heiberg archimedes method

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METRICAL SOLUTIONSDERIVED FROMMECHANICS

TREATISE BV ARCHIMEDES

RECENTLY DISCOVERED AND TRANSLATED FROM THE GREEK BYDR. J. L. HEIBERG

CHICAGOE OPEN COURT PUBLISHING COMPANY

LONDON AGENTSKEGAN PAUL, TRENCH, TRUBNER & CO., LTD.


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GEOMETRICAL SOLUTIONSDERIVED FROM

MECHANICSA TREATISE OF ARCHIMEDES

RECENTLY DISCOVERED AND TRANSLATED FROM THE GREEK BYDR. J. L. HEIBERG

PROFESSOR OF CLASSICAL PHILOLOGY AT THE UNIVERSITY OF COPENHAGEN

WITH AN INTRODUCTION BYDAVID EUGENE SMITH

PRESIDENT OF TEACHER S COLLEGE, COLUMBIA UNIVERSITY, NEW YORK

ENGLISH VERSION TRANSLATED FROM THE GERMAN BY LYDIA G. ROBINSONAND REPRINTED FROM "THE MONIST," APRIL, 1909.

CHICAGOTHE OPEN COURT PUBLISHING COMPANYLONDON AGENTS

KEGAN PAUL, TRENCH, TRUBNER & CO., LTD.1909


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VCOPYRIGHT BY

THE OPEN COURT PUBLISHING Co.1909


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INTRODUCTION.

IF there ever was a case of appropriateness in discovery,the finding of this manuscript in the summer of 1906was one. In the first place it was appropriate that the discovery should be made in Constantinople, since it was herethat the West received its first manuscripts of the other extant works, nine in number, of the great Syracusan. It wasfurthermore appropriate that the discovery should be madeby Professor Heiberg, facilis princeps among all workersin the field of editing the classics of Greek mathematics,and an indefatigable searcher of the libraries of Europefor manuscripts to aid him in perfecting his labors. Andfinally it was most appropriate that this work should appear at a time when the affiliation of pure and appliedmathematics is becoming so generally recognized all overthe world. We are sometimes led to feel, in consideringisolated cases, that the great contributors of the past haveworked in the field of pure mathematics alone, and thesaying of Plutarch that Archimedes felt that "every kindof art connected with daily needs was ignoble and vulgar" 1may have strengthened this feeling. It therefore assistsus in properly orientating ourselves to read another treatise from the greatest mathematician of antiquity that setsclearly before us his indebtedness to the mechanical applications of his subject.Not the least interesting of the passages in the manu-

1 Marcellus, 17.


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f2 GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS.script is the first line, the greeting to Eratosthenes. It iswell known, on the testimony of Diodoros his countryman,that Archimedes studied in Alexandria, and the latter frequently makes mention of Konon of Samos whom he knewthere, probably as a teacher, and to whom he was indebtedfor the suggestion of the spiral that bears his name. It isalso related, this time by Proclos, that Eratosthenes was acontemporary of Archimedes, and if the testimony of solate a writer as Tzetzes, who lived in the twelfth century,may be taken as valid, the former was eleven years thejunior of the great Sicilian. Until now, however, we havehad nothing definite to show that the two were ever acquainted. The great Alexandrian savant, poet, geographer, arithmetician, affectionately called by the students Pentathlos, the champion in five sports,

2 selected byPtolemy Euergetes to succeed his master, Kallimachos thepoet, as head of the great Library, this man, the mostrenowned of his time in Alexandria, could hardly havebeen a teacher of Archimedes, nor yet the fellow student ofone who was so much his senior. It is more probable thatthey were friends in the later days when Archimedes wasreceived as a savant rather than as a learner, and this isborne out by the statement at the close of proposition Iwhich refers to one of his earlier works, showing that thisparticular treatise was a late one. This reference beingto one of the two works dedicated to Dositheos of Kolonos,3and one of these (De lineis spiralibus) referring to anearlier treatise sent to Konon,4 we are led to believe thatthis was one of the latest works of Archimedes and thatEratosthenes was a friend of his mature years, although

8 His nickname of Beta is well known, possibly because his lecture roomwas number 2.8 We know little of his works, none of which are extant. Geminos and

Ptolemy refer to certain observations made by him in 200 B. C, twelve yearsafter the death of Archimedes. Pliny also mentions him.


