to prove it on the basis of such a happy use has made is to try togo through this use; either finding some other law just as fruitful, but more clear, or found-ing on the principles of an ordinary general equilibrium theory, from which then the virtualvelocities becomes just a corollary. So the state in which Lagrange had brought the sciencewas not a demonstration of the principle of virtual velocities, which might be sought imme-diately. The Mécanique analytique [Méchanique analitique], as the author conceived it, isbasically what it should be, and the demonstration of the principle of virtual velocities is notlacking at all, because if one tried to put it at the beginning of this book in a general and welldeveloped way, the work would be made, i.e. this demonstration would include already allthe mechanics.It should therefore be considered that Lagrange placed himself with a single shot on one ofthe high points of science in order to discover some general rule to solve, or at least to put inthe form of equations all problems of mechanics, and this objective has been fully achieved.But to form the science itself, one has to produce a theory that dominates equally all pointsof view. One needs to go straight, not to the obscure principle of virtual velocities, but to theclear rule that can be extracted from the solution of problems. And this natural and directsearch, which alone can satisfy our spirit, is the main purpose of the memoir that is going tobe read [195].6 (A.14.5)

6 pp. 427–430.


The above introduction to the Théorie générale, contains an attack on the Mécha-nique analitique much stronger than that of the introduction to the same work in1806, read by Lagrange. Poinsot argues that the principle of virtual velocities is ob-scure and unintuitive. The darkness comes from two factors: the first is contested byall opponents of the principle of virtual work, Stevin above all, that in the study ofequilibrium it does not make sense to consider a perturbation, a virtual movement.The second factor concerns the nature of infinitesimals, the notion of infinitesimalwas not so clear to Poinsot, or at least it was less clear than the concept of velocitythat Poinsot will choose. For this ‘darkness’, according to Poinsot, the virtual ve-locity principle cannot be assumed as a principle of mechanics, and any attempt toprove it does not make sense because it means replacing this principle with anotherprinciple, equally general but more clear, which makes the virtual velocity principleunnecessary. Moreover, there is no practical advantage to introduce it.

Poinsot’s position is incomprehensible to a modern reader, accustomed to hand-books of mechanics based on a highly formalized approach, in which axioms fallfrom nothing and there is no need for justification other than success in explain-ing mechanical phenomena. At the beginning of the discussion the equations of La-grange themselves are assumed [148]7 – or those of Hamilton – often as axioms, andthese equations are anything but intuitive. Poinsot’s position becomes clear whenexamined in the perspective of the epistemology of the XIX century, essentially inAristotelian style. A principle must be self-evident, and even if it is not required withAristotle, to be evident to the pure intellect, it must at least reflect the more immedi-ate experience. According to Poinsot who embraces Aristotle’s opinion, a principlecannot be proved, otherwise it is not a principle; according to other scholars of histime, a principle is not necessarily obvious, but it can be proved and this can be donestarting from metaphysical arguments – that is, with topics outside the science ofwhich the ‘principle’ is a principle – or from within the discipline, but with veryelementary arguments.

Poinsot’s criticism of the possibility to regard Lagrange’s virtual velocity law asa principle, therefore, appears to be partly reduced to a linguistic fact, all dependingon what it is understood by ‘principle’. Of different values is instead the criticism forwhich the demonstrations reported until now are unsatisfactory, or that the principleof virtual work is neither simple nor fundamental. This seems unfair.

The proofs of Lagrange, Fourier and Carnot, who do not depart from the ‘usual’principles of mechanics, are certainly very interesting. Lagrange connects the princi-ple of virtual velocities to the pulley, Fourier to the lever. Carnot starts instead by theimpact regarded as a phenomenon that can be characterized in a simple and obviousway. Even the demonstrations that originate from the usual principles of mechan-ics, such as Prony’s, do not seem less interesting than the demonstration reported byPoinsot himself. Then, in ease of use, if not in enunciation, the superiority of the vir-tual velocity principle compared to other seemingly simple principles is proved byits diffusion in treatises of mechanics. As to whether it is fundamental there seemsto be no doubt, with some difficulty in dealing with friction forces, but, however, thecriticism of Poinsot certainly was not referring to them.

7 p. 334.


14.2.1 Poinsot’s principles of mechanics

I now pass to the exposition of the Théorie générale de l’équilibre de systèmes,according to the text of 1806, beginning by enumerating the principles assumed.From this exposition it soon becomes clear that it would just turn against Poinsot thesame criticism of vagueness that he ascribes to his colleague scientists, because hedoes not set clearly and definitively the principles he uses. Although one can givethe excuse that some of them were already submitted in the Elements de statique of1803, also the others do not seem so obvious.

The first principle, which Poinsot gives as well known, an axiom, is generallycalled the solidification principle. For this principle, if constraints, both internal andexternal, are added to a system of bodies in equilibrium, the equilibrium is not altered[194].8 The principle was used by Stevin, Clairaut and Euler for the study of fluids(by Lagrange, Laplace and Ampère in previous chapters) and will be used later byPiola and Duhem (see Chapter 17) to obtain the indefinite equilibrium equations ofa three-dimensional continuum.

The second principle is presented as the fundamental property of equilibrium, itasserts that a necessary condition for the equilibrium of a system of bodies free fromexternal constraints is that all the forces applied at various points can be reduced toany number of pairs of collinear forces equal and opposite to each other. The condi-tion becomes sufficient for a body with invariant distances, i.e. a rigid body [194].9

The third principle is required by the second, even if not explicitly, and concernsthe possibility to decompose a force into other forces with the rule of the parallelo-gram and to move a force along its line of action. It is embarrassing that Poinsot doesnot state explicitly this principle, which, perhaps, is the most important and complex.He evidently takes it for granted even though it is difficult to argue that it is inher-ently more intuitive than the virtual velocity principle. Just as it is not very intuitiveto accept the second principle, for which the reduction of forces to a number of pairsof opposing forces to each other is a necessary condition for the equilibrium [194].10

The fourth principle concerns constrained material points moving on a surfaceand asserts the need of the orthogonality of the active forces to the surface for equi-librium:

In the equilibrium of systems, any force must be perpendicular to the surface of the curveon which its point of application would move if all the other points were considered as fixed[194].11 (A.14.6)

A fifth principle concerns the mechanical superposition for constraints, that is ifin a system of bodies or points there are more constraints, they are able to ab-sorb the sum of the forces that each constraint is capable of absorbing separately[194].12

8 p. 208.9 p. 209.10 p. 208.11 p. 234.12 p. 225.


Before considering in detail the constrained systems of material points, Poinsotexplores the consequences of the second and third principles, i.e. some necessaryconditions for equilibrium. Without going into detail he imposes the equivalencebetween the applied active forces Pi and a system of a pair of forces equal and oppo-site, an Ri j agent along the line joining the material points i j. Notice that Poinsot isnot using a principle of action and reaction, as for example Prony and Laplace did,but only takes an algebraic position because the lines joining the material points arein general only imaginary and do not represent rods, for example. For a system ofn material points there are 3n−6 possible connections and then 3n−6 componentsRi j to be considered as unknowns and 3n equations that express the equivalence be-tween the active forces Pi and their decomposition Ri j. So there is a surplus of sixequations which must be verified by the assigned active forces. Poinsot does not ex-hibit these equations of equilibrium, perhaps considering them irrelevant, perhapsbecause they are well exemplified in current treatises on mechanics as the cardinalequations of statics.

Fig. 14.2 illustrates the above for the case of four material points, where there are3×4−6 = 6 connections. The four forces P1,P2,P3,P4 are decomposed in the six –couples of – forces R12,R13, . . . ,R34, a priori unknowns. Among Pi and Ri j there arethe twelve equivalency equations of the kind:

X1 = R12 cosα12 +R13 cosα13 +R14 cosα14Y1 = R12 cosβ12 +R13 cosβ13 +R14 cosβ14Z1 = R12 cosγ12 +R13 cosγ13 +R14 cosγ14

· · ·X4 =−R14 cosα14−R24 cosα24−R34 cosα34

· · ·

(14.4)

whereαi j,βi j,γi j are the angles that the forces Ri j formwith axes x,y,z respectively,and Xi,Yi,Zi are the components of forces Pi along the same lines. By eliminating

P4

P3

P2P1

R13

R 34

R 23 2

R14

R 24

R12

1

4

3

Fig. 14.2. Decomposition of forces


Ri j the six cardinal equations of statics are achieved. Notice that the existence ofsolutions forRi j is only a necessary condition for equilibrium, as stated by the secondprinciple of Poinsot’s mechanics.

14.2.1.1 System of material points constrained by a unique equation

Then Poinsot passes to a more in-depth analysis of systems of constrained materialpoints with the use also of his first and fourth principles. For the sake of simplicityhe considers the system of four points shown in Fig. 14.3, the six mutual distancesof which, indicated by m,n, p,q,r,s, are subject to the equation of constraint:

L(m,n, p,q,r,s) = 0.

Applying the principle of solidification, the equilibrium conditions on the externalforces of a point, such as x1, do not change if the other three points are assumed asfixed. If m,n, p are the distances of x1 from the other three points, the condition ofconstraint takes the form, depending only on m,n, p:

L(m,n, p,q,r,s) = 0, (14.5)

in which the values of q,r,s are assigned.Relation (14.5) defines a surface, whose normal at x1 has as components in the di-

rections m,n, p the quantities L′(m),L′(n),L′(p), where the apex denotes the partialderivative with respect to the variable in parentheses. Applying the fourth princi-ple, i.e. the hypothesis of smooth constraints, when the point x1 is in equilibrium,it is necessary that the external force applied to it has the direction defined by thecomponents L′(m),L′(n),L′(p). The same holds true for the other points.

Consider now two points x1 and x3 of the system joined by the linem of Fig. 14.3.As mentioned above, for the equilibrium, the components of the forces applied tothe two points x1and x3 in the directions that connect them to other points must havecomponents in the form:

αL′(m), αL′(n), αL′(p) and βL′(m), βL′(q), βL′(r) (14.6)

x 1

x2

x3

x4

n

r

q

p

m

s

Fig. 14.3. Constrained material points


respectively, where α and β are constants of proportionality. But for the secondprinciple of Poinsot’s mechanics, it is necessary that the components of the forcesin the direction m which joins x1 and x3 are equal and opposite, so it should beα = β. It can be concluded that in order to have equilibrium, the components ofexternal forces in six directions should be proportional to the partial derivatives ofL′(m),L′(n),L′(p),L′(q),L′(r),L′(s). This result extends to any number of pointsconnected by a single condition of constraint.

After these considerations Poinsot encounters some difficulty in the use of thecomponents of forces in the global reference frame. His difficulties come from hav-ing delivered the conditions of constraints by means of the distances between pointsrather than by means of their coordinates with respect to the coordinate system, as itwould seem more natural, at least to a modern reader. The reasons for this are quitecomplex although may be not so interesting [197]. A few years later Cauchy [66]will resume the reasoning of Poinsot using constraint equations expressed by meansof the coordinates of the points.

I avoid referring to those aspects that do not have a very important conceptualvalue and, without giving the proof I pass to exposing the first conclusion of Poinsotwhich ensures that to have equilibrium in a system of any number of particles, subjectto a constraint of type L = 0, the components of the forces applied to each point xishould be proportional to the quantities:

∂L∂xi

,∂L∂yi

,∂L∂ zi

, (14.7)

with (xi,yi,zi) the Cartesian coordinates of the i-th point.This result was already obtained by Lagrangewith a different approach.Moreover

according to Poinsot, relations (14.7) provide the directions the external forces needto have so they are equilibrated, instead of according to Lagrange, being directionsof constraint forces that rise for the equilibrium.

Developing derivatives of (14.7) it is then:

∂L∂xi

=∂L∂ p

∂ p∂xi

+∂L∂q

∂q∂xi

+∂L∂ r

∂ r∂xi

+ etc.

∂L∂yi

=∂L∂ p

∂ p∂yi

+∂L∂q

∂q∂yi

+∂L∂ r

∂ r∂yi

+ etc.

∂L∂ zi

=∂L∂ p

∂ p∂ zi

+∂L∂q

∂q∂ zi

+∂L∂ r

∂ r∂ zi

+ etc.,

(14.8)

where p,q,r which represent the distances of the various points, should be consid-ered as functions of their Cartesian coordinates x1,y1,z1,x2, etc. in a fixed frame ofreference.


14.2.1.2 System of material points constrained by more equations

Poinsot can then turn to the case with more than one constraint condition:

L(m,n, p,q,r,s) = 0M(m,n, p,q,r,s) = 0N(m,n, p,q,r,s) = 0

etc.

(14.9)

In his words:

First, the mere fact that the points of the system are linked together by the first equationL = 0 the forces:

λ

√(∂L∂x

)2

+(

∂L∂y

)2

+(

∂L∂ z

)2

λ

√(∂L∂x′

)2

+(

∂L∂y′

)2

+(

∂L∂ z′

)2

λ

√(∂L∂x′′

)2

+(

∂L∂y′′

)2

+(

∂L∂ z′′

)2

&c.

can be applied to them, where λ designates any undetermined coefficient and being eachforce perpendicular to the surface L = 0, when one considers the three coordinates of thepoint of application as the only variables.

Second, because the points of the system are linked together by the second equationM = 0, it is still possible to apply the respective forces:

μ

√(∂M∂x

)2

+(

∂M∂y

)2

+(

∂M∂ z

)2

μ

√(∂M∂x′

)2

+(

∂M∂y′

)2

+(

∂M∂ z′

)2

μ

√(∂M∂x′′

)2

+(

∂M∂y′′

)2

+(

∂M∂ z′′

)2

&c.

with μ a new indeterminate coefficient, and each of these forces being perpendicular tothe surface represented by the equation M = 0, when the three coordinates of the point ofapplication are considered as the only variables.[…]It is clear that there will be equilibrium on the basis of all these forces, because there wouldbe equilibrium in particular in each group of each equation [194].13 (A.14.7)

Poinsot has implicitly accepted that more constraints working at the same time do notinteract with each other and that the overall effect is the sum of individual effects (it isthe fifth principle of his mechanics). He realizes that this fact is not very evident andtries to overcome a little below, in a passage that I do not refer for lack of space. He

13 pp. 223–224.


himself did not deem it good enough and then will return to this point in the Elementsde statique since the eighth edition of 1841 (for further clarification on the issue seethe work of Ampère on the virtual velocity principle examined in Chapter 13). EvenLagrange in the Théorie des fonctions analytique, made the same assumption ofPoinsot on the superposition of constraints but he did not feel the need to justify thefact.

The Théorie générale ends with the following theorem:

Whatever the equations governing the coordinates of various points of the system are, forequilibrium, each of them requires that to these points are applied forces, along their coor-dinates, proportional to the derivatives of these equations with respect to these coordinates,respectively.Thus, representing with L = 0, M = 0, etc. any equation between the coordinates x,y,z,x′,y′,z′, etc. of the different points, and with λ,μ, etc. any of the undetermined coefficients, thecomponents of the forces that must be applied to these points should satisfy:

X = λ(dLdx

)+μ

(dMdx

)+&c.

Y = λ(dLdy

)+μ

(dMdy

)+&c.

Z = λ(dLdz

)+μ

(dMdz

)+&c.

X ′ = λ(

dLdx′

)+μ

(dMdx′

)+&c.

Y ′ = λ(

dLdy′

)+μ

(dMdy′

)+&c.

Z′ = λ(

dLdz′

)+μ

(dMdz′

)+&c.

&c.

If the indeterminate λ,μ, etc. are eliminated from these equations, there will remain theequilibrium equations themselves, i.e. the relationships that must take place between theapplied forces and the coordinates of their points of application [194].14 (A.14.8)

The equations above allow the solution of the static problem. Given a set of forcesX , Y , Z, X ′, Y ′, etc. verify whether the system of material points is in equilibriumin a given configuration x, y, z, x′, y′, etc. This can be made as Poinsot suggests, bysolving the equations obtained by eliminating the indeterminate multipliers, or moresimply, by considering that previous equations define a linear system of algebraicequations in λ,μ,etc. with known coefficients. The linear systemmay be determined,undetermined or overdetermined; if it admits at least a solution then the system ofmaterial points is in equilibrium.

Note that Poinsot is establishing the mechanics of constrained bodies withoutreference to the concept of constraint reaction though it was there accepted at theÉcole polytechnique. He is evenmore rigorous than Lagrange and leaves no physicalmeaning to the ‘indeterminate’ ‘coefficients’ λ,μ, etc. that once known are generallyinterpreted as constraint forces.14 pp. 228–229.


The memoir ends with an interesting conclusion and two notes: the first regardsthe comments on the role of constraints, which is not particularly illuminating. Thesecond note concerns the demonstration of the virtual work principle and is of greatinterest, which is why I quote it in full.

14.3 Demonstration of the virtual velocity principle

In the demonstration which follows, Poinsot replaces, for the first time unequivo-cally, the virtual displacements (infinitesimal) with virtual velocities, that he alsocalls ‘actual’ to emphasize that there are no assumptions of smallness. Poinsot de-clares he wants to prove Lagrange’s virtual velocity principle; actually, however,because of his use of velocity instead of infinitesimal displacements he is going toprove a slightly different principle, which however is still a virtual work princi-ple. According to Poinsot, because this is almost an immediate consequence of themechanics he has developed, which takes into account the role of constraints, andfollows nearly immediately when the equations of equilibrium and constraints arewritten side by side, its proof has not a great value and also the principle in itself isof little interest. Personally I do not share Poinsot’s opinion and consider Lagrange’sproof very interesting and among the most cogent ever given.

Note IIDemonstration of the principle of virtual velocities. Identity of this principle with the generaltheorem object of the previous Memoir.In the Memoir we have been content to observe that from the theorem on the expressionof the general equilibrium of forces, one could easily switch to the principle of virtual ve-locities. But this principle is so famous in the history of mechanics that one cannot fail topoint out a few words with these steps. I am very happy to do this, since the principle ofvirtual velocities is not only a corollary of the general proposition stated above, but I thinkeven identical to it when one looks at it from his own point of view, and sets it out in acomprehensive manner. Let the system be defined by the following equations between thecoordinates of the bodies:

f (x,y,z,x′,y′,z′,&c.) = 0.

φ(x,y,z,x′,y′,z′,&c.) = 0.

&c.

Suppose to impress to all bodies any of the velocity that can actually occur without violatingthe terms of the constraints. The coordinates x,y,z,x′,y′,z′, &c., will vary with time t, ofwhich they must be considered functions, and because the impressed velocities:

dxdt

,dydt

,dzdt

,dx′

dt, &c. (A)

could be admissible by the constraints, as supposed, it will be necessary that they satisfy theequations:

f ′(x)dxdt

+ f ′(y)dydt

+ f ′(z)dzdt

+ f ′(x′)dx′

dt+ f ′(y′)

dy′

dt+&c. = 0

φ′(x)dxdt

+φ′(y)dydt

+φ′(z)dzdt

+φ′(x′)dx′

dt+φ′(y′)

dy′

dt+&c. = 0

&c.

(B)

14.3 Demonstration of the virtual velocity principle 349

obtained from the previous (A) and it will be sufficient to ensure that they meet them so thatthe constraint conditions are met.

Now if one multiplies these equations for the undetermined coefficients λ,μ, &c. andadds, it follows that the velocities satisfy the sole following condition, no matter λ,μ, &c.

[λ f ′(x)+μφ′(x)+&c.]dxdt

+[λ f ′(y)+μφ′(y)+&c.

] dydt

+

[λ f ′(z)+μφ′(z)+&c.]dzdt

+[λ f ′(x′)+μφ′(x′)+&c.

] dx′

dt+

[λ f ′(y′)+μφ′(y′)+&c.]dy′

dt+&c. = 0.

(C)

But the functions which multiply the velocities,

dxdt

,dydt

,dzdt

,dx′

dt, &c.

are nothing but (after what has been proven) the general expressions of the forces whichcan be balanced on the system. Assuming therefore that the forces X ,Y,Z,X ′,Y ′,Z′, &c.,are effectively balanced, it is:

Xdxdt

+Ydydt

+Zdzdt

+X ′dx′

dt+Y ′

dy′

dt+Z′

dz′

dt+&c. = 0. (D)

Instead of the three components X ,Y,Z, multiplied for the corresponding velocities:

dxdt

,dydt

,dzdt

it can be considered the resultant P, multiplied by the resulting velocity dx/dt, dy/dt, dz/dt,projected into the direction of P, which I will call ds/dt; the same can be done for the otherforces, and it will be:

Pdsdt

+P′ds′

dt+P′′

ds′′

dt+&c. = 0.

That is, if the forces are in equilibrium on any system, the sum of their products for thevelocities, one wants to give their bodies, whatever they may be, but allowed by their con-straints, will always be zero, by estimating these velocities along the directions of forces.One can see from this, it is possible to take any velocities of finite value, which are measuredby any straight lines that would be described simultaneously by the body if their links aresuddenly broken and each of them run away freely toward their part.Because of constraints among the bodies the velocities vary in eachmoment, when onewantsto measure these velocities using the spaces themselves that the bodies actually describe,these spaces should be taken infinitely small, otherwise they no longer would measure theimpressed velocities, and in this way one falls into the virtual velocities themselves, wherethe principle is to lose some of its clarity.In fact it follows from what we have said, that this beautiful property of equilibrium can bestated as follows:When the different bodies of a system run any of the movements which do not violate inany way the link established between them, i.e. the system is continuously in one of thoseconfigurations allowed by the constraint equations, it can be sure that the forces that willbe capable of being balanced in these configurations, when the system passes in them, aresuch that multiplied by the velocity of the bodies projected onto their directions, the sum ofall these products is necessarily equal to zero.In this way, the principle no longer maintains any trace of the ideas of the infinitely smallmovements and disturbance of the equilibrium, which seem extraneous to the issue and leavesome darkness in the spirit.


When there is equilibrium, it is clear that the principle holds for all the systems of velocitiesthat the points could have, passing through the configuration that is considered.But, when one wants to start from the principle enunciated in such a way that it ensuresthe equilibrium one should require that it holds for this infinite number of velocity systems.There is a plethora of conditions, and it is possible to show that it is enough to say that theequation (D) must be verified for all systems of velocity allowed by the constraint equations(B) or (bringing together, as we did above, all these equations in one (C) by means of inde-terminate λ,μ, &c.), it suffices to say that the equation (D) of the moments must be verifiedfor all systems of velocities:

dxdt

,dydt

,dzdt

,dx′

dt, &c.

But since, by definition, each of these systems of velocities must satisfy the equation (C),which amounts to say that all forces X ,Y,Z,X ′,Y ′,Z′, &c. that multiply the velocities:

dxdt

,dydt

,dzdt

,dx′

dt,

dy′

dt,dz′

dt, &c.

in the equation (D), have to be proportional to the functions:[λ f ′(x)+μφ′(x)+&c.

] dxdt

,[λ f ′(y)+μφ′(y)+&c.

] dydt

,[λ f ′(z)+μφ′(z)+&c.

] dzdt

,[λ f ′(x′)+μφ′(x′)+&c.

] dx′

dt,[λ f ′(y′)+μφ′(y′)+&c.

] dy′

dt,&c.

that multiply the same velocities as in the general equation (C), which requires for them theonly conditions of constraints. So the principle of virtual velocities well set out, i.e. where allthe ideas that one can make are clear: it is perfectly identical to the general theorem which isthe subject of this memoir. I say exactly the same thing, namely that for the equilibrium, thecomponents of the forces applied to bodies, by virtue of each equation must be proportionalto the derivatives of these [constraint] equations with respect to these coordinates, whichwas to be proved.15

Moreover, it would have been taken to recognize this identity by a description of the ordi-nary principle of virtual velocities, by making well aware of the true meaning that it needsto be given. In fact, the general problem of statics is not only to seek the relationship be-tween the forces which are in equilibrium, on the system, but rather [to seek] the generalexpression of the forces that may be continually equilibrated in any configurations where itcan go under the constraint equations. The general equation given by the principle of virtualvelocities is not, if I may speak so, the relation of an instant; it in no way should considersimply the equilibrium of the system in the configuration where it is, but also throughoutthe sequence of configurations where it can be, for it is this sequence of configurations thatcharacterizes its definition [emphasis added] So the equation of moments does not say thatone has to take the forces of a magnitude sufficient so that it is satisfied, but (since theseforces must vary with the configuration) [it says] how one must choose these functions ofthe coordinates, so that the equation of moments remain continually satisfied. Now, underthe constraint conditions themselves, one knows that between the velocities that the bodiescan simultaneously have, it must apply the linear equation (C), the coefficients of which arethe derivatives of functions given with respect to the coordinates by which this velocity isestimated. The equation of moments says that the forces of equilibrium must be represented

15 Here Poinsot’s writing is somewhat confused. He means that to prove the sufficiency of thevirtual work principle one should require the satisfaction of the equation of moments (D) for allpossible sets of virtual velocities. Comparing (D) with (C), which also apply to any set of virtualvelocity, one deduces the equality of their coefficients, the terms that multiply the virtual velocities– and thus the equation of moments.

14.3 Demonstration of the virtual velocity principle 351

by the derivative of these functions, therefore, to prove it, it is necessary to show how theseforces are actually equilibrated or it must look directly for what functions of the coordinatescan represent the forces of equilibrium, as we did from the beginning.This is why most of the demonstrations which trace back the principle of virtual velocitieseither to other principles or to the known law of some simple machine as the lever, &c.,seem to us more justifications than real demonstrations. All in fact, even the happiest, thatof Mr. Carnot, do not refer at all to the general definition of the system, as if the machinewas, so to speak, voila, and one does not see anything but the ropes where the powers areapplied. It may well be proved or made clear through some construction more or less simplethat if one perturbs a bit the equilibrium, these powers must be in a relationship with theextensions allowed to its ropes, but this cannot provide that the current ratios forces consid-ered as numeric values, and does not show at all its forms of expressions that are peculiar tothem.This disturbance of the balance would not know, in no event with which machine one has todo, and the same relationship between the applied forces, could occur even if the machineswere of quite different constitution, and each of them, however, imposes to the expressionof the forces that are generated, a different form that one should always see and find, if thedifficulty of the theorem were fully resolved. So the property of virtual velocities remainsnot less mysterious, and there is no real demonstration. I mean an open and clear explana-tion, where one sees not only that it works well but that it is a consequence of the generaldefinition of the considered system.It is perhaps in a similar way, and to get the equation of moments as an equation identicalthat Mr. Laplace considered only the equations representing the link of the various parts ofthe system, and has, moreover, used other principles besides the composition of forces andthe equality of action and reaction, which can be considered as elements of the equilibriumtheory. As it is, after all, either one wants to start from the principle of the virtual velocities tofollow its significance up to the end, or he directly attacks the problem of mechanics, whichis simpler, one is conducted to look for the functions of the coordinates that give the forcesof equilibrium in all the configurations that can be obtained in the system, in obedience tothe relationships between the coordinates of the different bodies. This is exactly the problemwe set ourselves, and our goal clear and distinct was to resolve it through the first principlesof statics and geometry [194].16 (A.14.9)

I do not see that the text of Poinsot needs comment. On the basis of his mechanicaltheory, in which the role of constraints is clearly explained, and on the basis of hisdefinition of virtual velocity, he can easily demonstrate a virtual work law which is avariant of Lagrange’s virtual velocity principle, both for its necessary and sufficientparts.

Poinsot maintains that the virtual velocity principle allows the study of the equi-librium not only in a given configuration: “but also in the entire sequence of config-urations where it can be, for it is this sequence of configurations which characterizesits definition”. In such a way it can also lead to solution of another interesting prob-lem of statics. Assigned a given configuration x,y,z,x′,y′, etc. find a set of forcesX ,Y,Z,X ′,Y ′, etc. for with the equilibrium is satisfied. This can be made by solvingthe equations obtained from the virtual work law by eliminating the indeterminatemultipliers.

16 pp. 237–241.

15

Complementary virtual work laws

Abstract. This chapter is devoted to a variant of VWL which goes under the nameof law of virtual forces. In the first part the formulation of the law by Cauchy ispresented and used to prove simple theorems of plane kinematics. In the final part afew theorems of spatial kinematics are enunciated.

From the demonstrations of reductionist type as that of Poinsot it should be clear thatany law of virtual work can be stated in dual form. One is the traditional principleof virtual work, for example in the form given to it by Lagrange, which studiesthe equilibrium of a system of constrained bodies subject to various forces fi byimposing the vanishing of the work for a system of infinitesimal displacements uicongruent with constraints; i.e.:

∑ fiui = 0, ∀ui. (15.1)

Another form can be achieved by focusing on a given set of infinitesimal displace-ments ui consistent with constraints, forcing the cancellation of virtual work for allsystems of balanced forces fi; i.e.:

∑ fiui = 0, ∀ fi. (15.2)

This second type of law goes by the name of the principle of complementary virtualwork or the principle of virtual forces. It is virtually ignored in the treaties of ra-tional mechanics for physicists and mathematicians while it is widely used in thoseaddressed to engineers, for whom the principle of virtual forces is an essential toolin the analysis of elastic structures [385].

In the following I cite only the applications of this principle to the study of rigidbody kinematics carried out by Cauchy.


354 15 Complementary virtual work laws

15.1 Augustin Cauchy formulation

Augustin Louis Cauchy was born in Paris in 1789 and diedin Sceaux in 1857. In mathematics he pioneered the study ofanalysis, both real and complex, and the theory of permuta-tion groups. He also researched in convergence and diver-gence of infinite series, differential equations, determinants,probability. In mechanics he contributed to the introductionof the concept of stress and made fundamental studies onelastic bodies, considered both as made up of particles andas continuous bodies. Laplace and Lagrange were visitors at

the Cauchy’s family home. Lagrange in particular seems to have taken an interest inyoung Cauchy’s mathematical education; he is said to have forecast Cauchy’s scien-tific genius while warning his father against showing him a mathematical text beforethe age of seventeen. In 1805 he entered the École polytechnique; in 1807 graduatedfrom the École polytechnique and entered the engineering school École des pontset chaussées which he left (1809?) to become an engineer, first at the works of theOureq Canal, then the Saint-Cloud bridge, and finally, in 1810, at the harbor of Cher-bourg. In 1816 he become professor at the École polytechnique. Cauchy was a verydevout Catholic and this attitude was already causing problems for him and for oth-ers. After the revolution of 1830 Cauchy refused to take the oath of allegiance andlost his chairs. When the revolution of 1848 established the second republic Cauchyresumed his academic position and was retained even when Napoleon III reestab-lished the oath in 1852, for Napoleon generously exempted the republican Aragoand the royalist Cauchy [290].

Cauchy gave no decisive contribution to the understanding of virtual work laws,but he used them enough and thus contributed to their spread. Among the workswhere Cauchy made use of virtual work laws they should be named: Sur un nouveauprincipe de mécanique of 1829 [67], which addresses issues of impact among bodiesand the Recherches des équations générales d’équilibre pour un système de pointsmatériels assujettis à des liaisons quelconques [66]. In this latter text, which dealswith the motion of systems of constrained particles Cauchy resumed, exposing it un-der a slightly different point of view, the discourse written by Poinsot in the Théoriegénérale de l’équilibre et du mouvement des systèmes several years before. The maindifference is the way he dealt with constraints, that rather than being defined by mu-tual relations among distances of material points, were defined by relations amongtheir coordinates with respect to a fixed system.

The memoir Sur le mouvements que peut prendre un système invariable, libreou assujetti à certaines conditions of 1827 [64] is however in my opinion the mostoriginal work. Here Cauchy applies, perhaps for the first time, the principle of virtualforces to determine the congruence of the acts of motion of a rigid body. The strengthof the memoir is not so much to obtain theorems of kinematics of rigid bodies, whichcan be obtained more easily and convincingly with purely geometric methods, but topresent an alternative way to use the law of virtual work. In the following I will refer

15.1 Augustin Cauchy formulation 355

briefly to the case of a plane rigid body, ignoring the long and difficult discussion ofCauchy on the three-dimensional rigid body.

Cauchy does not formulate clearly the principle of virtual forces, for which asystem of velocities, i.e. an act of motion, is congruent with constraints if and onlyif the work made against any system of balanced virtual forces is zero. In fact heuses only the necessary part of this principle: for an act of motion congruent withconstraints the virtual work of a system of balanced forces must be zero. The lack ofperception, by Cauchy and even by any not careful reader of his work, that he is usinga principle of virtual forces rather than of virtual velocities or displacements derivesfrom the coincidence of the necessary parts of the two principles. These necessaryconditions, set out for the acts of motion, are in the order:

For a congruent act of motion L = 0, for a system of balanced forces.For a system of balanced forces L = 0, for a congruent act of motion.

A modern reader however feels that Cauchy is using the principle of virtual forces,because he considers ‘real’ the velocities and virtual the forces. Regardless of hisawareness, Sur le mouvements que peut prendre un système invariable, libre auassujetti à certaines conditions should be considered as the first step towards theexplicit formulation ofmodern principles of virtual forces. Here’s what Cauchy says:

When an invariable system [a rigid body], free or subject to certain constraints, moves in thespace, there are among the velocities of the different points certain relationships that in manycases are expressed very simply and can be deduced from the formulas for the transformationof coordinates. I will show in this article, that the same relationships can be drawn by theprinciple of virtual velocities. This principle is usually used to determine the forces capableof maintaining equilibrium in a system of particles subject to given constraints, assumingas known the velocities that these points can have in one or more virtual motions of thesystem, i.e. in motions compatible with the constraints in question. But it is clear that onecan reverse the question [emphasis added], and after establishing the equilibrium conditionsthrough any method, or if you want through the consideration of some virtual motions, onecan use, to determine the nature of all other, the principle that we have mentioned.We add that it is useful in this determination, to replace the principle of virtual velocitieswith another principle to be drawn immediately from the first, and which is contained in thefollowing proposition:Theorem. Suppose that two system of forces are applied consecutively to points subject toany constraints. For these two systems of forces are equivalent, it will be necessary andsufficient that, in a whatsoever virtual motion, the sum of the moments of the forces of thefirst virtual system is equal to the sum of the moments of the forces of the second virtualsystem [64].1 (A.15.1)

The theorem referred to at the end of the quoted passage replaces a criterion of equi-librium of forces with a criterion of equivalence. With it the necessary part of theprinciple of virtual forces becomes: for a virtual congruent act of motion the mo-ments of two equivalent systems of forces must be equal. I will refer in the followingto this statement as Cauchy’s principle.

Note that the application of Cauchy’s principle and in particular the equation ofvirtual forces in general, it is necessary to presuppose a criterion of equilibrium or

1 pp. 94–95.


equivalence. If one does not want to fall in to a petition of principle, this criterionshould be other that that given by the equations of moments. In fact, if the equation ofmoments was used, the congruency of virtual displacements with constraints shouldbe taken for granted, but that is precisely the object of the principle of virtual forces.As a predefined criterion of equilibrium Cauchy adopts, without explicitly stating it,the rule of composition and decomposition of the forces and admits the possibilityof transport of forces along their lines of action.

15.1.1 Kinematics of plane rigid bodies

Based on his principle, Cauchy passes to an examination of the compatibility ofmotion for points of a rigid plane. The first theorem he proves is:

Theorem I. If at any time of the motion, two points of the not deformable system have zerovelocity, the velocities of all the other points will be reduced to zero [64].2 (A.15.2)

The theorem can be proved with the help of the following reasoning. Let A′ and A′′be the points the velocity of which is zero and ω be the velocity a third point Achosen arbitrarily. Apply to this last point a force P parallel toω, and using the ruleof the parallelogram of forces decompose P into two forces P′ and P′′ directed as thestraight lines AA′ and AA′′, as shown in Fig. 15.1a.

Since the translation along its line of action does not alter the conditions of equiv-alence between the forces, it is possible to assume the force P′ applied at point A′and the force A′′ at point P′′. The sum of the virtual moments of these two forceswill vanish as the velocity of A′ and A′′ are zero by assumption. Therefore, sincethe force P is equivalent in construction to the other two P′ and P′′, by virtue ofCauchy’s principle, it will have a virtual moment Pω equal to that of P′ and P′′. Butthis is zero because A′ and A′′ have zero velocity, so Pω = 0 and then ω must bezero, because P is not zero. Since A was chosen arbitrarily the theorem is proved.The above reasoning does not apply if point A were located on the line A′A′′. But,then, replacing the force P with two equivalent parallel forces P′ and P′′ applied, at

P

P'

P''A

A

A

a)

ω

ω

PP'

P''

A

b)

''

'

A''A'

Fig. 15.1. Two points with zero velocity

2 p. 96.


P

P'

P''A

A

A

'

''

ω

ω

''

'

Fig. 15.2. No points with zero velocity

A′ and A′′, it is still possible to recognize that the product Pω must be reduced tozero (see Fig. 10.1b).

The second theorem is proved in a similar manner:

Theorem II. If at any time of the motion, the velocity of all points of the not deformablesystem are different from zero, these velocities are all equal and directed along parallel lines[64].3 (A.15.3)

The proof is carried out by reductio ad absurdum. Letω′ andω′′ be the velocities ofany two points A′ and A′′ of Fig. 15.2; assume, by the absurd, that these velocitiesare not parallel and then the perpendiculars to their directions for the two points A′and A′′ meet at a third point A. Consider then a force P applied at A and directedin a random manner, equivalent to the system of two other forces P′, P′′ parallel toAA′ and AA′′ applied respectively to points A′ and A′′. If ω indicates the velocityof the point A, by virtue of Cauchy’s principle, it is:

Pωcos(P,ω) = P′ω′ cos(P′,ω′)+P′′ω′′ cos(P′′,ω′′), (15.3)

the first member of the equality being the virtual moment of the force P and thesecond member the virtual moment of the forces P′ and P′′ statically equivalent toP. Moreover, the lines AA′ andAA′′ are perpendicular to the direction of the velocityω′ and ω′′, and then the second member of the previous equality is canceled, so itis:

Pωcos(P,ω) = 0, (15.4)

which, because Pcos(P,ω) cannot be always equal to zero because P has arbitrarydirection, impliesω= 0. This is absurd becauseω �= 0 by assumption. This meansthatω′ andω′′ are parallel because the contrary gives the absurdity.

It remains to show that the velocities ω′ and ω′′ are equal to each other and tothose of all the other points. Cauchy considers first the case where the line A′A′′ isnot perpendicular to the directions of these velocity now supposed to be parallel.A generic force P directed along this line, can be thought as applied either at A′ orat A′′, forming two equivalent systems consisting of the only force P, as shown inFigs. 15.3a and 15.3b.

3 p. 96.


'

"P

a)

A"

P

b)

ω 'ω

ω ω

A'' A'''A'

Fig. 15.3. No points with zero velocity

For Cauchy’s principle the moments in the two situations are equal:

Pω′ cos(P,ω′) = Pω′′ cos(P,ω′′) (15.5)

and since the angles (P,ω′), (P,ω′′) are identical in construction it is ω′ =ω′′. Ifthe line A′A′′ is perpendicular to the direction of the velocity of A′ and A′′, chooseany point A outside of the line A′A′′ with velocity ω, then, reasoning as above,the equations ω′ =ω and ω′′ =ω can be obtained, from which is again obtainedω′ =ω′′.

The proof of the subsequent theorem is a bit more complicated:

Theorem III. If at any time of the motion one point in the rigid system has zero velocity,the velocity of a second arbitrarily selected point will be perpendicular to the radius vectorfrom the first point to the second and proportional to this vector radius [64].4 (A.15.4)

Let O be the only point on which the velocity is zero by assumption and ω thevelocity of another arbitrarily chosen point A, as shown in Fig. 15.4. If a force P isapplied to the point O directed along OA, its virtual moment will be zero (becausethe velocity of O is zero). Since moving P fromA to O gives another force equivalentto it, the virtual moment of this transported force will be zero for Cauchy’s principle,and then it is:

Pωcos(P,ω) = 0, (15.6)

from which observing that the quantities P andω are not zero it is:

cos(P,ω) = 0. (15.7)

So the angle (P,ω) is right and the velocity ω perpendicular to the radius vectorOA. It remains to show that the velocities vary in proportion to the distance from O.

P

A r ω

O

Fig. 15.4. Only one point with zero velocity

4 p. 98.


Q

A

Q'Q' r'

Q

'

A

r

O

ωω

'


Let now be ω′ the velocity of a third point A′, as shown in Fig. 15.5. This velocitywill be itself perpendicular to the radius vector OA′. Apply to points A and O twoforces equal to Q and perpendicular to OA (and therefore parallel to ω), forming acouple of forces, the first of which is directed in the same sense of the velocity ω.The sum of the virtual moments of the two forces of the couple is reduced to thevirtual moment of the first force, and consequently the product Qω, because in Othe velocity is zero. Apply also to the points A′ and O two new forces equal to Q′and perpendicular to the radius vector OA′ (and therefore parallel to ω′) such thatthey give a second couple equivalent to the first. For the equivalence between thetwo couples, if r and r′ designate the radius vectors OA and OA′ respectively, it is:

Q′r′ = Qr. (15.8)

Moreover, the virtual moment of the forces of the second couple must be equal tothe virtual moment of the forces of the first couple for Cauchy’s principle. So it is:

Q′ω′ = Qω. (15.9)

From this relation, combined with the previous one:

ω′

r′=ω

r. (15.10)

So the velocity of the points A and A′ are not only perpendicular to the radius vectorsr and r′, but also proportional to them. In the motion just seen, the velocity of a pointat unit distance from the centre O is called the angular velocity of the rigid planesystem around this same centre. If γ designates the angular velocity, the velocity ofpoint A, located at the end of the radius vector r, is defined by the relation:

ω= γr. (15.11)

Cauchy sums up his analysis with the comment:

Theorems I, II, III show all the relationships that may exist between the velocities of thematerial points rigidly linked to each other, contained in a fixed plane that may not ever getout. These theorems show that the velocities in question are always those which the systemwould have taken in the state of rest, or translated in the direction of a fixed axis, or turnedaround a fixed centre. We add 10 that a translation, parallel to a fixed axis, can be obtainedby a rotation around a fixed centre, where this centre is away at an infinite distance from


the origin of the coordinates. 20 that the centre of rotation is a point the position of whichvaries in general from time to time in the plane that is considered. It is for this reason thatwe designate the point at issue as the instantaneous centre of rotation [64].5 (A.15.5)

The section on the motion of plane systems concludes by studying the behavior ofthe instantaneous centre of rotation. The discussion is quite interesting but it is purelygeometric with no mechanical implications, thus I limit myself to the conclusion:

We will note that at the end of a designated time t, the different points on the moving surfaceof the space will occupy certain positions, and that one of them, the point O, for example, isthe instantaneous centre of rotation. In addition, it is clear that at this time [t] it is possible thatthrough O two separate curves pass, drawn to include, the first, all the points of the movingsurface and, the second, all the points of space, which later will become the instantaneouscentres of rotation [64].6 (A.15.6)

Cauchy proves that the arcs OA, OB, measured from the point O on the two curvesmentioned above differ by an amount of an order higher than the first. So thesetwo curves are tangent to each other. In addition, the first curve, delivered by themotion of the surface on which it is drawn, will cover a portion of the plane that willform by envelope the second curve. In the special case where one of the curves justconsidered reduces to a point, the same is true of the other. Then the instantaneouscentre of rotation keeps the same position not only in space but also on the movingsurface.

For the three-dimensional rigid body, I report only the statements of the theo-rems that Cauchy demonstrates, using a process similar to that used for the two-dimensional rigid body.

Theorem VI. Whatever the nature of the motion of a solid, the relationships between thevarious points will always be as those that they would occur if the body was kept so it couldonly turn around a fixed axis and slide along this axis [64].7 (A.15.7)

Theorem VII. Conceive a rigid body which moves in space by any means, and that at a givenmoment trace 1) in the body, 2) in the space, the different lines with which subsequently theinstantaneous axis of rotation of this rigid bodywill coincide.While the ruled surface, havingas generatrices the straight lines traced in the body, will be dragged by the motion of this, itwill constantly touch the ruled surface having as generatrices the straight lines traced in thespace, and consequently the second surface will be nothing but the envelope of the portionof space traveled by the first [64].8 (A.15.8)

Theorem VIII. Posited the same things as in the theorem VII, if the instantaneous axis ofrotation of the solid body becomes fix in the body it will become fix in the space and viceversa [64].9 (A.15.9)

5 p. 100.6 p. 101.7 p. 116.8 pp. 119–120.9 p. 120.

16

The treatises of mechanics

Abstract. This chapter is devoted to VWLs as presented in the main treatises of me-chanics where a reductionist approach is assumed. In the first part, the treatises ofSiméon Denis Poisson and Jean Marie Constant Duhamel are presented. In the sec-ond part the approach of Jean Marie Gustav Gaspard Coriolis is presented. To pointout the introduction of modern term virtual work, the introduction of the problemsassociated with friction and finally the change in the ontological status of virtualwork. From simple mathematical to physical magnitude.

After the publication of Lagrange’sMécanique analytique, following the reorganiza-tion of schools for higher education where regular courses in physics and mechanicsbegan to be taken by a large number of students, the first textbooks of mechanicsstarted to spread throughout Europe. Among the most famous were those of theprofessors of the École polytechnique in Paris, one of the first modern scientific in-stitutions.

A feature of these textbooks was the reproduction, in general terms, of the maintopics of mechanics. They were concerned with generally accepted principles andprocedures, and in their presentation often not even the names of authors and worksfrom which they are drawn were reported, as if they had become a common heritagewhich is unthinkable to criticize.

Given the high cultural and intellectual level of many of the writers, not infre-quently in these manuals there were exposures of levels at least equivalent to thatof publications in scientific journals. This was especially true for the laws of vir-tual work, because as of then, they were disclosed only through manuals. In thischapter I consider three authors who seemed to me the most significant: SiméonDenis Poisson, Jean Marie Constant Duhamel and Jean Marie Gustav GaspardCoriolis.


362 16 The treatises of mechanics

16.1 Siméon Denis Poisson

Siméon Denis Poisson was born in Pithiviers in 1781 anddied in Sceaux in 1840. Poisson went to study mathemat-ics at the École polytechnique in 1798. His teachers Laplaceand Lagrange became his friends. After he was graduated, in1802, he remained at the École Polytechnique as an assistantto Fourier, from whom he subsequently inherited the chair.In 1809 he was named professor at the newly founded facultyof sciences. His Traité de mécanique published in 1811 andagain in 1833 was a standard manual of mechanics. Poisson

took care of electricity with an important essay in 1812 and the theory of elasticitywith various memoirs, taking the molecular model of matter. Gugliemo Libri saidof him: “His only passion was the science, he lived and died for it” [290].

In his Traité de mécanique [200], after having set out the virtual work principleand verified it in some simple cases, Poisson reports two separate demonstrations.The first shows a clear influence of Laplace’sMécanique celeste, the second demon-stration instead reproduces that of Lagrange’s Mécanique analytique of 1798 and1811. In the following I give a somehow extensive sketch of the first demonstrationthat contains original and interesting ideas. Poisson demonstrates first the necessarypart of the virtual work principle, i.e. if there is equilibrium, the virtual work, orrather the total moment as he calls it, is zero for any virtual displacement. In orderto do this, he takes as a pre-existing criterion of equilibrium the rule of composi-tion of forces. From this criterion he proves the equation of the moments for a freematerial point and for material points constrained to move on a surface. In the lattercase the assumption of smooth constraints is implicit, assuming the orthogonality ofthe reaction force to the surface. Then Poisson extends his proof to systems of par-ticles constrained by inextensible wires that pass through apposite rings. There areno comments by Poisson on this choice of internal constraints, i.e. if he sees themas representative or not of every situation.

It is worth noting that Poisson uses the term virtual velocity meaning a vectorquantity:

The infinitely small straight lines [infinitesimal displacements] that describe the motion ofa point […], name that comes from the fact that they are considered simultaneously as thedistances traveled by the points of the system in the first instant, when the equilibrium isbroken [200].1 (A.16.1)

So he does not adopt Bernoulli’s definition of infinitesimal displacements in thedirection of forces. Poisson also seems influenced by the definition of Carnot’s geo-metric motions, since he raises the question of reversibility of virtual velocities. Hedeals only with bilateral constraints, believing that a virtual work principle can beformulated only with reference to them without commenting on the observations ofFourier on this topic.

1 p. 660.

16.1 Siméon Denis Poisson 363

The following comment on the role of constraint reactions is interesting:

The advantage of the virtual velocity principle is to furnish the equilibrium equations in eachparticular case, without the need to evaluate the internal forces. But because the demonstra-tion we are going to show is based on the consideration of these forces, of unknown value,here is the notation which we will use to represent them [200].2 (A.16.2)

Poisson writes first the equation of moments of a single material point, subject toactive forces, to external and internal constraint forces. Then he moves on to exam-ine the whole system, following a reasoning similar to that of Laplace’s Mécaniqueceleste for the analysis of the relative displacements of the material points, but de-veloped in a more rigorous way, with the clear awareness that the considerationsapply only for infinitesimal displacements. For the system of constrained materialpoints Poisson achieves the following equations of moments:

Pp+P′p′+P′′p′′+ etc.+[M,M′]δ(M,M′)+ [M,M′′]δ(M,M′′)+ etc.

+[M′,M′′]δ(M′,M′′)+ [M′,M′′′]δ(M′,M′′′)+ etc.+etc. = 0,

(16.1)

where P,P′,P′′,etc. are the active forces applied to the points of the system, p, p′p′′,etc. the components of virtual velocities in the direction of the forces, [M,M′],etc.represent the tension in the wires that connect M,M′,etc., δ(M,M′),etc. are the in-finitesimal variations of distance between m,m′,etc.

In the event that the relative changes in distance are zero, δ(M,M′), etc.= 0, as forrigid bodies, Poisson obtains the classical expression of the virtual work principle:

Pp+P′p′+P′′p′′+ etc. = 0. (16.2)

Then he shows that the same expression remains valid even if the distances betweenmaterial points vary, provided that the wires, always joining the material points,sliding without friction on the rings, maintain intact their whole length.

After the treatment of the necessary part Poisson passes to the sufficient part ofthe virtual work principle, i.e. that if the virtual work of the active forces is zerofor any virtual displacement then there is equilibrium. The demonstration that hedevelops has a historical interest and because of this I quote it in full:

It remains to prove that, conversely, when the equation (b) [equation (16.2)] applies to allinfinitesimal motions of the system of points M,M′,M′′, etc., the given force P,P′,P′′, etc.,are equilibrated.[…]Suppose for a moment that the equilibrium does not take place. The points M,M′,M′′, etc.or any part of them, are set in motion and at the beginning, simultaneously describe thestraight lines MN,M′N′,M′′N′′, etc. All these points can be reduced at rest by applyingappropriate forces, directed along these lines, in the opposite direction to the motions pro-duced. Therefore, denoting these forces with the unknowns R,R′,R′′, etc., the equilibriumwill be achieved among the forces P,P′,P′′, etc., R,R′,R′′, etc., so that if r,r′,r′′, etc., desig-nate the virtual velocities projected on the directions of these new forces R,R′,R′′, etc., for

2 p. 664.


the principle of virtual work we have demonstrated, it is:

Pp+P′p′+P′′p′′+ etc.+Rr+R′r′+R′′r′′+ etc. = 0

or simply:Rr+R′r′+R′′r′′+ etc. = 0 (c)

by virtue of equation (b) which is valid by assumption.Since this equation (c) applies to all motions compatible with the constraint conditions

of the system of points M,M′,M′′, etc.., it is possible to choose for the virtual velocitiesthe space actually described MN,M′N′,M′′N′′, etc. But since these spaces are valued in theextension of the directions of R,R′,R′′, etc. it follows that all the projections r,r′,r′′, etc..will be negative […]. Then, since all the terms in the equation (c) have the same sign, theirsum can not be zero, unless each term is zero, then:

R · MN = 0, R′ · M′N′ = 0, R′′ · M′′N′′ = 0, etc.

Now because the product R · MN is zero, it must be, or R = 0, or MN = 0, which meansin either case, the point M can not take any motion. The same applies to all other pointsand therefore the whole system is in equilibrium, what we proposed to demonstrate [200].3

(A.16.3)

The proof is made by reductio ad absurdum, assuming first that there is not equilib-rium and then by showing that any forces that should be applied to restore it mustall be zero, and then there was equilibrium. The demonstration seems convincing;it implicitly assumes that the forces to be applied to restore the equilibrium shouldbe all directed in the opposite direction to the motion allowed by assumption. This,although intuitive, is not demonstrable in the mechanics of reference considered byPoisson and it should be taken as an axiom.

