Sphere volume: Archimedes Scales

Archimedes (Archimedes of Sir­a­cusi, ancient Greek Ἀρχιμήδης, lat. Archimedes, c. 287 — c. 212 BC) con­sid­ered find­ing a rela­tion between vol­umes of a sphere and a cylin­der, cir­cum­scribed around it, his main math­e­mat­i­cal dis­cov­ery. It is not casual that a ball and a cylin­der were depicted on his grave.

Archimedes to Dositheus greet­ings! Shortly before this I for­warded to you some objects of my research, along with the proofs I have found […] I have now fin­ished some other the­o­rems I have thought of. some other the­o­rems which have occurred to me, of which the most remark­able are these: […] the cylin­der, which has as its base the largest cir­cle of a sphere and a height equal to its cross-sec­tion, is one and a half of a sphere; and its sur­face is one and a half of the sur­face of a sphere. These prop­er­ties undoubt­edly existed in the said fig­ures, but have not hith­erto been noticed by any­one who has stud­ied geom­e­try…

Archimedes. On the sphere and cylin­der.

When I was questor in Sicily I man­aged to track down his <Archimedes> grave. The Syra­cu­sians knew noth­ing about it, and indeed denied that any such thing existed. But there it was, com­pletely sur­rounded and hid­den by bushes of bram­bles and thorns. I remem­bered hav­ing heard of some sim­ple lines of verse which had been inscribed on his tomb, refer­ring to a sphere and cylin­der mod­elled in stone on top of the grave. And so I took a good look round all the numer­ous tombs that stand beside the Agri­gen­tine Gate. Finally I noted a lit­tle col­umn just vis­i­ble above the scrub: it was sur­mounted by a sphere and a cylin­der. I imme­di­ately said to the Syra­cu­sans, some of whose lead­ing cit­i­zens were with me at the time, that I believed this was the very object I had been look­ing for.

Cicero (c. 106 BC — c. 43 BC), Tus­cu­lan Dis­pu­ta­tions, Book V, Sec­tions 64—66.

(Trans­la­tion by Michael Grant in Cicero — On the Good Life, Pen­guin Books, New York, 1971, Pages 86—87)

Let’s look at unbal­anced scales. Imag­ine that there is a cylin­der on the one side of the scales, with a height equal to a base radius, and there is a cone and a half of a sphere on the other side of the scales, at the same dis­tance from a pen­du­lum as a cylin­der. So a radius of a cone base and a height are equal to the radius of a cylin­der, radius of a ball is equal to radius of a cylin­der.

Let’s start to col­lect these fig­ures by lay­ers, so that heights of lay­ers of each of three fig­ures are equal. It turns out that in сase of men­tioned rela­tions unbal­anced scales will always come to bal­ance. When fig­ures are fully col­lected, scales will be in bal­ance. Con­se­quently, vol­ume of a cylin­der is equal to a sum of vol­umes of a cone and a half of a sphere, if radiuses and heights of all three fig­ures are equal.

It is sur­pris­ing: on the one side of a scale there is a sim­ple fig­ure — a straight cir­cu­lar cylin­der, and on the other side there is also one of rel­a­tively sim­ple fig­ures — a straight cir­cu­lar cone, and a fig­ure, which bal­ances the scales is a sphere.

The case is that if we draw a plane, which is par­al­lel to bases of fig­ures, a square of a cir­cle, which is received in a sec­tion of a cylin­der, is equal to a sum of squares of cir­cles, which are received in a sec­tion of the given cone and sphere. It is not dif­fi­cult (in our times!) to check by a direct cal­cu­la­tion, that equal­ity of squares will be true for any posi­tion of a сut­ting plane.

Equal­ity of vol­umes leads from the men­tioned equal­ity of squares, like it is told now, accord­ing to the prin­ci­ple of Cav­a­lieri (it. Bonaven­tura Francesco Cav­a­lieri, lat. Cav­a­lerius, 1598—1647).

Rela­tion of vol­umes of a cylin­der and a cone was known before Archimedes:

Hav­ing, how­ever, now dis­cov­ered that the prop­er­ties are true of these fig­ures, I can­not feel any hes­i­ta­tion set­ting them side by side both with my for­mer inves­ti­ga­tions and with those of the the­o­rems of Eudoxus <Eudoxus Сnidus, ancient Greek Εὔδοξος, lat. Eudoxus, c. 408 BC — c. 355 BC> on solids which are held to be most irrefragably estab­lished, namely, that any pyra­mid is one third part of the prism which has the same base with the pyra­mid and equal height, and that any cone is one third part of the cylin­der which has the same base with the cone and equal height.

Archimedes. On the sphere and cylin­der.

Bal­ance of scales enables to express the vol­ume of a half of a sphere through a vol­ume of a cylin­der. Deduc­ing one third from vol­ume of a cylin­der — a vol­ume of a cone with the same base and height, one receives that vol­ume of a half of a ball is equal to $2/3$ of a cylin­der’s vol­ume.

Тем Con­se­quently, a rela­tion, described by Archimedes, has been proved: a vol­ume of a ball is equal to a vol­ume of a cylin­der, cir­cum­scribed around the sphere. It is inter­est­ing that, as Archimedes has set­tled, squares of their sur­faces are in the same rela­tions.

From Archimedes’ cor­re­spon­dence one can deter­mine a clear for­mula for a sphere’s vol­ume. In case of a cylin­der, cir­cum­scribed around a sphere with a radius $R$, a square of its basis is equal to $\pi R^2$, and its height is equal to $2R$. Con­se­quently, vol­ume of a cylin­der is equal to: $(\pi R^2)\cdot (2 R)=2 \pi R^3$. Mul­ti­ply­ing on a coef­fi­cient $2/3$, one will receive a for­mula for a vol­ume of a sphere: $4/3 \cdot \pi R^3$

Now, how­ever, it will be open to those who pos­sess the req­ui­site abil­ity to exam­ine these dis­cov­er­ies of mine. They ought to have been pub­lished while Conon was still alive <Сonon Samian, ancient Greek Κόνων, Latin Conon, c. 280 BC — c. 220 BC>, for I should con­ceive that he would best have been able to grasp them and to pro­nounce upon them the appro­pri­ate ver­dict; but, as I judge it well to com­mu­ni­cate them to those who are con­ver­sant with math­e­mat­ics, I send them to you with the proofs writ­ten out, which it will be open to math­e­mati­cians to exam­ine.

Archimedes. On the sphere and cylin­der.