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INTRODUCTION. 3

one of long standing-. The statement that the preliminarypropositions were sent "some time ago" bears out this ideaof a considerable duration of friendship, and the idea thatmore or less correspondence had resulted from this communication may be inferred by the statement that he saw,as he had previously said, that Eratosthenes was "a capablescholar and a prominent teacher of philosophy," and alsothat he understood "how to value a mathematical methodof investigation when the opportunity offered." We have,then, new light upon the relations between these two men.the leaders among the learned of their day.A second feature of much interest in the treatise is theintimate view that we have into the workings of the mindof the author. It must always be remembered that Archimedes was primarily a discoverer, and not primarily a compiler as were Euclid, Apollonios, and Nicomachos. Therefore to have him follow up his first communication of theorems to Eratosthenes by a statement of his mental processes in reaching his conclusions is not merely a contributionto mathematics but one to education as well. Particularlyis this true in the following statement, which may well bekept in mind in the present day: "I have thought it wellto analyse and lay down for you in this same book a peculiar method by means of which it will be possible for youto derive instruction as to how certain mathematical questions may be investigated by means of mechanics. And Iam convinced that this is equally profitable in demonstrating a proposition itself; for much that was made evidentto me through the medium of mechanics was later provedby means of geometry, because the treatment by the formermethod had not yet been established by way of a demonstration. For of course it is easier to establish a proof if onehas in this way previously obtained a conception of thequestions, than for him to seek it without such a preliminary notion. . . .Indeed I assume that some one among the


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4 GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS.investigators of to-day or in the future will discover by themethod here set forth still other propositions which havenot yet occurred to us." Perhaps in all the history ofmathematics no such prophetic truth was ever put intowords. It would almost seem as if Archimedes must haveseen as in a vision the methods of Galileo, Cavalieri, Pascal,Newton, and many of the other great makers of the mathematics of the Renaissance and the present time.The first proposition concerns the quadrature of theparabola, a subject treated at length in one of his earliercommunications to Dositheos. 5 He gives a digest of thetreatment, but with the warning that the proof is not complete, as it is in his special work upon the subject. He has,in fact, summarized propositions VII-XVII of his communication to Dositheos, omitting the geometric treatment of propositions XVIII-XXIV. One thing that hedoes not state, here or in any of his works, is where theidea of center of gravity started. It was certainly a common notion in his day, for he often uses it without definingit. It appears in Euclid s7 time, but how much earlier wecannot as yet say.

Proposition II states no new fact. Essentially it meansthat if a sphere, cylinder, and cone (always circular) havethe same radius, r, and the altitude of the cone is r and thatof the cylinder 2r, then the volumes will be as 4 : i : 6,which is true, since they are respectively %?rr3 , Yz r3 , and27rr3 . The interesting thing, however, is the method pursued, the derivation of geometric truths from principlesof mechanics. There is, too, in every sentence, a littlesuggestion of Cavalieri, an anticipation by nearly two thousand years of the work of the greatest immediate precursorof Newton. And the geometric imagination that Archi-

5 TeTpa.yt*}vi(T/J.bsc Kevrpa papwv, for "barycentric" is a very old term.7 At any rate in the anonymous fragment DC levi et ponderoso, sometimesattributed to him.


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INTRODUCTION. 5medes shows in the last sentence is also noteworthy as oneof the interesting features of this work : "After I had thusperceived that a sphere is four times as large as the cone . . .it occurred to me that the surface of a sphere is four timesas great as its largest circle, in which I proceeded from theidea that just as a circle is equal to a triangle whose base isthe periphery of the circle, and whose altitude is equal toits radius, so a sphere is equal to a cone whose base is thesame as the surface of the sphere and whose altitude isequal to the radius of the sphere." As a bit of generalization this throws a good deal of light on the workings ofArchimedes s mind.

In proposition III he considers the volume of a spheroid, which he had already treated more fully in one of hisletters to Dositheos,8 and which contains nothing new froma mathematical standpoint. Indeed it is the method ratherthan the conclusion that is interesting in such of the subsequent propositions as relate to mensuration. PropositionVdeals with the center of gravity of a segment of a conoid, andproposition VI with the center of gravity of a hemisphere,thus carrying into solid geometry the work of Archimedeson the equilibrium of planes and on their centers of gravity.9 The general method is that already known in thetreatise mentioned, and this is followed through proposition X.

Proposition XI is the interesting case of a segment ofa right cylinder cut off by a plane through the center ofthe lower base and tangent to the upper one. He showsthis to equal one-sixth of the square prism that circumscribes the cylinder. This is well known to us through theformula v= 2r2h/^ the volume of the prism being 4.r2h,and requires a knowledge of the center of gravity of thecylindric section in question. Archimedes is, so far as we

8 Ilepi Kwvoeidewv /ecu8 E7ri7re5wj> iffoppoiriwv rj Kevrpa ftapaJv iirur&wv.


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GEOMETRICAL SOLUTIONS DERIVED FROMMECHANICS.ARCHIMEDES TO ERATOSTHENES, GREETING:Some time ago I sent you some theorems I had discovered,writing down only the propositions because I wished you to findtheir demonstrations which had not been given. The propositionsof the theorems which I sent you were the following:

1. If in a perpendicular prism with a parallelogram 2 for basea cylinder is inscribed which has its bases in the opposite parallelograms2 and its surface touching the other planes of the prism,and if a plane is passed through the center of the circle that is thebase of the cylinder and one side of the square lying in the oppositeplane, then that plane will cut off from the cylinder a section whichis bounded by two planes, the intersecting plane and the one inwhich the base of the cylinder lies, and also by as much of thesurface of the cylinder as lies between these same planes ; and thedetached section of the cylinder is % of the whole prism.