This assumption may be false as it can be seen easily by the following example.Consider two material pointsm andm′ in Fig. 16.1 rigidly constrained to turn aroundthe point O.

It is clear that any rotational motion that leads m and m′ to move on the sameside can be balanced also by two forces one in one direction and the other in theopposite direction and not only by two forces in the same direction, both opposed tothe motions of m and m′. What matters is the total static moment of the forces that,

O

m

m'

v

v'

Fig. 16.1. Equilibrium of forces

3 pp. 670–672.

16.2 Jean Marie Duhamel 365

in the case of the figure must be clockwise. It is true that Poisson considers systemsconsisting of wires, but he also looks into the possibility of rigid motions, and thenthe above example should be valid.

From a logical point of view the asymmetry between the demonstrations of thenecessary and the sufficient parts of the principle should be underlined. The neces-sary part is proved as a theorem of a reference mechanics in which there is a pre-fixed criterion of equilibrium provided by the balance of forces in accordance withthe rule of the parallelogram. The sufficient part is instead proved by ignoring, in anuneconomic way, that criterion and the mechanics of reference on which it is based,introducing the principle of dynamic character, according to which motions can bedestroyed by forces acting in the opposite direction to them. A similar reasoning isalso found in many recent treatises of physics and rational mechanics (see § 2.2). It isinteresting to compare Poisson’s proof of the virtual work law with that of Poinsot.In the latter, where it is taken as a criterion of equilibrium of forces, there is a perfectsymmetry between the demonstration of the necessary or sufficient parts, without theexplicit use of dynamic principles.

16.2 Jean Marie Duhamel

Jean Marie Constant Duhamel was born in St. Malo in 1797and died in Paris in 1872. He entered the École polytechniquein 1814 to graduate in 1816. Except for one year, Duhameltaught continuously at the École polytechnique from 1830to 1869. He was first given provisional charge of the anal-ysis course, replacing Coriolis. He was made assistant lec-turer in geodesy in 1831, entrance examiner in 1835, profes-sor of analysis and mechanics in 1836, permanent examinerin 1840, and director of studies in 1844. The commission of

1850 demanded his removal because he resisted a program for change, but he re-turned as professor of analysis in 1851, replacing Liouville. Duhamel also taught atthe École normale supérieure and at the Sorbonne. He was known as a good teacher.He was elected to the Académie des sciences de Paris in 1840. Duhamel’s scientificcontributions were not fundamental; however, they were important. He worked onpartial differential equations and applied his procedures to the theory of heat, rationalmechanics and acoustics. Studies related to acoustic concerned vibration of stringsand air in cylindrical and conical tubes. The ‘principle of Duhamel’ in the theoryof differential equations derived from his work on the distribution of heat in a solidwith a temperature variable boundary [290, 354].

In his Cours de mécanique [97] Duhamel once again re-proposes Poisson’sdemonstration of the virtual velocity principle. Of some interest are his commentsin the introduction and some clarification in the text. In the introduction, Duhamelcriticizes the approach of those who replace the real system with an equivalent sys-tem, with no consideration for the true internal structure of the real system. Duhamelwrites:


Most Geometers regard as obvious that if the forces are in equilibrium in a system of points,subject to constraints that allow it to make certain infinitesimal motions, these same forceswill still be in equilibrium for the same system of points, subject to different constraintsthat allow the same motions. This principle […] has always seemed to me doubtful […] itappears to be based on a real confusion between physics and geometry.[…]Therefore we changed the proof of the principle of virtual velocities as derived by Ampèreand we adopted one that does not have the same problem and which is nothing but, after all,than that of Poisson’s Traité de mécanique [97].4 (A.16.4)

The principle of equivalence criticized by Duhamel, can be made more explicit asfollows: consider a system of particles subject to forces F and constraints L. If onereplaces the constraints L with other constraints l that enable the same virtual veloc-ity (the same infinitesimal displacements) for the configuration of the system wherethe forces are applied, then there is equilibrium of F on L if and only if there isequilibrium of F on l. Duhamel’s doubts about the validity of this principle appearto be well motivated; indeed, selecting the equality of virtual displacements as acriterion of equivalence between the two systems seems to have attained some cir-cularity, because the equivalence is established for quantities which are essential forthe provability of the virtual work principle, which is then somehow assumed. Thecriticism applies to the second demonstration of Fourier and to the demonstration ofLagrange in the Théorie des fonctions analytiques. And one can go even further, forexample, to Galileo and his demonstration of the law of the inclined plane from thatof the lever.

The following comment by Giovanni Vailati highlights the intimate connectionbetween the possibility of replacing the equivalent systems and the virtual work laws:

So, for what concerns Johann Bernoulli it is noteworthy that, by taking into account, in thefamous letter to Varignon (1717),5 the relations between the displacements [virtual], in-finitely small of the points of application of forces, he did nothing but at the end to applyand enunciate, in general form, a standard method which had been already frequently usedby his predecessors, among others, Leonardo da Vinci and Galileo, in their attempts to inferfrom the principle of the lever, that of the inclined plane, and to include it in that of a heavybody supported by two not parallel wires. This rule consist to substitute, as regards the equi-librium, two sets of constraints when they allow the same initial displacement. It, as Duhemnotes, is set out more explicitly by Descartes, in a letter to Father Mersenne (1638) […]. Onthe presence of similar considerations in the writings of Galileo see Mach (Mechanik, 4thed., pp. 25–26) [391].6 (A.16.5)

In Jouguet [341]7 it is pointed out that under certain constraint conditions the uncrit-ical application of the equivalence principle can lead to errors.

In the introduction of Duhamel’s book details are given that are interesting from ahistorical point of view, on the approximations involved in the use of infinitesimals:

If any system of points is in equilibrium, and if we consider an infinitely small displacementof all points, which is compatible with all the conditions of the constraints, the sum of the

4 pp. VI-VII.5 The date is wrong, it should be 1715.6 p. 267.7 pp. 170–174.

16.3 Gaspard Gustave Coriolis 367

virtual moments of all forces is zero, whatever this displacement is. And vice versa, if thiscondition is met for all virtual displacements, the system is in equilibrium.In this statement the infinitely small are treated in an ordinary way. The equation is exactonly considering the limits of the ratios, after dividing by any one of the infinitely smallquantities, in other words, the sum of the moments is infinitely small compared to the mo-ments themselves [97].8 (A.16.6)

These statements rebuff the embarrassment still existing in the second half of theXIX century on the use of the concept of infinitesimal displacements, ‘which aretreated in an ordinary way’.

16.3 Gaspard Gustave Coriolis

Gaspard Gustave Coriolis was born in Paris in 1792 and diedin Paris in 1843. He entered the École polytechnique in 1808and finished second among all students of that year. Aftergraduation he entered the École des ponts et chaussées. Heworked as an engineer in the district of Meurthe-et-Moselleand the Vosges mountains. Coriolis became professor of me-chanics at the École centrale des artes et manufactures in1829. Despite his reluctance to teach, in 1832 he accepteda position at the École des ponts and chaussées. Here he

worked with Navier, teaching mechanics. After the death of Navier, in 1836, hetook his chair and also replaced Navier at the Académie des sciences de Paris. Hecontinued to teach until 1838 at the École polytechnique, when he decided to stopteaching and to become director of studies. The name of Coriolis is famous for hiswork on the forces of drag, which showed that the laws of motion apply equally ina rotating frame of reference provided to add Coriolis’s forces [290]. Coriolis’s roleis very important for mechanics applied to machines also for his books De calcul del’effect de machines published in 1829 [78] and the Traité de mécaniques des corpssolides et du calcul de l’effect des machines, published in 1844 [80], with the latterconsidered a reworking of the first.

In the following I will first consider briefly the De calcul de l’effect de machines.In the more than twenty years which separate it from Carnot’s Principe fondamen-taux de l’équilibre et du mouvement, no important intermediate work was published[332]. Coriolis’s book is a didactic book so most parts are greatly extended; thoughconsidering machines in general it gave a lot of space to a particular kind of ma-chines; terms and concepts are clearly stated. In the first chapter of his book Coriolisdefines the main concepts of mechanics, among them that of force, mass and work.As for most scientists of the time, the ontology of force is no longer a problem forCoriolis, he is not interested in its status but only in its use; moreover force does notimply impact:

8 p. 193.


In what we are going to say the word force will apply only to what is analogous to weight,that is to what is called, in most cases, pressure, tension, and traction. With this meaningforce could not make the direction and the value of velocity to change sharply without itpasses through all the intermediate states [78].9 (A.16.7)

The mass of a body is defined as the ratio between force and acceleration and itsmeasure was given by its weight at an assigned sea level. The concept of work isconsidered the most important one for the study of machines in motion, while forceis important for machines in equilibrium. Coriolis introduces the word work (travail)to indicate what Carnot called moment of activity and others moment, mechanicalpower, quantity of action, energy, or even simply force.

These various and quite vague expressions were not suitable to spread easily. We proposethe appellation dynamical work, or simply work, for the quantity

∫Pds […]. This name will

not be confused with any other mechanical denomination. It seems suitable to give the rightidea of the thing, by maintaining its common meaning of physical work […] this name isthen suitable to designate the union of these two concepts, displacement and force [78].10

(A.16.8)

Coriolis uses the term work also in subsequent studies, particularly in the Mémoiresur la manière d’differéns établi les principes pour des systèmes de mécanique descorps, comme en des assemblages de considérant the molecules of 1835 [79]. It is ause that he definitely will consolidate with his workMécanique des corps solides of1844 where, in the preface, he writes:

I employed in this work some new nomenclature: I name work the quantity usually namedpuissance mécanique, quantité d’action ou effet dynamique, and I propose the name dy-namode for the unity of measure of this quantity. I introduced also one more little innovationby naming living force the product of the weight times the height associated to the height.This living force is one half of the product that today is associated to this name, i.e. the masstimes the square of speed [80].11 (A.16.9)

Notice Coriolis is introducing the factor 1/2 in front of the expression of livingforce (i.e. kinetic energy), because he suggests measuring the living force of a bodyof mass m and velocity v with the product mh, with h the height the body can reachif thrown upward with an initial velocity v (h = v2/2).

In a note to the passage quoted above, Coriolis writes:

This term work is so natural in the sense that I use it, which, though it has never been eitherproposed or approved as a technical expression, nevertheless it was used accidentally byMr. Navier in his notes on Belidor and Prony in his Mémoire sur les expériences de lamachine du Gros-Caillou [80].12 (A.16.10)

Although Coriolis’s texts were fundamental to the spread of the term work, again, atthe end of the XIX century propositions like: principle of virtual velocities, principleof moments and principle of virtual work, co-existed. See for this purpose a note by

9 pp. 2–3.10 p. 17.11 p. IX.12 p. IX.


Saint-Venant in his translation of Clebsh’s text on the theory of elasticity, where hespeaks of a “theorem of virtual work or virtual velocities [72].13 Notice that Coriolishimself used the term principle of virtual velocities.

It is also interesting the way Coriolis introduced machines in general, very closeto Carnot’s.

Here after we will use the name machine to indicate the mobile bodies to which we willapply the equation of living forces: in this sense a single body which moves is a machine,so has a more complicated system. In each particular case, once we will know by whatbodies in motion the machine is composed it will be enough to apply the principle previouslyestablished, to know precisely what are the masses which must be considered in the livingforces evaluation, and what are the motive and resistant forces which must be considered toevaluate the amount of work [78].14 (A.16.11)

In the following I report some reflections on the application of the virtual work prin-ciple taken from theMémoire sur la manière d’établir les différens principes de mé-canique pour des systèmes de corps, en les considérant comme des assemblages demolécules [79], a principle which is at the basis of his mechanics. Coriolis stands onthe ground of physical mechanics, advocated by Poisson, as opposed to the analyticalmechanics of Lagrange. In physical mechanics everything is reduced to a Laplacianmodel of material points, or molecules, unlike analytical mechanics, where the bod-ies are treated essentially in their geometric aspect, for example as rigid bodies.

According to Coriolis, concerning the statics of a particle, it is sufficient to assignthe law of composition of forces. To switch to statics of extended bodies or systemsof bodies it is necessary to add other principles, among which are those of action andreaction:

For statics and dynamics of the systems of bodies it is enough to lean on the principle ofequality between action and reaction. This principle is that if a molecule of a body producesa certain force of attraction or repulsion on a neighboring molecule, it likewise receives fromit a force equal and directly opposite, so that all the sets of molecules that make up a bodyis formed only by pairs of equal and opposite forces. It is only with the help of this startingpoint we are going to determine all the principles of mechanics [79].15 (A.16.12)

Coriolis begins his analysis of statics of extended bodies by introducing first thevirtual work principle for the single molecule:

If it is accepted that a point, to which a force P is applied, moves by an amount δs in anydirection, we will call element of virtual work the product of δs times the component of theforce16 in the direction of δs. Denoted by Pδs the angle of δs with the force P, the elementof virtual work will be

Pcos(Pδs)

[79].17 (A.16.13)

13 p. 577.14 p. 20.15 p. 94.16 Note that Coriolis is defining, as today, work as a product of displacement and the componentof force in the direction of motion and not of the force and the component of displacement in thedirection of the force, as was the tradition and the way he will define it later.17 p. 95.


In this passage and throughout the rest of his work, the definition of virtual displace-ment is a little ambiguous. Coriolis never says that it is an infinitesimal displacement.But in fact, he treats it that way. For example, in the case of two molecules m andm′ of Fig. 16.2, if R is the common force that they exchange, in accordance with theprinciple of action and reaction, Coriolis argues that the work done by the force R isprovided by the relation:

Rcos(Rδs)δs+Rcos(Rδs′)δs′ = Rδr, (16.3)

where δr is the change of distance between m and m′. But this relation is validonly if δs and δs′ are infinitesimally small as clear from Fig. 16.2. Because Coriolisstill considers virtual displacements that occur in real time, with which the forcesvary according to what was generally accepted before Poinsot, the force R movesin R as a result of the virtual displacements δs and δs′. If these are infinitesimal,the displacement δr can be considered to occur in the old direction of R and thenδr � cos(Rδs)δs+ cos(Rδs′)δs′, from which relation (16.3) immediately follows.

Coriolis continues his exposition by saying that each molecule inside a solid bodywill be in equilibrium under the action of external forces P and internal forces Rexchanged among the molecules, i.e. R+ S = 0, and then for each molecule, theelement of virtual work will be zero, and so will be the sum of the elements of virtualwork of all the molecules. Then it is:

∑Rδr+∑Pδp = 0, (16.4)

where δr represents the change of distance between the molecules and δp the com-ponent of the virtual displacement of the point of application of P in the direction ofP itself. In the case of an undeformable body δr = 0, so the previous relation gives:

∑Pδp = 0. (16.5)

It is assumed now that the virtual motions are limited to motions leaving the molecules inthe state of invariability of the mutual distances, then the distance δr will not change inthis motion and it will be δr = 0 and the equation above is reduced to: ∑Pδp = 0 [79].18

(A.16.14)

R s'

R m

R

m'

s

R

Fig. 16.2. Action and reaction of two molecules

18 p. 97.


Thus Coriolis found the equilibrium equation of the moments for the ‘solidified’body, which is only necessary for the equilibrium of a deformable body and sufficientfor a body that is actually rigid.

The above reasoning is similar to that developed by Laplace in the Mécaniqueceleste. Both Coriolis and Laplace apply the principle of action and reaction, butLaplace applies it to constrained points without any justification, Coriolis appliesit to free molecules, which in full respect of Newtonian mechanics exchange equaland opposite forces. In this context, the use of infinitesimal motion or velocity isimportant, not to take into account the conditions of constraint, which do not imposeany limit, but rather to simplify calculations.

After considering the equilibrium (and motion) of a single body, Coriolis passesto the examination of a system of bodies. Here the language is similar to that foundin the treatises of practical engineers [381] and those based on thermodynamics, toappear in a few years, see for example Chapter 18. The virtual work assumes a degreeof reality. It is more a physical quantity, observable and measurable in some way,than a purely mathematical definition as it appeared in the works of Lagrange andhis immediate successors:

If the equilibrium is obtained under the action of forces P, each molecule will be in equilib-rium and, taking into account all the molecular actions, it will be:

∑Rδr+∑Pδp = 0.

If now a virtual motion of each body is considered that leaves its invariability or solidity,and yet in this motion the bodies are left to slip or turn over each other with the freedomof motion allowed by the machine constitution, it is found that a large portion of virtualworks Rδr vanishes: it is that due to actions between molecules that have not switched awayduring the virtual motion, namely those belonging to the same body. In the equation aboveit will remain only the element of virtual work ∑Rδr coming from the actions among themolecules of adjacent bodies, when in the virtual motion these bodies do not move togetheras one system, but they slip or roll on each other. The actions R that remain will be only dueto molecules that are at a distance from the contact surface less than the extension of themolecular actions, or in other words, the radius of the sphere of action [79].19 (A.16.15)

This piece documents the way Coriolis conceives of virtual displacements. Theyare small possible displacements, and the virtual work is determined on the basis offorces that are assumed varying with them and not assumed frozen at the instant andthe point where to study the equilibrium, as Poinsot and Ampère did, but they arethough as function of the virtual displacements.

Coriolis then says that the virtual work between the molecules in the contact zonecan be calculated, assuming one of the bodies as fixed and considering for the othera virtual displacement equal for all the molecules, because of the small size of thecontact area. Furthermore he proposes to decompose the forces of each moleculeinto a tangential component and a normal component, so “the elementary work ofthe normal component will be zero because the angle that this component makeswith the virtual displacement is right. It will then only remain the element of workof the component in the direction of the sliding plane". Denoting with δ f the virtual

19 pp. 114–115.


displacement common to all molecules of the contact zone and with F the forcesexchanged among the molecules,20 the sum of the virtual work due to the action ofthe two bodies will be [79]:21

δ f ∑F cos(Fδ f ), (16.6)

with the summation extended only to the molecules in the area of contact which acton each other and δ f has been put into evidence because it is common. The equationof virtual work for all bodies will then be:

∑Pdp+∑δ f ∑F cos(Fδ f ), (16.7)

where the two summations of the internal forces are made with respect to the numberof bodies concerned and with respect to the molecules of the contact zone of eachbody. If it is assumed that the actions of two bodies in contact are reduced to a singlenormal force, i.e. if the tangential component vanishes:

F cos(Fδ f ) = 0. (16.8)

Then relation (16.7) reduces simply to:

∑Pdp = 0. (16.9)

Thus the principle of virtual work applies in this case between the only externalforces [80].22 But, Coriolis continues: “In fact, the sum F cos(Fδ f ) is not null andthen it is necessary to take this into account. The difficulty is to evaluate it. The di-rections and values of the actions F can only be seen by experimental consideration”[80].23

Next Coriolis examines the problems related to friction, using a language that wascompletely different from the classic terminology used for the virtual work laws. It ispossible, for example, by the nature of bodies, to suggest that there is no possibilityof sliding and that one body rolls on the another “then the virtual velocities becomezero for the points of contact […] so that the sum of the work due to this rolling iszero.” Finally he concludes:

We are led to realize that the principle of virtual velocities in the equilibrium of a machine,composed of more bodies, cannot take place without considering first the sliding friction,where the virtual displacements cause the slipping of the the bodies one on others, and finallythat the rolling when bodies cannot take that virtual motion without deformation near thecontact points.Frictions are recognized always, for experience, able to maintain equilibrium in a certaindegree of inequality between the sum of the positive work and the sum of the negative work,

20 There is something unclear in Coriolis’ text. He declares that F is the tangential component ofthe forces exchanged among the molecules. But in this case F and δ f would be parallel each otherand then cos(Fδ f ) = 1. I assume Coriolis got confused and attributed to F the meaning of wholeforce.21 p. 115.22 p. 116.23 p. 116.


here taking as negative the elements belonging to the smaller sum. It follows that the sumof the elements to which they give rise has precisely the value that can cancel the total sumand is equal to the small difference between the sum of the positive and negative elements[79].24 (A.16.16)

So friction contributes to the balance of the work by providing a negative term, since“for experience, it gives rise to a negative sum.”

In the classification of attempts to demonstrate the law of virtual work given inChapter 2, Coriolis should be placed among the reductionists since he frames thelaw of virtual work in the context of Newtonian physics. His mechanical theory is,essentially, the one I used in Chapter 2 to highlight the problems of the logic statusof the law of virtual work. Coriolis can prove quite easily the virtual work law for thesingle unconstrained rigid body, but he finds it difficult to deal with several bodiesconstrained together or with the outside world. In fact he uses the law of virtual workin the form T1 and sometimes in the form T2, when he mentions the possibility thatthe tangential component of the contact forces vanishes. But he never introduces thelaw of virtual work in the form T3, assuming the principle P2 of smooth constraints.This is because hewants to address the problemswith friction and not to limit himselfto a situation that he considers an ideal only.

24 p. 117.

17

Virtual work laws and continuum mechanics

Abstract. This chapter is devoted to the use of the VWL in the mechanics of con-tinuous media. In the first part, the pioneering works of Joseph Louis Lagrange andClaude-Louis Navier are presented. In the second part the use of the VWL in thetheory of elasticity by Alfred Clebsch is presented. In the final part the Italian schooltreatments of internal forces as reactions of constrain are presented.

17.1 First applications

The application of virtual work laws to continuous bodies goes back if not to an-cient Greek, at least to the XVI century, when Galileo Galilei applied it to studythe equilibrium of fluids in several cases, including that of communicating vessels,in the Discorsi intorno alle cose che stanno in sull’acqua e scritture varie in 1602[115]. More sophisticated applications arose in the XVIII century, see for exampleDiscorso intorno agli equilibri of Vincenzo Angiulli in 1770 [4], in which a law ofvirtual work is used in conjunction with the differential calculus.

17.1.1 Joseph Louis Lagrange

Amature application however must await Lagrange’sMécanique analytique in 1788[145]. Lagrange considers both mono-dimensional and bi-dimensional continua, un-der static and dynamic cases. Here, for the sake of space, I will present with somedetail only the equilibrium of flexible but inextensible wire, elastic and flexible wire[145],1 non-deformable solids, and incompressible fluids.

1 pp. 156–162.


376 17 Virtual work laws and continuum mechanics

17.1.1.1 Mono-dimensional continuum

For flexible and inextensible wires Lagrange writes first the expression of the virtualwork of the active external forces, assumed as distributed along the wire:

S(Xδx+Yδy+Zδζ)dm, (17.1)

in which X ,Y,Z are the components of the active forces for unitary mass, δx,δy,δzthe virtual displacements, S the integral symbol and dm the element of mass. Thequantities that appear under the integral sign should be considered as the function ofan abscissa ameasured in the unchanged configuration, which in the case of inexten-sible wire coincides with the curvilinear abscissa s on the deformed configuration.

Virtual displacements occurring in (17.1) cannot be arbitrary because the variouspoints while moving maintain the constraint of inextensibility. Indicating with dsthe length of the infinitesimal element of the wire, Lagrange writes the equation ofinextensibility in the form:

δds = 0. (17.2)

By following the technique he developed, introducing a multiplier λ, Lagrange canwrite the following equation of equilibrium, or with his terminology, the equationof moments:

S(Xδx+Yδy+Zδz)dm+λSδds = 0, (17.3)

where δx,δy,δz can vary freely. By making explicit δds as a function of δx,δy,δz:

δds =dxδx+dyδy+dzδz

ds(17.4)

and integrating, Lagrange gets the following differential equations of equilibrium[148]:2

Xdm−dλdxds

= 0, Ydm−dλdyds

= 0, Zdm−dλdzds

= 0. (17.5)

He comments on the result as follows:

As λδds can represent the moment of a force λ tending to vary the length of ds, the term Sλδds of the general equilibrium equation of the wire will represent the sum of the momentsof all forces λ that it can be assumed to act on all elements of the wire; in fact each elementresists for his inextensibility to the action of external forces and these resistance are usuallyconsidered as active forces [emphasis added] called tensions. So the quantity λ expressesthe tension of the wire [148].3 (A.17.1)

Lagrange will be also more clear about the meaning of the multiplier λ in the studyof the equilibrium of the elastic wire. In this case the moment of the external forcesis still given by the expression (17.1). Next to the active external forces Lagrangepresupposes the existence of active internal forces F due to the elasticity of the wire.The moment of these forces on an infinitesimal element of length ds is given by

2 p. 147.3 pp. 148–149.

17.1 First applications 377

Fδds, as a work of two equal and opposing forces acting in the direction of ds atthe ends of the element ds, which undergoes a relative displacement δds. Hence theequation of moments is [148]:4

S(Xδx+Yδy+Zδζ)+FSdsdm = 0, (17.6)

which is mathematically equivalent to (17.3). From this equation it can be seen thatthe multiplier λ of constraint conditions is identical with the tension F .

17.1.1.2 Three-dimensional continuum

Rigid bodiesIn the study of three-dimensional rigid solids. Lagrange imposes, as in the case ofthe inextensible wire, the constraint of rigidity, requiring this time that the mutualdistances of all points of the solid remain unchanged for any virtual displacement.

He gets a set of differential equations of the form in the Cartesian coordinates ofthe points (x,y,z):

dnxdnδx+dnydnδy+dnzdnδz = 0, (17.7)

of which only three are independent, for example those corresponding to n = 1,2,3.These expressions were already obtained by Euler in his work Decouvert d’un nou-veau principe de mécanique [101],5 in the case of motion of a body fixed to its centreof gravity.

By integration of equations (17.7) Lagrange gets the following expression of vir-tual displacements of a rigid body:

δx = δl− yδN + zδM,δy = δm+ xδN− zδL,δx = δn− xδM+ yδL,

(17.8)

that he comments:

Expressions found above for changes δx,δy,δz show that these variations are the result ofthe motions of translation and rotation that we considered in section III.[…]The previous analysis leads naturally to these expressions and testify with a more directand more general way than that of article 10 of section III, that when the different pointsof a system retain their relative positions, the system can have at any given instant onlytranslational motion in space and rotation around three orthogonal axes [148].6 (A.17.2)

Substituting expressions (17.8) into δx,δy,δz and ds in the moment equation:

S(Xδx+Yδy+Zδζ)+λSdsdm = 0, (17.9)

Lagrange obtains the cardinal equations of statics.

4 p. 157.5 pp. 197–201.6 pp. 187–188.


Indeformable fluidsTo illustrate the ideas of Lagrange on the constitution of fluids and the possibility ofapplying the principle of virtual velocities to study their balance, I quote two excerptsfrom the historical introduction to hydraulics of the Mécanique:

Although we ignore the internal composition of fluids, we cannot doubt that the particles thatcompose them are material and that, therefore, for them, as for solids, the general laws ofequilibrium apply. In fact, the main property of fluids, the only one that distinguishes themfrom solid bodies, is that all their parts cannot resist to the smallest force and can movebetween them with all possible ease, whatever the constraints and mutual action betweenthe parts. But, being this property easily translated into calculation, it follows that the lawsof equilibrium of fluids do not require a particular theory and that they must be a specialcase of the general theory of statics [145].7 (A.17.3)

Previous theories of equilibrium and pressure of fluids are, as we have seen, wholly inde-pendent of the general principles of statics, being based on empirical principles proper tofluids. This way of demonstrating the laws of hydrostatics, deducing from the experimentalknowledge of some of its properties that of the others, has been adopted by most modernwriters who made a science of hydrostatics completely different and independent from stat-ics. However, it is natural to connect these two sciences and having them depend on the sameprinciple. Now, among the different principles that can serve as a basis for the equilibriumand of which we have given a brief exposure in section I, it is clear that there is the principleof virtual velocities that applies naturally to the equilibrium of fluids [145].8 (A.17.4)

The use of the term particle at the beginning of the first passage suggests that La-grange, unlike Euler, admits an atomic structure for fluids. This fact is even clearerin the following considerations, which are located toward the end of the historicalintroduction:

Clairaut’s principle is nothing but a natural consequence of the principle of equality of pres-sure in all directions and we must recognize that this principle contains, in fact, the mostsimple and general properties that the experience allows to discover in the equilibrium offluids. But, in the search for the law of the equilibrium of fluids, is the knowledge of thisproperty essential? Cannot we directly derive this law from the nature of the fluids con-sidered as assembly of molecules loosely joined, independent of each other and perfectlymobile in all directions? [145].9 (A.17.5)

The atomic concept of matter does not prevent Lagrange from treating fluids as ifthey were continuous media and replacing, in the mathematical aspects, the particlesof matter with infinitesimals dm, resulting in expressions where integrals appearinstead of summations, easier to be treated. This approach of adopting a continuousmathematical model and a discrete physical model will be followed by the Frenchscientists of the XIX century.

Taking advantage of Euler’s studies on the concept of strain and pressure in fluids,Lagrange considers both compressible and incompressible fluids. I summarize belowthe case of equilibrium of incompressible fluids listed in Section VIII of part I of theMéchanique analitique, because it is the most simple and sufficiently representative.

7 p. 122.8 pp. 126–127.9 p. 129.


Let X ,Y and Z be the components of the specific forces acting on the elementarymasses dm, constituting the fluid, occupying the position x,y and z. The virtual workof these forces (the sum of the moments in the language of Lagrange) is given bythe integral: ∫

(Xδx+Yδy+Zδz) dm, (17.10)

where δx, δy, δz represent the virtual displacements of dm. Lagrange considers(17.10) as the expression of the virtual work of all forces acting on the fluid andbelieves implicitly zero the contribution of forces within the fluid, i.e. the pressure.For the justification of this viewpoint see the considerations in Chapter 10 on theapproach of the internal forces of the moon treated by Lagrange as a rigid body.

If the fluid particles were free to move, that is not constrained one to another, thebalance would be provided by the annulment of the integral (17.10) for each virtualvariation δx,δy,δz. In fact, since as assumed, the fluid is incompressible, its parti-cles are subject to the constraint of incompressibility and the virtual displacementsδx,δy,δz must meet this constraint, which can be represented by the relations:

L = dxdydz = const.; δL = δ(dxdydz) = 0, (17.11)

dxdydz being associated with the volume of the element dm.Following the theory of multipliers Lagrange can derive the following equation

of equilibrium: ∫(Xδx+Yδy+Zδz)) dm+λ

∫Ldm = 0, (17.12)

where λ is a Lagrange multiplier and where now δx,δy,δz can vary arbitrarily.From now on, for Lagrange it is only a matter of applications of his calculus of

variation which leads to recognition of the identity:

δ(dxdydz) = dxdydz

(∂δx∂x

+∂δy∂y

+∂δz∂ z

). (17.13)

He achieves this result both directly, by developing consistently the variation ofdxdydz, or with the use of kinematic relations already obtained by Euler [104],10

that in an elementary parallelepiped with sides dx,dy,dz the variation of their sizeis expressed by the relation, correct up to infinitesimals of second order:

dx

(1+

∂δx∂x

), dy

(1+

∂δy∂y

), dz

(1+

∂δz∂ z

), (17.14)

that with a modern language, using the axial components εx,εy,εz of the tensor ofdeformation, and making reference to Fig. 17.1, can be rewritten as:

dx(1+εx) , dy(1+εy) , dz(1+εz) . (17.15)

10 p. 286.


(1 +εz)dz

(1 +εy)dy

(1 +εx)dx

dy

dz

dx

x

y

z

P

Fig. 17.1. Deformation of an infinitesimal parallelepiped

Lagrange finally reaches the equilibrium equations of incompressible fluids, alreadyobtained by Euler [103]11 and acknowledges that the multiplier λ represents thescalar value of the pressure in the individual points of coordinates x,y and z:

dλdx

= ΔX ,dλdy

= ΔY,dλdz

= ΔZ, (17.16)

where Δ is the density of the fluid.After Lagrange, the application of the laws of virtual work to continuous bod-

ies is still considering them as aggregates of molecules, and then as if it were a setof material points. For the mathematical aspects use was made of both the discretemodel, in which the total virtual work is represented by summations, and the contin-uous model, in which it is represented by integrals. Applications are not limited tostatic problems, but also to dynamic ones, with the use of the living force concept.Of some interest are the considerations on the phenomena of shock. A brief reviewof the memoirs where continuous bodies are studied as aggregates of molecules canbe found in the works of Clebsch with commentary by Saint Venant [72],12 Cauchyin 1829 [67] and Coriolis in 1835 [79].

11 p. 23012 pp. 577–582.


17.1.2 Navier’s equations of motion

The most interesting work of the early XIX century, whichconcerns the application of virtual work laws to continu-ous mechanics, is however Navier’s Mémoire sur la lois del’équilibre et du mouvements des corps solides élastiques of1821, published in 1827 [173].

Claude Louis Navier was born in Dijon in 1785 and diedin Paris in 1836. In 1802 he enrolled at the École polytech-nique and in 1804 continued his studies at the École nationaledes ponts et chaussées, from which he graduated in 1806. He

directed the construction of bridges at Choisy, Asnières and Argenteuil in the De-partment of the Seine, and built a footbridge to the Île de la Cité in Paris. In 1824,Navier was admitted into the Académie des sciences. In 1831 he became Chevalierof the Legion of Honor and succeeded Cauchy as professor of calculus and me-chanics at the École polytechnique [290]. Navier formulated the general theory ofelasticity in a mathematically usable form. His major contribution however remainsthe Navier-Stokes equations (1822), central to fluid mechanics.

In the second part of Mémoire sur la lois de l’équilibre et du mouvements descorps solides élastiques, Navier, who is inspired explicitly by Lagrange’sMécaniqueanalytique, writes the equation of virtual work of internal and external force – theequation of moments in Navier’s terminology – acting on an elastic body thoughtof as an aggregate of particles, or molecules, that attract or repel with an elasticforce varying linearly with their relative displacements and obtains the equations ofequilibrium.

To determine the virtual work of the internal forces Navier focuses his attentionfirst on a material point M. The virtual work made on M is that of the internal forceexerted onM by all pointsM′ of the continuum in a general virtual motion. As virtualmotion Navier considers the variation δ f of the relative displacement f of M andM′ from the virgin state. Since the constitutive law is assumed linear elastic, theforce exerted on M by M′ is given by ε f , where ε is a constant depending on themechanical properties of a continuum, and its moment by ε fδ f , or as Navier notes,1/2εδ f 2, that with a modern language represents the variation of elastic potentialenergy.

If the expression of 1/2εδ f 2 is integrated by varyingM′ on all points of the elasticcontinuum, the total virtual work ofM is obtained. Expressing the relative displace-ment f by means of the absolute displacement (x,y,z) of the point of the continuumfrom the virgin state (a,b,c), after some mathematics, the following expression re-sults:

12εδ

[3dx2

da2+

(dxdb

+dyda

)2

+2dxda

dydb

+(dxdc

+dzda

)2

+2dxda

dzdc

+

3dy2

db2+

(dydc

+dzdb

)2

+2dydb

dzdc

+(dxdc

+dzda

)2

+3dz2

dc2

].

(17.17)


Developing the variation δ, adding the virtual work of the volume forces of com-ponents X ,Y,Z and surface forces of components X ′,Y ′,Z′, and integrating on thewhole continuum, the following global equation of virtual work is obtained:

0 = ε∫ ∫ ∫

dadbdc⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

3dxdaδdxda

+dxdbδdxdb

+dxdbδdyda

+dydaδdxdb

+dydaδdyda

+dxdaδdydb

+dydbδdxda

+

dxdaδdxdc

+dxdcδdzda

+dzdaδdxdc

+3dydbδdydb

+dzdaδdzda

+dxdaδdzdc

+dzdcδdxda

+

dydcδdydc

+dydcδdzdb

+dzdbδdydc

+dzdbδdzdb

+dydbδdzdc

+dxdcδdydb

+3dzdcδdzdc−

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭−∫ ∫ ∫

dadbdc(Xδx+Yδy+Zδz)− ∫ ∫ds(X ′δx′+Y ′δy′+Z′δz′) .

(17.18)

Here the triple integrals are extended on the volume and the double to the surface sof the elastic body.

Navier, starting from the variational equation (17.18), uses an approach that willbe followed by many scholars. Performing an integration by parts – which for uscorresponds to the application of Green’s formula – and considering arbitrary virtualdisplacements, obtains the indefinite equilibrium equations (for internal points of thecontinuum) and boundary equations. By way of example I only offer the indefiniteequilibrium equations:

−X = ε(3d2xda2

+d2xdb2

+d2xdc2

+2d2y

da db+2

d2zda dc

)−Y = ε

(d2yda2

+3d2ydb2

+d2ydc2

+2d2x

da db+2

d2zdb dc

)−Z = ε

(d2zda2

+d2zdb2

+3d2zdc2

+2d2xda dc

+2d2ydb dc

).

(17.19)

17.2 Applications in the theory of elasticity 383

17.2 Applications in the theory of elasticity

17.2.1 Alfred Clebsch

Perhaps the first organic appearance of a virtual work lawfor three-dimensional continua, in particular linear elastic,which uses the concept of tension developed by Cauchyin his works of the period 1823–1828 [267], is the Theo-rie der Elastizität fester Körper by Alfred Clebsch in 1862[71, 386, 387].

Rudolf Friedrich Alfred Clebsch was born in Königsbergin 1833 and died in Göttingen in 1872. He was a Germanmathematician who made important contributions to alge-

braic geometry and invariant theory. He attended the university of Königsberg andwas habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe. In 1866,he and Paul Gordan published a book on the theory of Abelian functions. On thewhole, to a modern reader a century later, the book may seem old fashioned; but itmust be remembered that it appeared long before Weierstrass’ more elegant lectureson the same subject. His cited book on elasticity may be regarded as marking the endof a period. In it he treated and extended problems of elastic vibrations of rods andplates. His interests concerned more the mathematical than the experimental side ofthe physical problems [290].

In 1883 Adhémar Jean Claude Barré de Saint-Venant (1797–1886), more thaneighty years old, translated Clebsch’s work on elasticity into French and publishedit as Théorie de l’élasticité des corps solides [72] making accessible Clebsch’s con-tribution to European scholars. Because of the importance of this book for the appli-cation of virtual work laws to continua I translate the entire § 16:

Evaluation of work for a small deformation of a body. Relation deduced among the thirty-sixfactors that serve to define the behavior of a crystalline substance, or all non-isotropic solids.Imagine that a body subject to the action of any forces, undergoes a sequence of changes ofshape so that the coordinates (x,y,z) of its points, become x+u, y+v, z+w parallel to x,y,z.The work produced throughout the body for this displacement is obtained by multiplyinga [generic] element of its volume dxdydz for the components in the directions x,y,z of theforces acting to drive that volume respectively for small paths δu,δv,δz and then adding thethree products and integrating their sum for the whole extension or for all the elements ofthe body. Now, the three components of forces, agent in the unit volume element dxdydzare only the second member of equation (5) of § 14, page 54, i.e.:

∂ txx∂x

+∂ txy∂y

+∂ txz∂ z

+X

in x direction and two similar quadrinomials in the directions y and z. The elementary workproduct, when the points run through the spaces the projections of which on the axes x,y,zare δu,δv,δw, is therefore expressed as:

δW = δU +δV


where:δU =

∫ ∫ ∫(Xδu+Yδv+Zδw)dxdydz

represents the work of external forces acting within the body and:

δV =∫ ∫ ∫

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

δu

(∂ txx∂x

+∂ txy∂y

+∂ txz∂ z

)+δv

(∂ tyx∂x

+∂ tyy∂y

+∂ tyz∂ z

)+δw

(∂ tzx∂x

+∂ tzy∂y

+∂ tzyz∂ z

)

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭dxdydz

represents the work that derives from the reciprocal actions of its particles, either from trac-tion or compression forces acting on its surface.Let us now consider one of the nine terms of the latter triple integral, for example:∫ ∫ ∫

δu∂ txx∂x

dxdydz.

By integrating partially with respect to x, i.e. for the small portion of the body containedin a channel [a cylinder] which is considered infinitely thin, the section dydz of which isconsidered as constant, as well as the coordinates y and z. If the second term

∂δu∂x

is replaced with δ∂u∂x

which is the same, this integration by part gives the expression:∫ ∫[txxδu]dydz−

∫ ∫ ∫txxδ

∂u∂x

.

The square brackets mean that, instead of the expression txxδu they contain, one must put thedifference of this expression at the opposite ends [the two terminal surfaces] of the channelin question.Denote now with dσ,dσ′ the elements that the channel cuts on the [external] surface of thebody and with p,q,r, the angles that the normal to dσ, directed toward the outside of thebody, makes with the coordinate axes x,y,z, and finally with p′,q′,r′, the same angles of thenormal to dσ′. If dσ is the front end of the channel, namely the one located on the positiveside of x, and dσ′ is its rear end, cos p is necessarily positive and cos p negative, so it is:

dydz = dσcos p =−dσ′ cos p′.

The difference of the limit values of txxδudydz therefore becomes the sum of the values thatthe expression txxδudσcos p takes at the end of the channel. Instead of extending the doubleintegral above to the ends of all channels that can be carried out in a similar way within thebody, it is possible to integrate over the set of elements dσ, which include the elements dσ′.Then: ∫ ∫

[txxδu]dydz−∫

txxδudσcos p.

[…]So in all cases, the considered term δV will be replaced by:∫

txxδudσcos p−∫ ∫ ∫

txxδ∂u∂x

dxdydz

where the first term is extended to the whole surface of the body. If it is made the same forall the terms δV , it is obtained:

δV = δU1−δU2

17.2 Applications in the theory of elasticity 385

where δU1 represents all the simple integrals [double integral] and δU2 all the triplicateintegrals, like that of the binomial expression of txx we have written. And it is:

δU1 =∫

(txx cos p+ txy cosq+ txz cosr)δudσ

+∫

(tyx cos p+ tyy cosq+ tyz cosr)δudσ

+∫

(tzx cos p+ tzy cosq+ tzz cosr)δudσ.

Now the expressions in brackets are those that because of the equations (25) are equivalent tothe component T cosω, T cosκ, T cosρ, of the tensile forces T applied on the body surface[as shown in another paragraph]; δU1 is then the work of the external tensile forces and itis:

δU1 =∫T (cosωδu+ cosκδv+ cosρδw)dσ.

Similarly it will be found that the eight terms of δU2 other than that containing:

txx∂u∂x

and which we wrote, are affected, under the integral, by other differential quotients:

∂u∂x

,∂u∂y

,∂v∂x

, . . .

of the displacements u,v,w and by the other components tyy, . . . of the tensions. Hence,introducing the elementary deformations13 in place of:

∂u∂x

,∂u∂y

, . . . ,∂u∂y

+∂v∂x

after the expressions (28) it is:

δU2 =∫

(txxδ∂x + tyyδ∂y + tzzδ∂z + tyzδgyz + tzxδgzx + tyxδgyx)dxdydz.

For the total work is is then:δW = δU +δU1 +δU2

where δU and δU1 represent the work of external forces acting respectively on the pointson the body surface. As a result δU2 is necessarily the work of the internal forces resultingfrom the molecular actions [72].14 (A.17.6)

At this point Clebsch abandons the exposition of the virtual work law because hisobjective was not in fact to obtain the equations of equilibrium which could be ob-tained trivially by requiring the vanishing of the forces acting on the element ofvolume, but rather to derive the expression of the work of internal forces and thenmove to the expression of the elastic potential upon which to make reasonings re-garding the constitutive relationships. The virtual work principle is used as a bridgeto link the ‘physical’ approach of the constitutive relationship, based on the mechan-ical intuition of tension and strain, to the ‘purely’ mathematical approach developedby Green in his work of 1839 on the law of reflection and refraction of light, in

13 The symbolism of Clebsch-Saint-Venant is still used with regard to the components of a stresstensor; it was however abandoned with respect to the components of a deformation tensor.14 pp. 57–64.


which to the internal forces are imposed the only restriction to be conservative andtherefore with a potential [278].

The treatment of Clebsch is quite modern, although now it is preferred to use theformalism of the vector and tensor calculus, the concepts of gradient and divergence,which makes the discussion much more compact. With a greater satisfaction for amodern reader, Clebsch’s exposition could have been concluded by showing explic-itly that equating to zero the total work δW = δU+δU1+δU2, under the condition ofequilibrium of stresses and congruence of deformations, the expression is obtainedimmediately:∫ ∫ ∫

(Xδu+Yδv+Zδw)dxdydz+∫

T (cosωδu+ cosκδv+ cosρδw)dσ

=∫

(txxδ∂x + tyyδ∂y + tzzδ∂z + tyzδgyz + tzxδgzx + tyxδgyx)dxdydz. (17.20)

One characteristic perhaps surprises in Clebsch’s treatment. His mathematicalpassages would have been much easier if he had directly used the formulas ofGreen. They were obtained more or less simultaneously by George Green (1783–1841) who published them at his own expense in 1828 [128] and Michel Ostro-gradsky who presented them at the Academy of sciences in St. Petersburg, also in1828 [177]. Then Clebsch surely knew them. The only justification for their non-use is perhaps the fact that they were not yet well known and Clebsch could assumethat his text would have been more understandable if he had not made reference tothem.

It is appropriate to quote a comment by Saint Venant on an alternative possibilityfor the expression of virtual work of the internal forces:

In 1858, Mr. Kirchhoff has been kind enough to indicate to me a simple and direct way toaccount for the sextinôme composition given above about the internal or molecular workU2for the unit of volume of an element. Let dx,dy,dz be the three sides, parallel to x,y,z ofthis rectangular element: 10, if the dilatation ∂x already suffered in the x direction is to beincreased, the two equal and opposed faces yz will stretch of ∂xdx, the normal componentsof the tension exerted by the matter surrounding these faces produce the work yztxxxδ∂x,this is basically the work txxδ∂x per unit of volume. 20 if, one of the two sides is assumed asfixed, the distortion gxy is increased by δgxy, and there is as a displacement xδgxy of on oneside over the other, so that the shear stress txy, acting on the unit of surface in the direction yof this distortion, produces a work txyxδgxy. There are then two other faces on which it actsa tension tyx equal to txy; i.e. the faces xz. In this motion, they rotate around the two sidesin the z direction of the two faces yz that remained motionless. But the tensions tyx act inthe sense x and not in the sense y which is that of the motion and have nothing to add tothe work yztxyxδgxy of the stress txy, work that is only txygxy, for unit of volume. Now thework of the six tensions over the faces of the element, so that there is equilibrium after thissmall displacement, has to be equal to the molecular work within the element. So this workper unit volume has its own value the expression of the sextinôme (txxδ∂x + · · ·+ txyδgxy)defining δU2 [72].15 (A.17.7)

In Appendix V of Navier-Saint Venant’s text [174]16 there is a strange reversal ofexposure. In the body of the text, the expression of the work of internal forces is

15 p. 61.16 pp. 712–715.

17.3 The Italian school 387

determined exactly in the way Saint Venant attributes it to Kirchhoff. Then, in afootnote, Navier-Saint Venant refers to an alternative way, attributed to Lamé,17

exactly what Clebsch shows in the body of the text to find the expression of thework of internal forces. It is worth noting that Kirchhoff’s approach, to calculate thevirtual work of internal forces, which now seems completely natural, did not appearthat way to the founders of the theory of elasticity. There was still some uncertaintyon the ontological status to be attributed to the forces in general and in particular totensions. Today, with the prevalence, especially in the areas of application, of theinstrumentalist spirit of theories, there are no scruples and symbols that representmechanical quantities are handled with greater indifference.

In addition, the purely formal approach for the expression of the work of theinternal forces must have appeared more strict than the heuristic one provided byKirchhoff. Without thinking, however, that the expression of the forces acting onthe element of volume, or alternatively, the internal equilibrium equations, were ob-tained currently, at least in those days, with a heuristic approach, based on infinites-imals, no more stringent than that used by Kirchhoff to obtain an explicit equationof the virtual work of internal forces.

17.3 The Italian school

In Italy, at the beginning of the XVIII century the influence of Lagrange was rele-vant. To many Italian mathematicians and mechanicians, modernity was representedby Lagrange. This was partly because Lagrange, even after leaving Turin in 1766,had remained in contact with the Italian world of science, and partly because Italiansconsidered him Italian and this was a period of rising nationalistic feelings. VincenzoBrunacci (1768–1818), professor of ‘Matematica sublime’ (Calculus) in Pavia, wasone of themain supporters of Lagrange’s ideas. Alongwith the fashionable purism ofthe time, he accepted Lagrange’s reduction of the differential calculus to algebraicprocedures [146] and rejected the XVIII century concept of infinitesimal in bothcalculus and mechanics [53]. Brunacci transmitted these ideas to his pupils, includ-ing Ottaviano Fabrizio Mossotti (1791–1863), Antonio Bordoni (1788–1860) andGabrio Piola (1794–1850), the brightest Italian mathematicians of the first half ofthe XVIII century. From them the modern Italian schools of mathematics originate.For example Enrico Betti and Eugenio Beltrami, the highest rank Italian mathemati-cian of the second half of the XIX century were students, respectively, of Mossottiand Bordoni.

17 These are the developments in the Clapeyron theorem contained in [154], pp. 80–83.


17.3.1 Gabrio Piola

Count Gabrio Piola Daverio was born in Milan in 1794 in arich and aristocratic family and died in Giussano della Bri-anza in 1850. He studied mathematics at Pavia university as apupil of Vincenzo Brunacci [313]. Though Piola was one ofthe most brilliant mathematical physicists of the XIX cen-tury he has been for a long time neglected. His revival isdue mainly to Clifford Truesdell, a great estimator of Italianscholars, andWalter Noll, whowas well aware of the interac-tion between the Italian and German schools in the last year

of the 1800s.Piola’s work on continuum mechanics concerned fluids and solids. These last

were published in various years [187, 188, 189, 190, 191], with La meccanica de’corpi naturalmente estesi trattata col calcolo delle variazioni of 1833 [188], probablythe most relevant one.

Lagrange in his Mécanique analytique had applied the principle of virtual work,in conjunction with the calculus of variations, to the study of the internal forces toone-dimensional elastic continua and fluids. Piola generalizes the approach to three-dimensional elastic continua. In his papers, Piola questions the need to introduceuncertain hypotheses on the constitution of matter by adopting a model of corpus-cles and forces among them, as the French mechanicians did. Piola states that it issufficient to refer to evident and certain phenomena: for instance, in rigid bodies, theshape of the body remains unaltered. Then, one may use the ‘undisputed’ equationof balance of virtual work; only after one has found a model and equations basedexclusively on phenomena, Piola says, is it reasonable to look for deeper analyses:

Here is the great benefit of Analytical Mechanics. It allows us to put the facts about whichwe have clear ideas into equations, without forcing us to consider unclear ideas […]. Theaction of active or passive forces (according to a well known distinction by Lagrange) issuch that we can sometimes have some ideas about them; but more often there remain […]all doubts that the course of nature is different […]. But in the Analytical Mechanics theeffects of internal forces are contemplated, not the forces themselves; namely, the constraintequations which must be satisfied […] and in this way, bypassed all difficulties about theaction of forces, we have the same certain and exact equations as if those would result fromthe thorough knowledge of these actions [188].18 (A.17.8)

The value of the internal forces, whose legality of use Piola does not question, isdetermined bymeans of Lagrangemultipliers of appropriate conditions of constraint.

In La meccanica de’ corpi naturalmente estesi trattata col calcolo delle vari-azioni [188], Piola is inspired by Lagrange but follows an inverse path with respectto him; he takes for granted the global equation of the rigid body (17.8) which forLagrange was the terminal point; he derives them opportunely and obtains six dif-ferential equations which characterise locally the rigidity constraint.

The material points of a body are labelled by two sets of Cartesian coordinates.The first refers to axes called a,b,c, as made by Lagrange in the Mécanique analy-

18 pp. 203–204.


tique [149],19 rigidly attached to the body – reference configuration – and the secondto axes called x,y,z, fixed in the ambient space and to which the motion of the bodyis referred – current configuration. It was not difficult for Piola to prove the validityof relations: (

dxda

)2

+(dxdb

)2

+(dxdc

)2

=(dyda

)2

+(dydb

)2

+(dydc

)2

=(dzda

)2

+(

dzdb

)2

+(dzdc

)2

= 1,(dxda

)(dyda

)+

(dxdb

)(dydb

)+

(dxdc

)(dydc

)=

(dxda

)(dzda

)+

(dxdb

)(dzdb

)+

(dxdc

)(dzdc

)=

(dyda

)(dzda

)+

(dydb

)(dzdb

)+

(dydc

)(dzdc

)= 0.

