The Impact of Ballistics on Mathematics

The work of Robins and Euler in the eighteenth-centuryShawn McMurran, Cal State San Bernardino V. Frederick Rickey, USMA

A Very Brief History of Projectile Motion Aristotles Impetus NotionDaniel Santbech, Problematum astronomicorum et geometricorum (Basel 1561)

Da Vincis Arcs"Four Mortars Firing Stones into the Courtyard of a Fort" (c.1504)

TartagliaNova scientia, 1537

The Mariners Magazine, or [Samuel] Sturmys Mathematicall and Practical Arts (1669)

Straight violent motionmixt or crooked motionVertical descent of natural motion

150 yeas after Tartaglia

Tartaglias Trajectories

Projectile motion depicted in Nova Scientia (1537) by Niccol Tartaglia (c.1500-1557)

Galileos Parabolic PathsFolios 116v and 117r, vol. 72, Galilean manuscripts, 1608

In Art"Judith Slaying Holofernes" (c. 1620), by Artemisia Gentileschi (1593-1652)

Benjamin Robins 1707 - 1751

Born 1707AutodictatHad an important advisorA clear writer

Biography of RobinsStudied on his ownMet Dr. Henry PembertonMoved to LondonStudied more mathematicsTraveled to the continentElected FRS, age 21Became a teacherFrontispiece to Sprat's History of the Royal Society

Authors Robins StudiedApolloniusArchimedesFermat HuygensDeWittSluseGregoryBarrowNewtonTaylorCotes

Mediation on experiments made recently on the firing of cannon.

Eulers first paper on cannon, E853, written 1727,published 1862.

A polemic against Euler, 1739Too algebraic Uses infinitesimals

Called to a Public Employment A Very Honorable Post

Sir Robert Walpole was prime minister, 1721 1742

He was reluctant to attack Spain

Robins wrote three anonymous pamphlets in favor of war

And became secretary of a secret tribunalAn interlude:

From Teacher to Professor ?Robins hoped to be the first professor of mathematics at WoolwichPlanned a course on fortifications and gunneryWalpole displayed his displeasure with Robinss previous attacksMr. Derham became the first professor of mathematics at Woolwich, served 1741 -1743.

Mathematics at Woolwich, 1741That the second Master shall teach the Science of Arithmetic, together with the principles of Algebra and the Elements of Geometry, under the direction of the Chief Master.

That the chief Master shall further instruct the hearers in Trigonometry and the Elements of the Conick Sections.

To which he shall add the Principles of Practical Geometry and Mechanics, applied to raising and transporting great Burthens;

With the Knowledge of Mensuration, and Levelling, and its Application to the bringing of water and the draining of Morasses;

And lastly, shall teach Fortification in all its parts.

But no calculus

Preface55 pages

Ch I: Internal ballistics65 pages

Ch2: External ballistics30 pages

Total: 150 pp

Published 1742

Euler 1745Frederick the Great asks about the best book on gunneryEuler magnanimously recognizes RobinsEuler starts researching Robinss resultsEuler adds annotations 2400

English translation of Euler, 1777From Eulers Preface Some are of the opinion that fluxions are applicable only in such subtle speculations as can be of no practical use. . . But what has been just now said of artillery is sufficient to remove this prejudice.

English translation of Euler, 1777More from Eulers Preface It may be affirmed, that things which depend on mathematics cannot be explained in all their circumstances without the help of fluxions, and even this sublime part of mathematics has met with difficulties which it has not fully mastered.

Postulates for motion of a projectile in a vacuumPostulate 2: If the Parabola, in which the Body moves, be terminated on a horizontal Plain, then the Vertex of the Parabola will be equally distant from its two Extremities.

Postulate 4: If a Body be projected in different Angles, but with the same Velocity, then its greatest horizontal Range will be, when it is projected in an Angle of 45 with the horizon.

Although these properties are demonstrated in many books, we shall here investigate them from the first principles of motion, partly to give clear insight into this matter; partly, and chiefly, the better to enable us to determine the real track described in the air.

Euler, p. 271

Remark III: Shot in a vacuum

Historical QuestionsDid anyone, before Euler, give an analytic (calculus based) derivation of the equations of projectile motion?

When was this derivation cleaned up and presented in textbooks?

Robins gives experimental evidence to confute the 7 postulates posed in Proposition V.

For example, according to postulate 5 in Prop V:A musket ball of an inch in diameter that has an initial velocity of 1700 feet per second at an angle of 45 should have a horizontal range of about 17 miles.Actual range:

Less than half a mile

Eulers RemarksOneDerives equations of motion for a shot in a horizontal line (7 pages) TwoDerives equations of motion for a vertical shot (10 pages)

ThreeDerive equations of motion for a shot made under an oblique angle with the horizon and compare his results with the conclusions of Robinss experiments (9 pages)

Remark II: Vertical ShotTime of ascent:

Time of descent:

Where a is given by:

To illustrate the accuracy of his formula, Euler chooses an example given by Daniel Bernoulli in the Petersburg Comentarii:

Flight time reported by Bernoulli: 34 seconds

Flight time predicted by Eulers formulae:Ascent: 13.75 secondsDescent: 20.11 seconds Total flight time:33.87 seconds!

Mathematics at Woolwich, 1772The Elements of EuclidTrigonometry applied to Fortification, and the Mensuration of Superficies and SolidsConic Sections. Mechanics applied to the raising and transporting heavy bodies, together with the use of the lever pulley, wheel, wedge and screw, &c.The Laws of Motion and Resistance, Projectiles, and Fluxions.