2. If in a cube a cylinder is inscribed whose bases lie in opposite parallelograms2 and whose surface touches the other four planes,and if in the same cube a second cylinder is inscribed whose baseslie in two other parallelograms2 and whose surface touches thefour other planes, then the body enclosed by the surface of thecylinder and comprehended within both cylinders will be equal to% of the whole cube.These propositions differ essentially from those formerly discovered ; for then we compared those bodies (conoids, spheroidsand their segments) with the volume of cones and cylinders but noneof them was found to be equal to a body enclosed by planes. Eachof these bodies, on the other hand, which are enclosed by two planesand cylindrical surfaces is found to be equal to a body enclosed

8 This must mean a square.


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8 GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS.by planes. The demonstration of these propositions I am accordinglysending to you in this book.

Since I see, however, as I have previously said, that you area capable scholar and a prominent teacher of philosophy, and alsothat you understand how to value a mathematical method of investigation when the opportunity is offered, I have thought it wellto analyze and lay down for you in this same book a peculiar methodby means of which it will be possible for you to derive instructionas to how certain mathematical questions may be investigated bymeans of mechanics. And I am convinced that this is equally profitable in demonstrating a proposition itself ; for much that was madeevident to me through the medium of mechanics was later provedby means of geometry because the treatment by the former methodhad not yet been established by way of a demonstration. For ofcourse it is easier to establish a proof if one has in this way previously obtained a conception of the questions, than for him to seek itwithout such a preliminary notion. Thus in the familiar propositionsthe demonstrations of which Eudoxos was the first to discover,namely that a cone and a pyramid are one third the size of thatcylinder and prism respectively that have the same base and altitude, no little credit is due to Democritos who was the first to makethat statement about these bodies without any demonstration. Butwe are in a position to have found the present proposition in thesame way as the earlier one ; and I have decided to write down andmake known the method partly because we have already talkedabout it heretofore and so no one would think that we were spreading abroad idle talk, and partly in the conviction that by this meanswe are obtaining no slight advantage for mathematics, for indeedI assume that some one among the investigators of to-day or in thefuture will discover by the method here set forth still other propositions which have not yet occurred to us.

In the first place we will now explain what was also first madeclear to us through mechanics, namely that a segment of a parabolais % of the triangle possessing the same base and equal altitude ;following which we will explain in order the particular propositionsdiscovered by the above mentioned method ; and in the last partof the book we will present the geometrical demonstrations of thepropositions.4

4 In his "Commentar," Professor Zeuthen calls attention to the fact thatit was already known from Heron s recently discovered Metrica that thesepropositions were contained in this treatise, and Professor Heiberg made thesame comment in Hermes. Tr.


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GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS. 91. If one magnitude is taken away from another magnitude andthe same point is the center of gravity both of the whole and of the

part removed, then the same point is the center of gravity of theremaining portion.

2. If one magnitude is taken away from another magnitude andthe center of gravity of the whole and of the part removed is notthe same point, the center of gravity of the remaining portion maybe found by prolonging the straight line which connects the centersof gravity of the whole and of the part removed, and setting offupon it another straight line which bears the same ratio to thestraight line between the aforesaid centers of gravity, as the weightof the magnitude which has been taken away bears to the weightof the one remaining [De plan, aeqml. I, 8].

3. If the centers of gravity of any number of magnitudes lieupon the same straight line, then will the center of gravity of all themagnitudes combined lie also upon the same straight line [Cf. ibid.i. si-

4. The center of gravity of a straight line is the center of thatline [Cf. ibid. I, 4].

5. The center of gravity of a triangle is the point in which thestraight lines drawn from the angles of a triangle to the centers ofthe opposite sides intersect [Ibid. I, 14].