(17.21)

To write down the balance equation Piola uses the technique developed by Lagrangein theMécaniqueanalytique, by equating to zero the virtual work of volume (density)forces, inertia forces included, integrated over the body volume:∫

da∫

db∫

dc ΓH[(

d2xdt2

−X

)δx+

(d2ydt2

−Y

)δy+

(d2zdt2

−Z

)δz

]= 0,

(17.22)where Γ is the mass density, H the Jacobian of the transformation from (a,b,c) to(x,y,z) and (δx,δy,δz), the virtual displacement of a material point of the body.

At this point Piola reminds the reader that the virtual displacements (δx,δy,δz) arenot free but they are constrained according to relations (17.21). To free (δx,δy,δz)from any constraints the Lagrange multiplier method can be used, by adding to theintegral on the left side of the variational equation (17.22) the integral of constraintrelations (17.21) each multiplied by appropriate Lagrangian multipliers (A, B, C, D,E, F) that, by reproducing the original Piola’s text, with S the symbold of integrals,gives:

SdaSdbSdc ·A{(

dxda

)(dδxda

)+

(dxdb

)(dδxdb

)+

(dxdc

)(dδxdc

)}

SdaSdbSdc ·B{(

dyda

)(dδyda

)+

(dydb

)(dδydb

)+

(dydc

)(dδydc

)}

SdaSdbSdc ·C{(

dzda

)(dδzda

)+

(dzdb

)(dδzdb

)+

(dzdc

)(dδzdc

)}

19 Section XI, art. 4.


SdaSdbSdc ·F{(

dxda

)(dδyda

)+

(dxdb

)(dδydb

)+

(dxdc

)(dδydc

)}+

{(dyda

)(dδxda

)+

(dydb

)(dδxdb

)+

(dydc

)(dδxdc

)}SdaSdbSdc ·E

{(dxda

)(dδzda

)+

(dxdb

)(dδzdb

)+

(dxdc

)(dδzdc

)}+

{(dzda

)(dδxda

)+

(dzdb

)(dδxdb

)+

(dzdc

)(dδxdc

)}SdaSdbSdc ·D

{(dyda

)(dδzda

)+

(dydb

)(dδzdb

)+

(dydc

)(dδzdc

)}+

{(dzda

)(dδyda

)+

(dzdb

)(dδydb

)+

(dzdc

)(dδydc

)}.

After lengthy calculations Piola arrives at the following balance equations in thereference configuration (x,y,z):

Γ[X−

(d2xdt2

)]+

dAdx

+dFdy

+dEdz

= 0

Γ[Y −

(d2ydt2

)]+

dFdx

+dBdy

+dDdz

= 0

Γ[Z−

(d2zdt2

)]+

dEdx

+dDdy

+dCdz

= 0, (17.23)

which compared with results by Cauchy [65, 267] and Poisson [199], gives a me-chanical meaning to the Lagrangian multipliers (A, B, C, D, E, F): they are the stresscomponents in an assigned coordinate system in the reference configuration.

Pierre Duhem in his course ofHydrodinamique, elasticity, acoustique of 1890–91[98], used a virtual work law to obtain the equilibrium equations in a way close tothat used by Piola, without any reference to him [268].

17.3.2 Eugenio Beltrami

Eugenio Beltrami was born in Cremona in 1836 and died inRome in 1900. He studied at the university of Pavia from1853 to 1856 where his teacher was Francesco Brioschi. In1863 he was offered the chair of geodesy at the university ofPisa by Enrico Betti. In Pisa hemet Bernhard Riemann. From1891 he was in Rome for the latest teaching [294]. Beltramidifferentials techniques influenced the birth of the tensor cal-culus, providing a basis for the ideas later developed by Gre-gorio Ricci-Curbastro and Tullio Levi-Civita. Some works

concern the mechanical interpretation of Maxwell’s equations. Beltrami’s contribu-tion to the history of mathematics is also important; in 1889 he brought to light the


work of Girolamo Saccheri [212] on parallel lines, compared his results with thoseof Borelli, Wallis, Clavius, Bolyai and Lobachevsky and gave himself an importantcontribution to non-Euclidean geometry [23].

Beltrami followed fairly closely the approach of Piola, making it more general.For Beltrami the equation of virtual work was a relation between dual variables,forces and deformations, taking mechanical meaning leaning to one another in forc-ing the cancellation of the virtual work. In his first organic paper on the theory ofelasticity Sulle equazioni generali dell’elasticità in 1880–1882 [24], Beltrami stud-ies the equations of elastic equilibrium in a space with constant curvature where abody with volume S and surface σ is present.

Beltrami’s work stems from the results obtained by Laméwith curvilinear coordi-nates and from some subsequent works by Carl Neumann and Borchardt. The lattersimplified Lamé’s calculations with the use of a potential function in curvilinear co-ordinates. According to Beltrami their approach, though it led to correct results, canbe improved. Lamé, Neumann and Borchardt formulated the problem in Cartesiancoordinates, implicitly assuming the Euclidian space. He instead proves directly theelastic equations of equilibrium without any assumption on the nature of space.

Beltrami has a more mature feeling with internal forces than Piola, or rather heis more ‘relaxed’ than him. He is not afraid to make explicit reference to them andassumes a position similar to that of Lagrange. The internal forces (tension) and thecomponents of the constraint conditions (deformation) are treated as dual variables,implicitly defined by the fact that their product is a virtual work, regardless of themetric adopted. His central idea lies in a suitable metrics and from it of suitable in-finitesimal strain measures; with reference to the infinitesimal element ds he writes:

ds2 = Q21dq

21 +ds2 +Q2

2dq22 +ds2 +Q2

2dq22, (17.24)

where q1,q2,q3 are curvilinear coordinates and Q1,Q2,Q3 are functions of q1,q2,q3(notice, the metrics will be Euclidian for Q1 = Q2 = Q3 = 1). Beltrami consid-ers six auxiliary quantities θ1,θ2,θ3,ω1,ω2,ω3, which are somehow related toq1,q2,q3,Q1,Q2,Q3 and allows him to write the equation [24]:20

δdsds

= λ21dθ1 +λ22dθ2 +λ22dθ3 +λ2λ3dω21 +λ1λ3dω2

2 +λ1λ2dω23. (17.25)

Here λ1,λ2,λ3 are the cosines of the angles that the linear elements dsmake with thecoordinate axes. Then he introduces the following expression for the virtual work:∫

(Θ1dθ1 +Θ2dθ2 +Θ3dθ3 +Ω1dω1 +Ω2dω2 +Ω3dω3)dS, (17.26)

where Θ1, Θ2, Θ3, Ω1, Ω2, Ω3 are not a priori specified functions of q1, q2, q3.Previous expressions of virtual work allow us to give a mechanical meaning to

the terms θ1,θ2,θ3,ω1,ω2,ω3, Θ1,Θ2,Θ3,Ω1,Ω2,Ω3. They are respectively strain

20 pp. 384–385.


and stress components; this is made clear also from the equilibrium equation derivedfrom the variational problem associated to the previous virtual work expression. Theequations obtained by Beltrami are coincident with those given by Lamé: Beltramisresults are however independent of Euclid’s V postulate.

17.3.3 Enrico Betti

Enrico Betti was born in Pistoia in 1823 and died in Soiana in1892. He studied at Pisa university as a student of OttavianoMossotti. In 1859 Betti was appointed professor in Pisa. Inthe following year Betti, along with Brioschi and Casorati,visited themathematical centres of Europe: Göttingen, Berlinand Paris making many important mathematical contacts. Inparticular in Göttingen Betti met and became friendly withGeorg Friedrich Bernhard Riemann (1826–1866). Back inPisa he moved in 1859 to the chair of analysis and higher

geometry. In 1865 Betti was appointed director of the Scuola normale in Pisa, a rolethat he maintained until his death. Since 1862 Betti was deputy and then senator ofthe Italian parliament [277].

Betti explored many aspects of mathematical physics; one of the most importantwas that regarding classical mechanics. At the beginning he assumed a mechanisticapproach, where force and not energy was the founding concept and the virtual workthe regulating law. In a first work on capillarity [42], Betti assumed bodies as formedbymolecules which are attracted to each other at short distance, repelled at very shortdistance, and which do not practically interact at larger, but still very short distances.In his memoirs on Newtonian forces Betti [41] declared his Newtonian ideology.Indeed he introduced a potential function, but only on mathematical grounds, as afunction from which forces can be obtained by derivation. Betti changed his attitudein a second memoir on capillarity [43], by giving the potential an energetic meaning.This change was once and for all in the 1874 Teoria della elasticità [44], where noreference is made to internal forces, even managing to avoid the explicit mention ofstress.

When Betti wrote the Teoria della elasticità, the theory of elasticity was alreadymature with known principles, though not completely shared. The exposition devel-ops then as in modern handbooks, following the axiomatic approach. Betti’s princi-ples are on one hand the concepts of potential energy and strains, on the other handthe principle of virtual work.

Betti cites the work ofWilliam Thomson (1824–1907) to give a thermodynamicalbasis to potential of elastic forces [226]. Thermodynamics however, at the time,concerned only homogeneous thermal processes, while in the continuum mechanicsheterogeneous processes are prevalent. To overcome this difficulty, after Thomson,Betti divides the continuum S into infinitesimal elements dS, each of them consideredas homogeneous. The whole potential energy is expressed as a summation of all theinfinitesimals. So if P is the density of elastic potential energy, the whole potential


energy for S is given by:

Φ =∫

PdS. (17.27)

ThenBetti, like Thomson andGreen, assumes thatP depends on infinitesimal strains,components of which he neglects any power higher than the second, by obtainingthe following quadratic expression:

P = ∑∑Arsaras, (17.28)

where Ars are constitutive constants and ar,as are the generic components of strain.In the derivation of the equations of equilibrium, as indeed throughout the book,

stresses are not introduced in any way. Betti writes down the equilibrium equationby means of the virtual work principle:

Let X ,Y,Z be the components of the accelerating forces [forces per unit mass] that act oneach point of the body, L,M,N the components of the forces acting on each point on thesurface of it, and ρ the constant density. Let any point of the body take a virtual motion anddenote by δu,δv,δw the changes that will take for his motion u,v,w. The work made in thismotion by the given forces will obviously be:∫

S(Xδu+Yδv+Zδw)dS+

∫σ(Lδu+Mδv+Mδw)dσ

being S the space occupied by the body and σ the surface. The work made by the elasticforces will be equal to the increase of the potential of the whole body:

Φ =∫

PdS

then, for the principle of Lagrange:

δΦ+∫S(Xδu+Yδv+Zδw)dS+

∫σ(Lδu+Mδv+Mδw)dσ= 0

[44].21 (A.17.9)

He then develops the variation δΦ to which he applies Green’s formula, without anycomment or reference, to obtain an expression of the work of internal forces wherestrains are replaced by δu,δv,δw. He obtains in this way three internal equilibriumequations and three more boundary equations. For the sake of compactness I willwrite down only one of the first kind and another of the second:

ρX =ddx

dPda

+ddy

dP2dh

+ddz

dP2dg

L =dPdaα+

dP2dhβ+

dP2dgγ.

(17.29)

Here x,y,z are the Cartesian coordinates of a generic point and a,h and g are thecomponents of strain [44].22 Notice that these equations, as in Navier, depend onstrains – as P is a function of strains – instead of on stresses as usual.

21 pp. 20–21.22 p. 19.

18

Thermodynamical approach

Abstract. This chapter is devoted to the use of the VWL where to virtual work isgiven amechanical meaning. In the first part, notes onmechanics derived fromPierreDuhem’s thermodynamics are presented. In the second part the derivation of a VWLfrom the law of conservation of energy is shown.

In the second half of the XIX century, mechanics no longer seemed to be theparadigm for all of physics. Not so much because of internal reasons due for ex-ample to the presence of contradictions and vagueness, but for external reasons as toits difficulty in explaining the new phenomena that were the subject of the nascentphysical disciplines such as thermodynamics and electromagnetism. In particular,the idea of force as a fundamental quantity of physics found itself in difficulty.For many scientists it had to be replaced, or at least allied with energy. And themechanistic explanatory model had to be replaced by a less challenging one at themetaphysical level, on the example of thermodynamics. Among the promoters ofthis tendency to be remembered are William John Macquorn Rankine (1820–1872)[204] Ernst Mach, Hermann Ludwig Ferdinand von Helmholtz (1821–1894), PierreMaurice Marie Duhem (1858–1916), Wilhelm Ostwald (1853–1932) [371]. In par-ticular, the latter author was a supporter of the energetics that took the form of aphilosophical movement in which energy was seen as a substance.

In this chapter there is no claim to comment on philosophical and scientific mat-ters but only to see how the concepts of work and energy of thermodynamics canprovide a new foundation of the laws of mechanics, in particular the laws of virtualwork. To do this I will refer here only to the contribution of Duhem, beginning witha summary of his ideas about mechanics:

The attempt that aims to reduce all Physics to rational Mechanics, which was always a futileattempt in the past, is it intended to pass a day? A prophet alone could answer affirmatively ornegatively to this question. Without prejudging the direction of this response, it seems wiserto abandon, at least provisorily, these fruitless efforts toward the mechanical explanation ofthe Universe.


396 18 Thermodynamical approach

We will try to formulate general laws for bodies to which all physical properties must obey,without assuming a priori that these properties are all reducible to geometry and local move-ment. The body of this general laws no longer will reduce to rational Mechanics.[…]Rational Mechanics must therefore result from the body of general laws that we propose tohold; it must be what one gets when applying these general laws to particular systems whereone considers only the figure of bodies and their local motion.The code of general laws of Physics is now known by two names: the name of Thermody-namics and the name of Energetics [99].1 (A.18.1)

18.1 Pierre Duhem’s concept of oeuvre

Pierre Maurice Marie Duhem was born in Paris in 1861and died in Cabrespine in 1916. He was one of the great-est scientists, philosophers and historians of his period.Duhem had the misfortune of being an enemy of thechemist Marcellin Berthelot (1827–1904) who becameminister of public instruction, precluding him from abrilliant academic career. As a physicist, he champi-oned ‘energetics’, holding generalized thermodynamicsas foundational for physical theory, that is, thinking thatall of chemistry and physics, including mechanics, elec-

tricity, and magnetism, should be derivable from first principles of thermodynamics.As a historian he was a supporter of the continuous development of science; in par-ticular he gave great importance to medieval science. As a philosopher of sciencehe argued that a scientific theory does not require a justification that goes beyondthe control of its internal consistency and accuracy in the prediction of experimentalresults. Duhem’s interests fell roughly into periods. Thermodynamics and electro-magnetism predominated between 1884 and 1900, although he returned to them in1913–1916. He concentrated on hydrodynamics from 1900 to 1906. His interest inthe philosophy of science was mostly in the period 1892–1906, and in the historyof science from 1904 to 1916, although his earliest historical papers date from 1895[290].

Duhem did not directly place a virtual work law as a principle of mechanics butrather as a theorem derived from the principle of conservation of energy rooted in ageneral physics called by him either thermodynamics or energetics. The term ther-modynamics refers directly to the history of mechanical practice. Its two most basicprinciples, that of Sadi Carnot – to transform heat into work, there must be bodiesat different temperatures – and the principle of conservation of energy, were dis-covered by studying the power of steam. The name of energetics is due to Rankine,energy being the first quantity to be defined, on which most other notions are based.

Below I will present Duhem’s ideas as reported in the Traité d’énergétique ou dethermodynamique générale [99]. More precisely, for simplicity, I only will refer to

1 pp. 2–3.

18.1 Pierre Duhem’s concept of oeuvre 397

the first volume devoted to non-dissipative systems. The text of Duhem representsone of the first attempts to establish a physical theory on a modern axiomatic basis.The author believes that the principles of a physical theory do not require any justi-fication beyond the assessment of the internal consistency. He considers, however,that the principles cannot be chosen at random, but that to arrive at a satisfactorytheory one must benefit from formulations of past similar principles; in this way thehistory of science becomes an integral part of science.

The most important and original concepts of Duhem in his energetic approachto mechanics, after those of space and time, are that of virtual transformation andactivity (oeuvre). Below in commenting on the introduction of these concepts I willtake for granted some basic notions of thermodynamics including that of state.

18.1.1 Virtual transformations

Virtual transformations are defined as operations performed completely in the mind,which submit to a mathematical scheme that serves to represent the system underexamination, imagining a continuous succession of states. The only requirement thetransformations should satisfy is the respect of constraints affecting the essence ofthe system, while experimental laws may be violated. They occur in a hyperuranictime, or even better without time.

Consider a continuous sequence of states of the same system, we fix the attention on thesedifferent states in the order that allows us to switch between them continuously. To identifythis intellectual operation to whichwe submit all themathematical schemes used to representthe set of concrete bodies, we say we impose on the system a virtual change.[…]Changes in the numerical values of the variables used to define the state of the system mustbe compatible with the conditions that logically result from the definition of the system,but only with these conditions. And in particular, the changes in numerical values may wellcontradict the experimental laws that govern the system of all the concrete bodies that ourabstract mathematical system has the duty to represent [99].2 (A.18.2)

Duhem uses another term to describe changes that meet the conditions of constraintsbut not necessarily the experimental laws and take place in time: ideal transforma-tions:

Do not confuse an ideal variation with virtual variation; virtual variation is composed ofvirtual configurations of the system that do not succeed in time, so that the change of con-figuration which is a virtual variation is not related to motion. In the virtual variation thevariation of velocity has no reason to exist [99].3 (A.18.3)

Regarding velocity Duhem introduces the difference between local velocities, whichare velocities in the standard meaning, i.e. the derivative of space with respect totime, and general velocities, defined as the quantity which, once the state of the sys-tem is known, allows one to evaluate all the local velocities and the derivative with

2 pp. 46–47.3 p.84.


respect to time of all the state variables. For example for the mechanical system con-stituted by a rigid body, the general velocities are furnished by the three componentsof the velocity of the center of gravity and of the three Eulerian angles.

Ideal transformations, when restricted to mechanics, coincide with virtual dis-placements as conceived before Lazare Carnot, virtual transformations with Carnot’sgeometric motion (at least for bilateral constraints). In virtual transformations itmakes no sense to talk about velocity (in the physical sense), in ideal transforma-tion it does. From the above it is evident that ideal transformations are also virtualtransformations (just ignore time); the contrary is not true.

The example Duhem presents for virtual transformations concerns a mixture ofhydrogen, oxygen and water. Suppose that the state of a system is defined by thetemperature θ and a variable x, which expresses the percentage of water vapour.There is an experimental law that at chemical equilibrium x relates to θ; be it x =f (θ); in the system there are no conditions of constituent constraints. In a virtualvariation it is possible to leave from a pair x0,θ0 that satisfies the empirical law andto vary x above or below the value x0 while maintaining θ= θ0. This transformationis purely intellectual and may not happen in practice.

18.1.2 Activity, energy and work

The definition of activity is much more complex. It in fact is not an explicit defini-tion, but rather an implicit one as usual in the modern axiomatic theories. Duhemformulates this ‘definition’:

Thus, when a system is transformed in the presence of external bodies, we admit that theseexternal bodies contribute to the transformation, either by causing it or facilitating it orblocking it, and this contribution we call the activity in the transformation of the system, bythe bodies outside that system [99].4 (A.18.4)

The activity has not necessarily a mechanical nature. It can have any nature, forexample, maybe the administration of electrical current.

However Duhem himself recognizes that this definition is “too obscure, vagueand mostly impregnated with anthropomorphism" [99].5 To eliminate these defectshe declares that the activity should be considered as a scalar physical quantity to berepresented with an appropriate algebraic symbol to perform calculations. He thenmakes a number of stipulations/conventions on how to assign it a numerical value.I quote here the first stipulation only to give an idea of the level of abstraction onwhich Duhem moves.

First convention. The mathematical symbol which should represent the activity, in a real orideal transformation, will be defined any time the nature of the system and of the transfor-mation it undertook is known. It will not change if the place and time of the system and theexternal bodies, where the transformation has occurred, change [99].6 (A.18.5)

4 p. 81.5 p. 81.6 p. 87.


Notice that Duhem’s ‘conventions’ refer to ideal and real changes but not the virtualones, because they only take into account the general velocity.

Although Duhem declares that his conventions should be considered as arbi-trary and therefore do not require any justification, their reading makes it manifestthat in introducing the concept of activity Duhem is generalizing that of mechan-ical work. In particular, he assumes the additivity and the independence of paths:if G1,G2, . . .Gn are the activities carried out on a system after the transformationsM1,M2, . . .Mn and if a unique transformation M is imagined that starts from the ini-tial state to the final due to these transformations, the overall activity, G, is the sumof all activities, G = G1 +G2 + · · ·+Gn, and does not depend on the order in whichthe transformations occur but only on initial and final states.

Note that this rule applied to a mechanical system subject to internal or externalforces, when activity is identified with work, requires the field of forces to be con-servative. Duhem does not give particular emphasis to this fact, or better he does noteven make it explicit.

Once the requirements which an activity must satisfy are defined, Duhem intro-duces the concept of total energy. IfG(e0,μ0,e,μ) is the activity required of a systemto pass from one arbitrarily chosen reference e0 and global velocity μ0 to the genericstate e and global velocity μ, the total energy of the system in the state e and globalvelocity is μ, as in the expression:

E(e,μ) = G(e0,μ0,e,μ). (18.1)

He proceeds by saying that the following relation holds true:

G(e1,μ1;e2,μ2) = E(e2,μ2)−E(e1,μ1); (18.2)

i.e. the activity to go from (e1,μ1) to (e2,μ2) is equal to the difference between thetotal energies.

At this point Duhem can formulate a principle of conservation of energy, whichhe qualifies as a hypothesis:

Principle of conservation of energy. When any system, isolated in the space, undergoes awhatever real variation, the total energy of the system maintain an invariable value [99].7

(A.18.6)

The principle of conservation of energy is not a convention such as all those intro-duced to define the activity; it receives its validity from the experience. The principleof conservation of energy does not apply to all the ideal changes but only to thosethat are also real.

The total energy principle formulated above seems to Duhem too general; he thensubmits it to two restrictions that exclude its validity from important fields of physics,including electrical systems. The first restriction requires that the total energy of thesystem does not change as a result of a simple translation in space [99].8 The second

7 p. 93.8 p. 97.


restriction requires that the total energy is composed of two terms [99]:9

U(e,μ) =U(e)+K(λ). (18.3)

The first termwhich depends only on the state e, is named potential energy or internalenergy; the second which depends on the local velocity λ (understood in the classi-cal sense) of all system components, is named kinetic energy. For systems made ofinfinitely small parts, Duhem attains the traditional representation for kinetic energy:

K =12

∫(u2 + v2 +w2)dm, (18.4)

where u,v,w are the components of velocity of the mass element dm of the system.With these restrictions the previous theorem of the total energy takes the more tra-ditional form:

Restricted form of the principle of conservation of energy. In any real variation of an isolatedsystem the following equality:

U +12

∫Mu2 + v2 +w2 dm = const.

is verified [99].10 (A.18.7)

Consider now two independent mechanical systems Sa and Sb. The system Sa,taken alone in the configuration A, has the potential energy U(A), the system Sbtaken alone in the configuration B, has the potential energyU(B). The two systemsSa and Sb, taken together to form the system S, have the total potential energy U ,different from the sum of potential energies of Sa and Sb. In all generality it can beassumed that:

U =Ua(A)+Ub(B)+Ψab(A,B) (18.5)

where Ψab(A,B) is called mutual potential energy.Consider then a virtual infinitesimal variation δa, generic but limited only to the

system Sa, while the state of the system Sb remains unchanged. There is the followingvariation of the total energy induced by bodies external to S:

δaU = δaUa(A)+δaΨab(A,B). (18.6)

The quantity:

δaL =−δaΨab(A,B) (18.7)

represents the activity accomplished by the system Sa on the system Sb in the virtualinfinitesimal variation δa; to it Duhem refers as the infinitesimal virtual work that Sbmakes on Sa.

The work as defined above is very similar to the activity, because it is furnishedby a variation of the energy. It however represents the variation of a limited part of

9 pp. 97–98.10 p. 113.


energy. To reach the concept of force it is enough to consider that the infinitesimalvirtual work can always be expressed as:

δL =−δΨab(A,B) = ∑ fkδqk (18.8)

where qk are the variables that define the system configuration and fk = ∂Uab/∂qbappropriate coefficients. Duhem refers to these coefficients as the actions of Sa onsystem Sb in order to reserve the term force for when qk have the meaning of dis-placements. In this case work is mechanical work.

Note how Duhem has introduced the concept of mechanical work. This is not aprimitive quantity, but one derived from the activity, which has a primitive character.It is difficult to agree that the concept of activity could be more primitive than thatof work.

18.1.3 Rational mechanics

18.1.3.1 Free systems

After these preliminaries Duhem proceeds to particularize the laws of energetics tomechanics. He does it in Chapter 5, entitled: La mécanique des solides invariableset la mécanique rationelle [99].11 The first consideration Duhem makes is that fora mechanical system, for example consisting of two bodies, the potential energiesU(A) andU(B) are constant, then the potential energy of the system can be providedonly by the mutual potential energy, i.e.:

U = Ψ. (18.9)

In case external forces act on the system, the principle of conservation of energy isno longer valid; it is replaced by Duhem with the following principle:

T +τ−δΨ = 0, (18.10)

where T is the virtual work made by the external forces, τ the virtual work madeby the forces of inertia. On the basis of relation (18.10) Duhem can affirm:

Comparison of the conditions […] provides the following statement, which is the principleof d’Alembert.To obtain for each moment the laws of motion of a system of rigid bodies without passiveresistance, simply require that the system remains in equilibrium if it is placed motionless inthe state is passing in that moment and submit it not only to external actions that are actuallycarrying on it, when it is in this state, but also to the fictitious external action equivalent tothe inertia actions in the body at that time [99].12 (A.18.8)

In case equilibrium is concerned, the virtual work of the inertia forces is zero, there-fore the previous relation provides:

T −δΨ = 0. (18.11)

11 pp. 183–246.12 p. 242.


This expression represents a law of virtual work:

For a system subjected to bilateral constraints [with no passive resistance], it is enough forthe equilibrium that the virtual work of the external forces is at most equal to the increasein potential energy [i.e. the virtual work of internal forces][99].13 (A.18.9)

18.1.3.2 Constrained systems

The introduction of constraints on the components of the system Sa does not signif-icantly modify the framework outlined above. Duhem says to consider constraintsin a purely geometric way (first restriction) “All constraints have this common char-acter and to define them, it is useless to appeal to any notion alien to Geometry ”[99].14 But he is not consistent and equips them with the property of the absence ofpassive resistance (second restriction), in the sense that:

The compulsion that the system receives from the constraints, in the case of its actual dis-placement is less than the compulsion that it receives from all the virtual displacementsoriginated from the same state: ∫

MN2dm <

∫PN

2dm.

That is what we mean when we say that the studied constraints have no passive resistance[99].15 (A.18.10)

Here, as shown in Fig. 18.1, M is the position that a material point A of mass dmof a system reaches starting from a state S0 with the real motion in an infinitesimalinterval of time h. N is the position A would have reached if the system were freefrom constraints. P is the position of A in a virtual motion starting from the state S0.The two integrals are respectively the compulsions of the real motion and that of thevirtual motion.

The quoted proposition is seen by Duhem from one hand as a definition of con-straints without passive resistance, on the other hand as a principle of mechanics,which allows one to determine the motion of a system; the principle of minimal con-

M

N

S0

Pfree motion

virtual motionreal motion

Fig. 18.1. Free, virtual and real motions

13 pp. 241–242.14 p. 190.15 p. 195.


straints, attributed to Johann Carl Friedrich Gauss (1777–1855) who in a short paperof 1829 wrote:

The motion of a system of particles, connected in any way (the motions of which are con-strained by any external conditions), is made at any time with the widest possible agreementwith the free motion, or under the condition of minimum action, considering as measure ofthe action of the whole system in any infinitesimal interval of time, the sum of products ofdeviations of each point from its free motion by its mass.Let m,m′,m′′ and so on the masses of the points, a,a′,a′′ and so on their positions at timet; b,b′,b′′ and so on the positions they would take on if they were completely free after aninfinitesimal dt because of the forces acting on them during this time and of the velocities anddirections which they had at the instant t. The actual position c,c′,c′′ and so on will then bethose for which, under all conditions eligible for the system,m(bc)2+m′(b′c′)2+m′′(b′′c′′)2and so on is a minimum [126].16 (A.18.11)

Duhem acknowledges that the motion or the equilibrium position obtained in ac-cordance with the principle of minimal constraints coincide with what it would beobtained considering the two principles:

T +τ−δΨ = 0 (18.12)

for motion and:

T −δΨ = 0 (18.13)

for equilibrium. These principles are formally equivalent to those expressed by re-lations with (18.10) and (18.11), but now the various expressions that appear areinfinitesimals and correspond to infinitesimal virtual displacements which meet theconstraint condition (bilateral for simplicity).

Relation (18.13) is a virtual work principle for constrained systems, which canbe stated as follows:

A constrained mechanical system, with constraint deprived of passive resis-tance, is in equilibrium if and only if the external work, for all virtual infinites-imal displacements, equals the infinitesimal variation of the potential energy.In particular, if, as happens for example in simple machines, or more gen-erally in systems of rigid bodies, there is no change in potential energy, thetraditional formulation of virtual work is recovered: a mechanical system is inequilibrium if and only if the virtual work of the external forces is zero for all(infinitesimal) virtual displacements

T = 0. (18.14)

In a thermodynamic frame the law of virtual work is therefore obtained as a corollaryof the principle of conservation of energy and it is therefore not strictly a principle buta theorem. Its formulation is less general than the commonly adopted one because itexcludes non-conservative forces, but it is worthy of great respect.

16 pp. 26–27.

Appendix

Quotations

A.1 Chapter 1

1.1 Momento è la propensione di andare al basso, cagionata non tanto dalla gravità del mobile,quanto dalla disposizione che abbino tra di loro diversi corpi gravi; mediante il qual momentosi vedrà molte volte un corpo men grave contrapesare un altro di maggior gravità.

1.2 La mesme force qui peut lever un poids, par exemple de cent livres a la hauteur de deux pieds,en peut aussy lever un de 200 livres, a la hauteur d’un pied, ou un de 400 a la hauteur d’undemi pied, & ainsy des autres.

1.3 Poiché, sì come è impossibile che un grave o un composto di essi si muova naturalmente all’insu, discostandosi dal comun centro verso dove conspirano tutte le cose gravi, così è impossibileche egli spontaneamente si muova, se con tal moto il suo proprio centro di gravità non acquistaavvicinamento al sudetto centro comune.

1.4 L’equilibrio nasce da ciò, che le azioni delle potenze, che equilibrar si devono, se nascessero,sarebbero uguali, e contrarie; e perciò l’uguaglianza, e la contrarietà delle azioni delle potenzeè la vera causa dell’equilibrio.[…]L’equilibrio non è altro, che l’impedimento de’ moti, cioè degli effetti dell’azione dellepotenze, a cui non è meraviglia se corrisponde l’impedimento delle cause, cioè delle azionistesse.

1.5 Et en général je crois pouvoir avancer que tous les principes généraux qu’on pourroit encoredécouvrir dans la science de l’équilibre, ne seront que le même principe des vitesses virtuelles,envisagé différemment, & dont ils ne différeront que dans l’expression. Au reste, ce Principeest non seulement en lui même très simple & très général; il a de plus l’avantage précieux &unique de pouvoir se traduire en une formule générale qui renferme tous les problèmes qu’onpeut proposer sur l’équilibre des corps. [...] Quant à la nature du principe des vitesses virtuelles,il faut convenir qu’il n’est pas assez évident par lui-même pour pouvoir être érigé en principeprimitif.

1.6 Il faut encore remarquer qu’on suppose le système déplacé d’une manière quelconque, sansaucun égard à l’action des puissances qui tend à le déplacer; le mouvement qu’on lui donneest un simple change de position où le temps n’entre pour rien.

1.7 Nous sommes conduits ainsi à reconnaître que le principe des vitesses virtuelles dans l’équilibred’une machine composée de plusieurs corps solides ne peut avoir lieu qu’en considérant


406 Appendix. Quotations

d’abord les frottemens de glissement, lorsque les déplacemens virtuels peuvent faire lisser lescorps les us sur les autres, et en outre ceux de roulement lorsque les corps ne peuvent prendrede mouvement virtuel sans se déformer près des points de contact.Les frottemens étant reconnus par expérience toujours capables de maintenir l’équilibre dansde certaines limites d’inégalité entre la somme des élémens de travail positif et la somme desélémens de travail negatif, en prenant ici pour négatifs les élémens appartenant à la somme laplus petite; il s’ensuit que la somme des élémens auxquels ils donnent lieu a précisément lavaleur propre à rendre nulle la somme totale et se trouve égale à la petite différence qui existeentre les sommes des élémens positifs et des élémens négatifs.

A.2 Chapter 2

2.1 Quella comune facoltà di primitiva intuizione, per cui ognuno si convince facilmente di un sem-plice assioma geometrico, come per esempio, che il tutto sia maggiore della parte, non servecertamente per convenire della sopraccennata verità meccanica, la quale è tanto più complicatadi quello che sia uno degli ordinari assiomi, quanto il genio di quei grandi Uomini, che l’hannoammessa per assioma, supera l’ordinaria misura dell’ingegno umano; ed è in conseguenzanecessario per coloro che non ne restano appagati, il procurarsene una dimostrazione dipen-dentemente da estranee teorie […] ovvero riposarsi sulla fede d’uomini sommi, disprezzandol’usuale ripugnanza ad introdurre in Matematica il peso dell’autorità.

2.2 La dimostrazione avviene per assurdo. Si suppone che pur valendo la (*), il sistema si metta inmoto; ossia che almeno uno dei suoi punti, diciamo l’i-esimo, subisca, nel tempo dt, succes-sivo a t, uno spostamento dri, compatibile con i vincoli. Dato che il punto materiale in esameparte dalla quiete, necessariamente avremo: Fidri > 0; quindi, sommando tutti i lavori parzialirelativi agli altri punti del sistema che effettivamente si muova, avremo anche:

∑Fidri > 0, (1)

dato che la somma è formata tutta di termini non negativi e di cui almeno uno, per ipotesi, nonè nullo.Ma Fi = F(a)

i +Ri, per cui riscriviamo la (1) come:

∑(F(a)i +Ri) ·dri > 0. (2)

A questo punto si fa l”ipotesi dei vincoli lisci e si ottiene l’assurdo ∑F (a)i +Ri) ·dri > 0, perché

contro l’ipotesi.

2.3 En effet, on démontre que, si un point n’a d’autre liberté dans l’espace, que celle de se mouvoirsur une surface ou sur une ligne fixement arrêtée, il n’y peut être en équilibre, à moins que larésultante des forces qui le sollicitent se soit perpendiculaire a cette surface ou à cette lignecourbe.

2.4 La force de pression d’un point sur une surface lui est perpendiculaire, autrement elle pourraitse décomposer en deux, l’une perpendiculaire à la surface, et qui seroit détruite par elle, l’autreparallèle à la surface, et en vertu de la quelle le point n’aurait point d’action sur cette surface,ce qui est contre la supposition.

2.5 Or si l’on fait abstraction de la force P, et qu’on suppose que le corps soit forcé de se mouvoirsur cette surface, il est claire que l’action, o plutôt la résistance que la surface oppose au corpsne peut agir que dans une direction perpendiculaire à la surface.

A.3 Chapter 3 407

A.3 Chapter 3

3.1 ‘Ambiguità’ che del resto si potrebbe riguardare come un documento glottologico della pri-mordialità della credenza a una connessione, tra le diverse velocità compatibili dei vari punti lecui posizioni dipendono le une dalle altre, e la diversa facilità colla quale i punti stessi possonoessere mossi a parità delle altre condizioni.

3.2 Percioché, seAristotile risolve per qual cagione la leva lungamuove più facilmente il peso, diceavvenir ciò per la lunghezza maggiore dalla parte della potenza che muove; e ciò benissimosecondo il suo principio nel quale suppone, che quelle cose che sono in maggior distanza dalcentro, si muovano più facilmente e con maggior forza: del che reca egli la causa principalenella velocità secondo la quale il cerchio maggiore supera il minore. È vera dunque la causa,ma indeterminata; perciochè non so io per tanto, dato un peso, una leva, et una potenza, comeio li abbia da dividere la leva nel punto ove ella gira, acciochè la data potenza bilanci il datopeso. Ammesso dunque Archimede il principio d’Aristotile passò più oltre; nè si contentò chemaggiore fosse la forza dalla parte della leva più lunga, ma determinò quanto ella deve essere,cioè con qual proportione ella deve.

3.3 Il secondo principio è, che il momento e la forza della gravità venga accresciuto dalla velocitàdel moto; sì che pesi assolutamente eguali, ma congiunti con velocità diseguali, sieno di forza,momento e virtù diseguale, e più potente il più veloce, secondo la proporzione della velocitàsua alla velocità dell’altro. [...] Tal ragguagliamento tra la gravità e la velocità si ritrova intutti gli strumenti meccanici, e fu considerato da Aristotile come principio nelle sue Questionimeccaniche: onde noi ancora possiamo prender per verissimo assunto che pesi assolutamentediseguali, alternatamente si contrappesano e si rendono di momenti eguali, ogni volta che leloro gravità con proporzione contraria rispondono alle velocità de’ moti, cioè che quanto l’unoè men grave dell’altro, tanto sia in constituzione di muoversi più velocemente di quello.

3.4 Ma per non confonderci dichiarerò prima il principio del quale parla Aristotele, et insieme dellemachine delle quali habbiamo dato gli esempi, cioè delle taglie et della stanga. [...] Tutte leoperazioni delle machine adunque consistono nel movimento loro, et per conseguente l’istessamachina farà maggiore et minore effetto quanto più propinquo sarà il movimento che se lefarà fare al suo proprio. [...] Resta adunque per manifesto dalle dimostrazioni precedenti chequanto meno un peso obligato à muoversi in giro, ò una forza s’allontana dal centro, con tantomaggior velocità si moverà et la forza tanto maggiore effetto farà.

3.5 Il a dit que centre de gravité ou d’inclinaison est un point tel que, lorsque le poids est suspendupar ce point, il est divisé en deux portions équivalentes. A la suite de cela Archimède et les mé-caniciens qui l’ont imité, ont scindé cette définition, et ils ont distingué le point de suspensiondu centre d’inclinaison.

3.6 Quelques-uns ont pensé à tort que la proportion existant dans l’état d’équilibre n’était plus vraiedans le cas d’un fléau irrégulier. Supposons un fléau de balance n’ayant pas partout même poidsni même épaisseur, et fait de matière quelconque; il est en équilibre lorsqu’on le suspend aupoint γ; nous entendons ici par équilibre l’arrêt du fléau dans une position stable, quand bienmême il serait incliné dans un sens ou dans un autre. Suspendons ensuite des poids à des pointsquelconques du fléau; soient δ et ε ces points; le fléau reprend une position d’équilibre aprèsque les poids ont été suspendus; et Archimède a démontré que, dans ce cas encore, le rapportdes poids est égal au rapport inverse des distances respectives.

3.7 Il est nécessaire d’expliquer comment on soutient, comment on porte et transporte les corpsgraves, avec les développements convenables pour une introduction. Archimède a traité cettematière avec un art très sûr dans son livre appelé Livre des Supports.

3.8 Supposons deux cercles avant un même centreα; soient leurs diamètres les deux lignesβγ, δε;ces deux cercles sontmobiles autour du pointα, qui est leur centre commun, et perpendiculaires


au plan de l’horizon. Suspendons aux deux points β, γ deux poids égaux, désignés par η et ζ.Il est évident que les cercles ne penchent ni d’un côté ni de l’autre, puisque les deux poids ζ etη sont égaux et les distances βα, αγ égales. Faisons de βγ un fléau de balance mobile autourd’un point de suspension qui est le point α. Si nous transportons en ε le poids qui est appliquéen γ, le poids inclinera ζ vers le bas, et il fera tourner les cercles. Mais si nous augmentonsle poids θ, il fera de nouveau équilibre au poids ζ et le rapport du poids θ au poids ζ seraégal au rapport de la distance βα à la distance αε. Ainsi la ligne βε joue le rôle d’un fléau debalance mobile autour d’un point de suspension, qui est le point α. Archimède a déjà donnécette proposition dans son livre sur l’équilibre entre les poids.

3.9 Les cinq machines simples qui meuvent le poids se ramènent à des cercles montés sur un seulcentre; c’est ce que nous avons démontré sur les diverses figures que nous avons précédemmentdécrites. Je remarque pourtant qu’elles se réduisent encore plus directement à la balance qu’auxcercles; on a vu en effet que les principes de la démonstration des cercles ne nous sont, venusque de la balance; on démontre que le rapport du poids suspendu au petit bras de la balance,au poids suspendu au grand bras, est égal au rapport du grand bras au petit.

3.10 Imaginons au contraire un autre poids au point ζ, et fixons-y une poulie η; faisons entrer danscette poulie une corde, et attachons-en les deux extrémités à un support fixe, en sorte que lepoids ζ demeure suspendu. Chacun des deux brins de la corde sera tendu par la moitié dupoids; et si l’on délie l’un des deux bouts de la corde, celui qui est attaché au point κ, et qu’oncontinue à maintenir la corde dans la même position, on aura à porter la moitié du poids. Lepoids se trouve donc être double de la puissance qui le retient.

3.11 Cet instrument et toutes les machines de grande force qui lui ressemblent sont lents, parceque, plus est faible la puissance comparée au poids très lourd qu’elle meut, plus est long letemps que demande le travail. Il y a un même rapport entre les puissances et les temps. Parexemple, lorsqu’une puissance de 200 talents a été appliquée au tambour β, et quelle a misle poids en mouvement, il faut un tour entier de β pour que le poids se meuve de la longueurde la circonférence de l’arbre γ. Si le mouvement est donné à l’aide du tambour δ, il faut quel’arbre γ tourne cinq fois pour que l’arbre fasse un seul tour, puisque le diamètre du tambourest cinq fois celui de l’arbre γ et que cinq tours de γ valent un tour de β. Cette remarque serenouvelle pour la suite des organes du train, soit que nous fassions les arbres égaux entre euxainsi que les tambours, soit que nous leur donnions des rapports variés, connue ceux que nousavons choisis. Le tambour δ fait mouvoir le tambour β et les cinq tours que doit effectuer letambour δ prennent cinq fois le temps d’un seul tour; 200 talents, d’autre part, valent cinq fois40 talents. Ainsi le rapport du poids à la forcemotrice est égal à l’inverse du rapport d’ensembledes arbres et des tambours, quelque nombreux qu’ils soient. Cela achève la démonstration.

3.12 Cet instrument et toutes les machines de grande force qui lui ressemblent sont lents, parce que,plus est faible la puissance comparée au poids très lourd qu’elle meut, plus est long le tempsque demande le travail. Il y a un même rapport entre les puissances et les temps.[…]Le ralentissement de la vitesse a lieu aussi dans cette machine.

3.13 Supposons, par exemple, que le support. stable auquel le poids est suspendu soit α. La cordeest la ligne αβ. Menons la ligne αγ perpendiculaire sur la ligne αβ, et marquons sur la ligneαβ deux points quelconques que nous désignons par les lettres δ, ε. […] Donc, quand noustirons le poids à partir du point ε, il vient en κ, et quand nous le tirons à partir du point δ, ilvient en η. Ainsi on élève davantage le poids en partant du point δ qu’en partant du point ε;et pour porter le poids plus haut, il faut une plus grande force que pour le porter moins haut,parce que, pour le porter dans un lieu plus élevé, il faut un temps plus long.

3.14 Nous aurons recors à quelque puissance ou à quelque poids appliqué de l’autre côte, pour faired’abord équilibre au poids donné, afin qu’un excès de puissance l’emporte sur de poids et letire en haut.

A.4 Chapter 4 409

A.4 Chapter 4

4.1 I. Dico ergo quod omnium duorum spaciorum que duo mota secant in tempore uno, proportiounius ad alterum est sic ut proportio virtutis motus eius quod secat spacium unum ad virtutemmotus illius secantis spacium alterum.Et ponam ad illud exemplum. Dico duorum viatorum perambulat unus 30 miliaria et perambu-lat secundus 60 miliaria in tempore uno. Et notum est ergo quod virtus motus eius qui peram-bulat 60 miliaria dupla est virtutis motus eius qui perambulat 30 miliaria sicut spacium quodest 60 miliaria est duplum spacii quad est 30 miliaria.Hec est propositio recepta per se, inter quam et inter intellectum non est medium separans ea.

4.2 III. Cum ergo iam manifestum est istud, tunc dico quod omnis linea que dividitur in duassectiones diversas et extimatur quod linea suspendatur per punctum dividens ipsam, et quodduorum ponderum proportionalium sicut [invers]1 proportionalitas duarum partium linee uniusad comparem suam secundum attractionem suspenditur unum in extremitate unius duarumsectionum et secundum in extremitate altera, tunc linea equatur super equidistantiam orizontis.

4.3 II. Et post hoc dico quod omnis linea que dividitur in duas sectiones, et figitur punctum eiussecans et movetur linea tota penitus motu quo non redit ad locum suum, tunc ipsa facit accidereduos sectores similes duorum circulorum, medietas diametri unius quorum est linea longioret medietas diametri secundi est linea brevior et quod proportio arcus quem signat punctumextremitatis unius duarum linearum ad arcum quem signat punctum extremitatis linee secundeest sicut proportio linee revolventis illum arcum ad lineam secundam.

4.4 Iam diximus in duobus spaciis que secant duo mota in tempore uno quod proportio virtutismotus unius eorum ad virtutem motus alterius est sicut proportio spacii quod ipsum secat adspacium alterum, et punctum A apud motum linee iam secavit arcum AT, et punctum B iamsecavit etiam apud motum linee arcum BD et illud in tempore uno. Ergo proportio virtutismotus puncti B ad virtutemmotus puncti A est sicut proportio duorum spaciorum que secueruntduo puncta in tempore uno, unius ad alterum, scilicet proportio arcus BD ad arcum AT. Et hecproportio iam ostensum est quod est sicut proportio linee GB ad lineam AG.

4.5 Cuius hec est demonstratio, secabo ex BG longiore quod sit equale AG breviori quod sit GE.Si ergo suspendantur super duo puncta A. E duo pondera equalia, equidistabit linea AE ori-zonti, quoniam virtus motus duorum punctorum est equalis, secundum quod ostendimus, donecsi inclinaverimus punctum A ad punctum T sufficiet cum eo pondus quod est ad punctum Adonec redeat ad locum suum, et sit arcus AT. Et quando permutabimus pondus ex punctoE ad punctum B, et si voluerimus ut linea remaneat super equidistantiam orizontis est nobisnecesse ut addamus in pondere quod est apud A additionem aliquam donec sit proportio eiustotius ad pondus quod est apud B sicut proportio BG ad AG. Quoniam virtus puncti B super-fluit super virtutem puncti A per quantitatem superfluitatis BG super AG, secundum quod iamostendimus, pondus ergo quod est apud punctum fortioris est minus pondere quod est apudpunctum debilioris secundum quantitatem qua proportionatur arcus arcui. Cum ergo est apudpunctum B pondus et est apud A pondus secundum et est proportio ponderis a ad pondus bsicut proportio BG ad AG, equidistat linea ab orizonti.

4.6 Je dis que si l’on suspend ab au point g et que l’on applique à ses deux extrémités, en a et b,deux poids proportionnels et équivalents à ses deux segments, [ab] sera parallèle à l’horizon.En effet, prenons sur le [côté] le plus long ag un [segment] gd égal à gb. Si on applique en d unpoids égal au poids appliqué en b, [ab] sera parallèle à l’horizon. Si on incline alors vers le bas[le poids qui est en d], le poids qui est en b le soulèvera et lui fera parcourir l’arc dd égal à l’arcbb, car gd est égal à gb. Si nous déplaçons alors le poids du point d au point a, [celui-ci étantdans la position] inférieure, et que nous voulions le soulever jusqu’à [la position] supérieure

1 In the Latin text here there is an “editing” mistake, corrected by Moody and Clagett.


de a, il nous faudra augmenter le poids qui est en b de telle sorte que le rapport du [poids] total[en b] au poids qui est en a soit égal au rapport de l’ara aa à l’arc dd. lesquels sont parcourusen même temps alors qu’ils sont inégaux. Or ce rapport est égal au rapport de l’un des deuxsegments de la droite a l’autre.

4.7 Si l’axe est pesant et s’il est divisé en deux segments inégaux, on augmente l’épaississeur dusegment le plus court jusqu’a ce que l’axe soit parallèle a l’horizon. […] On est alors ramenéau cas déjà traité de la ligne dépourvue de poids.

4.8 Considérons un levier où la puissance est α où la résistance est β; cette résistance se trouvaità une certaine distance du point d’appui, supposons que la puissance α la puisse mouvoir etlui faire décrire en un temps δ l’arc γ; elle pourra également mouvoir le poids α/2, placé à unedistance double du point d’appui, car dans les même temps δ, et le lui fera parcourir l’arc 2γ.Il faut donc la même puissance pour mouvoir un certain poids, place a une certaine distancedu point d’appui, et pour mouvoir un poids moitié placé à une distance double. De là, on tireaisément la justification de la théorie du levier donnée dan les Questions mécaniques.

4.9 Table 4.1S1 Omnis ponderosi motum esse ad medium virtutemque ipsius esse potentia ad inferi-

ora tendendi et motui contrario resistendi.S2 Quod gravius est velocius descendere.S3 Gravius esse in descendendo, quanto eiusdem motus ad medium rectior.S4 Secundum situm gravius esse, cuius in eodem situ minus obliquus descensus.S5 Obliquiorem autem descensum in eadem quantitate minus capere de directo.S6 Minus grave aliud alio secundum situm, quod descensum alterius sequitur contrarioS7 Situm aequalitatis esse aequalitatem angulorum circa perpendiculum, sive recti-

tudinem angulorum, sive aeque distantiam regulae superficiei Orizontis.

4.10 Omnis ponderosi motum esse admedium virtutemque ipsius esse potentia ad inferiora tendendivirtutem ipsius, sive potentia possumus intelligere longitudinem brachij librae, aut velocitereius quem probatur ex longitudine brachij librae, et motui contrario resistendi.

4.11 Patet ergo quod major est violentia in motu secundum arcum maiorem, quam secundum mi-norem; alias enim non fieret motus magis contrarius. Cum ergo apparet plus in descensuadquirendum impedienti, patet quia minor erit gravitas secundum hoc. Et quia secundum situ-ationem gravium sic fit, dicatur gravitas secundum situm in futuro processo.Ita enim, sillogizando de motu tamquam motus sit causa gravitatis vel levitatis, potius per mo-tum magis contrarium concludimus causam huiusmodi contrarietatis esse plus contrariam, idest, plus habere violentie. Quod quidem grave descendat, hoc est a natura; sed quod per lineamcurvam, hoc est contra naturam, et ideo iste descensos est mixtus ex naturali et violento. Inascensu vero ponderis, cum ibi nihil sit secundum naturam, debet arguì sicut de igne, quoniamnihil naturaliter ascendit. De igne enim arguitur in ascensu, sicut de gravi in descensu; ex quosequitur quod grave, quanto plus sic ascendit, tanto minus habet de levitate secundum situm,et sic plus habet de gravitate secundum situm.

4.12 Table 4.2

P1 Inter quaelibet gravia est virtutis, et ponderis eodem ordine sumpta proportio.P2 Cum equilibris fuerit positio equalis, equis ponderibus appensis ab equalitate non

discedet: et si a rectitudine separetur, ad aequalitatis situm revertetur. Si vero inequaliaappendantur, ex parte gravioris usque ad directionem declinare cogetur.

P3 Omne pondus in quamcunque partem ab aequalitate discedat, secundum situm fit levius.P4 Cum fuerint appensorum pondera equalia, non faciet nutum in equilibri appendiculorum

inequalitas.P5 Si brachia librae fuerint inequalia, equalibus appensis ex parte longiore nutum faciet.P6 Si fuerint brachia libre proportionalia ponderibus appensorum, ita ut in breviori gravius

appendatur, eque gravia erunt secundum situm appensa.