Now some calculus!

The Impact of Ballistics on Mathematics: Calculus was taught in artillery schools Piedmont-Savoy in 1750sRoyal Artillery and Military Academy, TurinPrussian Artillery CorpsFrench regimental school at AuxonneAustralian Artillery AcademyEcole Polytechnique, 1794West Point, 1807, 1813, 1823 to date.

Bonaparte read Robins / Euler in French.

Bonaparte rightly said that many of the decisions faced by the commander-in-chief resemble mathematical problems worthy of the gifts of a Newton or an Euler.

Carl von Clausewitz, Vom Kriege, 1832

I have recently received two treatises that you kindly sent me, one on cannons and the force of powder, the other containing the theory of the motion of planets and comets. For this double gift I am deeply grateful. I have now nearly finished reading the first of these books, but I have relied completely on the correctness of your computations and haven't verified them myself, for many of them seem too complicated to me.

Johann Bernoulli to Euler, 23 September 1745

NewtonA Treatise of the System of the World (published posthumously, 1729)

Robins wrote a polemic against Johann Bernoulli, 1728

A polemic against Berkeley, 1735Robins wrote in defense of Newton

*J*http://eee.uci.edu/clients/bjbecker/RevoltingIdeas/quodlibet4.html

******One of the heroes of our story wasborn in 1707,self-taught, an important advisor (for Euler it was Johann Bernoulli - for Robins it was some aristocrat), and a clear writer, so of course I must be talking about...Benjamin Robins, invented the ballistic pendulum to measure muzzle velocity

*Dr. Henry Pemberton (1694-1771) - a mathematician and physician, had exchanged letters with Newton re the Principia, important scientific connection for Robins(Initially Newton would not correspond with Pemberton, believing him to be the source of untruths about Newton)At around age 30, Pemberton was invited to superintend the editing of the third edition of the Principia mathematica (For Fred: Pemberton also wrote A short history of trigonometry)The frontispiece to Sprat's History of the Royal Society depicts the president of the Royal Society and Francis Bacon sitting on either side of a bust of Charles II. The newly restored king, Charles, is labelled "Author and Patron of the Royal Society," and Francis Bacon, holding the seal of England, is labelled "Renovator of Arts." Bacon points to artillery and mathematical instruments while on the side of the president is a shelf of books. In the background are more scientific instruments, including an air pump. The sweep of hand motions leads the observer to circle between books and instruments. http://www.princeton.edu/~his291/Sprat.html*As Deborah phrased it, some of the authors he cut his mathematical teeth on - a pretty impressive listDeWitt and Sluse - Dutch mathematicians who wrote sections of the Latin edition of Descartes Geometry.*An early paper of Euler - has a high Enestrome number because it wasnt published until 1862 - one of the earliest places we see e in print (also in Mechanica 1736)We write for the number whose logarithm is 1, e, which is ...*Robins and Jurin argue about foundations of Eulers Mechanica (1736)We suspect that this may be how Euler first heard of Robins - but Euler puts any hard feelings aside and still acknowledges the merits of Robins work*He didnt criticize only mathematicians, Late 1730s, Robins wrote three not very anonymous pamphlets supporting the warWalpole was PM from 1721-1742

*He didnt criticize only mathematicians, Late 1730s, Robins wrote three not very anonymous pamphlets supporting the warWalpole was PM from 1721-1742

*Royal Military Academy at Woolwich est. 1741 to train engineers and artillery officers.Walpole prevented Robins from becoming professorAn aside - nicknamed The Shop - first building was a converted workshop in Woolwich Arsenal. From which we have the phrase Talking Shopmeaning "to discuss subjects not understood by others"*Something missing from the list?*In spite of the brush-off, Robins published his book anyway to show that he did know something and should have had the job.The important part of the title for this talk is the consideration of air resistance Not the first to consider it, but he did have a lot of good empirical data.*Enter Euler - between 1742 and 1745, Frederick asks Euler about the best book on gunneryEuler adds annotations - 300 pages of them - where he makes a compelling case for the use of calculusNew Foundations of Artillery, current book price 2400 euro = 3200 US*Euler starts making his case for the use of calculus in the preface of the English Translations (copy at West Point - Fred Rickey finessed the library into purchasing)**Examples of the 7 postulates - Robins goal: He points out that if any one postulate fails for a projectile, then that projectile must deviate from a parabolic pathRobins intends to demonstrate by experiment than none of the postulates hold for a projectile in air.*Woops. Should be integriert**As an example of Eulers contributions Ill briefly discuss Proposition VI in Chapter II of Robins book**Note: These are not our standard derivations - pretty complicated and use a lot of series approximations***The Shop, p. 29. So - lets take a look at Mathematics at the Royal Military Academy at Woolwich 31 years after our previous look*http://www.carolinaarts.com/606scstatemuseum1.html***The history of science contains many famous quarrels and polemics,Early in his career (here he is only 21) Robins starts writing his fair share of polemical attacks:Johann Bernoulli - 1724 discours sur les loix de la communication du mouvement - thought experiments - e.g. the number of springs compressed by a moving ball is proportional to the square of its velocity.*Berkeley lived 1685-1753 (namesake of Berkeley, CA, settled in RI) Berkeley had attacked Newtons foundations of the calculusRobins polemic was in defense of Newton