6. The center of gravity of a parallelogram is the point whereits diagonals meet [Ibid. I, 10].

7. The center of gravity [of a circle] is the center [of thatcircle].8. The center of gravity of a cylinder [is the center of its axis].9. The center of gravity of a prism is the center of its axis.10. The center of gravity of a cone so divides its axis that the

section at the vertex is three times as great as the remainder.11. Moreover together with the exercise here laid down I willmake use of the following proposition:If any number of magnitudes stand in the same ratio to thesame number of other magnitudes which correspond pair by pair,and if either all or some of the former magnitudes stand in any

ratio whatever to other magnitudes, and the latter in the same ratioto the corresponding ones, then the sum of the magnitudes of thefirst series will bear the same ratio to the sum of those taken fromthe third series as the sum of those of the second series bears tothe sum of those taken from the fourth series [De Conoid. I],


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IO GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS.

Let a(3y [Fig. i] be the segment of a parabola bounded by thestraight line ay and the parabola a/3y. Let ay be bisected at 8, 8/?ebeing parallel to the diameter, and draw a(3, and /3y. Then thesegment a/?y will be % as great as the triangle a/2y.From the points a and y draw a 1 1 S/?e, and the tangent y ;

produce [y/? to K, andmake *0 = y*] . Think ofy0 as a scale-beam withthe center at K and let jube any straight line whatever

1 1cS- Now since y/?a

is a parabola, y a tangent and yS an ordinate,then c(3 = /3S; for this indeed has been proved inthe Elements [i. e., ofconic sections, cf. Quadr.parab. 2]. For this rea-son and because a and

Fig. 1. ILV= V, and * =

ica. And because ya:a=^ : o ( for this is shownin a corollary, [cf. Quadr. parab. 5]), ya:a| = y/c:*v; and y/c = K0,therefore OK:KV-H^\ o- And because v is the center of gravity ofthe straight line j*, since juv = v, then if we make r^ = |o and asits center of gravity so that rO - 6rj, the straight line rOrj will be inequilibrium with ^ in its present position because Ov is divided ininverse proportion to the weights rrj and i^, and OK : KV = /* : r;r ; therefore K is the center of gravity of the combined weight of the two.In the same way all straight lines drawn in the triangle ay||e8 arein their present positions in equilibrium with their parts cut off bythe parabola, when these are transferred to 6, so that K is the centerof gravity of the combined weight of the two. And because thetriangle ya consists of the straight lines in the triangle ya and thesegment a(3y consists of those straight lines within the segment ofthe parabola corresponding to the straight line |o, therefore thetriangle ay in its present position will be in equilibrium at thepoint K with the parabola-segment when this is transferred to asits center of gravity, so that K is the center of gravity of the combined


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GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS. II

weights of the two. Now let y* be so divided at x that y* = 3*x;then x will be the center of gravity of the triangle ay, for thishas been shown in the Statics [cf. De plan, aequil. I, 15, p. 186,3 with Eutokios, S. 320, 5ff.]. Now the triangle ay in its presentposition is in equilibrium at the point K with the segment pay whenthis is transferred to as its center of gravity, and the center ofgravity of the triangle ay is x \ hence triangle ay : segm. a/3y whentransferred to as its center of gravity =OK:KX- But 0K = 3*x;hence also triangle ay = 3 segm. a/?y. But it is also true that triangleay = 4Aa/?y because K = KO. and aS = 8y; hence segm. a/?y = % the

triangle a/?y. This is of course clear.It is true that this is not proved by what we have said here ;

but it indicates that the result is correct. And so, as we have justseen that it has not been proved but rather conjectured that theresult is correct we have devised a geometrical demonstration whichwe made known some time ago and will again bring forwardfarther on.

n.That a sphere is four times as large as a cone whose base is

equal to the largest circle of the sphere and whose altitude is equalto the radius of the sphere, and that a cylinder whose base is equalto the largest circle of the sphere and whose altitude is equal to thediameter of the circle is one and a half times as large as the sphere,may be seen by the present method in the following way:

Let aftyS [Fig. 2] be the largest circle of a sphere and ay and /38its diameters perpendicular to each other ; let there be in the spherea circle on the diameter /2S perpendicular to the circle a/2yS, andon this perpendicular circle let there be a cone erected with itsvertex at a ; producing the convex surface of the cone, let it becut through y by a plane parallel to its base ; the result will be thecircle perpendicular to ay whose diameter will be c. On thiscircle erect a cylinder whose axis = ay and whose vertical boundaries are eX and 77. Produce ya making a6 = ya and think of y0 asa scale-beam with its center at a. Then let /JLV be any straight linewhatever drawn ||/?8 intersecting the circle a/?y8 in and o, thediameter ay in cr, the straight line ae in TT and a in p, and on thestraight line fj.v construct a plane perpendicular to ay ; it will intersect the cylinder in a circle on the diameter \*v ; the sphere a/?y8, ina circle on the diameter |o ; the cone ae in a circle on the diameter


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12 GEOMETRICAL SOLUTIONS DERIVED FROM MECHANICS.

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heiberg archimedes method

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