A.4 Chapter 4 411

P7 Si duo oblonga per totum similia, et quantitate, et pondere equalia, appendantur ita utalterum dirigatur, alterum orthogonaliter dependeat, ita etiam ut termini dependentis etmedii alterius eadem sit a centro distantia, secundum hunc situm eque gravia fient.

P8 Si inequalia fuerint brachia libre, et in centro motus angulum fecerint, si termini eo-rum ad directionem hinc inde equaliter accesserint, equalia appensa in hac dispositioneaequaliter ponderabunt.

P9 Equalitas declinationis identitatem conservat ponderis.P10 Si per diversarum obliquitatum vias duo pondera descendant, fueritque declinationum et

ponderum una proportio eodem ordine sumpta, una erit utriusque virtus in descendendo.

4.13 Table 4.3Inter quaelibet gravia est velocitatis in descendendo et ponderis eodem ordine sumpta propor-tio, descensus autem et contrarii motus proportio eadem sed permutata.INTER QUAELIBET DUO GRAVIA EST VELOCITATIS IN DESCENDENDO PROPRIE, ET PONDERIS EO-DEMORDINE SUMPTA PROPORTIO, DESCENSUS AUTEM ET CONTRARII MOTUS PROPORTIO EADEM

SED PERMUTATA.Inter quaelibet gravia est virtutis, et ponderis eodem ordine sumpta proportio.

4.14 Inter quaelibet gravia est virtutis, et ponderis eodem ordine sumpta proportio.Sint pondera ab et c, levius c. Descendatque ab, in D, et c, in E; itemque pellatur ab, sursumin F, et c, in H. Dico ergo quod AD ad CE, sicut ab ponderis ad c pondus, quanta enim vir-tus ponderosi tanta descendendi velocitas. Atqui compositi virtus ex virtutibus componentiumcomponitur. Sit ergo a aequale c, que igitur virtus a eadem et c. Si ergo proportio ab ad c,minor quam virtutis ad virtutem, erit similiter proportio ab ad a, minor proportio quam virtutisab ad virtutem a. Ergo virtutis ab, ad virtutem b, minor proportio quam ab ad b; similium ergoponderum minor et maior proportio, quam virtutum. Et quia hoc inconveniens erit, utrobiqueeadem; ideoque ab ad c, sicut AD ad CE, et econtrario sicut CH ad AF.

4.15 Ponatur item quod submittatur ex parte b, et ascendat ex parte c, dico quoniam redibit ad ae-qualitatem. est enim minus obliquus descensus a, ad aequalitatem, quam a, b, versus e. Suman-tur enim sursum arcus aequales, quantumlibet parvi qui sint c, d, et b, g, et ductis lineis adaequidistantiam aequalitatis, quae sint, c, z, l, et d, m, n. Item b, k, m, g, y, t, dimittatur orthog-onaliter descendens diametrum quae sit f, r, z, m, a, k, y, e, erit quod z, m, maior k, y, quiasumpto versus f, arcu ex eo quod sit aequalis c, d, et ducta ex transverso linea x, r, s, erit r,z,minor z, m, quod facile demonstrabis. Et quia r, z, est aequalis k, y, erit z, m, maior k, y. Quiaigitur quilibet arcus sub c, plus capiat de directo quam ei aequalis sub b, directo est descensusa, c, quam a, b, et ideo in altiori situ gravius erit c, quam b, redibit ergo ad aequalitatem.

4.16 Propositio VISi fuerint brachia librae proportionalia ponderibus appensorum, ita ut in breviori gravius ap-pendatur, eque grauia erunt secundum situm appensa. Sit ut prius regula ACB, appensa a et b;sitque proportio b ad a, tamquam AC ad BC. Dico quod non mutabit in aliquam partem libra.Sit enim ut ex parte B, descendat, transeatque in obliquum linea DCE, loco ACB. Et appensad, ut a, et e, ut b, et DG, linea orthogonaliter descendat, et EH ascendat, palam autem quoniamtrianguli DCG et ECH similes sunt, quare proportio DC ad CE que DG, ad EH. Atqui DC adCE sicut b, ad a ; ergo DG ad EH, sicut b ad a. Sit igitur CL equalis CB, et CE, et l, equumb, in pondere, et descendat perpendicularis LM. Quia igitur LM et EH constant esse equales,erit DG ad LM sicut b ad a, est sicut l, ad a. Sed, ut ostensum est, a et l, proportionaliter sehabent ad contrarios motus alternatim. Quod ergo sufficit attollere a in D, sufficiet attollere l,secundum LM. Cum ergo equalia sint l, et b, et LC equale CB, l, non sequetur b, contrariomotu, neque a sequetur b, secundum quod proponitur.

4.17 Sit centrum c, brachia ac, longius bc, brevius, et descendat perpendiculariter ceg supra quamperpendiculariter cadant hinc, inde ag. et be, aequales.[…]Pertranseant enim aequaliter a , et be, ad k, et z, et super eas fiant portiones circulorum mbhz,


kxal, et circa centrum c, fiat commune proportio kyaf, similis, et aequalis portionis mbhz, etsint arcus ax, al, aequales sibi atque similes arcubusmb, bh. Itemque ay, af. Si ergo ponderosiusest a, quam b, in hoc situ descendat a, in x, et ascendat b, in m, ducantur igitur lineae zm, kx,yk, fl, et mp, super zbp stet perpendiculariter etiam xe, et fd, super kad, et quia mp, aequaturfd, et ipsa est maior xt, per similes triangulos erunt mp, maior xt, quia plus ascendit b, adrectitudinem, quam a, descendit. quod est impossibile, cum sint aequalia

4.18 Sit linea abc, aequedistans orizonti, et super eam orthogonaliter erecta sit bd, a qua descendanthinc, inde lineae da, dc, sitque dc, maioris obliquitatis proportione igitur declinationum diconon angulorum, sed linearum usque ad aequedistantem resecationem, in qua aequaliter sumuntde directo. Sit ergo e, pondus super dc, et h, super da, et sit e, ad b, sicut dc, ad ad. Dico eapondera esse unius virtutis in hoc situ, sit enim dk, linea unius obliquitatis, cum dc, et pondussuper eam. ergo aequale est e, quae sit g. Si igitur possibile est, descendat e, in l, et trahat h,in m, sitque gn, aequale hm, quod etiam aequale est e. Et transeat per g et h, perpendicularis,super db; sitque ghy, et ab l, tl, t; et tunc super ghy, nz, mx, et super lt, erit er. Quia igiturproportio nz, ad ng, sicut dy ad dg, et ideo sicut db, ad dk, et quia similiter mx, ad mh, sicutdb, ad da. Erit propter aequalem proportionalitatem perturbata mx, ad nz, sicut dk, ad da, ethoc est sicut g, ad h. Sed quia e, non sufficit attollere g, in n, nec sufficiet attollere h in m. Sicergo manebunt.

A.5 Chapter 5

5.1 Non è dubbio che se a una semplice fune si attacca un peso, poniamo il caso di mille libbre,che tutta la fatica e forza non sia unitamente da quella fune sostenuta, che poi se la detta funesarà raddoppiata e a quella una taglia d’un raggio appesa dove penda quel peso, che la funenon sia per avere il doppio meno di fatica e il doppio meno di forza non basterà ad alzare quelpeso; or che sarà poi se ci saranno più taglie? […] Se’l primo raddoppiamento leva la metàdel peso, il secondo al quale resta la metà, leverà via la metà di quella metà che sarà la quartaparte di tutto il peso, et dalla quarta parte della detta forza di prima sarà il peso levato.

5.2 Il segreto di tutti gli inventori delle machine de’ Molini, et altro è di cercare solo, come si disse,di poter accompagnare la forza con la velocità, cosa in vero difficilissima; perche dovendosiun istessa potenza multiplicare in molte, che possino l’una doppo l’altra alzare, overo portareun peso, è necessario, che similmente si multiplichi il tempo, come per essempio saria se sidovesse trasportare un peso di mille libre da un luogo all’altro, con la semplice forza d’un solohuomo, il quale ne porterà solo una parte che sarà cinquanta libre.

5.3 Table 5.3

Diffinitione III La vertu d’un corpo grave se intende, e piglia per quella potentia, chelui ha da tendere, over di andare al basso, e anchora da resistere al motocontrario, cioe à che il volesse tirar in suso.

Diffinitione IIII Li corpi se dicono de vertu, over potentia, equali, quando che quelli intempi eguali di moto pertransiscono spacii eguali.

Diffinitione XIII Un corpo si dice essere piu grave, over men grave d’un’altro, secondo illuoco, over sito, quando che la qualita del luoco dove che lui se riposa, egiace, lo fa essere piu grave dell’altro anchor che fusseno simplicementeegualmente gravi.

Diffinitione XIIII La gravita d’un corpo se dice essere nota, quando che il numero delle libre,che lui pesa nesia noto, over altra denomination de peso.

Diffinitione XVII Piu obliquo se dice essere quel descenso, d’un corpo grave, il quale in unamedesima quantita, capisse manco della linea della direttione, overamentedel descenso retto verso il centro del mondo.

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Petitione II Simelmente adimandamo, che ne sia concesso quel corpo, ch’è di mag-gior potentia debbia anchora discendere piu velocemente, et nelli moticontrarii, cioe nelli ascensi, ascendere piu pigramente, dico nella libra.

Petitione III Anchora adimandamo, che ne sia concesso un corpo grave esser in el dis-cendere tanto piu grave, quanto che il moto di quello è piu retto al centrodel mondo.

Petitione VI Anchora adimandamo, che ne sia concesso, niun corpo esser grave in semedesimo.

5.4 Table 5.4

I La proportione della grandezza di corpi de un medesimo genere, e quella della lorpotentia è una medesima.

II La proportione della potentia di corpi gravi de uno medesimo genere, e quella dellalor velocita (nelli descensi) se conchiude esser una medesima, anchor quella delli lormoti contrarii (cioe delli lor ascensi) se conchiude esser la medesima, ma trasmutati-vamente.

III Se saranno dui corpi simplicemente eguali di gravita, ma ineguali per vigor del sito,over positione, la proportione della lor potentia, e quella della lor velocita necessaria-mente sara una medesima. Ma nelli lor moti contrarii, cioe nelli ascensi, la proportionedella lor potentia, e quella della lor velocita se afferma esser la medesima, ma trasmu-tativamente.

IIII La proportione della potentia di corpi simplicemente equali in gravita, ma inequali pervigor del sito, over positione, e quella delle lor distantie dal sparto, over centro dellalibra, se approvano esser equali.

V Quando, che la positione de una libra de brazzi equali sta nel sito della equalita, enella istremita de l’uno, e l’altro brazzo vi siano appesi corpi simplicemente equali ingravita, tal libra non se separara dal detto sito della equalita, e se per caso la sia daqualche altro peso in luno de detti brazzi imposto separata dal detto sito della equalita,overamente con la mano, remosso quel tal peso, over mano, tal libra de necessitaritornara al detto sito della equalita.

VI Quando che la positione d’una libra de bracci eguali sia nel sito della egualita, e chenella istremita dell’uno e l’altro brazzo vi siano appesi corpi simplicemente inegualidi gravita, dalla parte dove sara il piu grave sara sforzata à declinare per fin alla lineadella direttione.

VII Se li brazzi della libra saranno ineguali, et che nella istremita di cadauno de quelli visiano appesi corpi simplicemente eguali in gravita dalla banda del piu longo brazzotal libra fara declinatione.

VIII Se li brazzi della libra saranno proportionali alli pesi in quella imposti, talmente, chenel brazzo piu corto sia appeso il corpo piu grave, quelli tai corpi, over pesi serannoequalmente gravi, secondo tal positione, over sito.

XIIII La egualita della declinatione è una medesima egualita de peso.XV Se dui corpi gravi descendano per vie de diverse obliquita, e che la proportione delle

declinationi delle due vie, e della gravita de detti corpi sia fatta una medesima, toltaper el medesimo ordine. Anchora la virtu de luno, e laltro de detti dui corpi gravi, inel descendere sara una medesima.

5.5 Siano li dui corpi .a.b. e .c. de uno medesimo genere, e sia .a.b. maggiore, e sia la potentiadel corpo .a.b. la .d.e. e quella de corpo .c. la .f. Hor dico che quella proportione, che è dalcorpo .a.b. al corpo .c. quella medesima è della potentia .d.e. alla potentia .f. Et se possibile èesser altramente (per l’aversario) sia che la proportione del corpo .a.b. al corpo .c. sia menore diquella della potentia .d.e. alla potentia .f. Hor sta del corpo .a.b. (maggiore) compreso una parteeguale al corpo .c. menore, quale sia la parte .a. e perche la vertu, over potentia del compositoè composta dalla vertu di componenti. Sia adunque la vertu, over potentia della parte .a. la.d. e la vertu, over potentia del residuo .b. de necessita sara la restante potentia .e. et perche


la parte .a. è tolta egual al .c. la potentia .d. (per il converso della .7. diffinitione) sara egualealla potentia .f. e la proportione de tutto il corpo .a.b. alla sua parte .a. (per la seconda partedella .7. del quinto di Euclide) sara, si come quella del medesimo corpo .a.b. al corpo .c. (peresser .a. egual al .c.) e similmente la proportione della potentia .d.e. alla potentia .f. sara, sicome quella della detta potentia .d.e. alla sua parte .d. (per esser la .d. egual alla .f.). Adunquela proportione de tutto il corpo .a.b. alla sua parte .a. sara menore di quella di tutta la potentia.d.e. alla sua parte .d. Adunque eversamente (per la .30. del quinto di Euclide) la proportionedel medesimo corpo .a.b. al residuo corpo .b. sara maggiore di quella di tutta la potentia .d.e.alla restante potentia .e. la qual cosa saria inconveniente, e contra la opinion dell’aversario, ilqual vol che la proportione del maggior corpo al menore sia menore, di quella della sua potentiaalla potentia del detto menore. Adunque destrutto l’opposito rimane il proposito.

5.6 Propositione VIIISe li brazzi della libra saranno proportionali alli pesi in quella imposti, talmente, che nel brazzopiu corto sia appeso il corpo piu grave, quelli tai corpi, over pesi seranno equalmente gravi,secondo tal positione, over sito.[…]Sia come prima la regola, over libra .a.c.b. e vi siano appesi .a. e .b. et sia la proportione del.b. al .a. si come del brazzo .a.c. al brazzo .b.c. Dico, che tal libra non declinara in alcunaparte di quella, e se possibil fusse (per l’aversario) che declinar potesse, poniamo che quelladeclini dalla parte del .b. e che quella discenda, e transisca in obliquo, si come sta la linea.d.c.e. in luoco della .a.c.b. e attaccatovi .d. come .a. e .e. come .b. e la linea .d.f. descendaorthogonalmente, e simelmente ascenda la .e.h. […] e sia posto .l. equale al .b. in gravita, edescenda el perpendicolo .l.m. Adunque perche eglie manifesto la .l.m. e la .e.h. esser equale,la proportione della .d.f. alla .l.m. sara si come delle simplice gravita del corpo .b. alla simplicegravita del corpo .a. over della simplice gravita del corpo .l. alla simplice gravita del corpo .d.[…] Onde se li detti dui corpi gravi, cioe .d. e .l. fusseno simplicemente equali in gravita, stantipoi in li medesimi siti, over luochi, dove, che al presente vengono supposti, el corpo .d. sariapiu grave del corpo .l. secondo elsito (per la .4. propositione) in tal proportione, qual é di tuttoil brazzo .d.c. al brazzo .l.c. e per che il corpo .l. è simplicemente (dal presupposito) piu gravedel corpo .d. secondo la medesima proportione (cioe, si come la proportione del brazzo. d.c.al brazzo .l.c. adunque li detti dui corpi .d. e .l. nel sito della equalita veneranno ad essereegualmente gravi [...]. Adunque sel corpo .b. (per l’aversario) è atto ad ellevare il corpo .a. dalsito della equalita per fin al ponto .d. el medesimo corpo .b. saria anchora atto, e sofficiente adellevare il corpo .l. dal medesimo sito della equalita per fin al ponto, dove che al presente è, elqual consequente é falso, e contra alla quinta propositione […] distrutto adunque l’opposito,rimane il proposito.

5.7 Propositione XVSe dui corpi gravi descendano per vie de diverse obliquita, e che la proportione delle declina-tioni delle due vie, e della gravita de detti corpi sia fatta una medesima, tolta per el medesimoordine. Anchora la virtu de luno, e laltro de detti dui corpi gravi, in el descendere sara unamedesima.[…]Sia adunque la lettera .e. supposta per un corpo grave posto sopra la linea .d.c. e un’altro lalettera .h. sopra la linea .d.a. e sia la proportione della simplice gravita del corpo .e. alla simplicegravita del corpo .h. si come quella della .d.c. alla .d.a. Dico li detti dui corpi gravi esser intai siti, over luochi di una medesime virtu, over potentia. Et per dimostrar questo, tiro la .d.k.di quella medesima obliquita, ch’è la .d.c. e imagino un corpo grave sopra di quella equale alcorpo .e. elqual pongo sia la lettera .g. ma che sia in diretto con .e.h. cioe equalmente distantedalla .c.k. Hor se possibel è (per l’aversario) […]. Anchora la proportione della .m.x. alla .n.z.sara si come quella della .d.k. alla .d.a. e quella medesima (dal presupposito) e dalla gravita delcorpo .g. alla gravita del corpo .h. perche il detto corpo .g. fu supposto esser simplicemente,egualmente grave con el corpo .e. adunque tanto quanto, che il corpo .g. è simplicemente piugrave del corpo .h. per altro tanto il corpo .h. vien à esser piu grave per vigor del sito del

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detto corpo .g. e pero si vengono ad egualiar in virtu, over potentia, e per tanto quella virtu,over potentia, che sara atta à far ascendere luno de detti dui corpi, cioe à tirarlo in suso, quellamedesima sara atta, over sofficiente à fare ascendere anchora l’altro, adunque sel corpo .e.(per l’aversario) è atto, e sofficiente à far ascendere il corpo .h .per fin in .m. el medesimocorpo .e.s aria adunque sofficiente à far ascendere anchora il corpo .g. à lui equale, e inequaledeclinatione, la qual cosa é impossibile per la precedente propositione, adunque il corpo .e.non sara de maggior virtu del corpo .h. in tali siti, over luochi, ch’é il proposito.

5.8 Post haec videndum est de ponderibus quae in libra constituuntur. Sit igitur libra, cuius trutinasit appensa in A, et finis ubi iunguntur latera lancis B, et lanx CD, et manifestum est quod CDmovetur circa B, velut centrum quoddam, quia CD non potest separari ab ipso B: et sit angulusABC, et ABD rectus.

5.9 Dico quod pondus in C constitutum erit gravius, quam si lanx collocetur in quocunque alioloco, ut puto quod constitueretur lanx in F. Ut autem cognoscamus quod C sit gravius in eo situ,quam in F […] Ideo ergo duplici ratione magis gravabit pondus lance posita ad perpendiculumcum trutina, quam in quoque alio loco […]. Primum igitur sic declaratur. Manifestum est instateris, et in his, qui pondera elevant, quod quanto magis pondus ae trutina, eo magis gravevidetur [...] manifestum est, libram quanto magis descendit versus C exA, tanto gravius pondusreddere, et eo velocius moveri: at ex C versus Q, contraria ratione pondus reddi levius, etmotum segniorem, quod et experimentum docet.

5.10 Secundum vero sic demonstratur. Quia enim CE est aequalis FG, sumatur CH aequalis CE[...] igitur BN maior OF, et ideo BM maior OP. Dum igitur libra movetur ex C in E pondusdescendit per BM lineam, seu propinquius centro redditur quam esset in C, et dum moveturper spatium arcus FG, descenditque per OP, et BM, maior est OP. Igitur supposito etiam quodin aequali tempore transiret ex C in K, et ex F in G, adhuc velocius descendit ex C, quam exF. Igitur gravius est in C, quam in F.

5.11 Quartum subtilitatis exemplum est in trochleis. Sed quia temporum proportio est, ut poten-tiarum per binos orbiculos, quadruplo per ternos, sexcuplo lentius trahet [...] quo sit ut puerille vix in unius hore spacio idem pondus his trochleis trahet, quod sexcuplo robustior vir,unica, superius existes, levare potest illico fune.

5.12 Propositio quadragesima quintaRationem staterae ostendereSi ergo ponantur loco lineae bd in e et f, et sit proportio e b ad bf, ut g ad h, dico, quod eritaequilibrium, per eandem enim h movebitur in k, scilicet ut perveniat in rectam a d, si enimnon esset suspensum h, moveretur in recta e h per eandem, quia ergo retinetur, movetur perobliquam hk, et sumatur in propinquum punctum (m ?) in be, et n in aequali distantia in ef,quia ergo eb totum movetur eadem vi in singulis partibus, quia a pondere h, et in h moveturper hk in m per mp, ergo qualis est proportio magnitudinis hk ad mp, talis est vis in mp ad vimin hk, et ita in b erit pene infinita.

5.13 Il centro della grandezza di ciascun corpo è un certo punto posto dentro, dal quale se con laimaginatione s’intende esservi appeso il grave, mentre è portato sta fermo, et mantiene quelsito, che egli haveva da principio, ne in quel portamento si và rivolgendo.Questa diffinitione delcentro della gravezza insegnòPappoAlessandrino nell’ottavo libro delle raccolteMatematiche.Ma Federico Commandino nel libro del centro della gravezza de’ corpi solidi dichiarò l’istessocentro in questa maniera descrivendolo: Il centro della gravezza di ciascuna figura solida èquel punto posto dentro, d’intorno al quale le parti di momenti eguali da ogni parte si fermano.Perochè se per tale centro sarà condotto un piano, che seghi in qual si voglia modo la figura,sempre la dividerà in parti che peseranno egualmente.


5.14 La bilancia egualmente distante dall’orizzonte, et che habbia nelle stremità pesi eguali, etugualmente distanti dal centro collocato in essa bilancia. Se ella indi sarà mossa, o non, dunqueella sarà lasciata, rimarrà.[…]Dico primariamente, che la bilancia DE non si muoverà, et rimarrà in quel sito. Hor perciochepesi AB sono eguali, sarà il centro della grandezza della magnitudine composta delli due pesiA et B in C. Per la qual cosa l’istesso punto C sarà il centro della bilancia, et il centro dellagravezza di tutto il peso. Et percioche il centro della bilancia che è C, mentre la bilancia A Binsieme co’ pesi si muove in DE, rimane immobile, non si muoverà ne anche il centro dellagravezza, che è l’istesso C.

5.15 Hor perche dicono che il peso posto in D in quel sito è più grave del peso posto in E nell’altrosito dal basso: mentre i pesi sono in DE, non sarà il punto C più centro della grandezza, im-perochè non stanno fermi se sono attaccati al C, ma sarà nella linea CD per la terza del primodi Archimede delle cose, che pesano ugualmente. Non sarà già nella CE per essere il peso Dpiù grave del peso E: sia dunque in H, nel quale se saranno attaccati, rimaranno. Et perciocheil centro della gravezza de’ pesi congiunti in AB stà nel punto C: ma de’ pesi posti in DE ilpunto è H: mentre dunque i pesi AB si muovono in DE, il centro della grandezza C moverassiverso D, et s’appresserà più da vicino al D, il che è impossibile, per mantenere i pesi unamedesima distanza fra loro: peroche il centro della gravezza di ciascun corpo stà sempre nelmedesimo sito per rispetto al suo corpo.

5.16 Pongansi le cose istesse, et da i punti DE siano tirate le linee DHEK a piombo dell’orizonte, etsia un’altro cerchio LDM, il cui centro sia N, il quale tocchi FDG nel punto D, et sia eguale adFDG. […]Ma la proportione dell’angoloMHD all’angolo HDG èminore di qual si voglia altraproportione, che si trovi trà la maggiore, et minore quantità: Adunque la proportione de i pesiDE sarà la minima di tutte le proportioni, anzi non sarà quasi neanche proportione, essendola minima di tutte le proportioni. […] dalle quali troveremo sempre la proportione minore ininfinito: et così segue, che la proportione del peso posto in D al peso posto in E non sia tantopicciola, che non si possa ritrovarla sempre minore in infinito. Et perche l’angolo MDG sipuote dividere in infinito, si potrà anche dividere quel più di grandezza che ha il D sopra lo Ein infinito.

5.17 Percioche il peso posto in L libero, et sciolto si muoverebbe verso il centro del mondo per LS,et il peso posto in D per DS . Ma perche il peso messo in L grava tutto sopra LS, et quello che èin D sopra DS, il peso in L graverà più sopra la linea CL, che quello, che sta in D sopra la lineaDC. Adunque la linea CL sosterrà più il peso, che la linea CD, et nel mondo istesso quanto più ilpeso sarà da presso ad F, si dimostrerà più esser sostenuto dalla linea CL per cotesta cagione,peroche sempre l’angolo CLS sarebbe minore, la qual cosa etiandio è manifesta; perché sele linee CKL, et LS s’incontrassero in una linea, il che aviene in FCS, all’hora la linea CFsosterrebbe tutto il peso, che è in F, et lo renderebbe immobile, né havrebbe niuna gravezza intutto nella circonferenza del cerchio.

5.18 Se dunque il peso posto in E è più grave del peso posto in D, la bilancia DE non starà giamai inquesto sito, la qual cosa noi habbiamo proposto di mantenere, ma si moverà in FG. Alle qualicose rispondiamo che importa assai, se noi consideriamo i pesi overo in quanto sono separatil’uno dall’altro, overo in quanto sono tra loro congiunti: perche altra è la ragione del peso postoin E senza il congiungimento del peso posto in D, et altra di lui con l’altro peso congiunto, sifattamente che l’uno senza l’altro non si possa movere. Imperoche la diritta, et naturale discesadal peso posto in E, in quanto egli è senza altro congiungimento di peso, si fa per la linea ES,ma in quanto egli è congiunto col peso D, la sua naturale discesa non sarà più per la linea ES,ma per una linea egualmente distante da CS percioche la magnitudine comporta de i pesi ED,et della bilancia DE il cui centro della gravezza è C, se in nessun luogo non sarà sostenuta, simuoverà naturalmente in giù nel modo che si trova, secondo la grandezza del centro per la lineadiritta tirata dal centro della gravezza C al centro del mondo S, finche il centro C pervenga nel

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centro S […] Ma se i pesi posti in ED sono l’un l’altro fra se congiunti, et gli considereremoin quanto sono congiunti, sarà la naturale inclinazione del peso posto in E per la linea MEK,percioche la gravezza dell’altro peso posto in D fa si, che il peso posto in E non gravi soprala linea ES, ma nella EK. Il che fa parimente la gravezza del peso posto in E, cioè, che’l pesoposto in D non gravi per la linea reta DS, ma secondo DH impedirsi ambedue l’uno l’altro, chenon vadino a propri luoghi […]. Adunque il peso posto in D non moverà in su il peso postoin E. Dalle quali cose segue che i pesi posti in DE, in quanto tra loro sono congiunti, sonoegualmente gravi.

5.19 Sia la leva AB, il cui sostegno C, et sia il peso D attaccato al punto B, et sia la possanza inA movente il peso D con la leva AB. Dico lo spatio della possanza in A allo spatio del pesoessere cosi come CA à CB.Ma sia la leva AB il cui sostegno B, & la possanza movente in A, & il peso in C. Dico lo spatiodella possanza mossa allo spatio del peso trasportato essere, come BA a BC.

5.20 CorollarioDa queste cose è manifesto, che maggiore proportione ha lo spatio della possanza, che moveallo spatio del peso mosso, che il peso alla medesima possanza. Percioche lo spatio della pos-sanza allo spatio del peso ha la medesima proportione, che il peso alla possanza, che sostiene ildetto peso. Ma la possanza, che sostiene è minore della possanza che move, però il peso havràproportione minore alla possanza che lo move, che alla possanza, che lo sostiene. Lo spatiodunque della possanza che move allo spatio del peso haurà proportione maggiore, che il pesoall’istessa possanza.

5.21 Proportio ponderis in C ad idem pandus in F erit quemadmodum totius brachii .BC. ad partem.B.U.

5.22 Ad cuius rei evidentiam imaginemur filum .F.u. perpendiculare, & in cuius extremo .U. penderepondus, quod erat in .F. unde clarum erit quod eundem effectum gignet, ac si fuisset in .F. […]Idem assero si brachium esset in situ .e.B. […] Quia tantum est quod ipsum sit appensum filo Qpendet ab .u. quantum quod ab ipso liberum appensum fuisset .e. brachii .B.e. & hoc procederetab eo quod partim penderet a centro .B. & si brachii esset in situ .B.Q. totum pondus centro .B.remaneret appensum, quem admodum in situ B.A. totum dicto centrum anniteretur.

5.23 Et quamuis appellem latus .BC. orizontale, supponens illud angulum rectum cum .C.O: facere,unde angulus C.B.Q. sit ut minor sit rector, ob quantitatem unius anguli equals ei, quem duae.C.O. et B.Q. in centro regionis elementaris constitutu, hoc tamen nihil refert, cum dictus an-gulus insensibilis sit magnitudinem.

5.24 Ex iis quae nobis hucusque sunt dicta, facile intelligi potest, quantitatis B.u. quae fere perpen-dicularis es a centro .B. ad lineam .F.u. inclinationis, ea est, quae non ductis in cognitionemquantitatis virtutis ipsius F in huiusmodi situ constituens videlicet linea .F.u. cum brachio .F.B.angulum acutum.

5.25 Ut hoc tamen melius intelligamus, imaginemur libram .b.o.a. fixam in centro .o. ad cuius ex-trema sint appensa duo pondera, aut duae virtutes moventes .e. et .c. ita tamen, linea inclina-tionis .e. idest .be. faciat angulum rectum cum .o.b. in puncto .b. linea vero inclinationis .c.idest .a.c. faciat angulum acutum, aut obtusum cum .o.a. in puncto .a. Imaginemur ergo lin-eam .o.t. perpendicularem lineae .c.a. inclinationis […] secetur deinde imaginatione .o.a. inpuncto .i. ita ut .o.i. aequalis. sit .o.t. & puncto .i. appensum sit a pondus aequale ipsi .c. cuiusinclinationis linea parallela sit linea inclinationis ponderis .e. supponendo tamen pondus autvirtutem .c. ea ratione maiorem esse ea, quae est .e. qua .b.o. maior est .o.t. absque dubio ex 6lib. primi Archi. de ponderibus .b.o.i. non movebitur situ, sed si loco .o.i. imaginabimur .o.t.consolidatam cum .o.b. & per lineam .t.c. attractam virtute .c. similiter quoque contingent ut.b.o.t.; communi quadam scientiam, non moveatur situ.


5.26 Sed in secunda parte quinte propositionis non videt vigore situs eo modo, quo ipse disputat,nulla elicitur ponderis differentia quia si corpus .B. descendere debet per arcum .IL. corpus .A.ascendere debet per arcum .V.S. Haec autem quinta propositio Tartalea est secunda quaestio aIordano proposita.

5.27 Momento è la propensione di andare al basso, cagionata non tanto dalla gravità del mobile,quanto dalla disposizione che abbino tra di loro diversi corpi gravi; mediante il qual momentosi vedrà molte volte un corpo men grave contrapesare un altro di maggior gravità: come nellastadera si vede un picciolo contrapeso alzare un altro peso grandissimo, non per eccesso digravità, ma sì bene per la lontananza dal punto donde viene sostenuta la stadera; la quale,congiunta con la gravità del minor peso, gli accresce momento ed impeto di andare al basso, colquale può eccedere il momento dell’altro maggior grave. È dunque il momento quell’impeto diandare al basso, composto di gravità, posizione e di altro, dal che possa essere tal propensionecagionata.

5.28 Momento, appresso i meccanici, significa quella virtù, quella forza, quella efficacia, con laquale il motor muove e ’l mobile resiste; la qual virtù depende non solo dalla semplice gravità,ma dalla velocità del moto, dalle diverse inclinazioni degli spazii sopra i quali si fa il moto,perché più fa impeto un grave descendente in uno spazio molto declive che in un meno.Il secondo principio è, che il momento e la forza della gravità venga accresciuto dalla velocitàdel moto: sì che pesi assolutamente eguali, ma congiunti con velocità diseguali, sieno di forza,momento e virtù diseguale, e più potente il più veloce, secondo la proporzione della velocitàsua alla velocità dell’altro. Di questo abbiamo accomodatissimo esemplo nella libra o staderadi braccia disuguali, nelle quali posti pesi assolutamente eguali, non premono e fanno forzaegualmente, ma quello che è nella maggior distanza dal centro, circa il quale la libra si muove,s’abbassa sollevando l’altro, ed è il moto di questo che ascende, lento e l’altro veloce: e taleè la forza e virtù che dalla velocità del moto vien conferita al mobile che la riceve, che ellapuò compensare altrettanto peso che all’altro mobile più tardo fosse accresciuto; sì che, sedelle braccia della libra uno fosse dieci volte più lungo dell’altro, onde nel muoversi la libracirca il suo centro, l’estremità di quello passasse dieci volte maggiore spazio che l’estremità diquesto, un peso posto nella maggiore distanza potrà sostenerne ed equilibrarne un altro diecivolte assolutamente più grave che non egli è; e ciò perché, muovendosi la stadera, il minorpeso si moveria dieci volte più velocemente che l’altro.

5.29 Avendo noi mostrato come i momenti di pesi diseguali vengono pareggiati dall’essere sospesicontrariamente in distanze che abbino la medesima proporzione, non mi pare di doversi pas-sar con silenzio un’altra congruenza e probabilità, dalla quale ci può ragionevolmente essereconfermata la medesima verità. Però che, considerisi la libra AB divisa in parti diseguali nelpunto C, ed i pesi, della medesima proporzione che hanno le distanze BC, CA, alternatamentesospesi dalli punti A, B: è già manifesto come l’uno contrapeserà l’altro, e, per conseguenza,come, se a uno di essi fusse aggiunto un minimo momento di gravità, si moverebbe al bassoinnalzando l’altro; sì che, aggiunto insensibile peso al grave B, si moveria la libra discendendoil punto B verso E, ed ascendendo l’altra estremità A in D. E perché, per fare descendere ilpeso B, ogni minima gravità accresciutagli è bastante, però, non tenendo noi conto di questoinsensibile, non faremo differenza dal poter un peso sostenere un altro al poterlo movere. Ora,considerisi il moto che fa il grave B, discendendo in E, e quello che fa l’altro A, ascendendoin D; e troveremo senza alcun dubbio, tanto essere maggiore lo spazio BE dello spazio AD,quanto la distanza BC è maggiore della CA; formandosi nel centro C due angoli, DCA edECB, eguali per essere alla cima, e, per conseguenza, due circonferenze, BE, AD, simili, eaventi tra di sé l’istessa proporzione delli semidiametri BC, CA, dai quali vengono descritte.Viene adunque ad essere la velocità del moto del grave B, discendente, tanto superiore alla ve-locità dell’altro mobile A, ascendente, quanto la gravità di questo eccede la gravità di quello;né potendo essere alzato il peso A in D, benché lentamente, se l’altro grave B non si muove inE velocemente, non sarà maraviglia, né alieno dalla costituzione naturale, che la velocità delmoto del grave B compensi la maggior resistenze del peso A, mentre egli in D pigramente si

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muove e l’altro in E velocemente descende. E così, all’incontro, posto il grave A nel punto D el’altro nel punto E, non sarà fuor di ragione che quello possa, calando tardamente in A, alzarevelocemente l’altro in B, ristorando, con la sua gravità, quello che per la tardità del moto vienea perdere. E da questo discorso possiamo venire in cognizione, come la velocità del moto siapotente ad accrescere momento nel mobile, secondo quella medesima proporzione con la qualeessa velocità di moto viene augumentata.

5.30 Tal ragguagliamento tra la gravità e la velocità si ritrova in tutti gli strumenti meccanici, e fuconsiderato da Aristotele nelle sue Questioni meccaniche: onde noi ancora possiamo prenderper verissimo assunto che pesi assolutamente diseguali, alternativamente si contrappesano e sirendono di momenti uguali, ogni volta che le loro gravità con proporzione contraria rispondonoalle velocità dei lor moti.

5.31 Un’altra cosa, prima che più oltre si proceda, bisogna che sia considerata; e questa è intornoalle distanze, nelle quali i gravi vengono appesi: per ciò che molto importa il sapere comes’intendano distanze eguali e diseguali, ed in somma in qual maniera devono misurarsi. [...]Ma se, elevando la linea CB e girandola intorno al punto C, sarà trasferita in CD, sì che lalibra resti secondo le due linee AC, CD, gli due eguali pesi pendenti dai termini A, D non piùpeseranno egualmente sopra il punto C; perché la distanza del peso posto in D è fatta minordi quello che era mentre si ritrovava in B. Imperò che, se considereno le linee per le quali idetti gravi fanno impeto, e discenderebbono quando liberamente si movessero, non è dubbioalcuno che sariano le linee AG, DF, BH: fa dunque momento ed impeto il peso pendente dalpunto D secondo la linea DF; ma quando pendeva dal punto B, faceva impeto nella linea BH;e perché essa linea DF più vicina al sostegno C di quello che faccia linea BH, perciò doviamointendere, gli pesi pendenti dalli punti A, D non essere in distanze eguali dal punto C, ma sìbene quando saranno constituiti secondo la linea retta ACB.

5.32 E per amplissima confermazione e più chiara esplicazione di questo medesimo, considerisi lapresente figura (e, s’io non m’inganno, potrà servire per cavar d’errore alcuni meccanici prat-tici, che sopra un falso fondamento tentano talora imprese impossibili), nella quale al vasolarghissimo EIDF, vien continuata l’angustissima canna ICAB, ed intendasi in essi infusal’acqua sino al livello LGH; la quale in questo stato si quieterà, non senza meraviglia di alcuno,che non capirà così subito come esser possa, che il grave carico della gran mole dell’acquaGD, premendo abbasso, non sollevi e scacci la piccola quantità dell’altra contenuta dentro allacanna CL, dalla quale gli vien contesa ed impedita la scesa. Ma tal meraviglia cesserà, se noicominceremo a fingere l’acqua GD essersi abbassata solamente sino a QO, e considereremopoi ciò che averà fatto l’acqua CL. la quale, per dar luogo all’altra che si è scemata dal livelloGH sino al livello QO, doverà per necessità essersi nell’istesso tempo alzata dal livello L sinoin AB, ed esser la salita LB tanto maggiore della scesa GQ, quant’è l’ampiezza del vaso GDmaggiore della larghezza della canna LC, che in somma è quanto l’acqua GD è più della LC.Ma essendo che il momento della velocità del moto in un mobile compensa quello della grav-ità di un altro, qual meraviglia sarà se la velocissima salita della poca acqua CL resisterà allatardissima scesa della molta GD?

5.33 SAGR. Ma credete voi che la velocità ristori per l’appunto la gravità? Cioè che tanto sia ilmomento e la forza di un mobile verbigrazia, di quattro libbre di peso, quanto quella di un dicento, qualunque volta quello avesse cento gradi di velocità e questo quattro gradi solamente?SALV. Certo sì, come io vi potrei con molte esperienze mostrare: ma per ora bastivi la con-fermazione di questa sola della stadera, nella quale voi vedrete il poco pesante romano allorapoter sostenere ed equilibrare la gravissima balla, quando la sua lontanaza dal centro sopra ilquale si sostiene e volgesi la stadera, sarà tanto maggiore dell’altra minor distanza dalla qualepende la balla, quanto il peso assoluto della balla è maggior di quel del romano. E di questo nonpoter la gran balla co ’l suo peso sollevar il romano, tanto men grave, altra non si vede poteresser cagione che la disparità de i movimenti che e quella e questo far dovrebbero, mentre laballa con l’abbassarsi di un sol dito facesse alzare il romano di cento dita.


5.34 SAGR. Voi ottimamente discorrete, e nonmettete dubbio alcuno nel concedere, che per piccolache sia la forza del movente, supererà qualsivoglia gran resistenza, tutta volta che quello piùavanzi di velocità, ch’ei non cede di vigore e gravità. Or venghiamo al caso della corda: esegnando un poco di figura, intendete per ora, questa linea ab, passando sopra i due punti fissie stabili a, b, aver nelle estremità sue pendenti, come vedete, due immensi pesi c, d, li quali,tirandola con grandissima forza, la facciano star veramente tesa dirittamente, essendo essa unasemplice linea, senza veruna gravità. Or qui vi soggiungo e dico, che se dal mezzo di quella, chesia il punto e, voi sospenderete qualsivoglia piccolo peso, quale sia questo h, la linea ab cederà,ed inclinandosi verso il punto f, ed in consequenza allungandosi, costringerà i due gravissimipesi c, d a salir in alto: il che in tal guisa vi dimostro. Intorno a i due punti a, b, come centri,descrivo 2 quadranti, eig, elm; ed essendo che li due semidiametri ai, bl sono eguali alli dueae, eb, gli avanzi fi, fl saranno le quantità de gli allungamenti delle parti af, fb sopra le ae, eb,ed in conseguenza determinano le salite de i pesi c, d, tutta volta però che il peso h avesseauto facoltà di calare in f: il che allora potrebbe seguire, quando la linea ef, che è la quantitàdella scesa di esso peso h, avesse maggior proporzione alla linea fi, che determina la salita dei due pesi c, d che non ha la gravità di amendue essi pesi alla gravità del peso h. Ma questonecessariamente avverrà, sia pur quanto si voglia massima la gravità de i pesi c, d, e minimaquella dell’h.

5.35 Dei quali inganni parmi di avere compreso essere principalmente cagione la credenza, che idetti artefici hanno avuta ed hanno continuamente, di poter con poca forza muovere ed alzaregrandissimi pesi, ingannando, in un certo modo, con le loro machine la natura; instinto dellaquale, anzi fermissima constituzione, è che niuna resistenza possa essere superata da forza, chedi quella non sia più potente. La quale credenza quanto sia falsa, spero con le dimostrazionivere necessarie, che averemo nel progresso, di fare manifestissimo.[…]Ora, assegnata qual si voglia resistenza determinata, e limitata qualunque forza, e notata qual-sivoglia distanza, non è dubbio alcuno, che sia per condurre la data forza il dato peso alladeterminata distanza; perciò che, quando bene la forza fusse picciolissima, dividendosi il pesoin molte particelle, ciascheduna delle quali non resti superiore alla forza, e tranferendoseneuna per volta. Avrà finalmente condotto tutto il peso allo statuito termine: né però nella finedell’operazione si potrà con ragione dire, quel gran peso essere stato mosso e traslato da forzaminore di sé, ma sì bene da forza la quale più volte averà reiterato quel moto e spazio, che unasol volta sarà stato da tutto il peso misurato. Dal che appare, la velocità della forza essere statatanta volte superiore alla resistenza del peso, quante esso peso è superiore alla forza; poichéin quel tempo nel quale la forza movente ha molte volte misurato l’intervallo tra i termini delmoto, esso mobile viene ad aver passato una sol volta: né perciò si deve dire, essersi super-ata gran resistenza con piccola forza, fuori della costituzione della natura. Allora solamente sipotria dire, essersi superato il naturale instituto, quando la minor forza trasferisse la maggioreresistenza con pari velocità di moto, secondo il quale essa cammina; il che assolutamente af-fermiamo essere impossibile a farsi con qual si voglia machina, immaginata o che immaginarsi possa. Ma perché potria tal ora avvenire che, avendo poca forza, ci bisognasse muovere ungran peso tutto congiunto insieme, senza dividerlo il pesi, in questa occasione sarà necessarioricorrere alla machina: col mezzo della quale si trasferirà il peso proposto nell’assegnato spaziodalla data forza.[…]E questa deve essere per una delle utilità che dal mecanico si cavano, annoverata: perché inverospesse volte occore che, avendo scarsità di forza, ma non di tempo, ci occorre muovere granpesi tutti unitamente. Ma ci sperasse e tentasse, per via di machine far l’istesso effetto senzacrescere tardità al mobile, questo certamente rimarrà ingannato, e dimostrerà di non intenderela natura delli strumenti mecanici e le ragioni delli effetti loro.

5.36 E qui si deve notare (il che anco a suo luogo si anderà avvertendo intorno a tutti gli altri stru-menti mecanici) che la utilità, che si trae da tale strumento, non è quella che i volgari mecanicisi persuadono, ciò è che si venga a superare, ed in un certo modo ingannare, la natura, vincendo

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con piccola forza una resistenza grandissima con l’intervento del vette; perché dimostreremo,che senza l’aiuto della lunghezza della lieva si saria, con la medesima forza, dentro al medes-imo tempo, fatto il medesimo effetto. Imperò che, ripigliando la medesima lieva BCD, dellaquale sia C il sostegno, e la distanza CD pongasi, per esempio, quintupla alla distanza CB, emossa la lieva sin che pervenga al sito ICG, quando la forza avrà passato lo spazio DI, il pesosarà stato mosso dal B in G; e perché la distanza DC si è posta esser quintupla dell’altra CB, èmanifesto, dalle cose dimostrate, poter essere il peso, posto in B, cinque volte maggiore dellaforza movente, posta in D. Ma se, all’incontro, porremo mente al camino che fa la forza daD in I, mentre che il peso vien mosso da B in G, cognosceremo parimente il viaggio DI esserquintuplo allo spazio BG: in oltre, se piglieremo la distanza CL eguale alla distanza CB, postala medesima forza, che fu in D, nel punto L, e nel punto B la quinta parte solamente del pesoche prima vi fu messo, non è alcun dubbio, che, divenuta la forza in L eguale a quasto peso inB, ed essendo eguali le distanze LC, CB, potrà la detta forza, mossa per lo spazio LM, trasferireil peso a sé uguale per l’altro eguale intervallo BG; e che reiterando cinque volte questa medes-ima azione, trasferirà tutte le parti del detto peso al medesimo termine G. Ma il replicare lospazio ML niente per certo è di più o di meno che il misurare una sol volta l’intervallo DI,quintuplo di esso LM: adunque il trasferire il peso da B in G non ricerca forza minore, o minortempo, o più breve viaggio, se quella si ponga in D, di quello che faccia di bisogno quando lamedesima fosse applicata in L. Ed insomma il commodo, che si acquista dal benefizio dellalunghezza della lieva CD, non è altro che il potere muovere tutto insieme quel corpo grave, ilquale dalla medesima forza, dentro al medesimo tempo, con moto eguale, non saria, se non inpezzi, senza il benefizio delle vette, potuto condursi.

5.37 Finalmente non è da passare sotto silenzio quella considerazione, la quale da principio si disseesser necessaria d’avere in tutti gl’instrumenti mecanici: cioè, che quanto si guadagna di forzaper mezo loro, altrettanto si scapita nel tempo e nella velocità. Il che per avventura non potriaparere ad alcuno così vero e manifesto nella presente speculazione; anzi pare che qui si mutli-plichi la forza senza che il motore si muova per più lungo viaggio che il mobile. Essendo chese intenderemo, nel triangolo ABC la linea AB essere il piano dell’orizonte, AC piano elevato,la cui altezza sia misurata dalla perpendicolare CB, un mobile posto sopra il piano AC, e adesso legata la corda ADF, e posta in F una forza o un peso, il quale alla gravità del peso Eabbia la medesima proporzione che la linea BC alla CA; per quello che s’è dimostrato, il pesoF calerà al basso tirando sopra il piano elevato il mobile E, né maggior spazio misurerà dettograve F nel calare al basso di quello che si misuri il mobile E sopra la linea AC.

5.38 Ma qui però si deve avvertire che, se bene il mobile E averà passata tutta la linea AC nel tempomedesimo che l’altro grave F si sarà per eguale intervallo abbassato, niente di meno il grave Enon si sarà discostato dal centro comune delle cose gravi più di quello che sia la perpendicolareCB; ma però il grave F, discendendo a perpendicolo, si sarà abbassato per spazio eguale a tuttala linea AC. E perché i corpi gravi non fanno resistenza a i moti transversali, se non in quantoin essi vengono a discostarsi dal centro della terra, però, non s’essendo il mobile E in tutto ilmoto AC alzato più che sia la linea CB, ma l’altro F abbassato a perpendicolo quanto è tutta lalunghezza AC, però potremo meritamente dire, il viaggio della forza F al viaggio della forza Emantenere quella istessa proporzione, che ha la linea AC alla CB, cioè il peso E al peso F.Moltoadunque importa il considerare per quali linee si facciano imoti, emassime ne i gravi inanimati:dei quali i momenti hanno il loro total vigore e la intiera resistenza nella linea perpendicolareall’orizonte; e nell’altre, trasversalmente elevate o inchinate, servono solamente quel più omeno vigore, impeto, o resistenza, secondo che più o meno le dette inclinazioni s’avvicinanoalla perpendicolar elevazione.

5.39 Il che avendo dimostrato, faremo passaggio alle taglie, e descrivendo la girella inferiore ACB,volubile intorno al centro G, e da essa pendente il peso H, segneremo l’altra superiore EF;avvolgendo intorno ad ambedue la corda DFEACBI, di cui il capo D sia fermato alla tagliainferiore, ed all’altro I sia applicata la forza; la quale dico che, sostenendo o movendo il pesoH, non sentirà altro che la terza parte della gravità di quello. Imperò che, considerando la


struttura di tal machina, vederemo il diametro AB tener il luogo di una lieva, nel cui termineB viene applicata la forza I, nell’altro A è posto il sostegno, dal mezzo G è posto il grave H, enell’istesso luogo applicata un’altra forza D; sì che il peso vien fermato dalle tre corde IB, FD,EA, le quali con eguale fatica sostengono il peso. Or, per quello che di già si è speculato, sendole due forze eguali D, B applicate l’una al mezzo del vette AB, e l’altra al termine estremo B,è manifesto ciascheduna di esse non sentire altro che la terza parte del peso H: adunque lapotenza I, avendo momento eguale al terzo del peso H, potrà sostenerlo e muoverlo. Ma peròil viaggio della forza I sarà triplo al camino che farà il peso, dovendo la detta forza distendersisecondo la lunghezza delle tre corde IB, FD, EA, delle quali una sola misurerà il viaggio delpeso.

5.40 È la presente speculazione stata tentata ancora da Pappo Alessandrino nel’8° libro delle sueCollezioni Matematiche; ma, per mio avviso, non ha toccato lo scopo, e si è abbagliato […].Intendasi dunque il cerchio AIC, ed in esso il diametro ABC, ed il centro B, e due pesi egualimomenti nelle estremità A, C; sì che, essendo la linea AC un vette o libra mobile intorno alcentro B, il peso C verrà sostenuto dal peso A.Ma se c’immagineremo il braccio della libra BCessere inchinato a basso secondo la linea BF, in guisa tale però che le due linee AB, BF restinosalde insieme nel punto B, allora il momento del peso C non sarà più eguale al momento delpeso A, per esser diminuita la distanza del punto F dalla linea della direzione che dal sostegnoB, secondo la BI, va al centro della terra. Ma se tireremo dal punto F una perpendicolare allaBC, quale è la FK, il momento del peso in F sarà come se pendesse dalla linea KB.

5.41 Vedesi dunque come, nell’inclinare a basso per la circonferenza CFLI il peso posto nell’estre-mità della linea BC, viene a scemarsi il suo momento ed impeto d’andare a basso di mano inmano più, per esser sostenuto più e più dalle linee BF, BL.[…]Se dunque sopra il piano HG il momento del mobile si diminuisce dal suo totale impeto, qualeha nella perpendicolare DCE, secondo la proporzione della linea KB alla linea BC o BF; es-sendo, per la similitudine de i triangoli KBF, KFH, la proporzione medesima tra le linee KF,FH che tra le dette KB, BF, concluderemo, il momento integro ed assoluto che ha il mobilenella perpendicolare all’orizzonte, a quello che ha sopra il piano inclinato HF, avere la medes-ima proporzione che la linea HF alla linea FK, cioè che la lunghezza del piano inclinato allaperpendicolare che da esso cascherà sopra l’orizonte. Sì che, passando a più distinta figura,quale è la presente, il momento di venire al basso che ha il mobile sopra il piano inclinato FH,al suo totale momento, con lo qual gravita nella perpendicolare all’orizonte FK, ha la medes-ima proporzione che essa linea KF alla FH. E se così è, resta manifesto che, sì come la forzache sostiene il peso nella perpendicolare FK deve essere ad esso eguale, così per sostenerlo nelpiano inclinato FH basterà che siano tanto minore, quanto essa perpendicolare FK manca dallalinea FH. E perché, come altre volte s’è avvertito, la forza per muover il peso basta che insen-sibilmente superi quella che lo sostiene, però concluderemo questa universale proposizione:sopra il piano elevato la forza al peso avere la medesima proporzione, che la perpendicolaredal termine del piano tirata all’orizonte, alla lunghezza d’esso piano.

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A.6 Chapter 6

6.1 Ex ijs omnibus, quae hactenus de centro gravitatis dicta sunt, perspicuum est, unum quod [que]grave in eius centro gravitatis proprie gravitare, veluti nomen ipsum centri gravitatis idipsummanifeste praeseferre videtur. ita ut tota vis, gravitasque ponderis in ipso gravitatis centrocoacervata, collectaque esse, ac tanquam in ipsum undique fluere videatur.Nam ob gravitatem pondus in centrum universi naturaliter pervenire cupit; centrum vero gravitatis (exdictis) est id, quod proprie in centrum mundi tendit. in centro igitur gravitatis pon-dus proprie gravitat. Praeterea quando aliquod pondus ab aliqua potentia in centro gravitatissustinetur; tunc pondus statim manet, totaque ipsius ponderis gravitas sensu percipitur. quodetiam contingit, si susteneatur pondus in aliquo puncto, a quo per centrum gravitatis ducta rectalinea in centrum mundi tendat. hoc nam [que] modo idem est, ac si pondus in eius centro grav-itatis proprie sustineretur. Quod quidem non contingit, si sustineatur pondus in alio puncto.ne [que] enim pondus manet, quin potius antequam ipsius gravitas percipi possit, vertitur uti[que] pondus, donec similiter a suspensionis puncto ad centrum gravitatis ducta recta linea inuniversi centrum recto tramite feratur.

6.2 Dicimus autem centrum gravitatis uniuscuiusque corporis punctum quoddam intra positum, aquo si grave appensummente concipiatur, dum fertur quiescit; et servat eam, quam in principiohabebat positionem: neque in ipsa latione circumvertitur.

6.3 Centrum gravitatis uniuscuiusque solidae figurae est punctum illud intra positum, circa quodundique partes aequalium momentorum consistunt. Si enim per tale centrum ducatur planumfiguram quomodocunque secans semper in partes aequeponderantes ipsam dividet.

6.4 Poiché, sì come è impossibile che un grave o un composto di essi si muova naturalmente all’insu, discostandosi dal comun centro verso dove conspirano tutte le cose gravi, così è impossibileche egli spontaneamente si muova, se con tal moto il suo proprio centro di gravità non acquistaavvicinamento al sudetto centro comune.

6.5 Suppositiones, et definitionesI. Ponatur eam esse centri gravitatis naturam, ut magnitudo libere suspensa ex quolibet suipuncto nunquam quiescat nisi cum centrum gravitatis ad infimum suae sphaerae punctum per-venerit.VI. Aequalia gravia ex aequalibus distantijs aequiponderant, sive libra ad horizontem par-allela fuerit, sive inclinata. Et gravia eandem reciproce rationem habentia, quam distantiae,aequiponderant, sive libra sit ad horizontem parallela, sive inclinata.

6.6 Vulgatissima est etiam apud gravissimos viros obiectio illa, videlicet. Archimedem supposu-isse aliquod falsum, dum fila magnitudinum ex libra pendentium consideravit tanquam interse parallela, cum tamen re vera in ipso terrae centro concurrere debeant. Ego vero, (quod paceclarissimorum virorum dictum sit) crediderim fundamentum Mecanicum longe alia rationeesse considerandum. Concedo si Fisicae magnitudines ad libram libere suspendantur, quod filamaterialia suspensionum convergentia erunt; quandoquidem singula ad centrum terrae respi-ciunt. Verum tamen si eadem libra, licet corporea, consideretur non in superficie terrae, sed inaltissimis regionibus utra orbem solis; tum fila (dummodo adhuc ad terrae centrum respiciant)multo minus convergentia inter se erunt; sed quasi aequidistantia. Concipiamus iam ipsam li-bram Mecanicam ultra stellatam libram firmamenti in infinitam distantiam esse provectam;quis non intelligit fila suspensionum iam non amplius convergentia, sed exacte parallela fore?

6.7 Tunc itaque falsum dici poterit fundamentum Mecanicum, nempe fila librae parallela esse,quando magnitudines ad libram appensae Fisicae sint, realesque, et ad terrae centrum con-spirantes. Non autem falsum erit, quando magnitudines (sive abstractae, sive concretae sint)non ad centrum terrae, neque ad aliud punctum propinquum librae respiciant; sed ad aliquodpunctum infinite distans connitantur.


6.8 Non me latet auctorum controversiam, circa libram inclinatam, an redeat, maneatve supponerecentramagnitudinum in ipsa libra esse collocata [il corsivo è nostro]. Nos tamen, quia in libello,semper considerabimus magnitudines infra libram appensas, maluimus rei nostrae servire,quam aliorum controversiae demonstrationem accomodare.

6.9 Quando noi ammettiamo che i pesi della libbra abbiano inclinazione verso il centro della Terra[…] ne seguirà che non ci sia libbra orizzontale con braccia disuguali e con pesi in reciprocaproporzione della lunghezza delle braccia, sicché detti pesi facciano equilibrio.

6.10 Ora posto che B figuri il centro, ed AC una Libbra di braccia uguali con due pesi uguali nelleestremità A, C, i cui momenti o gravità son misurate dalle perpendicolari DF, DE, siccomedichiara Giov. Battista de’ Benedetti nel suo libro Delle speculazioni matematiche, capitoloIII ovvero IV; ne segue che il momento del peso in A, al momento del peso in C, sia recip-rocamente come la retta BC alla retta AB, cioè reciprocamente come la distanza dei pesi dalcentro della Terra. E qui abbiamo, non solamente che il peso più vicino al centro, mentre ènella Libbra, pesa più del meno vicino, ma sappiamo ancora in qual proporzione più pesa.

6.11 Propositio IIMomenta gravium aequalium super planis inaequaliter inclinatis, eandem tamen elevationemhabentibus, sunt in reciproca ratione cum longitudinibus planorum.

6.12 Scio Galileum ultimis vitae suae annis suppositionem illam demonstrare conatum, sed quiaipsius argomentatio cum lib. de Motu edita non est pauca haec de momentis gravium libellonostro praefigenda duximus; ut appareat quod Galilei suppositio demonstrari potest, et quidemimmediate ex illo Theoremate quod pro demonstrato ex Mechanicis ipse desumit in secundaparte sextae Propositionis de motu accelerato, videlicet., esse inter se ut sunt perpendiculapartium aequalium eorumdem planorum.

6.13 PraemittimusDuo gravia simul coniuncta ex se moveri non posse, nisi centrum commune gravitatis ipsorumdescendat. Quando enim duo gravia ita inter se coniuncta fuerint, ut ad motum unius motusetiam alterius consequatur, erunt duo illa gravia tamquam grave unum ex duobus compositum,sive id libra fiat, sive trochlea, sive qualibet alia Mechanica ratione, grave autem huiusmodinon movebitur unquam, nisi centrum gravitatis ipsius descendat. Quando vero ita constitutumfuerit ut nullo modo commune ipsius centrum gravitatis descendere possit, grave penitus insua positione quiescet: alias enim frustra moveretur; horizontali, scilicet latione, quae nequa?quam deorsum tendit.

6.14 Connectantur etiam aliquo imaginario funiculo per ACB ducto, adeo ut ad motum unius motusalterius consequatur.

6.15 Proposition ISi in planis inaequaliter inclinatis, eandem tamen elevationem habentibus, duo gravia constitu-antur, quae inter se eandem homologe rationem habeant quam habent longitudines planorum,gravia aequale momentum habebunt.

6.16 Duo ergo gravia simul colligata mota sunt, et eorum commune centrum gravitatis non descen-dit. Quod est contra praemissam aequilibrij legem.

6.17 Propterea magnitudines aequiponderabunt etiam dum ad libram AC suspenduntur: alias, simoverentur, commune centrum gravitatis ipsarum, quod demonstratum est esse in perpen-diculo DF, ascenderet. Quod est impossibile.

6.18 La potenza in A alla potenza in C, sta reciprocamente come la retta CB alla CA. La di-mostrazione […] dipende dalle velocità perché muovendosi così la stanga AC, radente le duelinee dell’angolo retto ABC, le velocità nelle quali sta costituito il punto A alla velocità nellaquale sta costituito il punto B, sta come la BC alla BA.

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6.19 Fra gli effetti della meccanica osservati uno se ne trova non avvertito ancora da alcuno cheio sappia e pur da esso possono derivare cognizioni di qualche momento e di molta curiosità.Supposta una muraglia verticale AE alla quale nel piano orizzontale EF sia normale la retta EF,e supposta inoltre la trave diritta BCF il di cui centro di gravità sia C, la quale coll’estremitàsuperiore B si appoggi alla muraglia accennata, e possa coll’estremità F scorrere liberamentesopra il Piano EF, si cerca la proporzione del peso della trave a quella forza, la quale applicatain F e spingendo direttamente per la direzione FE può equilibrare il momento della trave ascorrere in virtù del suo peso per le direzione EF. Si supponga la forza richiesta eguale ad unpeso attaccato nel punto Z a una corda di data lunghezza FEZ, la cui lunghezza chiameremo λ,e che passi per il punto E. Dal Centro di gravita C si abbassi sopra la FE la normale CG, e sia laragione di BF ad FC la stessa che la ragione di 1 ad x, avremo per la similitudine dei triangoliBE : CG = BF : FC = 1 : x e conseguentemente, posto P il peso della trave, sarà la distanzadell’orizzontale EF dal peso accennato, che può intendersi raccolto bel centro di gravità dellatrave, eguale a, P· CG, o veramente scrivendo invece di CG l’eguale x ·BE sarà la distanzadella retta EF dal peso P eguale ad x· P ·BE.

6.20 Sia il peso che si cerca attaccato al punto Z della corda FEZ eguale a Q, sarà la distanza delpeso accennato dall’orizzontale eguale a Q· ZE, e la distanza del centro di gravità comune deidue pesi P e Q sotto l’orizzontale eguale a Q · ZE − P · CG = Q · ZE − x · P · BE. Si intendadescritto il centro L e col raggio LM eguale a BF il quarto di una circonferenza circolare SMs.Sia la retta Ls parallela all’orizzontale e riponga l’ascissa LO eguale a BE e si tiri l’ordinataOM, la quale necessariamente sarà eguale a EF, e dal punto M si conduca la tangente allacirconferenza circolare nM alla quale sia parallela la retta LrR e si prolunghi l’ordinata MOfino che incontri la retta LrR in R, sarà la distanza dal centro di gravità comune dei due pesi Pe Q dalla retta EF, che si è dimostrata eguale a Q ·ZE − x · P· BE, eguale ancora a Q · λ − Q· EF −4 x · P · BE, cioè a dire eguale a Q · λ − Q · OM − x · P · LO.

6.21 Se i due pesi eguali A, B sono legati ad un filo, passato sopra una carrucola o altro sostegno,che possano scorrere questi staranno in equilibrio, dovunque si saranno situati.

6.22 Perché se si movessero tanto acquisterebbe l’uno che scendesse, quanto acquisterebbe l’ altroche salisse, essendo i loro modi eguali, e per linee perpendicolari. E se possibile si muovanodal sito A , B nel sito C, D; è manifesto che, giunti li centri di gravità in linea retta, il centrocomune di A, B verrà in mezzo, cioè in E, ed il centro di gravità di C, D verrà in mezzo,cioè in F; perché essendo CA, BD uguali tra toro e parallele, congiunte CD, AB si seganonella medesima proporzione e nel mezzo, onde il centro comune non si sarà mosso, e non avràacquistato niente, sicchè i gravi A, B non si moveranno dal loro sito, in che furono posti.

6.23 Ma se il peso B sarà maggiore del peso A, quello scenderà, perché il centro comune loro èfuori del mezzo della BA, come in E, più vicina al centro B, ed è in luogo può può scenderesempre per la linea perpendicolare EG.

6.24 Moveatur autem et ex semidiametro BE centro B portio circuli describatur EH, quae secet BGin H, et BF in I; Et quia EM semidiametro BK perpendicularis per B, centrum non transit,erit EM ipsa BK, hoc est, BI brevior. Abscindatur ex BI, ipsi EM aequalis LB. Erit igiturpunctum L infra punctum I, hoc est, ipso I, mundi centro propius. Necesse igitur erit ad hoc utmurus corruat, centrum gravitatis E facta circa B, conversione aliquando fieri in I, ut demumtransferri possit in H, sed I remotius est a mundi centro ipsis E, L, ascendet igitur grave contrasui naturam ex E in I, at hoc est impossibile; quod fuerat demonstrandum.

6.25 Sunt autem trianguli ABF, ACF, aequales et aequeponderantes. angulus vero AFC rectus. lung-atur EC, erit igitur maior EC, ipsa EF. Rotetur iraque triangulum circa punctum C, fiatque; EChorizonti perpendicularis, sit que GH, et per E horizonti parallela ducatur EK, moto igiturtriangulo, centrum gravitatis E translatum erit in H, sed KC aequalis est EF, minor autem ipsaCH, elevatur ergo centrum gravitatis ab E in H, nempe supra K, totum spatium KH. ex quaelevatione fit in motu difficultas.


A.7 Chapter 7

7.1 Vitruve fait mention de cette sorte de machine, dite des Grecs troclearum, la quelle a son mou-vement par le moiey de poulies [...] un bout sera attache à la mousse e l’autre bout servira pourtirer le fardeau, comme il se peut voir en la figure si l’on tire le dit bout de corde marqué G unpied en bas, le fardeau qui sera attaché à la mousse E en mesme temps levera un demi pied,et ce d’autant que la corde est passee double aux polies, ainsi si l’on tire 20. pieds de corde,le fardeau ne levera que 10 aussi un homme tirera aussi pesant avec cette machine, comme enseroient deux, si la machine estoit simple mais les deux hommes tireront en mesme temps ledouble de la hauteur savoir 20 pieds, avant que l’autre en aye tiré plus de dix, et si aux moussesil y avoit deux poulies, comme la figure M, la force seroit quadruple, mais aussi ne monteroitle fardeau que 5 pieds en tirant 20 pieds de corde.Les roues dentelees se font encores avec la mesme raison comme les precedentes, car en aug-ment tant la force, l’on augmente proportionnellement le temps [...] tellement qu’un hommeseul, fera autant de force tirant un fardeau par cette machine comme huit homme [...] ma aussisi les huit hommes son une heure à lever leur pois, l’homme sera huit heures à lever le sien.

7.2 Aux poids equilibres comme le plus pesant est au plus leger, ainsi l’espace du plus leger està l’espace du plus pesant, ainsi aussi est la perpendiculaire du mouvement du plus leger à laperpendiculaire du mouvement du plus pesant.

7.3 Car en mesme tens que le poids G descend du point C au point B, le poids D monte du pointA au point E et par consequent BC sera la perpendiculaire des poids G et EF du poids D:pourtant puisque D est à G, comme la perpendiculaire BC à la perpendiculaire EF, les poids Det G seront equilibres à raisons des leurs situations.

7.4 Maintenant par la 2. prop. nous avons veu que si CA est le bras d’une balance sur le quel soitle poids A retenue par la chorde CA qu’il ne glisse le long du bras CA, et comme CB est aCF, ainsi soit le poids A a la puissance Q ou E tirant par la chorde QA, cette puissance Q ouE tiendra la balance CA en equilibre, et la chorde QA estant attache au centre du poids A, labalance demeura deschargee, et le poids A sera soustenu partie par la puissance Q ou E, partepar le plain LN2 perpendiculaire à la balance CA; ou en la place du pla LN2 par la chorde CA,par le Scholie du 4 axiome.

7.5 Scholie VIII. [...] le poids est posé en A sur les chordes CA et QA soustenues par les puissancesC, Q ou K, E, le poids estant aux puissances comme les perpendiculaires CB et QG sont auxlignes CF et QD.

[…]

Si au dessus du poids A, dans sa ligne de direction, on prend quelque ligne comme AP, ilarrivera que si le poids A descend jusques en P, tirant avec soy les chordes et faisant monterles puissances KE, il y aura reciproquement plus grande raison du chemin que les puissancesferont en montant, au chemin que le poids fait en descendant, que du mesme poids aux deuxpuissances prises ensemble; ainsi les puissances monteroient plus à proportion, que le poidsne descendroit en les emportant, qui est contre l’ordre commun.Que si au dessus du poids A, dans sa ligne de direction, on prend une ligne, comme AV, que lepoids monte jusques en V, les chordes montants aussi emportees par les puissances KE qui de-scendent, il y aura reciproquement plus grande raison du chemin que le poids sera en montant,au chemin que les puissances seront en descendant, que les deux puissances prises ensemble,au poids; ainsi le poids monteroit plus à proportion que les puissances ne descendroient enl’emportant, e qui est encore contre l’ordre commun, dans lequel le poids ou la puissance quiemporte l’autre, sait toujours plus de chemin à proportion, que le poids ou la puissance quiest emportee. Or que les raisons des chemins que seroient les poids A et ses puissances enmontant, et descendant, soient telles que nous venons de dire, et contre l’ordre commun, on

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en, trouvera la demonstration dans nos Mechaniqes, car elle est trop longue pour estre miseicy. Partant le poids A en subsistant et demeurant en son lieu, par les raisons de la 3. Prop.demeure ainsi dans l’ordre commun, ce que nous voulions remarquer.

7.6 Qu’il ne faut ny plus ny moins de force, pour lever un cors pesant a certaine hauteur, que pouren lever un autre moins pesant a une hauteur d’autant plus grande qu’il est moins pesant, oupour en lever un plus pesant a une auteur d’autant moindre.[…]Ce qu’on accordera facilement, si on considere que l’effect doit tousiours estre proportionné al’action qui est necessaire pour le produire, et ainsy que, s’il est necessaire d’employer la forcepar la quelle on peut lever un poids de 100 livres a la hauteur de deux pieds, pour en lever una la hauteur d’un pied seulement, cela tesmogne que cetuy pese 200 livres.

7.7 Il faut sur tout considerer que j’ai parlé de la force qui sert pour lever un poids a quelquehauteur, la quelle force a toujours deux dimensions & non de celle qui sert en chasque pointpour le soutenir, la quelle n’a jamais qu’une dimension, en sorte que ces deux forces differerentautant l’une de l’autre q’une superficie differe d’une ligne. Car la mesme force que doit avoirun clou pour soustenir un poids de 100 livres un moment de tems, luy suffit pour soutenir unan durant, pourvu qu’elle ne diminue point. Mais la mesme quantité de cete force qui sert alever ce poids a la hauteur d’un pied ne suffit pas eadem numero pour le lever a la hauteur dedeux pieds, & il n’est pas plus clair que deux & deux font quatre, qu’il est clair qu’il y en fautemployer le double.

7.8 Car je ne dis pas simplement que la force qui peut lever un poids de 50 livres a la hauteur de4 pieds, en peut lever un de 200 livres a la hauteur d’un pied, mai je dis qu’elle le peut, sitant est quelle lui soit appliquée. Or est-il qu’il est impossible de l’y appliquer par le moyen dequelque machine ou autre invention qui face que ce poids ne se hausse que d’un pied, pendantque cete force agira en tout la longueur de quatre pieds, & ainsy qui transforme le rectanglepar lequel est representée la force qu’il faut pour lever ce poids de 200 livres a la hauteur d’unpied, en un autre qui soit egal & semblable a celuy qui represente la force qu’il faut pour leverun poids de 50 livres a la hauteur de 4 pieds.

7.9 Car c’est le mesme de lever 100 livres a la hauteur d’un pied, et derechef encore 100 a lahauteur d’un pied, que d’en lever 200 a la hauteur d’un pied, et le mesme aussy que d’en levercent a la hauteur de deux pieds.

7.10 L’invention de tous ces engins n’est fondée que sur un seul principe, qui est que la mesmeforce qui peut lever un poids, par exemple, de cent, livres a la hauteur de deux pieds, en peutaussy lever un de 200 livres, a la hauteur d’un pied, ou un de 400 a la hauteur d’un demi pied,& ainsy des autres, si tant est qu’elle luy soit appliquée.[…]Or les engins qui servent a faire cete application d’une force qui agist par un grand espace aun poids qu’elle fait lever par un moindre, sont la poulie, le plan incliné, le coin, le tour ou laroue, la vis le levier; et quelques autres.

7.11 La poulie. Soit ABC une chorde passée autour de la poulie D, a laquelle poulie soit attachéle poids E. Et premierement supposant que deux hommes soutienent ou haussent egalementchascun un des bouts de cete chorde, il est evident que si ce poids pese 200 livres, chascun deces hommes n’employera, pour le soutenir ou soulever, que la force qu’il faut pour soutenirou soulever 100 livres; car chascun n’en porte que la moitié. Faisons apres cela qu’A, l’un desbouts de cete chorde, estant attaché ferme a quelque clou, l’autre C soit derechef soutenu parun homme; & il est evident que cet homme, en C, n’aura besoin, non plus que devant, poursoutenir le poids E, que de la force qu’il faut pour soutenir cent livres: a cause que le clouqui est vers A y fait le mesme office que l’homme que nous y supposions auparavant. Enfin,posons que cet homme qui est vers C tire la chorde pour faire hausser le poids E; & il estevident que, s’il y employe la force qu’il faut pour Iever 100 liures a la hauteur de deux pieds,


il fera hausser ce poids E, qui en pese 200, de la hauteur d’un pied car la chorde ABC estantdoublée comme elle est, on la doit tirer de deux pieds par le bout C pour faire autant hausserle poids E que si deux hommes la tiroient, l’un par le bout A & L’autre par le bout C, chascunde la longueur d’un pied seulement. Il y a toutefois une chose qui empesche que ce calcul nesoit exact, a scavoir la pesanteur de la poulie, & la difficulté qu’on peut voir a faire couler lachorde & a la porter. Mais cela est fort peu a comparaison de ce qu’on leve, & ne peut estreestimé qu’a peu pres.

7.12 Ainsy donc, pour ne point faillir, de ce que le clou A soutient la moitié du poids B, on ne doitconclure autre chose sinon que, par cete application, l’une des dimensions de la force qui doitestre en C, pour lever ce poids, diminué de moitié, & que l’autre en suite devient double. Defaçon que, si la ligne FG represente la force qu’il faudroit pour soutenir en un point le poids B,sans l’ayde d’aucune machine, & le rectangle GH, celle qu’il faudroit pour le lever a la hauteurd’un pied, le soutien du clou A diminué de moitié la dimension qui est représentée par la ligneFG, & le redoublement de la chorde ABC fait doubler l’autre dimension, qui est representéepar la ligne FH; & ainsy la force qui doit estre en C, pour lever le poids B a la hauteur d’unpied, est representée par le rectangle IK. Et comme on sçait en Geometrie qu’une ligne estantadioustée ou ostée d’une superficie, ne l’augmente ny ne la diminué de rien du tout, ainsydoit on icy remarquer que la force dont le clou A soutient le poids B, n’ayant qu’une seuledimension, ne peut faire que la force en C, considérée selon ses deux dimensions, doive estremoindre pour lever ainsy le poids B que pour le lever sans poulie.

7.13 La levier. Et pour mesurer exactement qu’elle doit estre cete force en chasque point de la tignecourbe ABCDE, il faut scavoir qu’e!le y agit tout de mesme que Ii elle trainoit le poids fur unplan circulairement incliné, & que l’inclination de chascun des poins de ce plan circulaire sedoit mesurer par celle de la ligne droite qui touche le cercle en ce point. Comme par exemplequand la force est au point B, pour trouver la proportion qu’elle doit avoir avec la pesanteurdu poids qui est alors au point G, il faut tirer la contingente GM, & penser que la pesanteur dece poids est a la force qui est requise pour le trainer sur ce plan, & par consequent aussy pourle hausser suivant le cercle FGH, comme la ligne GM ella SM. Puis a cause que BO est triplede OG, la force en B n’a besoin d’estre a ce poids en G, que comme le tiers de la ligne SM esta la toute GM. Tout de mesme quand la force est au point D.

7.14 Plusieurs ont coustume de confondre la consideration de l’espace avec celle du tems ou de lavitesse, en sorte que, par exemple, au levier, ou, ce qui est le mesme, en la balance ABCD,ayant suppose que le bras AB est double de BC, & que le poids en C est double du poids en A,& ainsy qu’ils sont en equilibre, au lieu de dire que ce qui est cause de cet equilibre est que,si le poids C soulevoit ou bien estoit soulevé par le poids A, il ne passeroit que par la moitéd’autant d’effect que luy, il disent qu’il iroit de la moitié plus lentement, ce qui est une fauted’autant plus nuisible qu’elle est plus malaysée a reconnoistre; car ce n’est point la differencede la vitesse qui fait que ces poids doivent estre l’un double de l’autre, mais la difference del’espace.

7.15 Comme il paroist de ce que, pour lever, par exemple, le poids F avec la main iusques à G, iln’y faut point employer une force qui soit iustement double de celle qu’on y aura employéele premier coup, si on le veut lever deux fois plus viste; mais il y en faut employer une quisoit plus ou moins grande que la double, selon la diverse proportion que peut avoir cete vitesseavec les causes qui luy resistent; au lieu qu’il faut une force qui soit justement double pour lelever avec mesme vitesse deux fois plus haut, a sçavoir jusques a H. Je dis qui soit justementdouble, en contant qu’un & un sont justement deux: car il faut empolyer certaine quantité decete force pour lever ce pois d’F jusques a G, & derechef encore autant de la mesme force pourle lever de G jusques a H.

7.16 Et au contraire, prenant un eventail en vostre main, vous le pourrez hausser, de la mesmevistesse qu’il pourrait descendre de soy mesme dan l’air, si vous le laissez tomber, sans qu’ilvous y faille employer aucune force, excepté celle qu’il faut pour le soustenir; mais pour le

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hausser ou baisser deux fois plus viste, il vous faudra employer quelque force qui sera plusque double de l’autre, puis qu’elle estoit nulle.

7.17 Or la raison qui fait que je reprens ceux qui se servent de la vitesse pour expliquer la forcedu levier, & autres semblantes, n’est pas que je nie que la mesme proportion de vitesse ne s’yrencontre tousiours; mais pouce que ceste vitesse ne comprend pas la raison pour laquelle laforce augmente ou diminue, comme fait la quantité de l’espace, & qu’il y a plusieurs autreschoses à considerer touchant la vitesse, qui ne sont pas aysées à expliquer.

7.18 Pour ce qu’a écrit Galilee touchant la balance & le levier, il explique fort bien quod ita sit,mais non pas cur ita sit, comme je fais par mon Principe. Et pour ceux qui disent que le devoisconsiderer la vitesse, comme Galilée, plutost que l’espace, pour rendre raison des Machines,je croy, entre nous, que ce sont des gens qui n’en parlant que par fantaisie, sans entendre rienen cette matiere.

7.19 Que la pesanteur relative de chaque cors, ou ce qui est le mesme, la force qu’il faut employerpour le soutenir & empescher qu’il ne descende, lors qu’il est en certaine position, se doitmesurer par le commencement du mouvement que devroit faire la puissance qui le soustient,tant pour le hausser que pour le suivre s’il s’abaissoit. En sorte que la proportion qui est en-tre la ligne droite que descriroit ce mouvement, & celle qui marqueroit de combien ce corss’approcheroit cependant du centre de la terre, est la mesme qui est entre la pesanteur absolute& la relative.

7.20 Soit AC un plan incliné sur l’horizon BC, et qu’AB tende a plomb vers le centre de la terre.Tous ceux qui escrivent des Mechaniques assurent que la pesanteur du poids F, en tant qu’ilest appuié sur ce plan AC, a mesme proportion a sa pesanteur absolue que la ligne AB a laligne AC.[…]Ce qui n’est pas toutefois entierement vray, sinon lorsqu’on suppose que les cors pesans tendenten bas suivant des lignes paralleles, ainsy qu’on fait communement, lors qu’on ne considereles Mechaniques que pour les rapporter a l’usage; car le peu de difference que peut causerl’inclination de ces lignes, entant qu’elles tendent vers le centre de la terre, n’est point sensible.[…]Et pour scavoir combien il pese en chascun des autres points de ce plan au regard de cetepuissance, par exemple au point D, il faut tirer une ligne droite, comme DN, vers le centre dela terre, et du point N, pris a discretion en cete ligne, tirer NP, perpendiculaire sur DN, quirencontre AC au point P. Car, comme DN est a DP, ainsy la pesanteur relative du poids F enD est a sa pesanteur absolue.

7.21 Notez que le dis commencer a descendre, non pas simplement descendre, a cause que ce n’estqu’au commencement de cete descente a laquelle il faut prendre garde. En sorte que si, parexemple, ce poids F n’estoit pas appuié au point D sur une superficie plate, comme est supposéeAD C, mais sur une spherique, ou courbée en quelque autre facon, comme EDG, pourvu que lasuperficie plate, qu’on imagineroit la toucher au point D, sur la mesme que ADC, il ne peseroitny plus ny moins, au regard de la puissance H, qu’il fait estant appuié sur le plan AC. Car, bienque le mouvement que seroit ce poids, en montant ou descendant du point D vers E ou vers Gsur la superficie courbe EDG, sust tout autre que celuy qu’il seroit sur la superficie plate ADC,toutefois, estant au point D sur EDG, il seroit determiné a se mouvoir vers le mesme costé ques’il estoit sur ADC, a scavoir vers A ou vers C. Et il est evident que le changement qui arrive ace mouvement, sitost qu’il a cessé de toucher le point D, ne peut rien changer en la pesanteurqu’il a, lorsqu’il le touche.

7.22 Itaque theoriae magis insistendum puto, in qua si quis exercitatus fuerit, nullo negotio illam inopus educere poterit, idque sine periculo fiet, cum vulgo non pateat. Alioqui periculum est, ne siparticularia tradantur iis contenti homines, ut fieri solet universalem cognitionem & causaruminquisi tionem negligant, pereatque scientia.


7.23 Theorema IDuarum virium connexarum, quarum (si moveantur) motus erunt ipsis >antipeponjwz propor-tionales neutra alteram movebit, sed equilibrium facient.

7.24 Ut igitur hunc tractatulum concludamus, ac velut in summam contrahamus: In motibus ciendistria sunt consideranda. Vis qua motum ciere volumus, vis quam movere volumus, & motumquo movere volumus: duo enim quælibet ex illis tertium determinant. Si enim vi parva vimmagnammovere volumus, id nonnisi parvo motu facere possumus: si vero vim aliquammagnomotu movere velimus, vi magna movente ad id opus est [...]: ut puta, si libra una centum librasmovere velimus, oportet motum illius, motu huius centuplo maiorem esse. Si vero velimuslibra una aliam vim ita movere, ut ea centuplo citius moveatur, quam librae illius pondus,illam centesimam tantum librae unius partem esse necesse est: si vero libram unam ita moverevelimus, ut centuplo citius moveatur, quam vis quae illam movebit, vi centum libris maioread id opus erit. Neque patitur natura sibi in his vim fieri: si enim eiusmodi proportio aliquomodo infringi posset, statim daretur a>ut’wma >end’elecez, vel ut vocant, motus perpetuus inperpetua materia.

7.25 D’où il paroît qu’un vaisseau plein d’eau est un nouveau principe demécanique, et unemachinenouvelle pour multiplier les forces à tel degré qu’on voudra, puisqu’un homme, par ce moyen,pourra enlever tel fardeau qu’on lui proposera.Et l’on doit admirer qu’il se rencontre en cette machine nouvelle cet ordre constant qui se trouveen toutes les anciennes; savoir, le levier, le tour, la vis sans fin, etc., qui est, que le chemin estaugmenté en même proportion que la force. Car il est visible, comme une de ces ouverturesest centuple de l’autre, si l’homme qui pousse le petit piston, l’enfonçoit d’un pouce, il nerepousseroit l’autre que de la centième partie seulement: car comme cette impulsion se fait acause de la continuité de l’eau, qui communique de l’un des pistons à l’autre.

7.26 Je prends pour principe, que jamais un corps ne se meut par son poids, sans que son centre degravité descende. D’où je prouve que les deux pistons figurés en la figure 7, sont en équilibreen cett sorte; car leur centre de gravité commun est an point qui divise la ligne, qui joint leurscentres de gravité particuliers, en la proportion réciproque de leurs poids; qu’ils se meuventmaintenant, s’il est possible: donc leurs chemins seront entre eux comme leurs poids récipro-quement, comme nous avons fait voir: or, si on prend leur centre de gravité commun en cetteseconde situation, on le trouvera précisément au même endroit que la première fois; car il setrouvera toujours au point qui divise la ligne, qui joint leurs centres de gravité particuliers, en laproportion réciproque de leurs poids; donc à cause du parallélisme des lignes de leurs chemins,il se trouvera en l’intersection des deux lignes qui joignent les centres de gravité dans les deuxsituations: donc le centre de gravité commun sera au même point qu’auparavant: donc les deuxpistons considérés comme un seul corps, se sont mus, sans que le centre de gravité communsoit descendu; ce qui est contre le principe: donc its ne peuvent se mouvoir: donc its seront enrepos, c’est-à-dire, en équilibre; ce qu’il falloit démontrer.

7.27 I Definitio. Statica est quae ponderis et gravitatis corporum rationes, proportiones, et qualitatesinterpretatur.

7.28 Ex his consequens est nullum corpus sive solidum in rerumNatura esse, ut mathematice loquar,praeter Globum, quod est suae gravitatis centro cogitatione suspensum, quemlibet datum situmretinet, sive per quod planum quodlibet ipsum corpus in partes situ aequipondias dividit, verumpropter varios et infinitos situs, varia etiam et infinita gravitatis centra erunt.

7.29 Verum, quandoquidem discriminem illud, in iis quae ab hominibus ponderantur, nullum,saltem inobservabile est, iugum enim aliquot milia longum esse debet, antequam deprehendiposset, perpendiculares parallelas habendas esse concedi nobis postulamus.

7.30 BN ducatur, secans AC continuatam in N, consimiliter D O secans continuatam LI, hoc est, la-tus columnæ in O, ut angulus IDO aequalis sit angulo CBN. Appendatur quoque ad DO pondus

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P oblique attollens, quod (amotis M, E ponderibus) columnam in suo situ conservet. Quia veroDL & BA, item DI & BC latera triangulorum DLI & BAC homologa sunt, hujusmodi conclu-sio inde deducitur. Quemadmodum BA ad BC: ita sacoma lateris B A ad anti sacoma laterisBC (per 2 consectarium) item quemadmodum DL ad DI: ita sacoma lateris DL ad antisacomalateris DI, hoc estita M ad E. sed homologa latera triangulorum similium ABN, LDO sunt AB& DL, item BN, & DO. Itaque ut supra, quemadmodum BA ad B N: ira sacoma B A ad antisacoma B N (per 1 consectarium) Et quemadmodum DL ad DO: ita illius sacoma ad hujus antisacoma, id est, M ad P. si linea BN à puncto B aliovorsum; A scilicet versus, ultra BC fuissetducta, etiam recta DO à D ultra DI cecidisset, hoc est, ut nunc citra: ita tunc ultra cecidisset,& praecedens demonstratio etiam isti situi accommoda fuisset, hoc est, quemadmodum BA adBN ita sacoma lateris BA, ad anti sacoma lateris BN esset: & quem-admodum DL ad DO: itasacoma lateris DL, ad anti sacoma lateris DO. hoc est M ad P. Ut ista proportio non tantumin exemplis valeat, in quibus linea attollens, ut DI, perpendicularis est axi, sed etiam in aliiscuiusmodi cunque sint anguli.

7.31 Si columna, & duo pondera oblique extollentia situ aequilibria sunt, erit quemadmodum lineaoblique extollans, ad lineam rectè extolletem: ita ponderum quodque obliquum ad suumpondusrectum.

7.32 Causam aequilibritatis situs non esse in circulis ab extremitatibus radiorum descriptis.Cur pondera aequalia in aequalibus radiis situ aequiponderent, communi notione scitur: at nonperinde patet causa aequamenti ponderum inaequalium in radiis disparibus, quique ponderibussuis reciproce proportionales sint hanc veteres circuli de circinatis a radiorum extremitatibus inesse crediderunt, quemadmodum apud Aristotelem in Mechanicis eiusque spectatores viderelicet, quod falsum esse hoc pacto redarguimus.Quiescen nullum describit circulum,Duo situ aequilibria quiescunt,Itaque duo situ aequilibria nullum describunt circulum.Et consequenter nullus erit circulus; atqui sublato circulo etiam causa tollitur quae ipsi subest,quae causa aequilibrium situs in circuli hic non latet.

7.33 Ipsique globi ex sese continuum et aeternum motum efficient, quod est falsum.

7.34 Propositio. Ponderum trochleis sublime tractorum formas inquirere.Priusquam rem ipsam exordimur generaliter intelligito, et cogitatione concipito, datum pondushic constitui a trochlea infima cum pondere ipsi alligato: praeterea differentiam gravitatis quaea funibus existit, nullius momentia nobis nunc aestimari. I Exemplum ponderum quae rectaattolluntur.Esto in primo hoc diagrammate trochlea A, ex qua dependet pondus B, funis CD F, cuius duaepartes CD, FE parallelae sint, et utraque horizonti perpendicularis, Quibus positis, totoquepondere B ita è duabus istis partibus CD, FE suspenso, ututraque pars pari potentia afficiatur,etiam singulis propter orbiculi volubilitatem cedet semissis ponderis B. quamobrem si quismanu sua funem in F sustineat, is ferret gravitatem dimidii ponderis B, ex quo liquet, curetiam unica trochlea facilius, quam sine ea pondus attollatur.

7.35 Notato autem hic illud Staticum axioma etiam locum habere: ut spatium agentis, ad spatiumpatients: Sic potentia patientis, ad potentiam agentis.

7.36 Sit [nodus] in E. Ergo PE raccourci de DEQE de AESE enlongé de BE

Optortet DE in p + AE in q − BE in t = 0.


7.37 CA in P − CD in R − CO in S = 0

sin TEP in P − sin TER in R − sin TES in S = 0sin PER in R − sin PES in S − sin PET in T = 0sin RES in S − sin RET in T − sin REP in P = 0sin SET in T + sin SEP in P − sin SER in R = 0

ou

p =f r− ct

er =

es+atd

s =ap−br

et =

dp− f ttb

.

7.38 Gravitas, est vis motrix, deorsum; sive ad Centrum Terrae.Quodnam sit, in consideratione physica, Gravitatis principium, non hic inquisimus. Nequeetiam, an Qualitas dicit debeat, aut, corporis Affectio; aut, quo alio nomine censeri par sit.Sive enim ab innata qualitate in ipso gravi corpore; sive a communi circumstatium vergentia adcentrum; sive ab electrica vel magnetica Terrae facultate, quae gravia ad se alliciat; et effluviissuis, tamquam catenulis, attrahat; sive alias undecunque; (de quo non est ut hic moveamuslitem) sufficit ut Gravitas nomine, eam intelligamus, quam sensu deprehendimus, Vim deorsummovendi, tum ipsum Corpus grave, tum quae obstant minus efficacia impedimenta.Per pondus intelligo gravitatis mensuram.

7.39 Prop. IGravia, caeteris paribus, gravitant in ratione Ponderum. Et, universaliter, ViresMotrices, quae-libet, agunt pro Virium ratione.

7.40 Prop. IIGrave, quatenus non impeditur, Descendit; seu propius ad Terrae Centrum appropinquat.Et universaliter, vis quaevis Motrix, secundum Directionem suam, quatenus non impeditur,procedit.

7.41 Prop. IIIGrave tantumdemDescendit, quanto sit Terra Centro propius: Tanto ascendit, quanto remotius.Et Universaliter, Cuiusvis Vis Motricis progressus tantus est, quantum secundum directionemsuam movetur; Regressus, Contra.

7.42 Prop. VGravium Descensus, invicem comparati, in ea ratione pollent, quae ex Ponderum ratione etratione Altitudinum Descensuum componitur. Atque Ascensus similiter.Adeoque; Sit Pondera sint aequalia, in ratione Altitudinum: si Altitudines sint aequales; inratione Ponderum: si Pondera et Altitudines, vel utraque sint aequalia, vel sint reciproce pro-portionalia; Aequipollent.Et, universaliter, ViriumMotricium, quarumcunque Progressus Regressive, pollent in Ratione,qua ex ratione Virium, et Progressuum Regressuumve secundum lineam Directionis ViriumAestimatorum, componitur.

7.43 Prop. VIConjunctis invicem Descensu et Ascensu; si praepollet Descensus, pro Descensu simpliciterhabendi sunt: Pro Ascensu vero, si Ascensu Praepollet: (ET quidem utrubique tanto, quantaest praepollentia:) Sin aequipollent, pro Neutro.Si vero, vel plures Ascensus conjuncti sint, vel plures Descensus: tantundem simul pollentatque eorundem summa.

7.44 Exempli gratia […] puta, Descensus Ponderis 2P per Altitudinem 3d, cum Descensu Ponderis3P per altitudinem 2D comparatus; Aequipollebit, (propter 2× 3 = 3× 2;) adeoque quae sicmovenda sunt, Aequiponderabunt. At descensus Ponderis 2 P per Altitudinem 4 D, Descensui

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Ponderis 3 P per Altitudinem 2D, praepollebit, (propter 2× 4 > 3× 2;), Adeoque, quod sicmovendum erit, praeponderabit. Et in aliis similiter.

7.45 Prop XVLineae curvae Declivitas, in singulis respective punctis; eadem est atque rectae ibidem contin-gentis. Et superficiei curvae, eadem atque ibidem contingenti plani. Quod aliis perinde atquegravium motibus accommodabitur.

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8.1 Hinc Vis quoque duplex: alia elementaris, quam et mortuam appello, quia in ea nondum exis-tit motus, sed tantum solicitatio ad motum, qualis est globi in tubo, aut lapidis in funda, etiamdum adhuc vinculo tenetur; alia vero vis ordinaria est, cum motu actuali conjuncta, quam vocovivam. Et vis mortuae quidem exemplum est ipso vis centrifuga itemque vis gravitatis seu cen-tripeta, vies etiam qua Elastrum tensum se restituere incipit. Sed in percussione, quae nascitera gravi iam aliquamdiu cadente, ant ab arcu se aliquamdiu restituente, aut a simili causa vis estviva, ex infinitis vis mortua impressionibus continuatis nata.

8.2 Eodem modo etiam fit, ut gravi descendente, si fingatur ei quovis momento nova aequalisquedari celeritatis accessio infinite parva, vis mortuae simul et vivae aestimatio observetur, nempeut celeritas quidem aequabiliter crescat secundum tempora, sed vis ipsa absoluta secundumstatis seu tempore quadrata, id est secundum effectus. Ut ita secundum analogiam Geometriaeseu analysis nostrae solicitationes sint ut dx, celeritates ut x, vires ut xx seu ut

∫xdx.

8.3 Et est à propos de considerer que l’équilibre consiste dans un simple effort (conatus) avant lemouvement, et c’est que j’appelle la force morte qui a la mesme raison à l’égard de la forcevive (qui est dans le mouvement mesme) que le point à la ligne. Or, au commencement dela descente, lorsque le mouvement est infiniment petit, les vitesses ou plutôt les élémens desvitesses sont comme les descentes, au lieu qu’après l’accélération, lorsque la force est devenuevive, les descentes sont comme les carré des vitesses.

8.4 Je suppose deux lignes droites quelconques données AC, BD, que je prends pour deux rangsde petits ressorts égaux & également bandez. je suppose de plus, que deux boules égales com-mencent à se mouvoir des points C & D, vers F & J, lorsque les ressorts commencent à sedilater: Soient CML, DNK deux lignes courbes, dont les appliquées GM, HN expriment lesvitesses acquises aux point G & H: je nomme BD = a , l’abscisse DH = x, sa differentielle HP,ou NT = dx, l’apliquée HN= v, sa differentielle dv: Je prends ensuite les abscisses CG, CE dela courbe CML, telles qu’elles soient aux abscisses de la courbe DNK, comme AC est à BD,ou, ce qui est, la même chose, je fais BD : AC = DH: CG = DP: CE. Suposant donc AC = na,on aura CG = nx, GE = ndx; soit enfin l’apliquée GM = z. Tout ceci supposé, je raisonne ainsi.2. Les boules étant parvenues aux points G &H, chaque ressort, tant de ceux qui étoient resser-rez dans l’intervalle AC, que de ceux qui l’étoient dans l’interval BD, sera dilaté également,parce que AC : CG = BD : DH; chacun de ces ressorts aura donc perdu, de part & d’autre, unepartie égale de son élasticité, & il leur en restera par conséquent à chacun également. Doncles pressions & les forces morte, que les boules en reçoivent, sont aussi égals entr’elles: jenomme cette pression p. Or l’accroissement élementaire de la vitesse en H, je veux dire la dif-ferentielle TO, ou dv, est, par la loi connue de l’acceleration, en raison composée de la forcemotrice, ou de la pression p, & du petit tems que le mobile met parcourir la differentielle HP,ou dx, lequel tems s’exprime par HP : HN = dx : v; On aura donc dv = pdx : v, & partantvdv = pdx, ce qui donne par l’intégration 1

2 vv =∫pdx. Par la même raison on a dz = p× GE

: GM = p×ndx : z, par conséquent zdz = npdx; & en intégrant 12 zz = n

∫pdx, d’où il suit que

vv : zz =∫pdx : n

∫pdx = 1 : n = a : na = BD : AC. Or B D est à AC, comme la force vive

acquise en H est à la force vive acquise en G. Donc ces deux forces sont entr’elles comme vv à


zz; ainsi les forces vives des corps égaux en masses ont comme les quarrez de leurs vitesses, &les vitesses elles mêmes sont en raison sousdoublée, ou comme les racines quarrées des forcesvive.C.Q.F.D.

8.5 La distinction que vous faites entre la force des poids et celle des vents n’est point une raisond’admettre le principe de statique pour ceux-là et de le rejeter pour ceux-ci, car cette distinctionne regarde que les causes productrices des forces. Or il n’est pas question de savoir comment lesforces sont produites, il suffit qu’elles soient existantes; de quelque cause qu’elles proviennent,elles feront toujours la même impression, la même action, par conséquent le même effet pourvuque ces forces soient appliquées de la même manière.

8.6 Je me sers ici du mot de puissance an lieu de celui de force, afin de me rendre plus intelligibleen faisant voir que la force des vents n’a rient de singulier pour la distinguer sur un autre genrede puissance continuellement et uniformément appliquée.

8.7 Je suis peut être un des plus zelés defenseurs de la composition des forces, commeVous l’aurezvi dans mon livre et en ben d’autres occasions; mais permettez moi que je Vous dise que Vousabusez ici de ce grand principe de Mechanique: Vous n’en faites pas une bonne application ànotre sujet; pour Vous le faire voir, voyons en quoi consiste ce principe; c’est principalementen deux cas: le premier est, lorsque deux forces mortes agissantes ensemble, mais suivant dif-férentes directions, elles en font naitre une troisième moyenne; le second de ces cas est, lorsquedeux forces vivantes s ’appliquent immediatement et dans un moment suivant differentes di-rections sur un corps mobile, qui lui imprimeroient chacune séparément de certaines vitesses,ces forces produiront dans le mobile, si elles agissent ensemble, une vitesse moyenne, qui seracomme dans le cas des forces mortes la diagonale du parallelogramme.[…]Pour en venir à notre sujet, le premier de nos deux cas n’y fait rien, car il ne s ’agit pas ici deforces mortes; le second n’y scauroit être appliqué non plus, car le vaisseau n’est pas poussépar le vent comme une bille par un seul choc instantané, mais par une force continuellementappliquée.

8.8 Cependant comme le vent agit tout autrement sur la voile par sa continuation, on peut consid-érer son action comme des bouffées reiterées à tout moment, dont chacun ajoute un nouveaudegrez infiniment petit de vitesse au vaisseau, jusqu’à ce que la vitesse totale du vaisseau soitsi grande que le vent ne lui en puisse plus rien ajouter, ce qui arrive quand le vaisseau, commeje l’ai, dit, fuit le vent de toute la vitesse absolue du vent.

8.9 Dans la demonstration que vous faites de l’equilibre des poids vous dites que les puissancesou les forces sont comme les produits des masses par les vitesses; cela est tres vrai dans unbon sens, mais prenez garde si dans l’application que faites à l’equilibre des trois voiles vousne confondez pas la puissance ou la force avec l’energie de la puissance ou de la force; et sivous ne confondez pas la vitesse actuelle du vent la quelle multipliée par la masse produit laforce absolue, avec la vitesse virtuelle, laquelle étant multipliée avec la force absolue produitle momentum ou l’energie de cette force.

8.10 J’entends par vitesse virtuelle la seule disposition à se mouvoir que les forces ont dans unparfait equilibre, où elles ne se meuvent pas actuellement. Ainsi dans votre figure 1 qui est icila seconde, si ce poids B inseparable de la ligne MB, est en equilibre avec les poids N et O,sa vitesse virtuelle est la petite ligne BP, et les vitesses virtuelles de N et O sont CP et RP; etalors le produit du poids B par BP, ce qui fait l’energie du poids B, est egal aux deux prodnitsdu poids N par CP, et du poids O par RP lesquels sont leur energies; C’est pourquoy eviterl’equivoque, an lieu de dire que leurs puissances ou les forces sont comme les produits desmasses par leurs vitesse vous auriez peut-être mieux fait de vous exprimer ainsi, les energiesdes puissances ou des forces sont comme les produits de ces puissances ou de ces forces parles vitesses virtuelles.

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8.11 Le point essentiel en pouvant être mis sur une demi page; mais c’est là justement où M. Renause trompe grossierement, en ce qu’il confond les forces des vents avec les energies des forces,oubliant que pour avoir l’energie ou ce que les Latins appellent momentum de la force du vent,il ne suffit pas de prendre, comme il fait, le quarré de la vitesse du vent, ce qui ne donneroitque la simple force du vent, mais qu’il faut multiplier ce quarré de la vitesse avec sa vitessevirtuelle, c’est à dire avec l’eloignement du centre d’appui, autour duquel la force appliquéetend à se mouvoir.

8.12 Concevez plusieurs forces différentes qui agissent suivant différentes tendances ou directionspour tenir en équilibre un point, une ligne, une surface, ou un corps; concevez aussi que l’on im-prime a tout le système de ces forces un petit mouvement, soit parallèle a soi-même suivant unedirection quelconque, soit autour d’un point fixe quelconque: il vous sera aise de comprendreque par ce mouvement chacune de ces forces avancera on reculera dans sa direction, a moinsque quelqu’une ou plusieurs des forces n’ayent leurs tendances perpendiculaires a la directiondu petit mouvement; auquel cas cette force, ou ces forces, n’avanceroient ni ne reculeroientde rien; car ces avancemens ou reculemens, qui sont ce que j’appelle vitesses virtuelles, nesont autre chose que ce dont chaque ligne de tendance augmente ou diminue par le petit mou-vement; et ces augmentations ou diminutions se trouvent, si l’on tire une perpendiculaire al’extremite de la ligne de tendance de quelque force, la quelle perpendiculaire retranchera dela meme ligne de tendance, mise dans la situation voisine par le petit mouvement, une petitepartie qui sera la mesure de la vitesse virtuelle de cette force. Soit, par exemple, P un pointquelconque dans le système des forces qui se soutiennent en équilibre; F, une de ces forces,qui pousse ou qui tire le point P suivant la direction FP on PF; Pp, une petite ligne droite quedécrit le point P par un petit mouvement, par le quel la tendance FP prend la direction fp, quisera ou exactement parallele a FP, si le petit mouvement du système se fait en tous les pointsdu système parallèlement a une droite donnée de position; ou elle fera, étant prolongee, avecFP, un angle infiniment petit, si le petit mouvement du système se fait autour d’un point fixe.Tirez donc PC perpendiculaire sur fp, et vous aurez Cp pour la vitesse virtuelle de la force F, ensorte que F × Cp fait ce que j’appelle Energie. Remarquez que Cp est on affirmatif on négatifpar rapport aux autres: il est affirmatif si le point P est poussé par la force F, et que l’angleFPp soit obtus; il est négatif, si l ’angle FPp est aigu; mais au contraire, si le point P est tire,Cp sera négatif lorsque l’angle FPp est obtus; et affirmatif lorsqu’il est aigu. Tout cela étantbien entendu, se forme cette Proposition generale: En tout équilibre de forces quelconques, enquelque maniere qu’elles soient appliquées, et suivant quelques directions qu’elles agissent lesunes sur les autres, on mediatement, on immediatement, la somme des Energies affirmativessera egale a la somme des Energies négatives prises affirmativement.

8.13 Donc nous aurons A × pm = B× pn + C × 0 = B× pn c’est a dire A : B = pn : pm = sinus del’angle pPn : sinus de l’angle pPm.

8.14 J’appelle vitesses virtuelles, celles que deux ou plusieurs forces mises en équilibre acquirent,quand on leur imprime un petit mouvement; ou si ces forces sont déja enmouvement. La vitessevirtuelle est l’élement de vitesse, que chaque corps gagne ou perde, d’une vitesse déja acquise,dan un tems infiniment petit, suivant sa direction.

8.15 Deux agens sont in équilibre, ou ont des momens égaux, lorsque leurs forces absolues sont enraison reciproque de leurs vitesses virtuelle; soit que les forces qui agissent l’une sur l’autresoins en mouvement, ou en repose.C’est un principe ordinaire de Statique & Méchanique; que je ne m’arrêterai pas à démontrer:j’aimemieux l’employer à faire voir la maniere dont le mouvement se produit par la force d’unepression qui agit sans interruption, & sans autre opposition que celle qui vient de l’inertie dumobile.

8.16 Me confirma encore dans l’opinion ou j’étais qu’il faut entrer dans la génération de l’équilibrepour y voir en soi, & pour y reconnaitre les propriétez que tous les autres principes ne prouvent,tout au plus, que pour nécessité de conséquence.


8.17 Pour préparer l’imagination aux mouvemens composez, coincevons le point A sans pesanteuruniformement mû vers B le long de la droite AB, pendant que cette ligne se meut uniformementvers CD le long de AC en demeurant toujours parallele a elle-même, c’est à dire, faisant l’angletoujours le même quelconque avec cette ligne immobile AC: de ce deux mouvemens com-mencez en même tems, soit la vitesse du premier a la vitesse du second, comme les cotez AB,AC, du parallelogramme ABCD, le long des quels ils se font, Quel que soit ce parallelogrammeABCD, je dis que par le concourse des deux forces productrices de ces deux mouvemens dansle mobile A, ce point parcourra la diagonale AD de ces parallelogramme, pendant le tems quechacune d’elles lui en auroit fait parcourir seule chacun des cotes AB, AB, correspondans.

8.18 Definition XXIILes produit de chaque poids ou puissance absolue par sa distance à l’appui du Levier auquel elleest appliquée s’appelle en Latin Momentum […] nous ne laisserons pourtant pas de l’appelleraussi moment, pour nous moins éloigner du langage ordinaire. La raison de ce nom vient sansdoute de ce que ces produits son égaux ou inégaux comme les impressions de deux puissancessur un Levier.

8.19 Les formules contraire de momens en seront toujours égales entr’elles, c’est à dire que lasomme de leurs momens conspirans à faire tourner le Levier en un sens sur son appui seratoujours alors égale à la somme des conspirans à le faire tourner en sens contraire sur cetappui, ainsi qu’on l’à déjà vu dans le Corol. 9 du Th. 21.

8.20 Votre project d’une nouvelle Mécanique fourmille d’un grand nombre d’exemples, dontquelque uns à juger par les figures paroissent assez compliquées; mai je vous deffie de m’enproposer un à votre choix, que je ne resolve sur le champe et comme en jouant par ma dite regle.

8.21 Celle que vous pretendez substituer à la mienne et qui est fondée sur la composition des forces,n’est elle meme qu’un petit corollaire de la regle d’energie. J’ai donc raison d’appeller le grandet le premier principe de statique sur le quel j’ai fondé ma regle qui est que dans chaque equi-libre il y a une egalité d’energie des forces absolues, c’est à dire entre le produict des forcesabsolues par les vitesses virtuelles.

8.22 Je vous prie d’y panser, vous y trouverez sans doute un fond inepuisable pour enrichir lamecanique et pour en rendre l’etude incomparablement plus commode et plus aisée qu’ellen’a eté pur le passé, le traité complet de cette science que vous promettez depuis si longtemsne pourra que paroitre d’autant plus estimable, qu’on le verra fondé sur un principe aussi uni-verselle, aussi simple, aussi intelligible et aussi certain que celui dont il s’agit ici et dont jevous ai montré tant d’avantages.

8.23 Mais la mecanique de cette même proposition, & de la generale que vous ajoutez dans votrederniere, bien loin d’être le grand & le premiere principe de statique, ne me paroit pouvoir êtrequ’un corollaire des mouvemens composée, ou de quelqu’autre principe, qui demontre cetteproposition, c’est a dire votre egalité des sommes d’energies, en deduisant des directions don-née par lui, ou par supposition, les chemine instantanées de M. Descartes, que vous appellezvitesses virtuelles, qui avec les puissances trouvées d’ailleurs, à la supposition de leur equi-libre, son tout ce qui entre dans cette egalité de sommes des energies, de laquelle on aouroitdroit de douter si elle n’etoit pas prouvés par quelqu’un de ces principes.

8.24 Les Cartesiens conformément à la lettre que je viens de citer (art. 1.) de leur Maistre, avoientdesja deduit de son principe la même egalité de Momens, ou d’energie, ou de quantités demouvement, que vous employez pour deux puissances en equilibre sur les machines simples,& dans le fluides, par les commencemens de mouvement que M. Descartes prescrit dans cettelettre; mais vous etes le seul, que je sçache, qui ait étendu cette égalité d’energies à tant depuissances qu’on voudra supposer en équilibre entr’elles suivant des directions quelconques.Cette Remarque est fort belle; mais (comme j’ay desja dit) elle suppose équilibre entre despuissances donnée & de directions données sans le prouver.

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8.25 L’équilibre du non équilibre, ils ne font proprement qu’une demonstration ab abdsurdo.

8.26 J’ai peur de tomber dans une logomachie si j’entreprens de m’entendre sur tous ce que vous medites touchant ma regle des energie, que j’ai pretendue être generale pour toute la mecanique,tant des fluides que des solides [...] evitons donc la logomachie et ne prenez pas le change,il ne s’agissoit uniquement que d’etablir la verité et l’universalité de ma regle des energiescontre votre objection; que cette regle soit un principe ou un corollaire d’un autre, qu’importe,il suffit qu’elle soit vraie, generale et commode, sans exception, uniforme et facile à en fairel’application; avantage que la composition des forces n’a pas.

8.27 Vous me citez la lettre de M. Descartes pour me prouver que cet Autheur avoit deja l’idéed’expliquer l’équilibre des puissances par l’egalité des energies en considerant leurs cheminsinstantanée, que j’appelle vitesses virtuelles, je respons, que je ne me vans pas d’être le premierinventeur de cette idée, non plus que vous vous vanterez d’être celui d’expliquer les équilibrespar la composition des forces.

8.28 Vous ferez de ma regle d’energie ce que vous voudrez en l’ajoutant ou en ne l’ajoutant pas àvotre mechanique; je vous permete l’un comme l’autre; mais de prétendre qu’elle soit un corol-laire du principe de la composition des mouvemens ou forces, j’aurois peutetre encore de quoifaire valoir les raisons données dans mes precedentes pour en prouver le contraire, si je voulaism’engager dans une dispute qui nous couteroit du temps et de la peine: ainsi j’aime mieux vouslaisser le plaisir de croire que le principe de la composition deive préceder celui d’energie quede hazarder une longue et ennuyeuse contestation, il suffit que le dernier pouvant être appliquéaux fluides comme aux solides, soit plus general qua le premier qui ne sert que pour les solides,outre que celui ci demanderoit encore un autre principe dont il doit être deduit, puisque la com-position des forces n’est pas claire par elle même comme un axiome. Il est donc ce me sembleraisonnable que le principe d’energie comme le plus general et pour le moine aussi clair purlui même que le principe de la composition contienne celui ci comme le moins general.

A.9 Chapter 9

9.1 Per nome di potenza dunque non altro intendiamo, che la pura, e semplice pressione, o siaquello sforzo, che fa la gravità, o altra forza contro qualche ostacolo invincibile, come è perl’appunto quello, che fa una palla di piombo contro una tavola immobile, oppure contro lamano che la sostiene.

9.2 Sicché se una palla, a cagion d’esempio di piombo, sarà collocata sopra di una tavola immo-bile, la gravità, che in essa risiede, sarà forza soltanto premente, e perciò forza morta. Ma sesi rimuoverà l’ostacolo, cioè la tavola sottoposta, nella palla si indurrà tosto cangiamento distato. […] i meccanici per loro metodo si immaginarono la potenza dar al corpo un impulso, ilquale però appena nato, fosse dall’invincibile ostacolo distrutto, e così secondo il metodo deimatematici si rappresentarono la forza morta sotto l’idea di un impulso infinitamente picciolo[...] Ma poiché i meccanici più chiara idea formar potessero dell’azion della potenza, siccomes’avevano rappresentata la potenza sotto l’idea di un impulso, che nel procinto del suo nascereresta per l’invincibile ostacolo estinto, e distrutto, così, rimosso l’ostacolo invincibile, con-cepirono tutti gli impulsi […] conservarsi nel corpo medesimo, e quindi si avvisarono l’azionedella potenza non esser, che la somma di tutti gli impulsi accumulati, e conservati nel corpo.Quel tanto poi d’energia che per l’azion della potenza si genera nel corpo [...] viene chiamataforza viva.

9.3 Imperocché s’egli è vero come si è detto, che la potenza considerar si deve come un impulsominore di ogni altro dato, e che l’azion della potenza è la somma di tutti gli impulsi comuni-cati al corpo, e nel corpo stesso conservati, quella dovrà certamente esser la proporzione dellapotenza e l’azion della potenza, che passa tra una quantità infinitesima, ed una quantità finita.


Imperocché dalle cose sopra divisate apparisce, che agendo la potenza nel corpo, a cui è ap-plicata, genera in esso la forza viva, e che questa produce il cambiamento di stato. Sicché laforza viva considerarsi deve come un effetto dell’azion della potenza e come causa del can-giamento di stato che nel corpo si induce; e poiché in questo caso si parla di cause intere, etotali, avrà luogo l’assioma Ontologico, che le cause debbono essere proporzionali agli effettie gli effetti alle cause. Quindi nascono due modi di misurar la forza viva; cioè, o con misuraril di lei effetto, che è il cangiamento di stato, o con misurar la di lei causa, che è l’azione dellapotenza.

9.4 Sicché col dire degli Antichi, che la causa degli equilibri consiste nell’uguaglianza de’ mo-menti, non altro sembran aver detto, che l’equilibrio dipende dall’uguaglianza di quelle quan-tità, dall’uguaglianza delle quali l’equilibrio dipende.

9.5 L’equilibrio nasce da ciò, che le azioni delle potenze, che equilibrar si devono, se nascessero,sarebbero uguali, e contrarie; e perciò l’uguaglianza, e la contrarietà delle azioni delle potenzeè la vera causa dell’equilibrio.[…]L’equilibrio non è altro, che l’impedimento de’ moti, cioè degli effetti dell’azione dellepotenze, a cui non è meraviglia se corrisponde l’impedimento delle cause, cioè delle azionistesse.

9.6 Quindi stabiliamo un principio, cioè un criterio generale per conoscere quando tra le potenzesucceder debba l’equilibrio, ed egli è quello che si contiene nel seguente teorema: Le potenzesaranno in equilibrio qualora trovansi in tali circostanze, che se nascesse un moto infinites-imo, le di loro infinitesime azioni sarebbero uguali. E tal principio deve aver luogo in tutti gliequilibri.

9.7 Perché la celebre controversia delle forze vive, che consiste nel definire se quelle si deb-bano misurar per la massa moltiplicata per la velocità, oppure per la massa moltiplicata peril quadrato della velocità stessa, riducesi a quest’altra quistione, cioè se l’azione della potenzadebba esser proporzionale al tempo piuttosto che allo spazio.

9.8 Non potendosi dunque l’azion della potenza misurare per la potenza moltiplicata pel tempo,uopo è rivolgersi allo spazio. In tutti gli equilibri conosciuti si trova vero, come si vedrà neiseguenti capitoli, che facendosi un moto infinitesimo, le potenze sono in ragion reciproca de’loro rispettivi spazietti d’accesso, o di ricesso dal centro delle potenze stesse […] Sicché sel’azione della potenza si misurerà per la potenza moltiplicata per lo spazio, per cui la potenzaagendo trasporta il corpo, facendolo avvicinare fino al centro, o dal centro facendolo allon-tanare, si salverà negli equilibri l’uguaglianza tra le minime azioni delle potenze […]. Dunquel’azione della potenza dee veracemente misurarsi per la potenza moltiplicata per lo spazio se-condo il metodo dei Leibniziani.

9.9 Le potenze sono in equilibrio, qualora trovansi in tali circostanze costituite, che facendosi unmoto infinitesimo, onde alcune potenze si avvicinino al suo centro, alcune altre dal suo centro siallontanano, la somma dei prodotti positivi delle potenze moltiplicate per gli rispettivi spaziettid’accesso o di recesso, sia uguale allo somma de’ simili prodotti negativi.

9.10 Dico dal principio delle azioni dedursi, che qualora nella verga ABC si ha l’equilibrio, lapotenza Z sia alla potenza X come CN : CM, cioè, che vale l’equazione Z · CM = X · CM.

9.11 I centri delle potenze Z, X sian i punti Z, X. Si concepisca ora nella verga ACB nascer unmoto infinitesimo, cosicché i punti A, B descrivendo gli archetti Aa, Bb vengano in a, b. Dalpunto b al punto X si tiri la retta bX, e dal punto a al punto Z la retta aZ; indi col centro Z,e coll’intervallo aZ intendasi descritto l’archetto aF che incontri la AZ in F, e similmente colcentro X, […] Fatto ciò è manifesto esser AF lo spazietto di accesso al centro della potenza Z,e bG lo spazietto di recesso dal centro della potenza X. Il principio delle azioni richiede, cheavendosi nella verga ABC l’equilibrio, sia la potenza Z alla potenza X come bG : AF.

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9.12 Si noti in secondo luogo, che facendosi comparazione tra il principio dell’equivalenza, equello delle azioni, debbono amendue stimarsi egualmente fecondi, ed estesi, con quellalor differenza, che in alcun casi con maggiore facilità, ed eleganza si adopra il principiodell’equivalenza, in altri casi poi riesce più comodo, ed opportuno l’adoperare il principiodelle azioni.È finalmente con attenzion da notarsi, che il metodo della composizione, e risoluzion delleforze non è il vero metodo della natura, ma è un metodo che si han formato i Geometri per lapiù facile e spedita solutione de’ lor problemi. La natura nelle sue opere non va’ giammai acomporre, e risolvere le forze, ma adopera sempre azioni, le quali essendo uguali, e contrarie,fan sì, che si producano gli equilibri.

9.13 Nella puleggia stabile, poiché si abbia equilibrio richiedesi tra la potenza, e il peso la ragiond’uguaglianza. Sia AB una puleggia stabile, che abbia intorno a sé la fune EABD, alla di cuiestremità D sia attaccato il peso P, all’altra estremità E sia applicata la potenza, che il pesostesso sostiene. Dico, che per aversi l’equilibrio in quella macchina bisogna, che la potenzaapplicata in E sia al peso P affatto uguale.

9.14 Si facci un moto infinitesimo secondo la direzione della potenza applicata in E, cosichèl’estremità E della fune giunga in G, mentre l’estremità D giungerà in H. Egli è troppo mani-festo, che è EG lo spazietto d’accesso al centro della potenza, e DH lo spazietto di recesso dalcentro del peso. Sicché acciò s’abbia l’equilibrio tra la potenza, e il peso, convien, che quellastia a questa come DH : EG. Ma è DH = EG; poiché supponendosi, che la fune non praticaalcuna distrazione, ma che resti sempre della stessa lunghezza, sarà la lunghezza DAE ugualealla lunghezza HAG; onde, resteranno DH, EG uguali tra di loro. Dunque nella puleggia per-ché si abbia l’equilibrio, richiedesi che la potenza sia uguale al peso. Che è quel, che bisognavadimostrare.

9.15 Sia GHPQ un sifone qualunque, se in un braccio di esso GH si verserà una quantità di flu-ido omogeneo […]. Posto che sarà in equilibrio il fluido versato nel sifone, nell’un braccio enell’altro del sifone stesso si troverà elevato alla medesima altezza.

9.16 Solamente io avvertirò che il famoso teorema dell’incomparabile Giovanni Bernoulli, il qualeè stato dimostrato in tutte le macchine dal dottissimo sig. Varignon, non è altro che unaconseguenza dell’equalità delle azioni contrarie, che è necessaria in ogni equilibrio. Il teo-rema Bernoulliano è il seguente. In ogni equilibrio di quante e quali potenze si vogliano, inqualunque maniera applicate, e agenti per qualsiasi direzione, la somma delle energie positiveè uguale alla somma delle energie negative, purché come affermative si prendano. Per nomed’energia il sig. Bernoulli altro non intende se non se il prodotto della potenza e della veloc-ità virtuale della stessa potenza; la quale sarà positiva, se seguita la direzione della potenza,sarà negativa, se seguita la direzione opposta. E chi è che non veda, che la velocità virtualedella potenza è proporzionale allo spazio, per cui il corpo, o la potenza si avvicina al cen-tro delle forze; ovvero se le potenze siano corde elastiche alla contrazione o diffrazione dellecorde. Dunque l’energia bernoulliana è la stessa, o almeno proporzionale a quella che per noichiamasi azione della potenza.

9.17 Per dichiarare siccome si distinguano le potenze e l’azioni loro, io concepisco un corpo gravesospeso da un filo, che gl’impedisce di discendere, e d’avvicinarsi alla terra. Fin ora altro nonintendo, ch’una potenza di gravità applicata al corpo, a cui è contraria l’elasticità del filo, chela contrasta, e non le lascia produrre effetto di sorte alcuna. Io tronco il filo, e levo l’elasticitàcontraria alla gravità. Ora oltre la potenza intendo, ch’essa successivamente e continuamentereplica i suoi impulsi o sollecitazioni contro il corpo, il quale è obbligato di cangiare stato.La somma e l’aggregato di cotai impulsi si vuol chiamare l’azione di tal potenza; e l’effettoossia la mutazione di stato non alla potenza, ma all’aggregato dei suoi impulsi è proporzionale.Tre quantità pertanto si vogliono distinguere, cioè la potenza considerata in se stessa, la qualpressione ancor si suol chiamare; l’azione, che è l’aggregato dei suoi impulsi, onde la potenzaspinge il corpo; la quale è in ragion composta della potenza e del numero degli impulsi; e


l’effetto, ossia la mutazione di stato, che soffre il corpo, il quale effetto è in proporzione nondella potenza, ma della sua azione.

9.18 Sicché dunque sebben la forza centrifuga non ha propriamente altra cosa, che l’inerzia delcorpo in alcune circostanze considerata, non è inutile l’introdurla ne’ raziocini, ne si dee bandirdalla fisica: anzi sarà profittevole il fissar le sue leggi, e si riconosceranno per veri, e belli i teo-remi prodotti intorno a cotal forza dal dotto, e profondo Cristian Ugenio.Similmente risponderò io intorno alla forza viva. Essa non è per verunmodo distinta dalla forzad’inerzia, anzi è la medesima forza d’inerzia da alcune particolari condizioni modificata: con-tuttociò sarà utile il considerarla con questo nome, e il fissarne le leggi, che in molte quistioni,e ricerche potranno essere di non picciolo giovamento.

9.19 Per mettere in buona vista il nostro metodo, e l’uso del principio, non devo omettere una os-servazione, che sembrami importante. Quando non sia possibile, se non se un movimento,come avviene a’ corpi, che si raggirano intorno a un asse, allora se concepito un minimo movi-mento, la azioni spontanee e sforzate misurate dallo spazio di accesso e di recesso si ritrovanoeguali, ma senza cautela si deduca l’equilibrio. Ma quando liberi sieno più movimenti e in piùdirezioni, se concependo un qualche movimento ad arbitrio, io ritrovo come sopra l’egualitàdelle azioni, non posso affermare un equilibrio pieno e compito, ma soltanto pronunciare chequel movimento è impossibile, e che in quella direzione equilibrate sono le potenze.

9.20 Mi sono ancora servito di qualche parte delle ricerche, che io avea già fatte vent’anni addie-tro sulla gran Cupola di S. Pietro in Roma, e principalmente dalla teoria che mi condusse aconoscere la forza con cui un cerchio di ferro spinto in fuori da forza applicata perpendicolar-mente in tutti i suoi punti, resiste, trovandola maggiore di quella che sarebbe la stessa sprangadi ferro tirata direttamente nella direzione della sua lunghezza un poco più che a sei doppi, cioèin proporzione della circonferenza del circolo al raggio, d’onde poi il Marchese Polini ricavòl’idea di quella esperienza, in cui un filo di seta ottagono, tirato in fuora in tutti gli angoli peresser rotto, ebbe bisogno di una forza incirca a sei doppi maggiore, che quando un altro filosuo compagno era tirato direttamente.

9.21 Due sono le forze, che spingono in fuori verso l’imposta gi, cioè il peso del cupolino, e ilpeso de’ costoloni con gli spicchi delle cupole; e due parimente le forze, che resistono a talespinta, cioè le catene circolari, o cerchi LL, ed il sostegno […] ridotto a due distinti, il primode quali è il tamburo HI, col pezzo interior della base CDF; il secondo i contraforti mGF collaparte esteriore ABE della base medesima […]. Il distacco, delle parti quanto fosse difficile, eche resistenza, abbia fatto non è possibile l’esaminarlo a minuto. Dipende esso in gran partedalla qualità del cemento, e dalla diligenza del lavoro. Per metter’in conto le forze, e vedere sequeste stanno in equilibrio convien’in prima determinare la quantità assoluta delle medesime,e poi quello che da’ Mecanici chiamasi il Momento. Per avere la quantità assoluta della forza,con cui agisce da una parte il Cupolino, e la volta della Cupola co’ costoloni per spingere, edall’altra la base, il tamburo, i contrafforti per ritenere la spinta, conviene averne il peso.

9.22 Supposto questo principio, in primo luogo parci, che l’energia di una catena di ferro, curvatain cerchio debba crescere sopra quella forza assoluta, che avrebbe se distesa fosse in dirittura,in quella medesima proporzione, che ha la circonferenza del circolo al raggio, cioè poco piùche a sei doppj. Imperocchè, si concepisca distribuita una forza per tutta la circonferenza diun cerchio, che da essa venga costretto a distendersi, e dilatarsi fino all’atto di rompersi, eduna verga di ferro uguale distesa in dirittura venga tirata da un’altra forza, come farebbe unpeso attaccatole verticalmente, che la riduca al medesimo estremo. In questo secondo caso ladiscesa del peso nel tender le fibre di quella sarebbe uguale alla somma delle tensioni di tuttequante le fibre disposte lungo la stessa verga, ma nel primo dilatandosi il cerchio, e crescendocosì la sua circonferenza, la forza che lo costringe, a dilatarsi non si avanzerebbe, se non quantocresce il raggio del circolo, mentre la somma delle tensioni delle medesime fibre disposte ingiro sarebbe uguale all’accrescimento di tutta quanta la circonferenza.

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10.1 Theoriae oscillationum, quas adhuc Auctores pro corporibus dederunt solidis, invariatum par-tium situm in illis ponunt, ita ut singula communimotu angulari ferantur. Corpora autem, quaeex filo flexili suspenduntur, aliam postulant theoriam, nec sufficere ad id negotium videnturprincipia communiter in mechanica adhiberi solita, incerto nempe situ, quem corpora inter sehabeant, eodemque continue variabili.

10.2 Sed quod omnibus scriptis, quae sine analysi sunt composita, id potissimum Mechanicisobtingit, ut Lector, etiamsi de veritate eorum, quae proferuntur, convincatur, tamen non satisclaram et distinctam eorum cognitionem assequatur, ita ut easdem quaestiones, si tantillumimmutentur, proprio marte vix resolvere valeat, nisi ipse in analysin inquirat easdemquepropositiones analytica methodo evolvat. Idem omnino mihi, cum Neutoni Principia et Her-manni Phoronomiam perlustrare coepissem, usu venit, ut, quamvis plurium problematum so-lutiones satis percepisse mihi viderere, tamen parum tantum discrepantia problemata resol-vere non potuerim.

10.3 La composition des forces suffit comme l’on fait pour démontrer l’équilibre du levier, & ré-ciproquement cette dernière proposition une fois prouvée, on peut facilement en déduire lacomposition des forces. Elle nous fournit d’ailleurs une démonstration fort-simple du principedes vitesses virtuelles, qu’on peut avec raisons considérer comme le plus fécond & le plusuniversel de la Mécanique: tout les autres en effet s’y réduisent sans peine, le principe de laconservation des forces vives, & généralement, tous ceux que quelques Géomètres on imag-inés pour faciliter la solution de plusieurs Problèmes, n’en sont qu’une conséquence purementgéométrique, ou plus tost ne sont que ce même principe réduit en formule.

10.4 La force est donc une cause quelconque de mouvement. Sans connaitre la force en elle-même,nous concevons encore très clairement qu’elle agit suivant une certaine direction, et avec unecertaine intensité.

10.5 On entend en général par force ou puissance la cause, quelle qu’elle soit, qui imprime ou tend àimprimer dumouvement au corps auquel on la suppose appliquée; & c’est aussi par la quantitédu mouvement imprimé, ou prêt à imprimer, que la force ou puissance doit s’estimer. Dans1’état d’équilibre la force n’a pas d’exercice actuel; elle ne produit qu’une simple tendanceau mouvement; mais on doit toujours la mesurer par l’effet qu’elle produiroit si elle n’étoitpas arrêtée. En prenant une force quelconque, ou son effet pour l’unité, l’expression de touteautre force n’est plus qu’un rapport, une quantité mathématique qui peut être représentée pardes nombres ou des lignes; c’est sous ce point de vue que l’on doit considérer les forces dansla Méchanique.

10.6 On ne peut cependant s’empêcher de reconnaitre que le principe du levier a seul l’avantaged’être fondé sur la nature de l’équilibre considéré en lui-même, et comme un état indépen-dant du mouvement: d’ailleurs il y a une différence essentielle dans la manière d’estimer lespuissances, qui se font équilibre dans, ces deux principes. De sorte que, si l’on n’était pasparvenu, à les lier par les résultats, on aurait pu douter avec raison s’il était permis de sub-stituer au principe fondamental du levier celui qui résulte de la considération étrangère desmouvements composes.

10.7 De même que le produit de la masse et de la vitesse exprime la force finie d’un corps enmouvement, ainsi le produit de la masse et de la force accélératrice que nous avons vu êtrereprésentée par I’élément de la vitesse divisé par l’élément du temps, exprimera, la forceélémentaire on naissante; et cette quantité, si on Ia considère comme la mesure de l’effortque le corps peut faire en vertu de la vitesse élémentaire qu’il a prise, ou qu’il tend à prendre,constitue ce qu’on nomme pression; mais si on la regarde comme la mesure (le la force oupuissance nécessaire pour imprimer cette même vitesse, elle est alors ce qu’on nomme force.


10.8 C’est un principe généralement vrai en Statique que, si un système quelconque de tant decorps ou de points que l’on veut, tirés chacun par des puissances quelconques, est en équili-bre, et qu’on donne à ce système un petit mouvement quelconque, en vertu duquel chaquepoint parcoure un espace infiniment petit, la somme des puissances, multipliées chacune parl’espace que le point où elle est appliquée parcourt suivant la direction de cette même puis-sance, sera toujours égaie à zéro.

10.9 Dans la question presente, si l’on imagine que les lignes X ,Y,Z,R,R′ deviennent, en variantinfiniment peu la position de la Lune autour de son centre

X +δX , Y +δY, Z+δZ, R+δR, R′+δR′

il est facile de voir que les différences

δX , δY, δZ, δR, δR′

exprimeront les espaces parcourus en même temps pat le pointα dans des directions opposéesa celles des puissances

αd2Xdt2

, αd2Ydt2

, αd2Zdt2

, αTR2 , α

SR′2

dm

qui sont censées agir sur ce point; on aura donc, pour les conditions de l’équilibre, l’équationgénérale∫

L

[αd2Xdt2

(−δX)+αd2Ydt2

(−δY )+αd2Zdt2

dm(−δZ)+αTR2 (−δR)+α

SR′2

(−δR′)]

savoir, en changeant les signes,∫L

(αd2Xdt2δX +α

d2Ydt2δY +α

d2Zdt2δZ

)+T

∫LαδRR2 +S

∫LαδR′

R′2.

10.10 Le principe de Statique que je viens d’exposer n’est, dans le fond, qu’une généralisation decelui qu’on nomme communément le principe des vitesses virtuelles, et qui est reconnu depuislongtemps par les Géomètres pour le principe fondamental de l’équilibre. M. Jean Bernoulliest le premier, que je sache, qui ait envisagé ce principe sous un point de vue général etapplicable à toutes les questions de Statique, comme on le peut voir dans la Section IX de lanouvelle Mécanique de M. Varignon, ou cet habile Géomètre, après avoir rapporté d’aprèsM. Bernoulli le principe dont il s’agit, fait voir, par différentes applications, qu’il conduit auxmêmes conclusions que celui de la composition des forces.

10.11 Ensuite en ayant égard aux équations de condition, donnes par la nature du système proposé,entre les coordonnées des différens corps, on réduira les variations de ces coordonnées au pluspetit nombre possible, ensorte que les variations restantes soient tout-à-fait indépendantesentr’elles & absolument arbitraires. Alors on égalera à zéro la somme de tous les termesaffectés de chacune de ces dernières variations; & l’on aura toutes les équations nécessairespour la détermination du mouvement du système.

10.12 1. Le principe donné parM. d’Alembert réduit les lois de la Dynamique à celles de la Statique;mais la recherche de ces dernières lois par les principes ordinaires de l’équilibre du levier,ou de la composition des forces, est souvent longue et pénible. Heureusement il y a un autreprincipe de Statique plus général, et qui a surtout l’avantage de pouvoir être représenté parune équation analytique, laquelle renferme seule les conditions nécessaires pour l’équilibred’un système quelconque de puissances. Tel est le principe connu sous la dénomination deloi des vitesses virtuelles; on l’énonce ordinairement ainsi: Quand des puissances se fontéquilibre, les vitesses des points où elles sont appliquées, estimées suivant la direction de ces

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puissances, sont en raison inverse de ces mêmes puissances. Mais ce principe peut être rendutrès-général de la manière suivante.2. Si un système quelconque de corps, réduits a des points et tirés par des puissances quel-conques, est en équilibre, et qu’on donne à ce système un petit mouvement quelconque envertu duquel chaque corps parcoure un espace infiniment petit, la somme des puissances mul-tipliées chacune par l’espace que le point où elle est appliquée parcourt suivant la directionde cette puissance est toujours égale a zéro.

10.13 Pour avoir les valeurs des variations ou différences

δp,δq,δr, . . . ,δp′,δq′,δr′, . . .

on différentiera à l’ordinaire les expressions des distances p,q.r, . . . , p′,q′,r′ mais en regar-dant les centres des forces comme fixes.

10.14 De plus, en ayant égard à la disposition mutuelle des corps, on aura une ou plusieurs équationsde condition entre les variables x,y,z,x′,y′,z′ par le moyen desquelles on pourra exprimertoutes ces variables par quelques-unes d’entre elles, ou bien par d’autres variables en moin-dre nombre et telles, qu’elles soient entièrement indépendantes et répondent aux différentsmouvements que le système peut recevoir.

10.15 Ceux qui jusqu’à présent ont écrit fur le Principe des vitesses virtuelles, se sont plutôt attachésà démontrer la vérité de ce principe par la conformité de ses résultats avec ceux des principesordinaires de la Statique, qu’à montrer l’usage qu’on en peut faire pour résoudre directementles problèmes de cette Science. Nous nous sommes proposé de remplir ce dernier objet avectoute la généralité dont il est susceptible, & de déduire du Principe dont il s’agit, des for-mules analitiques qui renferment la solution de tous les problèmes sur l’équilibre des corps,à-peu-près de la même manière que les formules des soutangentes, des rayons osculateurs,&, renferment la détermination de ces lignes dans toutes les courbes.

10.16 Si un système quelconque de tant de corps ou points que l’on veut, tirés chacun par despuissances quelconques, est en équilibre, et qu’on donne à ce système un petit mouvementquelconque, en vertu duquel chaque point parcoure un espace infiniment petit qui exprimerasa vitesse virtuelle, la somme des puissances multiplies chacune par l’espace que le point ouelle est appliquée parcourt suivant la direction de cette même puissance, sera toujours égaleà zéro, en regardant comme positifs les petits espaces parcourus dans le sens des puissances,et comme négatifs les espaces parcourus dans un sens opposé.

10.17 Et en général je crois pouvoir avancer que tous les principes généraux qu’on pourroit encoredécouvrir dans la science de l’équilibre, ne seront que le même principe des vitesses virtuelles,envisagé différemment, & dont ils ne différeront que dans l’expression. Au reste, ce Principeest non seulement en lui même très simple & très général; il a de plus l’avantage précieux& unique de pouvoir se traduire en une formule générale qui renferme tous les problèmesqu’on peut proposer sur l’équilibre des corps. Nous allons exposer cette formule dans touteson étendue; nous tâcherons même de la présenter d’une manière encore plus générale qu’onn’est pas fait jusqu’à présent, & d’en donner des applications nouvelles.

10.18 La loi générale de l’équilibre dans les machines, est que les forces ou puissances soiententr’elles réciproquement comme les vitesses des points où elles sont appliquées, estiméessuivant la direction de ces puissances.

10.19 On substituera ensuite ces expressions de dp,dq,dr,&, dans l’équation proposée, & il faudraque cette équation ait lieu, indépendamment de toutes les indéterminées, afin que l’équilibredu système subsiste en général & dans tous les sens. On égalera donc séparément à zero, lasomme des termes affectés de chacune des mêmes indéterminées; & l’on aura, par ce moyen,autant d’équations particulières, qu’il y aura de ces indéterminées; or il n’est pas difficile dese convaincre que leur nombre doit toujours être égal à celui des quantités inconnues dans la


position du système; donc on aura par cette méthode, autant d’équations qu’il en faudra pourdéterminer l’état d’équilibre du système.

10.20 Maintenant comme ces équations ne doivent servir qu’à éliminer un pareil nombre de dif-férentielles dans l’équation des vitesses virtuelles, après quoi les coefficiens des différen-tielles restantes, doivent être égalés chacun à zéro, il n’est pas difficile de prouver parla théorie de l’élimination des équations linéaires, qu’on aura les mêmes résultats si onajoute simplement à l’équation des vitesses virtuelles, les différentes équations de conditiondL = 0, dM = 0, dN = 0, &, multipliées chacune par un coefficient indéterminé, qu’ensuiteon égale à zéro la somme de tous les termes qui se trouvent multipliés par une même différen-tielle; ce qui donnera autant d’équations particulières qu’il y a de différentielles; qu’enfin onélimine de ces dernieres équations les coefficients indéterminés par lesquels on a multipliéles équations de condition.

10.21 Réciproquement ces forces peuvent tenir lieu des équations de condition résultantes de la na-ture du système donné; de manière qu’en employant ces forces, on pourra regarder les corpscomme entièrement libres & sans aucune liaison. Et de-là on voit la raison métaphysique,pourquoi introduction des termes λdL+μdM+&c., dans l’équation générale de l’équilibre,fait qu’on peut ensuite traiter cette équation comme si tous les corps du système étoient en-tièrement libres; c’est en quoi consiste l’esprit de la méthode de cette section.A proprement parler, les forces en question tiennent lieu des résistances que les corps de-vroient éprouver en vertu de leur liaison mutuelle, ou de la part des obstacles qui, par lanature du système, pourroient s’opposer à leur mouvement, ou plutôt ces forces ne sont queles forces mêmes de ces résistances, lesquelles doivent être égales & directement opposéesaux pressions exercées par les corps. Notre méthode donne, comme l’on voit, le moyen dedéterminer ces forces & ces résistances; ce qui n’est pas un des moindres avantages de cetteméthode.

10.22 Quant à la nature du principe des vitesses virtuelles, il faut convenir qu’il n’est pas assezévident par lui-même pour pouvoir être érigé en principe primitif; mais on petit le regardercomme l’expression générale des lois de l’équilibre, déduites des deux principes que nousvenons d’exposer. Aussi, dans les démonstrations qu’on a données de ce principe, on l’atoujours fait dépendre de ceux-ci, par des moyens plus on moins directs. Mais ii y a, enStatique, un autre principe général et indépendant du levier et de la composition des forces,quoique les mécaniciens l’y rapportent communément, lequel parait être le fondement natureldu principe des vitesses virtuelles; on peut l’appeler le principe des poulies.

10.23 On a objecté, avec raison, à cette assertion de Lagrange l’exemple d’un point pesant en équili-bre an sommet le plus élevé d’une courbe; il est évident qu’un déplacement infiniment petitle ferait descendre, et, pourtant, ce déplacement ne se produit pas. La première démonstrationrigoureuse du principe des vitesses virtuelles est due à Fourier (Journal de École Polytech-nique, tome II, an VII). Le même Cahier du Journal contient la démonstration que Lagrangereproduit ici.

10.24 Si donc on imprimant à chaque corps des forces égales et directement contraire a à celles-là,l’effet de ces forces serait détruit par la résistances dont nous venons de parler; par conséquent,le système devrait demeure en équilibre. […] Or, par le principe des vitesses virtuelles, lasomme des forces multipliée chaque par la vitesse que le point où elle est appliquée aurait,suivant la direction de la force, si on donnant au système un mouvement quelconque, doitêtre nulle dans le cas de l’équilibre […] on aura pour l’équilibre des forces dont il s’agit,l’équation:

−Π f ′(x)−Π f ′(y)−Π f ′(z)−Ψ f ′(ξ)−Ψ f ′(ν)−Ψ f ′(ζ)−&c. = 0

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f (x,y,z,ξ,ν,ζ) = 0 […]. Or, si on prend l’équation prime de cette équation, relativement autemps t, dont les variables x,y,z, ξ, &c., sont censées être fonctions, on a:

x′ f ′(x)+ y′ f ′(y)+ z′ f ′(z)+ξ′ f ′(ξ)+ν′ f ′(ν)+ζ′ f ′(ζ)+&c. = 0

et il est visible que cette équation ne peut subsister avec la précédent, indépendamment desvaleurs des vitesses x′, y′, &c., à moins qu’on n’ait Π = Ψ = &c.

10.25 Les fonctions primes de la même fonction, prises par rapport aux différentes coordonnées,sont toujours proportionnelles aux forces qui agissent suivant ces coordonnées, et qui dépen-dent de la condition exprimée par cette fonction.

10.26 Soient X ,Y,Z les forces appliquées à l’un des corps suivant les directions des coordonnéesx,y,z prolongées, Ξ,ϒ,Σ les forces appliques à un autre corps suivant le prolongement deses coordonnées, ξ,η,ζ, et X, Y, Z les forces appliquées a un troisième corps suivant leprolongement de ses coordonnées x, y, z; on aura, par ce qu’on vient de démontrer,

X = ΠF ′(x)+ΨΦ′(x), Y = ΠF ′(y)+ΨΦ′(y), Z = ΠF ′(z)+ΨΦ′(z)Ξ = ΠF ′(ξ)+ΨΦ′(ξ), ϒ = ΠF ′(η)+ΨΦ′(η), Σ = ΠF ′(ζ)+ΨΦ′(ζ)X = ΠF ′(x)+ΨΦ′(x), Y = ΠF ′(y)+ΨΦ′(y), Z = ΠF ′(z)+ΨΦ′(z)

et de là on tirera immédiatement

Xx′+Yy′+Zz+Ξξ′+ϒη′+Σζ′+Xx′+Yy′+Zz′ =ΠF(x,y,z,ξ,η,ζ,x, y, z)′+ΨΦ(x,y,z,ξ,η,ζ,x, y, z)′.

Le second membre de cette équation est évidemment nul, en vertu des équations de condition,puisque les quantités indéterminées Π, Ψ, se trouvent multipliées par les fonctions primes deces équations; donc on aura

Xx′+Yy′+Zz+Ξξ′+ϒη′+Σζ′+Xx′+Yy′+Zz′ = 0

équation générale du principe des vitesses virtuelles pour I’équilibre des forces X ,Y,Z,Ξ,ϒ,Σ, X, Y, Z, dans laquelle les fonctions primes x′,y′,z′, ξ′, … expriment les vitessesvirtuelles des points auxquels soit appliquées les forces X ,Y,Z, Ξ … estimées suivant les di-rections de ces forces.Au reste, on ne doit pas être surpris de voir le principe des vitesses virtuelles devenir uneconséquence naturelle des formules qui expriment les forces d’après les équations de condi-tion, puisque la considération d’un fil qui par sa tension uniforme agit sur tous les corps et yproduit des forces données suffit pour conduire à une démonstration directe et générale de ceprincipe, comme je l’ai fait voir dans la seconde.

10.27 On peut s’étonner que l’illustre auteur, ordinairement si soigneux de faire connaitre l’originedes idées qu’il expose, ne fasse ici aucune citation. Le passage qu’on vient da lire est, en effet,postérieur da sept années à la publication du célèbre Mémoire sur l’Equilibre et le mouve-ment des systèmes, dans lequel M. Poinsot se propose et résout précisément la même ques-tion, d’affranchir la mécanique du principe des vitesses virtuelles en cherchant directementles forces qui correspondent à une équation donnée. Ce Mémoire avait vivement frappé La-grange, comme le prouvent des notes autographes nombreuse placées par lui sur les margesd’un exemplaire qu’il m’a été permis de consulter. Je me bornerai à reproduire ici une de cesnotes, qui ne peut laisser subsister aucun doute sur la question de priorité.


10.28 Au reste le principe de Statique que je viens d’exposer, étant combine avec le principe deDynamique donné par M. d’Alembert, constitue une espèce de formule générale qui renfermela solution de tons les Problèmes qui regardent le mouvement des corps.

10.29 Si maintenant on suppose le système en mouvement, & qu’on regarde le mouvement quechaque corps a dans un instant comme composé de deux; dont l’un soit celui que le corpsaura dans l’instant suivant, il faudra que l’autre soit détruit par l’action réciproque des corps,& par celle des forces motrices dont ils sont actuellement animés. Ainsi il devra y avoiréquilibre entre ces forces & les pressions ou résistances qui résultent des mouvemens qu’onpeut regarder comme perdus par les corps d’un instant à l’autre. D’où il suit que pour étendreau mouvement du système la formule de son équilibre il suffira d’y ajouter les termes ds cesdernieres forces.

10.30 Il est clair que le mouvement ou la vitesse du corps m dans l’instant dt ou peut être regardéecomme composée de trois autres vitesses exprimées par:

dxdt

,dydt

,dzdt

et dirigées parallèlement aux axes des x, y, z. Il est de plus évident que si le corps était libre etqu’aucune force étrangère n’agit sur lui, chacune de ces trois vitesses demeurerait constante;mais dans l’instant suivant elles se changent réellement en celles-ci

dxdt

+ddxdt

,dydt

+ddydt

,dzdt

+ddzdt

donc, si l’on regarde les vitesses précédentes comme composées de ces dernières et desvitesses

−ddxdt

, −ddydt

, −ddzdt

ou bien (en prenant di constant)

−d2xdt2

, −d2ydt2

, −d2zdt2

il s’ensuit que celles-ci doivent être détruites par l’action des forces qui agissent sur les corps.Mais ces vitesses sont dues à des forces accélératrices égales à

d2xdt2

,d2ydt2

,d2zdt2

et dirigées parallèlement aux axes des x, y, z (en exprimant, suivant l’usage reçu, la forceaccélératrice par l’élément de la vitesse divisé par l’élément du temps), on, ce qui revient aumême, à des forces égales à

d2xdt2

,d2ydt2

,d2zdt2

et dirigées en sens contraire.

[…]

D’ou il suit qu’il doit y avoir équilibre entre ces différentes forces et les autres forces quisollicitent les corps, et qu’ainsi les lois du mouvement du système se réduisent à celles de sonéquilibre; c’est en quoi consiste le beau principe de Dynamique de M. d’Alembert.

10.31 Si l’on imprime à plusieurs corps des mouvements qu’ils soient forcé de changer à cause deleur action mutuelle. il est claire qu’on peut regarder ces mouvements comme composés deceux que les corps prendront réellement, et d’autres mouvements qui sont détruites: d’ou ilsuit que ces deniers doivent être tels, que les corps animés de ces seuls mouvements se fassentéquilibre.

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10.32 Mais la difficulté de déterminer les forces qui doivent être détruites, ainsi que les lois del’équilibre entre ces forces, rend souvent l’application de ce principe embarrassant et pénible.[…]Si l’on voulait éviter les décompositions de mouvements que ce principe exige, il n’y auraitqu’à établir tout de suite l’équilibre entre les forces et les mouvements engendrés, mais prisdans des directions contraires. Car, si l’on imagine qu’on imprime à chaque corps, en senscontraire, le mouvement qu’il doit prendre, it est clair que le système sera réduit au repos;par conséquent, il faudra que ces mouvements détruisent ceux que les corps avaient reçus etqu’ils auraient suivis sans leur action mutuelle; ainsi il doit, y avoir équilibre entre tous cesmouvements, ou entre les forces qui peuvent les produire.Cette manière de rappeler les lois de la Dynamique à celles de la Statique est à la vérité moinsdirecte que celle qui résulte du principe de d’Alembert, mais elle offre plus de simplicité dansles applications; elle revient à celle d’Herman et d’Euler qui l’a employée dans la solution debeaucoup de problèmes de Mécanique, et on la trouve dans quelques Traités de Mécanique.sous le nom de Principe de d’Alembert.

10.33 Comme il s’agit présentement de déterminer le mouvement de la corde par les forces sollic-itantes, soit la force accélératrice, par laquelle le point M de la corde est accéléré vers l’axeAB = P, & il est clair que toutes ces forces, par lesquelles chacun des élemens de la cordeest pressé vers l’axe AB prises ensemble doivent être équivalentes à la force, par laquelle lacorde est actuellement tendue & qui nous avons posée AF = F; ou bien, si nous concevonsdes forces contraries & égales à P, appliquées suivant ML dans chacun des points M de lacorde, alors elles devront se trouver en équilibre avec la force qui tend la corde.

10.34 Pourquoi donc aurions-nous recours à ce principe dont tout le monde fait usage aujourd’hui,que la force accélératrice est proportionnelle à l’élément de vitesse? […]Nous n’examineronspoint si ce principe est de vérité nécessaire […] non plus, avec quelque Géometres, commede vérité contingente […] nous nous contenterons d’observer, que vrai ou douteux, clair ouobscure, il est inutile à la Méchanique, & que par conséquent il doit être banni.

10.35 Ce que nous appelons causes, même de la première espèce, n’est tel qu’improprement; ce sontdes effets desquels il résulte d’autres effets. Un corps en pousse un autre, c’est-à-dire ce corpsest en mouvement, il en rencontre un autre, il doit nécessairement arriver du changement àcette occasion dans l’état des deux corps, à cause de leur impénétrabilité; l’on détermine leslois de ce changement par des principes certains, & l’on regarde en conséquence le corpschoquant comme la cause du mouvement du corps choqué. Mais cette façon de parler estimpropre. La cause métaphysique, la vraie cause nous est inconnue.

10.36 Ainsi nous entendrons en général par la force motrice le produit de la masse qui se meutpar l’élement de sa vitesse, ou qui est la même chose, par le petit espace qu’elle parcurroitdans un instant donné en vertu de la cause qui accélere ou retarde son Mouvement; par forceaccélératrice nous entendrons simplement l’élément de la vitesse.

10.37 Probléme GénéralSoit donné un système de corps disposés les uns par rapport aux autres d’une manière quel-conque; & supposons q’on imprime à chacun de ces Corps unMouvement particulier, qu’il nepuisse suivre à cause de l’action des autres Corps; trouver le Mouvement que chaque Corpsdoit prendre.SolutionSoient A,B,C, &c. les corps qui composent le système, & suppose qu’on leur ait imprimé lesmouvemens a,b,c, &c. qu’ils soient forcés, à cause de leur action mutuelle, de changer dansle mouvemens a, b, c, &c. Il est clair qu’on peut regarder le mouvement a comme imprimé auCorp A comme composé du mouvement a, qu’il a prise, & d’un autre mouvement α; qu’onpeut de même regarder le mouvemens b,c, &c. comme composé de mouvemens b, β, c, κ,&c. d’ou il s’ensuit que le mouvement des corps A,B,C, &c. entr’eux auroit été le même, siau lieu de leur donner les impulsions a,b,c on leur donné à la fois les doubles impulsions


a, α; b, β, c; κ, &c. Or par la supposition, les corps A,B,C, &c. ont prix d’eux mêmes lesmouvemens a, b, c,&c. Donc les mouvemens α,β,κ, &c. doivent être tels qu’ils ne dérangerrien dans les mouvemens a, b, c, &c. c’est-à-dire que si les corps n’avoient reçu que les mou-vemens α,β,κ, &c. ces mouvemens auroient du se détruire, le système demeurer en repos.De là résulte le principe suivant, pour trouver le mouvement de plusieurs corps qui agissentles uns sur les autres. Décomposé le mouvemens a,b,c, &c. imprimés à chaque corps, chacunen deux autres a, α ; b, β; c, κ, &c., qui soient tels, qui si l’on n’eut imprimé aux corps que lesmouvemens a, b, c, &c. ils eussent pu conserver ces mouvemens sans se nuire réciproquement;& que si on ne leur eut imprimé que les mouvements α,β,κ, &c. le système fut demeure enrepos; il est claire que a, b, c seront les mouvemens que ces corps prendront en vertu de leuraction. Ce qu’il falloit trouver.

10.38 On voit encore qu’il a été inutile de rappeler le fameux principe de D’Alembert, qui réduitla Dynamique a la statique. En vertu de ce principe, si l’on décompose chaque mouvementimprimé en deux autres, dont l’un soit celui que le corps prendra réellement, tous les autresdoivent se faire équilibre entre eux; c’est-à-dire, que si l’on décompose chaque mouvementimprimé en deux autres dont l’un soit celui que le corps perd, l’autre sera celui qu’il prendra.Mais cela revient immédiatement à ce qu’ion vient de dire, savoir que le mouvement réel dechaque point est la résultant de son mouvement imprimé, et de la résistance qu’il éprouve parsa liaison avec les autres; ce qui est évident de soi-même. Ainsi le principe de D’Alembertn’est au fond que cette idée simple qu’on remarque à peine dans la suite du raisonnement, etqui ne revêt la forme d’un principe que par l’expression qu’on lui donne.

10.39 L’avantage du principe de D’Alembert consiste à trouver les lois du mouvement indépen-damment de la considération des résistances ou forces de tension qu’on employait avant lui.

10.40 Les forces de résistance dont en parle ne sont autre chose que les forces capables d’être enéquilibre sur le système, ce sont les mêmes que celles qui emplois D’Alembert. C’est, si l’onveut pour abréger, qu’on les appelle forces de résistance mutuelles.

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11.1 On a donné a cet opuscule le titre d’Essai sur les machines en général, premièrement, parceque ce sont principalement les machines qu’on y en vue, comme étant l’objet le plus im-portante de la mécanique; et en second lieu, parce qu’il n’y est question d’aucune machineparticulière, mais seulement des propriétés qui sont communes à toutes.

11.2 Parmi les philosophes qui s’occupent de la recherche des loix du mouvement, les uns font dela mécanique une science expérimentale, les autres, une science purement rationnelle; c’est-à-dire, que les premiers comparant les phénomènes de la nature, les décomposent, pour ainsidire, pour connoître ce qui, ont de commun, et le réduire ainsi a un petit nombre de faits prin-cipaux, qui servent en suite à expliquer tous les autres, et à prévoir ce qui doit arriver danschaque circonstance; les autres commencent par des hypothèses, puis raisonnant conséquem-ment à leurs suppositions, parviennent à découvrir les loix que suivirent les corps dans leursmouvements, si leur hypothèses étoient conformes à la nature, puis comparant leurs résul-tats avec les phénomènes, et trouvant qu’ils s’accordent, en concluent que leur hypothèse estexact, c’est à dire, que les corps suivent en effet les loix qu’ils n’avoient fait d’abord quesupposer.Les premiers de ces deux classes de philosophes, partent donc dans leurs recherches, desnotions primitives que la nature a imprimées en nous, et des expériences qu’elle nous offrecontinuellement; les autres partent de définitions et d’hypothèses; pour les premiers, les nomsde corps, de puissance, d’équilibre, de mouvement, répondent à des idées premières; ils nepeuvent ni ne doivent définir; les autres au contraire ayant tout a tirer de leur propre lands, sont

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obligé de définir ces termes avec exactitude, et d’expliquer clairement toutes leurs supposi-tions; mais si cette méthode paroit plus élégante, elle est aussi bien plus difficile que l’autre;car il n’y a rien da embarrassant dans la plupart des sciences rationnelles, et sur-tout danscelle-ci, que de poser d’abord d’exactes définitions sur les quelles il ne reste aucune ambigu-ité: ce seroit me jeter dans des discussions métaphysiques, bien au dessus de mes forces, quede vouloir approfondir toutes celles qu’on a proposées jusqu’ici: je me contenterai d’examinerla première et la plus simple.[…]Les deux loix fondamentales dont je suis parti (Xl), sont donc des vérités purement expérimen-tales; et je les ai proposées comme telles. Une explication détaillée de ces principes n’entroitpas dans le plan de cet ouvrage, et n’auroit peut-être servi qui embrouiller les choses: lessciences sont comme un beau fleuve, dont le cours est facile à suivre, lorsqu’il a acquis unecertaine régularité; mais si l’en veut remonter à la source, on ne la trouve nulle part, parcequ’elle est par-tout; elle est répandue en quelque sorte sur toute la surface de la terre; de mêmesi l’on veut remonter à l’origine des sciences, on ne trouve qu’obscurité, idées vagues, cerclesvicieux; et l’on se perde dans les idées primitif.

11.3 Les anciens établirent en axiome que toutes nos idées viennent des sens: et cette grande véritén’est plus aujourd’hui un sujet de contestation.

11.4 Cependant les sciences ne tirent pas toutes un même fonds de l’expérience: les mathémati-ques pures en tirent moins que toutes les autres; ensuite les sciences physico-mathématiques;ensuite les sciences.Il séroit sans doute satisfaisant de pouvoir assigner su juste dans chaque science, le point oùelle cesse d’etre expérimentale pour devenir entièrement rationnelle: c’est-à-dire, de pouvoirréduire au plus petit nombre possible les vérités qu’on est obligé de tirer de l’observation, etqui une fois établies, suffisent pour qu’étant combinées par le seul raisonnement, elles em-brassent toutes les ramifications de la science: mais cela paroit très-difficile. En voulant re-monter trop haut par le seul raisonnement, on s’expose à donner des définitions obscures, desdémonstrations vagues et peu rigoureuses. Il y a moins d’inconvénient à tirer de l’expérienceplus de données qu’il ne seroit peut-etre strictement nécessaire.[…]C’est donc dans l’expérience que les hommes ont puisé les premières notions de lamécanique.Cependant les lois fondamentales de l’équilibre et du mouvement qui lui servent de bases’offrent d’une part si naturellement à la raison, et de l’autre, elles se manifestent si clairementpar le faits les plus communs, qu’il semble d’abord difficile de dire, si c’est à l’une plutôtqu’aux autres que nous devons la parfaite conviction de ces lois.

11.5 Maintenant il s’agit d’établir sur ces faits, et sur les autres observations qui peuvent encores’offrir, des hypothèses qui se trouvent constamment d’accord avec ces observations, et quedès-lors on puisse regarder comme des lois générales de la nature.[…]Nous comparons ensuite les conséquences qui en résultent, avec les phénomènes, et si noustrouvons qu’ils s’accordent, nous conclurons que nous pouvons considérer ces hypothèsescomme les véritable lois de la nature.

11.6 Mon objet n’a pas été de les réduire au plus petit nombre possible; il me suffit qu’elles nesoient point contradictoires et qu’elles soient clairement entendues […] mais elles sont peut-être plus propre à confirmer les principes, en faisant voir comment ils ne sont, pour ainsi dire,que les mêmes vérités qui reparoissent toujours sous des formes différentes.

11.7 Connaissant le mouvement virtuel d’un système quelconque de corps, (c’est-â-dire, celuique prenderoit chacun de ces corps, s’il étoit libre) trouver le mouvement réel qui aura lieul’instant suivant, à cause de l’action réciproque des corps, en les considérant tels qu’ils exis-tent dans la nature, c’est-à-dire, comme doués de l’inertie commune è toutes les parties de lamatière.


11.8 Première loi. La réaction est toujours égale et contraire à l’action.Seconde loi. Lorsque deux corps durs agissent l’un sur l’autre, par choc ou pression, c’est-à-dire, en vertu de leur impénétrabilité leur vitesses relative, immédiatement après l’actionréciproque, est toujours nulle.

11.9 Cette Essai sur les machines n’étant point un Traité de mécanique, mon but n’est pasd’expliquer en détail, ni de prouver les loix fondamentales que je viens de rapporter; ce sontdes vérité que tout le monde sent très-bien.

11.10 Que l’intensité du choc ou de l’action qui s’exerce entre deux corps qui se rencontrent, nedépend point de leurs mouvements absolus, mais seulement de leur mouvement relatif.Que la force ou quantité demouvement qu’ils exercent l’un sur l’autre, par le choc, est toujoursdirigée perpendiculairement à leur surface commune au point de contingence.

11.11 Si un système de corps part d’une position donnée, avec un mouvement arbitraire, mais telqu’il eut été possible aussi de lui en faire prendre un autre tout-à-fait égal et directementopposé: chacun de ces mouvements sera nommé mouvement géométrique.

11.12 Tout mouvement, qui imprimé à un système de corps ne change rien à l’intensité de l’actionqu’ils exercent ou pourroient exercer les uns sur les autres si on leur imprimoit d’autres mou-vemens quelconques, sera nommé mouvement géométrique.

11.13 La théorie des mouvemens géométriques est très-importante; c’est, comme je l’ai déjà ob-servé ailleurs (Géométrie de position, page 337), une espece de science intermédiaire entre lagéométrie ordinaire et la mécanique. […] Cette science n’a jamais été traitée spécialement:elle est entièrement à créer, et mérite, tant par sa beauté en elle-même que par son utilité,toute l’attention des Savans.

11.14 Dans le choc des corps durs, soit que ce choc soit immediat, ou qu’il se fasse par le moyend’une machine quelconque sans ressort, il est constant qu’à l’égard d’un mouvement quel-conque géométrique:1 – Le moment de la quantité de mouvement perdue par tout le système, est égal à zéro.2 – Le moment de la quantité de mouvement perdue par une partie quelconque des corps dusystème, est égal au moment de la quantité de mouvement gagné par l’autre partie.3 – Lemoment de la quantité demouvement réelle du système générale, immédiatement aprèsle choc, est égal au moment de la quantité de mouvement du même système, immédiatementavant le choc.

11.15 Parmi tous les mouvements dont est susceptible un système quelconque de corps durs agis-sant les uns sur les autres, soit par un choc immédiat, soit par des machines quelconques sansressort, celui de ces mouvements qui aura lieu réellement, l’instant d’après, sera le mouve-ment géométrique, qui est tel que la somme des produits de chacune des masses, par le carréde la vitesse qu’elle perdra, est un minimum, c’est à dire, moindre que la somme des pro-duits de chacun de ces corps, par la vitesse qu’il auroit perdue, si le système eut pris un autremouvement quelconque géométrique.

11.16 Dans le choc de corps durs, soit qu’il y en ait de fixes, ou qu’ils soient tous mobiles (ou cequi revient au même) soit que ce choc soit immédiat, ou qu’il se fasse par le moyen d’unemachine quelconque sans ressort; la somme des forces vives avant le choc, est toujours égaleà la somme des forces vives après le choc, plus la somme des forces vives qui auroit lieu, sila vitesse qui reste à chaque mobile, étoit égale à celle qu’il a perdus dans le choc.

11.17 Lorsqu’un système quelconque de corps durs change de mouvement par degré insensibles;si pour un instant quelconque on appelle m la masse de chacun corps, V sa vitesse, p saforce motrice, R, l’angle compris entre les directions de V et p, u la vitesse qu’auroit m, sion faisoit prendre au système un mouvement quelconque géométrique, r l’angle formé par u

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et p, y l’angle formé par V et u, dt l’élément du temps; on aura ces deux équations

∑mV pdt cosR−∑mVdV = 0

∑mupdt cosr−∑mud(V cosy) = 0.

11.18 Théorème fondmentalPrincipe général de l’équilibre et du mouvement dans les machines.XXXIV. Quel que soit l’état de repos ou de mouvement où se trouve un système quelconquede forces appliquées à une machine, si l’on fait prendre tôt-à-coup un mouvement quelconquegéométrique, sans rien changer à ces forces, la somme des produits de chacune d’elles, par lavitesse qu’aura dans le premier instant le point ou elle est appliquée, estimée dans le sens decette force, sera égale à zéro.

11.19 Il ne sera peut-étre pas inutile de prévenir une objection qui pourroit se présenter à l’esprit deceux qui n’auroient pas fait attention à ce qui a été dit sur le vrai sens qu’on doit attacher aumot force: imaginons, par exemple, dira-t-on, un treuil à la roue et au cylindre duquel soientsuspendus des poids par des cordes; s’il y a équilibre, ou que le mouvement soit uniformele poids attaché à la roue, sera à celui du cylindre, comme le rayon du cylindre est au rayonde la roue; ce qui est conforme è la proposition. Mai il n’est pas de même lorsque la ma-chine prend un mouvement accéléré ou retardé; il paroit donc qu’alors les forces ne sont pasen raison réciproque de leurs vitesses estimées dans le sens de ces forces, comme il suivroitde la proposition. La réponse à cela est, que dans le case où ce mouvement n’est pas uni-forme, les poids en question ne sont pas les seules forces exercées dans le système, car lemouvements de chaque corps, changeant continuellement, il oppose aussi à chaque instant,par son inertie, une résistance à ce changement d’état; il faut donc aussi tenir compte de cetterésistance. Nous avons déjà dit, comment cette force doit s’évaluer, et nous verrons plus bas,comment on doit la faire entrer dans le calcul. En attendant, il suffit de remarquer que lesforces appliquées à la machine dont il est ici question, ne sont pas les poids même, mais lesquantité de mouvement perdues par ces poids, lesquelles doivent s’estimer par les tensionsdes cordons auxquels ils sont suspendus: or, que la machine soit en repos ou en mouvement,que ce mouvement soit uniforme ou non, la tension du cordon attaché à la roue, est à celle ducordon attaché au cylindre, comme le rayon du cylindre est au rayon de la roue, c’est-à-dire,que ces tensions sont toujours en raison réciproque des vitesses des poids qu’ils soutiennent;ce qui est d’accord avec la proposition. Mais ces tension ne sont pas égales aux poids; ellessont les résultantes de ces poids et de leurs forces d’inertie, lesquelles sont elles-mêmes lesrésultantes des mouvements actuels de ces corps, et des mouvements égaux et directementopposés à ceux qu’ils prendront réellement l’instant d’après.

11.20 Lorsque plusieurs poids appliqués à une machine quelconque, se font mutuellement équilibre,si l’on fait prendre à cette machine un mouvement quelconque géométrique, la vitesse ducentre de gravité du système, estimée dans le sens vertical sera nulle au premier instant.

11.21 Il y a deux manières d’envisager la mécanique dans ses principes. La première est de la con-sidérer comme la théorie des forces; c’est-à-dire des causes qui impriment les mouvemens. Laseconde est de la considérer comme la théorie des mouvemens eux-mêmes. La première estpresque généralement suivie, comme la plus simple; mais elle a le désavantage d’être fondéesur une notion métaphysique et obscure qui est celle des forces.

11.22 Si une force P se meut avec la vitesse u, et que l’angle formé par le concours de u et P soitz, la quantité Pcoszudt dans laquelle dt exprime l’élément du temp, sera nommé momentd’activité, consommé par la force P pendant dt.

11.23 Dans une machine dont le mouvement change par degrés insensibles, le moment d’activitéconsommé dans un temps donné par les forces sollicitante, est égal au moment d’activitéexercé en même temps par les forces résistantes.


11.24 Plusieurs corps soumis aux loix d’une attraction exercée en raison d’une fonction quelconquedes distances, soit par ces corps mêmes les uns sur les autres, soit par différents points fixes,étant appliqué à une machine quelconque; si l’on fait passer cette machine d’une positionquelconque donnée, à celle de l’équilibre, lemoment d’activité consommé dans ce passage parles forces attractives dont ces corps seront animés pendant ce mouvement, sera un maximum.

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12.1 Al rinascere delle Scienze Galileo investigò i Teorici Fondamenti dell’equilibrio, e del motoassoggettandoli alla guida della Geometria, e col Principio delle Velocità Virtuali sparse unanuova, universale radiazione, in tutte le macchine semplici e composte.[…]Infatti la Meccanica per mezzo del Principio delle Velocità Virtuali, unita alla Geometriapartecipò della medesima evidenza, e ne godè i privilegi, per tutta l’ampiezza, in cui potevaspaziare la sintesi. In seguito la nuova Geometria (la quale con rapido volo percorre lo spazio,che l’antica era obbligata a misurare con lento passo, e giunge ove quella non si sa che siamai penetrata) ha corrisposto alle più lusinghiere speranze, ed il Sig. La Grange il primonell’immortale sua Opera intitolata Meccanica Analitica, non solo mostrò che il Principiodelle Velocità Virtuali è dovuto a Galileo, ma rilevò ancora, che questo Principio ha il van-taggio di potersi tradurre in linguaggio algebraico, cioè di essere espresso per una formulaanalitica, onde tutte le risorse della analisi vi si applicano direttamente.Quel Principio dopo inventato da Galileo era rimasto, quasi negletto, come penderebbe inutileuna grande spada, fino a tanto che non nascesse un braccio atto a brandirla. Infatti il Sig. LaGrange padrone di tutto l’Ente matematico, ha saputo valutarne l’importanza, e la feconditàfacendo per mezzo di esso della Meccanica una scienza nuova a segno, che nella universaledottrina dell’equilibrio, e del moto dei solidi, e dei fluidi, tutti quei difficili Problemi, cheavevano condotto fino ad ora i Geometri per mille diverse spinosissime strade, sono ridottiad un procedere regolare ed uniforme. E per dare un’idea di quanto abbia progredito lo spiritoumano, si può dire, che il moto, e l’equilibrio dei Corpi Celesti, la figura di essi e le orbite,che descrivono, non richiamano in sostanza, per quanto appartiene alla Meccanica, a consid-erare altre leggi oltre quelle, che hanno luogo nel calcolare il moto, e l’equilibrio di un Vettedel primo genere quantunque le difficoltà di puro calcolo, e la moltitudine degli oggetti dacontemplare presentino un apparato più vasto, ed imponente.[…]Alcuni si sono occupati nel far vedere, che questo Principio è vero, mostrando la conformitàdei risultati di esso con quelli dedotti da altri metodi universalmente ammessi. Ma veramentenon se ne potesse ottenere altra autentica, saremmo ben lontani dallo scopo, a cui miranoordinariamente i Geometri; nella stessa guisa, che allor quando i seguaci di Leibnitz manca-vano di una convincente dimostrazione del Calcolo Infinitesimale, era debole appoggio peressi l’osservare l’uniformità de’ suoi resultati, con quelli della Geometria degli Antichi.[…]Quella comune facoltà di primitiva intuizione, per cui ognuno si convince facilmente di unsemplice assioma Geometrico, come per esempio, che il tutto sia maggior della parte nonserve certamente per convenire della sopraccennata verità meccanica, la quale è tanto piùcomplicata di quello che sia uno degli ordinari assiomi, quanto il genio di quei grandi Uo-mini, che l’hanno ammessa per assioma, supera l’ordinaria misura dell’ingegno umano; edè in conseguenza necessario per coloro, che non ne restano appagati, il procurarsene una di-mostrazione dipendentemente da estranee teorie, come è piaciuto al Riccati (che con qualchesoccorso tutto metafisico, si è ristretto presso a poco a questo caso particolare in alcune Let-tere stampate in Venezia nel 1772) ovvero riposarsi sulla fede d’uomini sommi disprezzandol’abituale ripugnanza ad introdurre in Matematica il peso dell’autorità. E se veramente questatiranna della ragione dovesse per una sol volta apparire nel Tempio d’Urania, non potrebbeseguir ciò con minore scandalo, che trovandosi essa in mezzo a Galileo, e a La Grange.

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12.2 Teorema. L’equazione delle forze avrà luogo egualmente che quella dei momenti, quando icorpi saranno stabiliti in linea retta, ed inoltre le forze comunque applicatevi, se non avrannole direzioni parallele tra loro, le avranno almeno tali, che sieno parallele le proiezioni di esse,fatte in un piano passante per la linea dei corpi.

12.3 Si potrà dunque concludere, che in ogni sistema, in cui l’equilibrio dipenda dalle equazionipunti (1), (2), (3), (4), (5), (6) [4.3–8], del § LXXI, la proprietà della somma dei momenti = 0è una proprietà necessaria e indivisibile dall’equilibrio.

12.4 Non è dunque possibile di negare, che qualunque volta abbia luogo l’equilibrio, esista neces-sariamente l’equazione dei momenti; ma gli è sicuro che qualunque volta esiste l’equazionedei momenti abbia sempre luogo l’equilibrio.

12.5 Potrebbe dubitarsi che oltre a queste sei equazioni se ne potessero dare altre.

12.6 J’ai lu votre Ouvrage avec plaisir. S’il a encore quelque chose à désirer dans la Mécanique,c’est le rapprochement, et la réunion des principes, qui lui servent de base, et peut-être mêmela démonstration rigoureuse et directe de ces principes. Votre travail est un nouveau servicerendu a cette science. Vous observez avec raison, qu’il y a des cas, où l’équation des vitessesvirtuelles a lieu aussi par rapport aux différences finies, le système alors en changeant desituation ne cesse par d’être en équilibre. Ces sortes d’équilibres tiennent le milieu entreles équilibres stables, où le système revient de lui mime à son premier état, lorsqu’il en estdérange, et les équilibres non stables, où le système, une fois dérangé de son état d’équilibre,tend à s’en éloigner de plus en plus.

12.7 J’ai donné une démonstration du principe des vitesses virtuelles tirée de 1’équilibre desmousses. Un principe si important ne peut-étre prouvé de trop de manières. Votre travailsur ce sujet a, outre son propre mérite, celui d’avoir fait éclore d’autres ouvrages, et on luidoit les Mémoires de Prony et de Fourier, et dont 1es auteurs ont dû vous faire hommage.

12.8 Se poi supponghiamo che siano più punti in qualunque modo insieme connessi, e muovansiancora all’intorno d’un asse qualsiasi, chi è che subito non ravvisi che la teoria di siffattomoto dal principio del vette dipende necessariamente.

12.9 Benché, come notammo, alcuni son d’avviso che una dimostrazione rigorosa della teoria delvette da Archimede e dopo di lui da altri uomini prestatissimi investigata lasci tuttor desideriodi sè.

12.10 Dunque dovremo essere ancor noi del sentimento di quelli che opinano avere il principiodi risoluzion delle forze e di composizione per invisibil compagnia l’infallibilità metafisica;quello del vette, il patrocinio soltanto della continua e costante esperienza, e finalmente quellodelle velocità virtuali che da’ due precedenti deducesi, non poter maggior grado di certezzaacquistare di quello che si ravvisa nel principio del vette.

12.11 Eclaircir le principe des vitesses virtuelle dans toute sa généralité tel qu’il a été énoncé parM. Lagrange: Faire voir, si ce principe doit être regardé, comme une vérité évident par la seulexposition du principe même, ou s’il exige une démonstration: fournir cette démonstrationdans le case qu’on la juge nécessaire.

12.12 Quocumque modo secum invicem connectantur duo puncta A et A′, si velocitates eorumvirtuales v, v′ sint semper intensionis aequales, vires P,P′ respective iis applicate et in rectisvelocitatum oppositae in aequilibrio constant.

12.13 Propositiones septem priores ex prolegomenis mechanices depromptae ut axiomata teneridebent; ultimam vero ut concedatur saltem postulamus. Caeterum eius evidentiam paucisdeclarare iuvat.


12.14 Libenter fateor Polyspastum apud antiquos, licet ab iis cognitum, uti , inter alia , ex Pappi col-lectionum libro 8 col1igitur, (erat enim Polyspaston tertia facultas mechanica apud Heronem)minus celebratum quam vectis et in novissimi temporibus, inter aequilibrii scientiae principia,praedicatum non fuisse nisi a solis fere Landen et Lagrange: ast ubi de delectu principii agi-tur, attendendum videtur ad ipsius evidentiam et foecunditatem praesertim: porro commodahaec in summo gradu prae se fert Polyspasti theoria at nemo non diffitebitur.

12.15 Dicet insuper aliquis forsan mancam aut incompletam esse nostram demonstrationem, ratuscum quibusdam proeliandum esse non solum ab aequilibrii inter vires hypothesi momen-torum aequationem dimanare, sed etiam reciproce, ex hypothesi momentorum aequationis,aequilibrium inter vires sequi: verum attendatur momentorum aequatione (6) (11°), quae inaequilibrio systematis Polyspastis instructi valet, exprimi evidenter aequilibrium adesse intervires; eandemque, propter propositionum concatenationem , aequationis momentorum signi-ficationem obtinere in omni systematum genere et liquebit aequationem hanc haberi debereut aequilibrium adesse declarantem non vero tantum ut aequilibrium Concomitantem.

12.16 Perciocché vedesi costretta a mettere in campo il ripiego di certo meccanico movimento fit-tizio infinitesimale, che diede occasione bensì alla scoperta d’insigni verità maravigliose, mache lascia nel tempo stesso sussistere tutt’ora il desiderio di una chiara semplice ed unicadimostrazione del vincolo primitivo, e necessario, che ad esso lega siffatte proprietà, di-mostrazione che può dirsi non ancora conseguita, se si considera l’incostanza, la compli-cazione, e l’oscurità dei tentativi, che per essa sono stati fatti.

12.17 Si plusieurs forces, ayant des directions quelconques, sont appliques à un système de corps oude points et se font équilibre; la somme de ces puissances multipliées chacune par la vitessequ’elle tend à imprimer au point auquel elle est appliquée est nécessairement égale à zero.On voit évidemment que cet énoncé rentre dans celui qui à été exposée ci-dessus, mais qu’ilen seulement degagé des quantités infiniment petits.

12.18 Queste riflessioni persuadono che sarebbe un cattivo filosofo chi si ostinasse a volere cono-scere la verità del principio fondamentale della meccanica in quella maniera che gli riescemanifesta l’evidenza degli assiomi. Però dovrà necessariamente mancare di questa evidenzail principio che assumerò […] il quale è lo stesso assunto da Lagrange nella parte terza dellateorica delle funzioni. Ma se il principio fondamentale della meccanica non può essere evi-dente, dovrà essere non di meno una verità facile a intendersi e a persuadersi.

12.19 Se un corpo attratto verso un punto fino a passare in linea retta nel tempo t lo spazio φ(t),qualora gli venga impresso un altro moto […] αt […] per l’azione simultanea dei due motinon percorre uno spazio espresso da φ(t)+αt ma da un’altra funzione del tempo.

A.13 Chapter 13

13.1 Pour la Patrie, pour les Sciences et la Gloire.

13.2 Je dois aussi indiquer aux élèves un ouvrage dont il leur sera très-utile de réunir la lectureet l’étude, aux instructions qu’ils reçoivent à l’École sur fa même matière c’est un, mémoireitalien publié à Florence en 1796, par M. Fossombroni, et intituléMemoria sul principio dellevelocità virtuali. Ce traité leur offrira une foule d’exercice très-profitables sur-tout à ceux quiveulent étudier la Mécanique analytique.

13.3 La démonstration précédent ne laisse rien à désirer pour la rigueur; mais l’équation desvitesses virtuelles, présentée de cette manière, offre une conséquence plutôt qu’une véritéfondamentale; et il est nécessaire, pour lui conserver le caractère de principe, de la déduirede théorèmes de Mécanique encore plus élémentaire et plus près des vérités de définition que

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ceux dont je me suis servi: c’est que je vais faire, en ne supposant que la composition despuissances appliquées à un point unique et de celle des puissances parallèles.

13.4 J’ai pensé aussi qu’il ne suffisait pas de prouver, d’une manière absolue, la vérité de la propo-sition, mais qu’on devait le faire indépendamment de la connaissance que nous avons desconditions de de l’équilibre dans les différentes espèces de corps, puisqu’il s’agit de consid-érer ces conditions comme des conséquences de la proposition générale. Cet objet se trouverempli par les démonstrations que nous allons rapporter; il nous semble qu’elles ne laissentrien à désirer sous le double rapport de l’étendue et de l’exactitude. Nous supposerons connuele principe de levier, tel qu’il est démontré dans les livres d’Archimède, ou ce qui revient aumême le théorème de Stevin sur la composition des forces, et quelques propositions qu’il estaisé de déduire des précédentes.

13.5 Si un corps est déplacé par une cause quelconque suivant une certaine loi, chacune des quan-tités qui varient avec sa position, comme la distance d’un de ses points à un point ou à unplan fixe, est une fonction déterminée du temps, et peut être considérée comme l’ordonnéed’une courbe plane dont le temps est l’abscisse, la tangente de l’angle que fait cette courbe àl’origine avec la ligne des abscisses, où la première raison de l’accroissement de l’ordonnéeà l’abscisse exprime la vitesse avec laquelle cette quantité commence a croitre, ou, pour nousservir d’une dénomination reçue, la fluxion de cette quantité.Le corps étant soumis à l’action de plusieurs forces, si l’on prend sur la direction de chacuneun point fixe dont la force tende a rapprocher le point du système où elle est appliquée, le pro-duit de cette force par la fluxion de la distance entre les deux points est le moment de la force:le corps peut être déplacé d’une, infinité de manières, et à chacune répond une valeur du mo-ment. Si l’on prend le moment de chaque force pour un même déplacement, la somme de tousces momens contemporains sera appelée le moment total, ou le moment des forces, pour cedéplacement. Nous distinguerons d’abord les déplacemens compatibles avec l’espèce et l’étatdu système, de ceux qu’on ne petit lui faire éprouver sans altérer les conditions auxquellesil est assujetti; et nous supposons ces conditions exprimée, autant qu’il est possible, par deséquations.Maintenant le principe des vitesses virtuelles consiste en ce que les forces qui sollicitent uncorps de quelque nature qu’il puisse être, étant supposées se faire équilibre, le moment totaldes forces est nul pour chacun des déplacemens qui satisfont aux equations de condition.JeanBernoulli considère au lieu des fluxions les accroissements naissans. Il faut alors regarderchacun des points du système comme décrivant un petit espace rectiligne d’un mouvementuniforme pendant un instant infiniment petit. Cet petit espace projeté perpendiculairementsur la direction de la force, est la vitesse virtuelle; et si on la multiplie par la force, le produitreprésent le moment. J’adopterai cette heureuse abréviation, et tous les procédés usités ducalcul différentiel.

13.6 Si l’on considère deux forces qui se font équilibre étant appliquées aux exterminés d’un filinextensible, il sera facile de connaitre leur moment total pour un déplacement compatibleavec la nature du corps en équilibre. Il suit de l’article précédent, que le moment est nultoutes lés fois que la distance est conservée; c’est-à-dire, lorsque l’équation de condition estsatisfaite. Pour tous les autres déplacemens possibles, le moment est positif, et le système enéquilibre ne peut être trouble de manière que le moment total soit négatif.

13.7 Au lieu de transformer, comme nous l’avons fait jusqu’ici, les forces qui sollicitent le sys-tème, nous substituerons à ce système, sur le quel elles agissent, un corps plus simple, maissusceptible d’être déplacé de la même manière.

13.8 Si l’on se contentait de substituer à chacune des forces un poids attaché a un fil renvoyé parune poulie fixe, on reconnaitrait que pour chaque déplacement du système en équilibre laquantité de mouvement des poids que s’élèvent est égale à celle des poids qui s’abaissent; etquoique cette remarque ne puisse pas être considérée comme une démonstration, néanmoinselle ramène le principe des vitesses virtuelles à celui de Descartes, ou au principe, employé


par Toricelli. Il est naturel de penser que Jean Bernoulli connaissait quelque constructionanalogue. On trouve les memes idées dans un ouvrage de Carnot, imprimé dès 1783, sous celitre: Essai sur les machines en général.

13.9 Le principe des vitesses virtuelles qui sen de base à cet admirable ouvrage, fut considérépar son auteur comme un fait dont il ne se proposait alors que de développer toutes les con-séquences; on s’est occupé depuis de la démonstration générale de ce principe M. Lagrangel’a ramené, d’une manière très-simple, au principe de l’équilibre des mousses, M. Carnotà celui de l’équilibre du levier. La démonstration de ce même principe a été déduite parM. Laplace, de considérations plus générales, mais trop abstraites peut-être pour être misesfacilement à la portée des commençans. Je me suis proposé de donner, autant qu’il me serapossible, la même généralité à une démonstration qui reposât uniquement sur la théorie de lacomposition et de la décomposition des forces appliques à un même point, et qui fut dégagéde la considération des quantités infiniment petite: tel a été le but que je me suis proposé dansles recherches que j’ai l’honneur de présenter à la classe.

13.10 Les lois de l’équilibre se déduisent, de la manière la plus rigoureuse, de quelques considéra-tions fort simples, lorsque les forces sont appliquées à un même point; mais elles deviennentplus difficiles à démontrer, surtout lorsqu’on se propose de les considérer dans toute leurgénéralité, dès que les forces agissent sur différens points: assujettis à des conditions qui con-tribuent à la destruction mutuelle des forces. La difficulté vient surtout de la nécessité de faireentrer, d’une manière générale, ces conditions dans le calcul. Il semble, au premier aspect,qu’on peut les considérer séparément, et ne supposer d’abord qu’une des conditions, puis uneautre, et ainsi de suite; mais un peu de réflexion fait voir qu’il faudrait alors pouvoir démon-trer à priori, que les effets produits par la réunion de plusieurs conditions, se composent deseffets qui résultent de chaque condition en particulier, sans qu’elles soient modifiées par leurréunion; vérité qui parait plutôt devoir une conséquence des équations de l’équilibre, qu’unmoyen de les obtenir.

13.11 Une autre simplification qu’on pourrait employer dans la recherche dont nous nous occupons,consiste à supposer successivement tous les points du système fixes, à l’exception de deuxd’entre eux, ce qui est d’autant plus commode que, par l’addition des équations ainsi obtenues,on compose précisément avec les dérivées partielles relatives à chaque variable, les dérivéestotales dont on a besoin; mais un exemple très-simple me parait suffisant pour faire voir quecette supposition n’est pas toujours admissible.

13.12 Le principe connu sous le nom de principe des vitesses virtuelles, se réduit à ce que si l’onfait une somme des momens de toutes les forces appliquées an système, en prenant avec dessignes contraires, ceux dont les forces et les projections tombent du même coté, et ceux dontles forces et les projections tombent de cotés opposés; que l’on ajoute à celle somme celledes équations déduites de toutes les conditions données, multipliées chacune par un facteurarbitraire; et réduites à contenir dans tous leurs termes les dérivées x′,y′,z′, à la premièrepuissance; qu’on égale séparément a zéro les quantités qui multiplient chaque dérivée, etqu’on élimine tous les facteurs arbitraires, l’équation ou les équations restantes expirerontmutes les conditions de l’équilibre.

13.13 Or, c’est un théorème d’algebre aisé à démontrer que l’équation résultant de cette éliminationest identiquement la même que celle qu’on obtiendrait en ajoutant à la somme des momensles équations des conditions multipliées par des facteurs arbitraires, en égalant séparément àzéro les quantities qui multiplient chaque dérivée, et en eliminant les facteurs.

13.14 Or la force du pression d’un point sur une surface lui est perpendiculaire, autrement ellepourroit se décomposer en deux, l’une perpendiculaire à la surface, et qui seroit détruit parelle, l’autre parallèle à la surface, et en vertu de laquelle le point n’auroit point d’action surcette surface, ce qui est contre la supposition.

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13.15 Deux points dont les masses m et m′, ne peuvent agir l’une sur l’autre, que suivant la droitequi le joint. A la vérité, si les deux points sont liés par un fil qui passe sur une poulie fixe,leur action réciproque peut n’être point dirigée suivant cette droite. Mais on peut considérerla poulie fixe, comme ayant à son centre, une masse d’une densité infinie, qui réagit sur lesdeux corps m et m′, dont l’action l’un sur l’autre n’est plus qu’indirecte.

A.14 Chapter 14

14.1 Les lignes aa′,bb′,cc′, &c, sont ce qu’on appelle dans les auteurs le vitesses virtuelles despoints a,b,c, &c., mais si l’on veut avoir le valeur des moments, on multiple les forces parces lignes estimées suivant les directions des forces, c’est à dire projetées sur elles. Il est doncconvenable, pour abréger, d’appeler ces projections elles-mêmes les vitesses virtuelles.

14.2 De cette manière on exclurait les idées de mouvemens infiniment petits et de perturbationsd’équilibre qui sont des idées étrangères à la question; et le principe des vitesses virtuellesparaitrait comme un simple théorème de géométrie dégagé de ces considérations qui laissenttoujours dans l’esprit quelque chose d’obscur. Mais il est bon d’observer que cette propriétéde l’équilibre dont nous nous occupons ne fut découverte que par la considération de ces pe-tites vitesses, par ce qu’elle s’offre naturellement lorsqu’on dérange unemachine en équilibre.Il semble que par ce dérangement on estime les énergies des puissances pour mouvoir la ma-chine. Lorsqu’un système est en équilibre, on connait bien la valeur absolue de chaque force,mais non pas l’effort quelle exerce su egard à sa position. En dérangeant un peu le système,on voit quels sont les mouvemens simultanés que peuvent prendre les points où les forcessont appliquées, quelques uns de ces points se mouvant du même cote que tirent les forces,les autres étant entrainés dans le sens contraire, et si l’on estime les énergies proportionnellesaux produits des forces par les vitesses des points d’application, on trouve que les énergiesqui obtiennent leur effet sont égales aux énergies vaincues.

14.3 Si un système libre de figure invariable est en équilibre en vertu de forces quelconques qui luisont appliquées, en supposant que les forces agissent toutes aux points de rencontre de leursdirections avec un plan situé comme on voudra, l’équation des moments aura lieu quel quesoit le déplacement fini qu’on donne au système.

14.4 Il faut encore remarquer qu’on suppose le système déplacé d’une manière quelconque, sansaucun égard à l’action des puissances qui tend à le déplacer; le mouvement qu’on lui donneest un simple change de position où le temps n’entre pour rien.

14.5 Le principe des vitesses virtuelles est connu depuis long-temps, aussi bien que la plupart desprincipes généraux de la mécanique. Galilée observa le premier, dans les machines, cettefameuse propriété des vitesse. virtuelles, c’est-à-dire, cette relation si connue qui existe entreles forces appliquées et les vitesses que prendraient leurs points d’application si l’on venaità troubler infiniment peu l’équilibre de la machine. Jean Bernouilli vit toute l’étendue de ceprincipe, et l’énonça avec cette grande généralité qu’on lui donne aujourd’hui. Varignon et laplupart des géomètres prirent soin de le vérifier dans presque toutes les questions de la sta-tique; et quoiqu’on n’en est point de démonstration générale, il fut universellement regardécomme une loi fondamentale de l’équilibre des systèmes.Mais jusqu’à M. Lagrange, les géomètres s’étaient plus appliqués à démontrer ou I’étendreles principes généraux de la science, qu’à en tirer une règle générale pour la solution desproblèmes; ou plutôt ils ne s’étaient pas encore propose ce grand problème qui est à lui seultonte la mécanique.Ce fut alors une heureuse idée de partir sur-le-champ du principe des vitesses virtuellescomme d’un axiome, et sans s’arrêter davantage à le considérer en lui-même de ne songerqu’à en tirer une méthode uniforme de calcul pour former les équations de l’équilibre et du


mouvement dans tous les systèmes possibles. On franchit par là toutes les difficultés de lamécanique: évitant pour ainsi dire, de faire la science elle-même, on la transforma en unequestion de calcul: et cette transformation, l’objet et le résultat de la Mécanique analytique,parut comme un exemple frappant de la puissance de l’Analyse.Cependant, comme dans cet Ouvrage, on ne fut d’abord attentif qu’à considérer ce beaudéveloppement de la Mécanique qui semblait sortir tout entière d’une seule et même for-mule, on crut naturellement que la science était faite, et qu’il ne restait plus qu’à chercherla démonstration du principe des vitesses virtuelles. Mais cette recherche ramena toutes lesdifficultés qu’on avait franchies par de principe même. Cette loi si générale, où se mêlent desidées vagues et étrangères de mouvemens infiniment petits et de perturbation d’équilibre, nefit en quelque sorte que s’obscurcir l’examen: et le livre de M. Lagrange n’offrant plus alorsrien de clair que la marche des calculs, on vit bien que les nuages n’avaient paru levés sur lecours de la Mécanique, que parce qu’ils talent pour ainsi dire rassembles à l’origine même decette science.Une démonstration générale du principe des vitesses virtuelles devait au fond revenir à établirla mécanique entière sur une autre base. Car la démonstration d’une loi qui embrasse tout unescience ne peut être autre chose que la réduction de cette science à une autre loi aussi générale,mais évidente, ou du moins plus simple que la première, et qui partant la rende inutile. Ainsi,par cela même que le principe des vitesses virtuelles renferme toute la mécanique, comme ila besoin d’une démonstration approfondie, il ne peut lui servir de base première. Chercherà le démontrer pour l’heureuse usage qu’on en a fait, c’est chercher à s’en passer pour cetusage même; soit en trouvant quelque autre loi aussi féconde mais plus claire, soit en fondantsur les principes ordinaires une théorie générale de l’équilibre, dont la propriété des vitessesvirtuelles ne devient plus alors qu’un simple corollaire. Ainsi, dans cet état où M. Lagrangeavait porté la science, ce n’était point la démonstration du principe des vitesses virtuelles qu’ilfallait chercher immédiatement. La Mécanique analytique, telle que l’auteur l’a conçue, estau fond ce qu’elle doit être: et la démonstration du principe des vitesses virtuelles n’ymanquepoint, puisque, si l’on essayait de la mettre à la tête de ce livre d’une manière générale et biendéveloppée, l’ouvrage se trouverait fait deux fois; je veux dire que cette démonstration com-prendrait déjà toute la mécanique.Il faut considérer que M. Lagrange s’est placé tout d’un coup sur un des points élevés de lascience, afin de découvrir quelque règle générale pour résoudre, ou du moins pour mettre enéquations, tous les problèmes de la mécanique; et cet objet est parfaitement rempli. Mais pourformer la science elle-même, il faut élever une théorie qui domine également tous les pointsde vue d’où l’on peut l’envisager. Il faut aller directement, non pas au principe obscur desvitesses virtuelles, mais à cette règle claire qu’on en a pour la solution des problèmes: et cetterecherche directe, la sente propre à satisfaire notre esprit, fait l’objet principal du Mémoirequ’on va lire.

14.6 Dans l’équilibre des systèmes, chaque force doit être perpendiculaire à la surface ou à lacourbe sur la quelle le point d’application n’aurait plus que la liberté se se mouvoir si tousles aitres points devenaient fixes.

14.7 Premièrement, par cela seul que les points du système sont liés entre eux par la premièreéquation L = const., on peut leur appliquer les forces respectives:

λ

√(∂L∂x

)2

+(

∂L∂y

)2

+(

∂L∂ z

)2

λ

√(∂L∂x′

)2

+(

∂L∂y′

)2

+(

∂L∂ z′

)2

A.14 Chapter 14 459

λ

√(∂L∂x′′

)2

+(

∂L∂y′′

)2

+(

∂L∂ z′′

)2

&c.

λ désignant un coefficient quelconque indéterminé, et chaque force étant normale à la sur-face représenté par l’équation L =const., lorsq’on y regarde les trois coordonnées du pointd’application comme seules variables; et l’on est sur que ces forces se feront équilibre sur lesystème.

En second lieu, parce que les points sont liés entre eux par la seconde équation M = const.,on peut leur appliquer encore les forces respectives

μ

√(∂M∂x

)2

+(

∂M∂y

)2

+(

∂M∂ z

)2

μ

√(∂M∂x′

)2

+(

∂M∂y′

)2

+(

∂M∂ z′

)2

μ

√(∂M∂x′′

)2

+(

∂M∂y′′

)2

+(

∂M∂ z′′

)2

&c.

μ étant un nouveau coefficient indéterminé; et chacune de ces forces étant normale à la surfacereprésenté par l’équationM =const. lorsqu’on y regarde les cordonnées du point d’applicationcomme seules variables.

[…]

Il est bien manifeste qu’il y aura équilibre en vertu de toutes ces forces, puisqu’il y auraitéquilibre en particulier dans chaque groupe relatif à chaque équation.

14.8 Quelles que soient les équations qui règnent ente les coordonnées des différens points dusystème, chacune d’elles pour l’équilibre, demande qu’on applique à ces points, le long deleurs coordonnées, des forces quelconques proportionnelles aux fonctions primes de cetteéquation, relativement à ces coordonnées respectives.Ainsi, en représentant par L= 0,M = 0, &c. des équations quelconques entre les coordonnéesx.y,z; x′,y′,z′, &c. des différens points, et par λ, μ, &c. des coefficients quelconques indéter-minés, on aura, pour les forces totales X ,Y,Z; X ′,Y ′,Z′, &c., qui doivent être appliquées àces points suivant leurs coordonnées:

X = λ(dLdx

)+μ

(dMdx

)+&c.

Y = λ(dLdy

)+μ

(dMdy

)+&c.

Z = λ(dLdz

)+μ

(dMdz

)+&c.

X ′ = λ(

dLdx′

)+μ

(dMdx′

)+&c.

Y ′ = λ(

dLdy′

)+μ

(dMdz′

)+&c.

Z′ = λ(

dLdz′

)+μ

(dMdz′

)+&c.

&c.


Si l’on élimine de ces équations les indéterminées λ,μ, &c., il restera les conditions del’équilibre proprement dites, c’est à dire les relations qui doivent avoir lieu entre les seulesforces appliquées et les coordonnées de leurs points d’application pour l’équilibre du système.

14.9 Note IIDémonstration du principe des vitesses virtuelles: Identité de ce principe avec le théorèmegénéral qui fait l’objet du Mémoire précédent.On s’est contenté d’observer dans le Mémoire, que du théorème où l’on est parvenu surl’expression générale des forces de l’équilibre, on pouvait passer aisément au principe desvitesses virtuelles. Mais ce principe est si célèbre dans l’histoire de la Mécanique, que je nepuis m’empêcher de marquer en peu de mots ce passage; et j’y reviens d’autant plus volon-tiers, que, non -seulement le principe des vitesses virtuelles est un corollaire de la propositiongénérale établie ci-dessus, mais qu’il me parait encore identique avec elle lorsqu’on le re-garde sous son vrai point de vue, et qu’on l’énonce d’une maniere complète.Soit le système défini par les équations suivantes entre les coordonnées des corps,

f (x,y,z,x′,y′,z′,&c.) = 0.

φ(x,y,z,x′,y′,z′,&c.) = 0. (A)

&c.

Supposons qu’on imprime a tous ces corps des vitesses quelconques qu’ils puissent avoiractuellement sans violer les conditions de la liaison; ses coordonnées x,y,z; x′,y′,z′, &c.varieront avec le temps t, dont il faudra les regarder comme fonctions; et, pour que les vitessesimprimées dx

dt ,dydt ,

dzdt ,

dx′dt , &c. soient permises par la liaison, comme on le suppose, il faudra

qu’elles satisfassent aux équations

f ′(x)dxdt

+ f ′(y)dydt

+ f ′(z)dzdt

+ f ′(x′)dx′

dt+ f ′(y′)

dy′

dt+&c. = 0

φ′(x)dxdt

+φ′(y)dydt

+φ′(z)dzdt

+φ′(x′)dx′

dt+φ′(y′)

dy′

dt+&c. = 0

&c.

(B)

tirées des précédentes (A); et il suffira qu’elles y satisfassent pour que les conditions de laliaison soient observées.Ou bien, si l’on multiplie ces équations par des coefficiens quelconques indéterminés, λ,μ,&c., et qu’on les ajoute, il suffira qu’elles satisfassent à la seule équation suivante, indépen-damment de λ,μ, &c.

[λ f ′(x)+μφ′(x)+&c.]dxdt

+[λ f ′(y)+μφ′(y)+&c.

] dydt

+

[λ f ′(z)+μφ′(z)+&c.]dzdt

+[λ f ′(x′)+μφ′(x′)+&c.

] dx′

dt+

[λ f ′(y′)+μφ′(y′)+&c.]dy′

dt+&c. = 0.

(C)

Or les fonctions qui multiplient les vitesses dxdt ,

dydt ,

dzdt ,

dx′dt , &c., ne sont autre chose (d’après

ce qui a été démontré ) que les expressions générales des forces capables d’être en équilibresur le système. Supposant donc des forces X ,Y,Z, X ′,Y ′,Z′, &c. qui se feraient actuellementéquilibre, on aurait:

Xdxdt

+Ydydt

+Zdzdt

+X ′dx′

dt+Y ′

dy′

dt+Z′

dz′

dt+&c. = 0. (D)

Au lieu des trois composantes X ,Y,Z, multipliées par les vitesses respectives dxdt ,

dydt ,

dzdt on

peut mettre la résultante P, multipliée par la vitesse résultante, projetée sur la direction de P,

A.14 Chapter 14 461

et que je nommerai dsdt ; et de même pour les autres; et l’on aura:

Pdsdt

+P′ds′

dt+P′′

ds′′

dt+&c. = 0

c’est-à-dire que si des forces se font équilibre sur un système quelconque, la somme de leursproduits par les vitesses, quelles qu’elles soient, qu’on voudra imprimer aux corps, mais queleur liaison permet, sera toujours égale à zéro, en estimant ces vitesses suivant les directionsdes forces.On voit par-là qu’on peut prendre des vitesses quelconque finies, que l’on mesurerait par desdroites quelconques qui seraient simultanément décrites par les corps, s’ils venaient tout-à-coup à rompre leur liaison et à s’échapper librement chacun de son coté.Quand on veut mesurer ces vitesses par les espaces mêmes que les corps décrivent réellement,comme elles varient à chaque instant par la liaison des corps, il faut prendre ces espacesinfiniment petits, sans quoi ils ne mesureraient plus les vitesses imprimées; et c’est ainsiqu’on tombe dans les vitesses virtuelles proprement dites, où le principe vient perdre unepartie de sa clarté.Il résulte, en effet, de ce que nous venons de dire, que cette belle propriété de l’équilibre peuts’énoncer de la manière suivante: Lorsqu’on voit suivre aux différens corps d’un système desmouvemens quelconque qui ne violent point la liaison établie entre eux, c’est- à-dire, quinous présentent continuellement le système dans des figures où les équations de conditionsubsistent, on put être sûr que les forces qui seraient capables de se faire équilibre sur unede ces figures, dans le moment où le système y passe, sont telles que, multipliées par lesvitesses actuelles des corps projetées sur leurs directions, la somme de tous ces produits estnécessairement égale à zéro.Le principe de cette manière n’offre plus aucune trace de ces idées de mouvemens infinimentpetits, et de perturbation d’équilibre: qui paraissent étrangères à la question, et qui laissentdans l’esprit quelque chose d’obscur.Lorsqu’il y a équilibre, il est clair que le principe a lieu pour tous les systèmes de vitesses queles points pourraient avoir en passant par la figure que l’on considère.Mais, quand on veut partir du principe, et l’énoncer de manière qu’il assure l’équilibre, faut-ildire qu’il a lieu pour ce nombre infini de systèmes de vitesses. Il y aurait surabondance deconditions, et l’on voit qu’il suffit de dire que l’équation (D) doit se vérifier pour autant desystèmes de vitesses que les équations de condition (B) en laissent d’indépendantes; ou bien(en réunissant comme on l’a fait ci-dessus, toutes ces équations en une seule (C), aumoyen desindéterminées, λ,μ, &c.), il suffit de dire que l’équation (D) des momens doit se verifier pourautant de systèmes de vitesses qu’il y a de vitesses dx

dt ,dydt ,

dzdt ,

dx′dt , &c. Mais comme chacun de

ces systèmes de vitesses doit satisfaire a l’équation (C), par hypothèse, cela revient à dire quetoutes les forces appliquées X ,Y,Z, X ′,Y ′,Z′, &c. qui multiplient les vitesses dxdt ,

dydt ,

dzdt ,

dx′dt ,

&c., dans l’équation (D), doivent être toutes proportionnelles aux fonctions[λ f ′(x)+μφ′(x)+&c.

] dxdt

,[λ f ′(y)+μφ′(y)+ etc..

] dydt

,[λ f ′(z)+μφ′(z)+&c.

] dzdt

,[λ f ′(x′)+μφ′(x′)+&c.

] dx′

dt,[

λ f ′(y′)+μφ′(y′)+&c.] dy′

dt,&c.

qui multiplient les mêmes vitesses dans l’équation générale (C). qui fait régner entre elles lesseules conditions que la liaison exige. Donc le principe des vitesses virtuelles, bien énoncé,c’est-à-dire avec toutes les idées qui peuvent le faire comprendre, est parfaitement identiqueavec le théorème général qui fait l’objet duMémoire. Il dit exactement la même chose; savoir,que, pour l’équilibre, les forces appliquées suivant les coordonnées des corps, en vertu dechaque équation, doivent être proportionnelles aux fonctions primes de cette équation rela-


tivement a ces coordonnés respectives: mais c’est là précisément ce qu’il fallait démontrer.Au reste, on serait encore conduit à reconnaitre cette identité en partant de l’énoncé ordinairedu principe des vitesses virtuelles, et se rendant bien compte, avant tout, du vrai sens qu’on ydoit attacher. En effet, le problème général de la statique n’est pas seulement de chercher lesrapports des forces qui se font actuellement équilibre, sur le système, mais bien l’expressiongénérale des forces qui peuvent s’y faire continuellement équilibre, dans toutes les figures oùil peut passer en vertu des équations de condition. L’équation générale donnée par le principedes vitesses virtuelles n’est donc pas, s’il est permis de parler ainsi, la relation d’un moment;elle ne doit pas simplement considérer l’équilibre du système dans la figure où il est, maisencore dans toute la suite des figures où il peut être, puisque c’est cette suite de figures quile caractérise et en constitue la définition. Ainsi l’équation des momens ne dit pas qu’il fautprendre pour les forces de tels nombres qu’elle en soit satisfaite, mais (puisque ces forcesdoivent varier avec la figure) qu’il faut choisir pour les représenter de telles fonctions descoordonnées, qu’elle demeure continuellement satisfaite, ou soit identique. Or, en vertu desconditions mêmes, on sait qu’il doit régner entre les vitesses simultanées que pourraient avoirles corps, une équation linéaire identique (C), dont les coefficiens sont les fonctions primesdes fonctions données par rapport aux coordonnées suivant lesquelles on estime ces vitesses.L’équation des momens dit donc que les forces de l’équilibre doivent être représentées parces fonctions; et par conséquent, pour la démontrer, il faut faire voir comment de telles forcesse font effectivement équilibre; ou bien il fallait chercher directement quelles fonctions descoordonnées peuvent représenter les forces de l’équilibre, comme nous l’avons fait d’abord.C’est pourquoi la plupart des démonstrations par lesquelles on a ramené le principe desvitesses virtuelles, ou à d’autres principes, ou a la loi connue de quelque machine simple, telleque le levier, &c., nous, paraissent bien plutôt des preuves que de véritables démonstrations.Toutes, en effet, même la plus heureuse, qui est de M. Carnot, e font sans rien emprunter dela définition générale du système, comme si la machine était, pour ainsi dire voilée, et qu’onn’en vît sortir que les cordons où sont appliquées les puissances. On peut bien prouver ourendre sensible par quelque construction plus ou moins simple, que si l’on trouble un peul’équilibre, ces puissances doivent être dans un certain rapport avec les allongements permisde ces cordons; mais cela ne peut offrir que les rapports actuels des forces considérées commenombres, et ne montre point du tout la forme d’expression qui leur est propre propre.Cette perturbation de l’équilibre n’apprendrait, dans aucun cas, à quelle machine on auraitaffaire, et les mêmes rapports pourraient s’offrir entre les forces appliquées, quoique les ma-chines fussent de constitution tout-à–fait différente, et que chacune d’elles imprimât pourtantà l’expression des forces qui lui conviennent, une forme différente qu’on y devrait voir etretrouver sans cesse, si la difficulté du théorème était entièrement consommée. Ainsi, la pro-priété des vitesses virtuelles n’en reste pas moins mystérieuse, et l’on n’a pas de véritabledémonstration, veux dire une explication ouverte et claire, où l ’on voie non-seulement quela chose se passe ainsi, mais qu’eIle est encore une suite de la définition général que soi-mêmeon a donnée au système que l’on considère.C’est peut-être par mie vue semblable, et pour arriver à l’équation des moment comme à uneéquation identique, queM. Laplace n’a considéré que les équations qui représentent la liaisondes parties du système, et n’a d’ailleurs employé d’autres principes que celui de la composi-tion des forces et de l’égalité entre I’action et la réaction; ce qu’on peut regarder comme lesélémens de la théorie de l’équilibre.Quoi qu’il en soit, au reste, soit qu’on veuille partir du principe des vitesses virtuelles pouren suivre jusqu’au bout la signification intime, soit qu’on attaque directement le problèmede la mécanique, ce qui est plus simple, on se trouve amené sur-le-champ à chercher quellessont les fonctions des coordonnées qui donnent les forces de l’équilibre dans toutes les figuresque peut affecter le système, en obéissant aux équations qui règnent entre les coordonnéesdes différens corps. Tel est exactement le problème que nous sommes proposé; et notre objetbien net et bien distinct a été de le résoudre par les premiers principes de la statique et de lagéométrie.

A.15 Chapter 15 463

A.15 Chapter 15

15.1 Lorsqu’un système invariable, libre ou assujetti à certaines conditions, se meut dans l’espace,il existe entre les vitesses des différents points certaines relations qui, dans beaucoup de cas,s’expriment très simplement, et que l’on déduit des formules relatives à la transformation descoordonnées. Je vais montrer, dans cet article, que les mêmes relations peuvent être tiréesdu principe de vitesses virtuelles. Ordinairement, on se sert de ce principe pour déterminerles forces capables de maintenir en équilibre un système de points matériels assujetti à desliaisons données, en supposant connues les vitesses que ces points peuvent acquérir dans unou plusieurs mouvements virtuels du système, c’est-à-dire dans des mouvements compatiblesavec les liaisons dont il s’agit. Mais il est clair qu’on peut renverser la question, et qu’aprèsavoir établi les conditions d’équilibre par une méthode quelconque, ou même, si l’on veut, parla considération de quelques-uns des mouvements virtuels, on pourra se servir, pour déter-miner la nature de tous les autres, du principe que nous venons de rappeler.Ajoutons qu’il est utile, dans cette détermination, de substituer au principe des vitessesvirtuelles un autre principe que l’en tire immédiatement du premier, ci qui se trouve renfermédans la proposition suivante.Théorème. Supposons que deux systèmes de forces soient successivement appliqués à despoints assujettis à des liaisons quelconques. Pour que ces deux systèmes de forces soientéquivalents, il sera nécessaire, et il suffira que, dans un mouvement virtuel quelconque, lasomme desmoments virtuels des force du premier système soit égale à la somme desmomentsvirtuels des forces du second système.

15.2 Théorème I. Si, à une époque quelconque du mouvement, deux points du système invariableont des vitesses nulles, les vitesses de tous les autres points se réduiront à zéro.

15.3 Théorème II. Si, à une époque quelconque du mouvement, les vitesses de tous les pointsdu système invariable sont différentes de zéro, ces vitesses seront toutes égales et dirigéessuivant des droites paralleés.

15.4 Théorème III. Si, à une époque quelconque du mouvement, un seul point du système invari-able a une vitesse nulle, la vitesse d’un second point choisi arbitrairement sera perpendiculaireau rayon vecteur mené du premier point au second, et proportionnelle à ce rayon vecteur.

15.5 Les théorèmes I, II et III indiquent toutes les relations qui peuvent exister entre les vitesses depoints matériels liées invariablement les uns aux autres, et compris dans un plan fixe dont ilsne doivent jamais sortir. Ces théorèmes prouvent que les vitesses dont il s’agit sont toujourscelles que présenterait le système pris dans l’état de repos, ou transporté parallèlement à unaxe fixe, ou tournant autour d’un centre fixe. Ajoutons: 1) que le mouvement de translation,parallèlement à un axe fixe, se déduit du mouvement de rotation autour d’un centre fixe,quand ce centre s’éloigne à une distance infinie de l’origine des coordonnées; 2) que le centrede rotation est un point dont la position, déterminée à chaque instant, varie en général d’unmoment à l’autre dans le plan que l’on considère. C’est pour celle raison que nous désigneronsle point dont il s’agit sous le nom de centre instantané de rotation.

15.6 Nous observerons d’abord que, à la fin d’un temps désigné par t, les différents points de lasurface mobile occuperont dans l’espace des positions déterminées, et que l’un d’eux, le pointO, par exemple, sera le centre instantané de rotation. De plus, il est clair que, à cette époque,on pourra faire passer par le point O deux courbes distinctes tracées de manière à comprendre,la première, tous les points de la surface mobile, et, la seconde, tous les points de l’espace quideviendront plus tard des centres instantanés de rotation.

15.7 Théorème VI. Quelle que soit la nature du mouvement d’un corps solide, les relations ex-istantes entre les différents points seront toujours celles qui auraient lieu, si le corps étaitretenue de manière à pouvoir seulement tourner autour d’un axe fixe et glisser le long de cetaxe.


15.8 Théorème VII. Concevons qu’un corps solide se meuve d’une manière quelcomque dansl’espace, et qu’à un instant donné on trace: 1) dans le corps; 2) dans l’espace, les différentesdroites avec lesquelles coïncidera successivement l’axe instantané de rotation de ce corpssolide. Tandis que la surface réglée, qui aura pour génératrices les droites tracées dans lecorps, sera entrainée par le mouvement de celui-ci, elle touchera constamment la surfaceréglée qui aura pour génératrices les droites tracées dans l’espace, et, par conséquent, la sec-onde surface ne sera autre chose que l’enveloppe de la portion de l’espace parcourue par lapremière.

15.9 Théorème VIII. Les mêmes choses étant posées que dans le théorème VII, si l’axe instantanéde rotation da corps solide devient fixe de position dans le corps, il sera fixe dans l’espace; etréciproquement.

A.16 Chapter 16

16.1 Les droites infiniment petites MN,M′N′,M′′N′′, etc., sont ce qu’on appelle les vitessesvirtuelles des points M,M′,M′′, etc.; dénomination qui provient de ce qu’elles sont consid-érées comme les espaces qui seraient parcourus simultanément par les points du système,dans le premier instant où l’équilibre viendrait à se rompre.

16.2 L’avantage du principe des vitesses virtuelles est de donner l’équation d’équilibre danschaque cas particulier, sans qu’on ait besoin de calculer ces forces intérieures; mais commela démonstration que nous allons donner est fondée sur la considération de ces forces, degrandeur inconnue, voici la notation dont nous ferons usage pour les représenter.

16.3 Il faut encore démontrer que, réciproquement, quand l’équation (b) a lieu pour tous les mou-vemens infiniment petits qu’on peut faire prendre au système des points M,M′,M′′, etc., lesforces données P,P′,P′′, etc., sont en équilibre.

[…]

Supposons pur un moment que l’équilibre n’ait pas lieu. Les points M,M′,M′′, etc., ou unepartie d’entre eux, se mettront en mouvement, et, dans le premier moment, ils décriront simul-tanément des droites telles que MN,M′N′,M′′N′′, etc.; on pourra donc réduire tous ces pointsau repos, en leur appliquant des forces convenables, diriges suivant les prolongements deces droites, en sens contraire des mouvemens produits; per conséquent, si nous désignons cesforces inconnues par R,R′,R′′, etc. l’équilibre aura lieu entre les forces P,P′,P′′, etc., R,R′,R′′,etc.; en sorte que r,r′,r”, etc., désignant les vitesses virtuelles projetées sur les directions deces nouvelles forces R,R′,R”, etc., on aura, d’apres le principe des vitesses virtuelles qui vientd’être démontré,

Pp+P′p′+P′′p′′+ etc.+Rr+R′r′+R′′r′′+ etc. = 0

ou simplement:Rr+R′r′+R′′r′′+ etc. = 0 (c)

en vertu de l’équation (b), qui a lieu par hypothèse.Cette équation (c) existant pour tous les mouvemens infiniment petits compatibles avec lesconditions du système des points M,M′,M′′, etc., nous pouvons choisir pour leurs vitessesvirtuelles les espaces réellement décrits MN,M′N′, M′′N′′, etc., dans un même instant; maiscomme ces lignes sont comptées sur les prolongemens des directions de R,R′,R′′, etc., ils’ensuit que toutes les projections r,r′,r′′, etc., seront négatives, et égales, abstraction faite dusigne, à ces mêmes lignes MN,M′N′,M′′N′′, etc. Alors, tous les termes de l’équation (c) étantde même signe, leur somme ne peut être nulle, moins que chaque terme ne soit séparément

A.16 Chapter 16 465

égal à zéro; on aura donc

R ·MN = 0, R′ ·M′N′ = 0, R′′ ·M′′N′′ = 0, etc.

or, pou que le produit R ·MN soit nul il faut qu’on ait, ou R = 0, ou MN = 0; ce qui signifie,dans l’un et l’autre cas, que le point M ne peut prendre aucun mouvement: il en est de mêmeégard de tous les autres points; par conséquent, le système entier est en équilibre; et c’est ceque nous nous proposions de démontrer.

16.4 La plupart des géomètres regardent comme évident que si des forces sont en équilibre sur unsystème de points, soumis à des liaisons qui leur permettent de prendre certains mouvements,ces mêmes forces seraient encore en équilibre sur le même système de points, soumis à des li-aisons différentes qui permettraient identiquement les mêmes déplacements. Ce principe […]nous avait toujours paru un peu hypothétique […] qui nous semble fondé sur une véritableconfusion de la Géométrie et de la Mécanique.[…]En conséquence, nous avons changé la démonstration du principe des vitesses virtuelles quenous avions empruntée à Ampère, et nous en avons adopté une qui n’offre pas le même in-convénient, et n’est autre chose au fond que celle qui se trouve dans le Traité de Mécaniquede Poisson.

16.5 Perciò che riguarda Giovanni Bernoulli è da notare che, col prendere in considerazione, nellacelebre lettera a Varignon (1717), le relazioni tra gli spostamenti (virtuali), infinitamentepiccoli, dei punti d’applicazione delle forze, egli non fece in fondo che applicare ed enunciare,in forma generale, una norma di metodo di cui era stato fatto già frequentemente uso dai suoipredecessori, tra gli altri da Leonardo da Vinci e da Galileo, nei loro tentativi di dedurre, dalprincipio della leva, quello del piano inclinato, e di far rientrare sotto quest’ultimo il caso di ungrave sostenuto da due fili non paralleli. Tale norma è quella che consiste nel riguardare comesostituibili, per quanto riguarda l’equilibrio, due sistemi di vincoli quando essi permettono glistessi spostamenti iniziali. Essa, come nota a proposito il Duhem, si trova enunciata, sotto laforma più esplicita, da Descartes, in una lettera al Padre Mersenne (1638) […]. Sulla presenzadi considerazioni analoghe negli scritti di Galileo è da vedere quanto dice il Mach (Mechanik,4° ediz., pag. 25-26).

16.6 Si un système quelconque de points est en équilibre, et que l’on conçoive un déplacementinfiniment petit de tous ses points, qui soit compatible avec toutes les conditions auxquellesil est assujetti, la somme des moments virtuels de toutes les forces est nulle, quel que soit cedéplacement. Et réciproquement, si cette condition a lieu pour tous les déplacement virtuels,le système est en équilibre.Dans cet énoncé, les infiniment petits sont considérés de la manière ordinaire. L’équationn’est exacte qu’en considérant les limites des rapports, après avoir divisé par l’une quelconquedes quantités infiniment petites; en d’autres termes, la somme des moments est infinimentpetite par rapport à ces moments eux-mêmes.

16.7 Dans ce qui nous allons dire, le mot de force s’appliquera doc seulement à ce qui est analoguesaux poids, c’est-à-dire à ce qu’on appelle, dans plusieurs cas, pression, tension, ou traction.En ce sens, une force ne peut jamais faire changer sensiblement la direction et la grandeurd’une vitesse sans le faire passer par tous les états intermédiaires.

16.8 Ces diverses expressions assez vagues ne paraissent pas propre à se répandre facilement. Nousproposerons la dénomination de travail dynamique, ou simplement travail, pour la quantité∫Pds […]. Ce nom ne fera confusion avec aucune autre dénomination mécanique; il parait

très propre à donner une juste idée de la chose, tout en conservant son acception communedans le sens de travail physique […] ce nome est donc très propre à designer la réunion deces deux éléments, chemin et force.


16.9 J’ai employé dans cet ouvrage quelques dénominations nouvelles: je désigne par le nom detravail la quantité qu’on appelle assez communément puissancemécanique, quantité d’actionou effet dynamique, et je propose le nom de dynamode pour l’unité de cette quantité. Jeme suispermis encore une légère innovation en appelant force vive le produit du poids par la hauteurdue à la vitesse. Cette force vive n’est que la moitié da produit qu’on a désigné jusqu’à présentpar ce nom, c’est-à-dire de la masse par le carré de la vitesse.

16.10 Ce mot de travail vient si naturellement dans le sens où je l’emploie, que, sans qu’il ait été niproposé, ni reconnu comme expression technique, cependant il a été employé accidentelle-ment par M. Navier, dans ses notes sur Bélidor, et par M. de Prony dans sonMémoire sur lesexpériences de la machine du Gros-Caillou.

16.11 Dorénavant nous nous servirons de la dénomination de machine pour designer les corps mo-biles auxquels nous appliquerons l’équation des forces vives: en ce sens. un seul corps qui semeut serait une machine tout comme un ensemble plus compliqué. Dans chaque cas partic-ulier, une fois qu’on saura bien de quels corps en mouvement se compose la machine dont onveut s’occuper, il suffira pour y appliquer les principes précédemment établis, de bien con-naître quelles sont les masses qui doivent entrer dans le calcul des forces vives, et quelles sontles forces mouvantes et résistants qui doivent entrer dans le calcul de la quantité de travail.

16.12 Pour passer à la statique et à la dynamique des systèmes des corps on n’aura besoin que des’appuyer sur le seul principe de l’égalité entre l’action et la réaction. Ce principe consiste ince que, si une molécule d’un corps produit une certaine force d’attraction ou de répulsion surune molécule voisine, elle recevra en même temps de celle-ci une force égale et directementopposée: en sorte que les forces qui se produisent dans l’ensemble des molécules qui formentun corps n’existent que par couples d’action égales et opposées. C’est à l’aide de ces seulspoints que nous allons donner tous les principes de mécanique.

16.13 Si l’on conçoit qu’un point auquel est appliquée une force P vienne à se déplacer d’une quan-tité δs dans une direction quelconque, nous appellerons élement de travail virtuel le produitde δs par la componente de la force dans la direction de δs; nous désignerons par Pδs l’anglede δs avec la force P, en sorte que l’élément de travail virtuel sera

Pcos(Pδs).

16.14 Si l’on suppose maintenant que les mouvemens virtuelles soient restreints à des mouve-mens opérée, en laissant l’ensemble des molécules dans l’état d’invariabilité des distancesmutuelles, qu’on peut appeler de solidification; alors les distances r ne variant pas dans cemouvement, on aura δr = 0, et l’équation ci-dessus se réduit à: ∑Pδp = 0.

16.15 L’équilibre ayant lieu sous l’action des forces extérieures P, chaque molécuIe sera en équili-bre, et l’on aura, en tenant compte de toutes le actions moléculaires R,

∑Rδr+∑Pδp = 0.

Si maintenant on prend un mouvement virtuel qui laisse à chaque corps son invariabilité deforme ou sa solidité, et que néanmoins dans ce mouvement on fasse glisser et rouler les corpsles uns sur les autres avec toute la latitude dans ces mouvemens que permet la constructionmême de la machine; il y aura une grande partie des élemens de travail virtuels Rδr qui s’eniront ce seront tous ceux qui sont dus des actions entre des molécules qui n’ont pas changé dedistance pendant le mouvement virtuel, c’est-à-dire entre celles qui appartiennent à un mêmecorps. Il ne restera donc dans l’équation ci-dessus que ceux des élémens de travail virtuel∑Pδr qui proviennent des actions entre les molécules de deux corps contigus, lorsque dansle mouvement virtuel ces corps ne se mouvront pas ensemble comme un seul système, maisqu’ils glisseront ou rouleront l’un sur l’autre. Les actions R qui resteront ainsi ne seront dues

A.17 Chapter 17 467

qu’à des molécules qui seront è une distance de la surface de contact qui sera moindre quel’étendue des actions moléculaires, ou en d’autres termes, que le rayon de la sphère d’activité.

16.16 Nous sommes conduits ainsi à reconnaître que le principe des vitesses virtuelles dans l’équi-libre d’une machine composée de plusieurs corps solides ne peut avoir lieu qu’en considérantd’abord les frottemens de glissement, lorsque les déplacemens virtuels peuvent faire lisserles corps les uns sur les autres, et en outre ceux de roulement lorsque les corps ne peuventprendre de mouvement virtuel sans se déformer près des points de contact.Les frottemens étant reconnus par expérience toujours capables de maintenir l’équilibre dansde certaines limites d’inegalité entre la somme des élémens de travail positif et la somme desélémens de travail negatif, en prenant ici pour négatifs les élémens appartenant à la sommela plus petite; il s’ensuit que la somme des élémens auxquels ils donnent lieu a préciément lavaleur propre à rendre nulle la somme totale et se trouve égale à la petite différence qui existeentre les sommes des élémens positifs et des élémens négatifs.

A.17 Chapter 17

17.1 Comme λδds la quantité peut représenter le moment d’une force λ tendante à diminuer lalongueur de l’élément ds le termeS λδds de l’équation générale de l’équilibre du fil représen-tera la somme des moments de toutes les forces λ qu’on peut supposer agir sur tous les élé-ments du fil: en effet chaque élément résiste par son inextensibilité à l’action des forces ex-térieures, et l’on regard communément cette résistance comme une force active qu’on nommetension. Ainsi la quantité λ exprimera la tension du fil.

17.2 Les expressions trouvé plus haut pour les variations font voir que ces variations ne sont queles résultats des mouvements de translation et de rotation que nous avons considérés en par-ticulier dans la section III.[…]L’analyse précédente conduit naturellement à ces expressions et prouve par là, d’une manièreencore plus directe et plus générale que celle de l’article 10 de la Section III, que lorsque lesdifférents points d’un système conservent leur position relative, le système ne peut avoir àchaque instant que des mouvements de translation dans l’espace et de rotation autour de troisaxes perpendiculaire entre eux.

17.3 Quoique nous ignorions la constitution interne des fluides, nous ne pouvons douter que lesparticules qui les composent ne soient matérielles, & que par cette raison les loix généralesde l’équilibre ne leur conviennent comme aux corps solides. En effet, la propriété principaledes fluides & la seule qui les distingues des corps solides, consiste en ce que toutes leursparties cèdent à la moindre force, & peuvent se mouvoir entr’elles avec tonte la facilité pos-sible, quelle que soit d’ailleurs la liaison & l’action mutuelle de ces parties. Or cette propriétépouvant aisément être traduite en calcul, il s’ensuit que les loix de l’équilibre des fluides nedemandent pas une théorie particulière, mais qu’elles. ne doivent être qu’un cas particulierde la théorie générale de la Statique.

17.4 Les théorie précédentes de l’équilibre & de la pression des fluides sont, comme l’on voit,entièrement indépendantes des principes généraux de la Statique, n’étant fondées que surdes principes d’expérience, particuliers aux fluides; & cette manière de démontrer les loix del’Hydrostatique, en déduisant de la connaissance expérimentale de quelques-unes de ces loix,celle de toutes les autres a été adopté depuis par la plupart des Auteurs modernes, & a fait del’Hydrostatique une science tout-à-fait différente, & indépendante de la Statique.Cependant il étoit importante de lier ces deux sciences ensemble, & le faire dépendre d’un seul& même principe. Or parmi les differens Principes qui peuvent servir de base à la Statique,& dont nous avons donné une exposition succinte dans la premier Section, il est visible qu’iln’y a que celui des vitesses virtuelles qui s’applique naturellement à l’équilibre des fluides.


17.5 Le principe de M. Clairaut n’est que une conséquence naturelle du Principe de l’égalitédes pression en tout sens. Aussi M. d’Alembert a-t-il déduit immédiatement de ce dernièreprincipe, les mêmes équations différentielles que M. Clairaut avoit trouvées par le sien; & ilfaut avouer que ce principe renferme en effet la propriété la plus simple & la plus généraleque l’expérience ait fait découvrir dans l’équilibre des fluides. Mais la connaissance de cettepropriété est-elle indispensable dans la recherche des loix de l’équilibre des fluides? Et nepeut-on pas dériver ces loix directement de la nature même des fluides considérées commedes amas de molécules très-déliées, indépendant les unes des autres, & parfaitement mobilesen tout sens?

17.6 Détermination du travail pour une petite déformation d’un corps. Relations qu’on en déduitentre les trente-six coefficients qui servent a definir la manière de se comporter dune sub-stance cristalline, ou de toute substance solide non isotrope.lmmaginons qu’un corps, soumis à l’action de forces quelconques, subisse une suite dechangements dans sa forme, en sorte que ses divers points (x,y,z) dont les coordonnées sontdevenues x+u,y+ v,z+w, continuent de se déplacer, et franchissent, pendant un temps in-finiment petit des espaces élémentaires δu,δv,δw parallèlement aux x, aux y et aux z. Letravail produit dans tout le corps par ce petit mouvement s’obtiendra en multipliant un élé-ment dxdydz de son volume par les composantes, dans les directions x,y,z des forces agissantsur l’unité de ce volume ci respectivement par les petits espaces parcourus δu, δv, δw, puisaoutant dans les trois produits et intégrant leur somme pour toute l’étendue ou pour tousles élémens du corps. Or, les trois composantes de forces agissant sur l’unité de volume del’élement dxdydz ne sont autre chose que les seconds membres des équations (50) du § 14,page 54, c’est-à-dire

∂ txx∂x

+∂ txy∂y

+∂ txz∂ z

+X

dans le sens X , et deux quadrinomes analogues dans les sens y et z. L’élément de travailproduit pendant que les points parcourent les espaces dont les projections suir les x,y,z, sontδu,δv,δw, se présente donc sous la forme

δW = δU +δV

oùδU =

∫ ∫ ∫(Xδu+Yδv+Zδw)dxdydz

représente le travail des forces extérieures agissant sur l’intérieur du corps, et

δV =∫ ∫ ∫

⎧⎪⎪⎪⎨⎪⎪⎪⎩δu

(∂ txx∂x + ∂ txy

∂y + ∂ txz∂ z

)+δv

(∂ tyx∂x + ∂ tyy

∂y + ∂ tyz∂ z

)+δw

(∂ tzx∂x + ∂ tzy

∂y + ∂ tzyz∂ z

)⎫⎪⎪⎪⎬⎪⎪⎪⎭dxdydz

représente le travail des tensions qui proviennent des actions réciproques de ces molécules,que des forces quelconques de pression et de traction pouvant solliciter sa surface.Considérons d ’abord un seul des neuf termes de cette dernière intégrale triple, par exemple∫ ∫ ∫

δu∂ txx∂x

dxdydz

integrant par rapport à x partiellement, à savoir pour la petite portion du corps qui est contenuedans un canal infiniment délié dont nous considérons la section dydz comme constante ainsique les cordonnées y et z. Cette integration par parties, si l’on replace, dans le second term

∂δu∂x

avec δ∂u∂x

A.17 Chapter 17 469

qui lui est identique, nous donne l’expression :∫ ∫[txxδu]dydz−

∫ ∫ ∫txxδ

∂u∂x

.

La parenthèse carré signifie qu’au lieu de l’expression txxδu qu’elle renferme, on doit mettrela différence des valeurs que prend cette expression aux deux extrémité du canal considéré.Désignons maintenant par dσ,dσ′ les élémens que découpe, sur la surface du corps, se canala ses extrémités, et par p,q,r les angles que forme avec les axes coordonnés la normale à dσmenée vers l’extérieur du corps, enfin par p′,q′,r′, les mêmes angles pour la normale à dσ.Si dσ est l’extrémité antérieure du canal, c’est-à-dire celle qui se trouve le plus du côté positifdes x, et dσ′ son extrémité postérieure, cos p est nécessairement positif, et cos p′ négatif, ensorte qu’on a

dydz = dσcos p =−dσ′ cos p′.

La différence des valeurs limites de txxδudydz devient donc la somme des valeurs que prendl’expression txxδudσcos p pour les extrémités du canal. Au lieu d’étendre l’intégrale doubleci-dessus aux extrémités de tous les canaux parallèles à l’axe des x que l’on peut mener sem-blablement dans l’intérieur du corps, il est évidemment possible d’intégrer directement pourl’ensemble des éléments dσ qui comprennent les éléments dσ′. Donc∫ ∫

[txxδu]dydz−∫

txxδudσcos p.

[…]Ainsi donc dans tous les cas, on replacera le terme de δV que nous avons considéré par∫

txxδudσcos p−∫ ∫ ∫

txxδ∂u∂x

dxdydz

où la première intégrale doit être étendue à toute la surface σ du corps.Si l’on fait de même pour tous les autres termes de δV , on obtient

δV = δU1−δU2

δU1 représentant l’ensemble des integrales simples et δU2 l’ensemble des intégrales triplescomme est celle de l’expression binôme en txx qu’on vient d’écrire. Et l’on a

δU1 =∫

(txx cos p+ txy cosq+ txz cosr)δudσ

+∫

(tyx cos p+ tyy cosq+ tyz cosr)δudσ

+∫

(tzx cos p+ tzy cosq+ tzz cosr)δudσ.

Or les expressions entre parenthèses sont précisément celles qui d’aprés les équations (25),équivalent aux composantes T cosω,T cosκ, T cosρ des forces de traction T appliquées àla surface du corps; δU1, n’est donc rien autre chose que le travail de ces forces de tractionextérieures, en sorte que

δU1 =∫T (cosωδu+ coskδv+ cosoδw)dσ.

On trouvera de moine que les huit termes de U2 autres que celui qui contient txxδ ∂u∂x , et que

nous avons écrit, sont affectés, sous le triple signe d’intégration, des incréments δ des autresquotients différentiels

∂u∂x

,∂u∂y

,∂v∂x

, . . .


des déplacements u,v,w ci des autres composantes tyy, . . . des tensions; d’où résulte, en met-tant, d’après les expressions (28) les déformations élémentaires ∂x,∂y, . . . ,gxy au lieu de

∂u∂x

,∂u∂y

, . . . ,∂u∂y

+∂v∂x

.

On a ainsi pour le travail total

δW = δU +δU1 +δU2

où δU et δU1 représentent les travaux des forces extérieures agissant respectivement sur lespoints intérieurs ci sur la surface du corps. Par conséquent−δU2 est nécessairement le travaildes forces internes qui procèdent des actions moléculaires.

17.7 M. Kirchhoff a eu l’obligeance de m’indiquer, vers 1858, une manière simple et directe de serendre compte de a composition sextinôme e l’expression ainsi donnée du travail interne oumoléculaire U2 pour l’unité de volume d’un élément. Soient dx,dy,dz les trois cotes trèspetits, parallèles aux x,y,z, de cet élément rectangle; 10 si la dilatation ∂x déjà subie parson cote x vient à être accrue de δ∂x, les deux faces opposées et égales yz s’éloignent dexδ∂x; les composantes normales de tension exercées par la matière environnante sur ces facesproduisent un travail yztxxxδ∂x; cela fait, par unité du volume xyz, le travail txxδ∂x; 20 si,l’une des deux faces opposées yz restant immobile, le glissement gxy vient à augmenter deδgxy, il y a un cheminement xδgxy de l’autre face parallèlement à celle-ci; en sorte que latension tangentielle txy qui agit par unité de sa surface yz dans le sens y de ce cheminement,produit un travail yztxyxδgxy. Il y a bien deux autres faces sur lesquelles agit une tension outyx égale a txy; ce sont les faces xz. Elles ont, dans ce mouvement pivoté autour des deuxcotes z de celle des deux faces yz qui est restée immobile; mais les tensions tyz s’y exercentdans le sens x et non dans le sens y qui a été celui du mouvement, elles n’ont donc rienajouté au travail yztxyxδgxy des tensions txy, travail qui est ainsi, seulement, txygxy par unitéde volume de l’élément. Or, le travail des six tensions sur les faces de l’élément doit, pourque l’équilibre ait lieu. après comme avant ces petits mouvements, être égal (au signe près)au travail moléculaire de l’intérieur de l’élément. Donc ce travail a bien pour grandeur, parunité de volume, le sextinôme (txxδ∂x + · · ·+ txyδgxy) de la parenthèse de l’expression de δU2.

17.8 Ecco il maggiore vantaggio del sistema della Meccanica Analitica. Esso ci fa mettere inequazione i fatti di cui abbiamo le idee chiare senza obbligarci a considerare le cagioni di cuiabbiamo idee oscure […]. L’azione delle forze attive o passive (secondo una nota distinzionedi Lagrange) è qualche volta tale che possiamo farcene un concetto, ma il più sovente ri-mane […] tutto il dubbio che il magistero della natura sia ben diverso […]. Ma nella M. A. sicontemplano gli effetti delle forze interne e non le forze stesse, vale a dire le equazioni dicondizione che devono essere soddisfatte […] e in tal modo, saltate tutte le difficoltà intornoalle azioni delle forze, si hanno le stesse equazioni sicure ed esatte che si avrebbero da unaperspicua cognizione di dette azioni.

17.9 Siano X, Y, Z le componenti delle forze acceleratrici che agiscono su ciascun punto del corpo;L,M,N, le componenti delle forze che agiscono su ciascun punto della superficie di esso, e ρ ladensità costante. Diamo ad ogni punto del corpo un moto virtuale e denotiamo con δu,δv,δwle variazioni che prenderanno per questo u,v,w. Il lavoro fatto in questo moto dalle forze datesarà evidentemente:∫

S(Xδu+Yδv+Zδw)dS+

∫σ(Lδu+Mδv+Mδw)dσ

A.18 Chapter 18 471

essendo S lo spazio occupato dal corpo eσ la sua superficie. Il lavoro fatto dalle forze elastichesarà uguale all’aumento del potenziale di tutto il corpo dato da:

Φ =∫

PdS

onde per il principio di Lagrange:

δΦ+∫S(Xδu+Yδv+Zδw)dS+

∫σ(Lδu+Mδv+Mδw)dσ= 0.

A.18 Chapter 18

18.1 La tentative qui se propose de réduire toute la Physique à la Mécanique rationnelle, tentativequi fut toujours vaine dans le passé, est-elle destinée à réussir un jour? Un prophète seulpourrait répondre affirmativement ou négativement à cette question. Sans préjuger le sens decette réponse, il parait plus sage de renoncer, au moins provisoirement, à ces efforts, stérilesjusqu’ici, vers l’explication mécanique de Univers.Nous allons donc tenter de formuler le corps des lois générales auxquelles doivent obéir toutesles propriétés physiques, sans supposer à priori que ces propriétés soient toutes réductiblesà la figure géométrique et au mouvement local. Le corps de ces lois générales ne se réduiraplus, dès lors, à la Mécanique rationnelle.[…]La Mécanique rationnelle doit donc résulter du corps de lois générales que nous nous pro-posons de constituer; elle doit etre ce qu’on obtient lorsqu’on applique ces lois général à dessystèmes particuliers où l’on ne tient compte que de la figure des corps et de leur mouvementlocal.Le code des lois générales de la Physique est connu aujourd’hui sous deux noms: le nom deThermodynamique et le nom d’Energétique.

18.2 Imaginons qu’une suite continue d’états d’unmême système isolé ait été formée: fixons notreattention sur ces divers états dans l’ordre qui permet de passer de l’un à l’autre d’une manièrecontinue; pour designer cette opération tout intellectuel à laquelle nous soumettons le schèmemathématique qui nous doit servir à représenter un ensemble de corps concrets, nous disonsque nous imposons au système une modification virtuelle.[…]Les variations des valeurs numériques des variables qui servent à définir un état du systèmedoivent être compatibles avec les conditions qui résultent logiquement de la définition dece système, mais avec ces conditions-là seulement. En particulier elles peuvent fort biencontredire aux lois expérimentales régissant l’ensemble de corps concrets que notre systèmeabstrait et mathématique a pour objet de représenter.

18.3 Il ne faut pas confondre une modification idéale avec une modification virtuelle; une mod-ification virtuelle se compose d’états du système qui ne se succèdent pas dans le temps; ensorte que le changement d’état qui constitue une modification virtuelle n’est pas lié à unmouvement; en la modification virtuelle, la notion de vitesse n’a point de place.

18.4 Ainsi donc quand un système se transforme en présence de corps étrangers nous considéronsces corps étrangers comme contribuant à cette transformation soit en la causant, soit en aidant,soit l’entravant; c’est cette contribution que nous nommons l’oeuvre accomplie, en une trans-formation d’un système par les corps étrangers à ce système.

18.5 Première convection. La symbole mathématique destiné a représenter le valeur de l’oeuvreaccomplie, en une modification réal ou idéale d’un système, sera déterminé toutes les foisqu’on connaitre la nature du système et la modification qu’il a subie; il ne changera pas si


l’on se borne à changer l’époque et le lieu où la modification a été produite ainsi quel lescorps étrangers en présence desquelles elle a été accomplie.

18.6 Principe de la Conservation de l’Énergie. Lorsqu’un système quelconque, isolé dans l’espace,éprouvé une modification réelle quelconque, l’énergie totale du système garde une valeurinvariable.

18.7 Forme restreinte du principe de la Conservation de l’Énergie. En toute modification réelled’un système isolé, l’égalité

U +12

∫Mu2 + v2 +w2 dm = const.

est vérifiée.

18.8 La comparaison de ces conditions […] fournit l’énoncé suivant, qui est celui du principe ded’Alembert: Pour obtenir, à chaque instant, les lois du mouvement d’un système de solidesassujettis à des liaisons sans résistance passive, il suffit d’écrire que le système demeurait enéquilibre si on le plaçait sans mouvement dans l’état qu’il traverse à cet instant, et si on lesoumettait non seulement aux actions extérieures qui s’exercent réellement sur lui au momentoù il se trouve en cet état, mais encore à des actions extérieures fictives équivalentes auxactions d’inertie qui le sollicitent à ce moment.

18.9 Dans le cas particulier où le système est assujetti exclusivement à des liaisons et bilatérales,il faut il suffit, pour l’équilibre, que le travail externe soit, tout déplacement en virtuel, égalà l’accroissement de l’énergie interne.

18.10 La contrainte que le système éprouve, de la part des liaisons, au cours de son déplacementréel est moindre que la contrainte qu’il éprouverait en toute autre déplacement virtuel issu dumême état: ∫

MN2dm <

∫PN

2dm

that is what we mean when we say that the studied constraints have no passive resistance.

18.11 Die Bewegung eines Systems materieller, auf was immer für eine Art unter sich verknüpfterPunkte, deren Bewegungen zugleich an was immer für äussere Beschränkungen gebundensind, geschieht in jedem Augenblick in möglich grösster Übereinstimmung mit der freienBewegung, oder unter möglich kleinstem Zwangen, indem man als Maass des Zwanges, dendas ganze System in jedem Zeittheilchen erleidet, die Summe der Producte aus dem Quadrateder Ablenkung jedes Punkts von seiner freien Bewegung in seine Masse betrachtet.Es seienm,m′,m“ u. s. w. dieMassen der Punkte; a,a′,a′′ u. s. w. ihre platze zur Zeit t; b,b′,b′′u. a. w. die Plätze, weiche sie, nach dem unendlich kleinen Zeittheilchen dt, in Folge derwahrend dieser Zeit auf sie wirkenden Kräfte und der zur Zeit t erlangten Geschwindigkeitenund Richtungen, einnehmen wurden, falls sie alle vollkommen frei waren. Die wirklichenPlätze c,c′,c′′ u.s.w. werden dann diejenigen sein, für welke, unter allen mit den Bedingungendes Systems vereinbaren, m(bc)2 +m′(b′c′)2 +m′′(b′′c′′)2 u.s.v. ein Minimum wird.

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Index

Abattouy, Mohammed, 68, 70, 482Al-Isfizari, 66, 71, 74, 136Alberti, Giovanni Battista, 95, 473Ampère, Andreé Marie, 1, 8, 9, 12, 13, 266,

290, 317, 318, 328–332, 339, 342, 347,366, 371, 473,

Angiulli, Vincenzo, 7, 12, 13, 16, 206,217–229, 231–233, 299, 375, 473

Apianus, Petrus, 75, 93, 473Araldi, Michele, 473Archimedes, 3, 34, 43, 45–54, 59, 64–67, 92,

93, 95–97, 103, 104, 108, 110, 114, 118,131, 137, 142, 160, 173, 178, 179, 199,225, 308, 322, 473, 475, 484, 486

Aristotle, 1–3, 10, 12, 34–38, 40–45, 51, 52,56, 63–66, 77, 90, 92, 95, 104, 105, 123,131, 135, 136, 185, 225, 258, 271, 341,473, 483, 486, 486, 487

Bagni, Giorgio, 482Bailhache, Patrice, 335, 482Baldi, Bernardino, 43, 92–94, 108, 154, 155,

473Barbaro, Daniele, 95–97, 474Barroso, Filho, 482Belhoste, Bruno, 482Bellucci, Giovanni Battista, 95, 474Beltrami, Eugenio, 387, 390–392, 474Benedetti, Giovanni Battista, 84, 91, 92, 94, 96,

116–119, 121, 143, 474Benvenuto, Edoardo, 482Bernoulli, Daniel, 239Bernoulli, Jakob, 239Bernoulli, Johann, v, 5–7, 10, 12, 13, 15, 16, 50,

187, 195, 199–204, 206–210, 213–215,217, 218, 220, 221, 225, 227, 228, 231,

233, 237–239, 242, 248–250, 252–254,256, 271, 299, 323, 328, 334, 337, 340,362, 366, 474

Bertoni, Giuseppe, 482Bertrand, Joseph Louis François, 252, 253,

263, 267Betti, Enrico, 387, 390, 392, 393, 474, 483Bevilacqua, 483Biringuccio, Vannoccio, 95, 474Bordoni, Antonio, 387Borelli, Alfonso, 5, 153, 217, 225, 391, 474Borgato, Maria Teresa, 482Boschiero, Luciano, 482Boscovich, Ruggiero Giovanni, 218, 224, 231,

233, 234, 236, 299, 475, 483, 485Bottecchia Dehò, Maria Elisabetta, 40, 473Bradwardwine, Thomas, 64Brioschi, Francesco, 390, 392Brown, Joseph Edward, 75, 77, 83, 475Brugmans, Anton, 51, 475Brunacci, Vincenzo, 387, 388, 475Buchner, Ferdinand, 482Boudri, Christian, 482Burzio, Filippo, 482

Camerota, Michele, 482Capecchi, Danilo, 482, 483Cardano, Girolamo, 91–94, 96, 104–107, 110,

131, 132, 157, 178, 190, 475Carnot, Lazare, 8–12, 16, 243, 244, 259, 297,

281–297, 317, 328, 329, 339, 341, 351,362, 367–369, 398, 475, 483–485

Carnot, Sadi, 297, 396Cauchy, Augustin Louis, 8, 248, 293, 318, 335,

353–360, 381, 383, 390, 475, 482Cavalieri, Bonaventura, 190

490 Index

Caverni, Raffaello, 103, 139, 483Ceccarelli, 485, 486Ceradini, Cesare, 483Cesariano, Cesare, 93, 475Clagett, Marshall, 56, 57, 68, 75, 77, 79, 483Clairaut, Alexis Claude, 475Clarke, John, 475Clarke, Samuel, 177, 475Clavius, Cristophorus, 120, 391Clebsch, Alfred, 375, 380, 383, 385–387, 475Clerke, Maxwell James, 177, 390Cockle, Maurice James Draffen, 483Commandino, Federico, 59, 92, 93, 96, 108,

109, 137, 141, 178, 180, 473, 475, 476Comte, August, 318, 483Coriolis, Gustave Gaspard, 10, 11, 158, 256,

293, 297, 361, 367–373, 380, 476Cusanus, Nicholas, 177

D’Alembert, Jean Baptiste le Ronde, 200, 212,237, 239–241, 243–247, 249, 251, 252,259, 268–271, 273–279, 286, 287, 291,294, 332, 401, 482, 476, 484

D’Ayala, Mariano, 482Da Cremona, Gerardo, 65, 68, 70Dal Monte, Guidobaldo, 12, 13, 30, 43, 56, 64,

84, 91–93, 96, 108–110, 112–116, 120,121, 132, 136, 137, 139, 159, 160, 179,187, 210, 476, 486

Damerow, Peter, 486De Brussel, Gerardus, 35, 477De Caus, Salomon, 476De Challes, Claude François Milliet, 176, 177,

476De Marchi, Francesco, 95, 476De Nemore, Jordanus, 12Descartes, René, 4, 5, 12, 13, 114, 127, 138,

152, 157, 159, 160, 164–174, 176, 177,187, 190, 192, 196, 199, 206, 210, 213,214, 225, 253, 275, 285, 292, 294, 328,366, 476

Di Giorgio, Francesco, 483Dijksterhuis, Eduard Jan, 50, 95, 184, 186, 483Drabkin, Israel Edward, 115, 483Drago, Antonino, vi, 23, 284, 290, 483, 484Drake, Stillman, 92, 104, 115, 483, 486Dugas, René, 157, 198, 484Duhamel, Jean Marie, 361, 365, 366, 476Duhem, Pierre Maurice Marie, 11, 32, 40, 44,

56, 66, 75–77, 84, 90, 95, 104, 126, 154,157, 159, 160, 184, 324, 342, 366, 390,395–403, 476, 484

Erasmus of Rotterdam, 177Euclid, 3, 9, 12, 34, 45–47, 63, 65–67, 69, 74,

83, 97, 100, 108, 178, 217, 391, 392, 474,487

Euler, Leonhard, 15, 200, 212, 218, 238–245,248, 251, 271–273, 286, 335, 342,377–379, 476

Ferriello, Giuseppina, 484Festa, Egidio, 131, 484Filoni, Andrea, 484Folkerts, Menso, 484Foncenex, Daviet de, 241, 299Fontana, Domenico, 95, 477Fossombroni, Vittorio, 9, 12, 16, 257, 263,

300–306, 309, 319, 320, 338, 477Fourier, Jean Baptiste Joseph, 8, 9, 12, 13, 262,

317, 319, 321–329, 341, 362, 366, 477Fraser, Craig, 484

Galilei, Galileo, 2–5, 12, 13, 30, 31, 43, 44,50, 58, 71, 72, 91, 93, 108, 120–132,137–140, 142, 144–148, 152, 160, 166,167, 169–171, 173, 179, 210, 218–220,223, 225, 226, 229, 239, 245, 253,256, 271, 300, 301, 340, 366, 375, 477,483–485

Galletto, Dionigi, 50, 241, 484Galluzzi, Paolo, 122, 484Gatto, Romano, vi, 92, 93, 484Gaukroger, S., 482Gauss, Carl Friedrich, 8, 324, 403, 477Gentile, Giuseppe, 484Gerardo da Cremona, 65Germain, Paul, 484Ghinassi, G., 484Giampaglia, Amedeo, 484Giardina, Giovanna Rita, 484Gille, Bernard, 484Gillispie, Charles Coulston, 282, 297, 485Giusti, Enrico, 56, 485Green, George, 382, 385, 386, 393, 477Gutas, Dimitri, 485

Hankins, Thomas, 485Helbing, Mario Otto, 485Helmholtz, Hermann Ludwig, 11, 395, 477Herigone, Pierre, 158, 159, 173, 477Hermann, Jacob, 238, 239, 271–273, 478Hero of Alexandria, 3, 4, 12, 33, 34, 45, 51–61,

65–67, 91–93, 95, 96, 104, 106, 109, 131,132, 168, 310, 473, 476–478, 484, 486

Høyrup, Jensen, 485Hunter, Michael, 485

Index 491

Huygens, Chrstiaan, 5, 6, 50, 127, 166, 171,178, 187, 188, 201, 204, 217, 225, 232,238, 478, 486

Huygens, Constantin, 164, 167, 174

Indorato, Luigi, 485

Jacobi, Carl, 263Jacquier, 478, 483Jammer, Max, 485Jaouiche, Khalil, 64, 68, 70, 73, 478Jordanus, 4, 12, 13, 63, 65, 66, 75–87, 89–93,

97–100, 102, 103, 107–112, 114–116,118, 119, 121, 124, 131, 132, 154, 157,159, 162, 484, 485

Jouguet, François, 366, 485

Khanikoff, Nicolas, 485Kirchhoff, Gustav, 386, 387Klein, Felix, 485Knorr, Wilbur Richard, 64, 68, 485Koetsier, Teun, 485Koyré, Alexandre, 485Kraft, Fritz, 38, 485

Lagrange, Giuseppe Lodovico, v, 7–9, 11–13,15–17, 22, 27, 31, 50, 149, 200, 201, 206,212, 228, 231, 237, 239–273, 277–279,282, 299, 300, 304–306, 308, 310–315,317, 319, 321, 322, 328, 329, 332, 333,335–337, 339–342, 345, 347, 348, 351,353, 354, 361, 362, 369, 371, 375–381,387–389, 391, 478, 480, 482, 483, 484,486

Laird, Walter Roy, 44, 485Lamé, Gabriel, 387, 391, 392, 478Lamy, Bernard, 5, 176, 177, 478Laplace, Pierre Simon, 8, 13, 17, 26, 328, 329,

332–335, 342, 343, 351, 354, 362, 363,371, 478

Le Seur, Thomas, 234, 483Leibniz, Gottfried Wilhelm, 6, 7, 186, 197–199,

202, 204, 210, 217–220, 224, 232, 233,245, 301, 478, 479

Leonico, Tomeo, 94, 479Libri, Guglielmo, 362, 485Lorch, Richard, 484Loria, Gino, 140, 485Lorini, Bonaiuto, 95, 96, 479Love, Augustus Edward Hough, 479

Mach, Ernst, 44, 48, 50, 166, 184, 187, 263,264, 277, 278, 366, 395, 485

Magistrini, Giovanni Battista, 311, 479

Maltese, Giulio, 485Manno, Salvatore Domenico, 290Marcolongo, Roberto, 95, 485Martinovic, 485Mascheroni, Lorenzo, 299, 479Maupertuis, Pierre Louis, 239, 258, 275, 292,

479Maurolico, Francesco, 93, 94, 479McCloskey, Michael, 485Mersenne, Martin, 120, 127, 152, 158, 160,

164, 166–169, 171, 174, 366, 479Merton, Robert King, 486Monge, Gaspard, 317, 318, 321Montfaucon, Bernard, 92, 479Moody, Edward, 68, 75, 77, 79, 479Moscovici, Serge, 486Mossotti, Ottaviano Fabrizio, 387, 392Mussini, Massimo, 486

Nagel, Ernest, 486Napolitani, Pier Daniele, 486Nastasi, Pietro, 485, 486Navier, Claude Louis Marie Henri, 297, 367,

368, 375, 381, 382, 386, 387, 393, 479Nenci, Elio, 154, 486Nernessian, Nancy, 486Newton, Isaac, 37, 138, 157, 187, 190,

193–199, 210, 217, 218, 227, 232,237–239, 243, 251, 258, 275, 276, 284,474, 476, 479, 482, 486

Ostrogradsky, Mikhail Vasilyevich, 8, 325,386, 479

Ostwald, Wilhelm, 395, 486

Pappus of Alexandria, 33, 34, 45, 50, 51,59–61, 64–66, 91, 96, 109, 114, 131, 132,137, 138, 141, 180, 310, 479, 480

Pardies, Ignace Gaston, 5, 176, 177, 480Pascal, Blaise, 124, 138, 175, 176, 480Pearson, Karl, 486Philoponus, John, 480Piaget, Jean, 486Piola, Gabrio, 12, 311–315, 342, 387–391, 480,

484Pisano, Raffaele, vi, 486Plutarcus, 480Poincaré, Henri, 29Poinsot, Louis, 1, 8–10, 12, 13, 16, 26, 27,

212, 243, 264–268, 277–279, 290, 304,307, 309, 318, 319, 321, 328–330, 332,334–339, 341–348, 350, 351, 353, 354,365, 370, 371, 480

Poisson, Simon Denise, 8, 24, 318, 361–366,369, 480

492 Index

Prony, Gaspard Clair François Riche de, 8,212, 263, 300, 317, 319–321, 328, 332,335–337, 341, 343, 368, 480

Pulte, Helmut, 486

Radelet, De Grave, Patricia, 201, 486Rankine, William John Macquom, 11, 395,

396, 480Renau, d’Elicagaray, Bernard, 6, 201–204, 480Renn, Jürgen, 486Riccardi, Geminiano, 311, 312, 480Riccati, Jacopo, 217, 230Riccati, Vincenzo, 7, 12, 13, 16, 206, 217–219,

221, 224, 225, 227–234, 271, 299, 301,306, 307, 480, 482, 485

Riemann, Bernhard, 390, 392Roberval, Gille Personne, 5, 126, 138, 152,

157, 160–163, 179, 190, 480Rohault, Jaques, 176, 177, 480Rose, Paul Lawrence, 486Roux, 481, 484Russo, Lucio, 486Ruta, Giuseppe, 483

Saccheri, Girolamo, 138, 217, 391, 481Saint Venant, Adhémar J.C. Barré, 297, 318,

369, 380, 383, 385, 386, 387Saladini, Girolamo, 300, 306–308, 481Scott, Wilson L., 486Servois, François Joseph, 12, 206, 263, 300,

308–310, 481Stevin, Simon, 12, 50, 56, 122, 131, 132, 138,

157–160, 171, 173, 178–187, 210, 225,322, 323, 341, 342, 481, 485, 486

Stigliola, Nicola Antonio, 91–93, 131, 481, 484Sturm, Charles François, 293, 481

Tartaglia, Niccolò, 12, 77–79, 83, 84, 91–93,97–103, 108, 110, 116, 118–120, 131,481

Tazzioli, Rossana, 483Thabi, Ibn Qurra, 68, 72Thabit, 12, 63, 65–72, 74, 75, 77, 81, 82, 84,

85, 106, 107, 482Thomson, William (Lord Kelvin), 392, 393,

481Timoshenko, Stephen Prokofievich, 486Tocci, Cesare, vi, 483Todhunter, Isaac, 486Torricelli, Evangelista, 4, 5, 12, 109, 124, 135,

138–154, 175, 176, 190, 192, 196, 218,220, 253, 285, 295, 299, 328, 481–484,486

Truesdell, Clifford Ambrose, 95, 388, 486

Vailati, Giovanni, 40, 44, 50, 366, 486Valerio, Luca, 93, 137, 138, 160, 481, 486Valla, Lorenzo, 93Varignon, Pierre, v, 5–7, 15, 174, 176, 177,

195, 199, 201, 203, 204, 206, 207,209–215, 227, 228, 231, 249, 250, 258,307, 309, 337, 340, 366, 481

Varro, Michel, 131, 173–175, 481, 482, 486Vassura, Giuseppe, 152, 486Venturi, Giovanni Battista, 95, 487Venturoli, Giuseppe, 481Vilain, Christiane, 40, 487Villalpando, Juan Bautista, 154, 481Viscovatov, B., 481Vitruvio, 93, 96, 481, 486

Wallis, John, 5, 12, 59, 187, 190–192, 206,220, 253, 391, 481

Webster, Charles, 189, 487Westfall, Richard Samuel, 198, 487Wiedemann, Eilhard, 68, 487Winter, Thomas Nelson, 38, 487Woepcke, Franz, 65, 487

End of printing: February 2012

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