Volumes: Lever scales

Lever scales enable you to calcu­late the ratio between the volumes of a pyramid and a prism, a cone and a cylinder, a sphere and a cylinder.

The key prin­ciple of seesaw scale is intu­itively obvious: to balance a heavy body one needs to put the light body farther from the fulcrum. Quan­ti­ta­tively, this prin­ciple is expressed in the fact that in the equi­lib­rium posi­tion the rela­tion­ship between the masses (there­fore, volumes, since the objects’ densi­ties are the same) is equal to inverse rela­tion­ship of the distances from the fulcrum to the bodies being weighed.

In the case given, the pyramid should be placed three times as far as the prism. The volume of the pyramid is one third of the volume of the prism with the equal base and height.

A cylinder and a cone with equal bases and equal heights can be repre­sented as prism and pyramid limits respec­tively. So the volume of a cone is equal to one third of the volume of a cylinder with the same base and height.

A ball and a cylinder. The radius of the base of the cylinder is equal to the radius of the sphere and the height of the cylinder is equal to the diam­eter of the sphere. With these dimen­sions, a sphere can be inscribed into the cylinder.

How do the volumes of a cylinder and a ball relate: how must the bodies be placed on a lever scale to bring them to equi­lib­rium?

It can be veri­fied that rela­tion­ship of scale arms (arm is the distance from the fulcrum to the point where the weighted object is placed) in equi­lib­rium will be $2:3$. Thus, the ball volume is equal to two thirds of the cylinder volume.

It is believed that out of his discov­eries, Archimedes most appre­ci­ated the finding of the rela­tion­ship between the volumes of the ball and the cylinder, thereby deter­mining the ball volume.

The formula for a ball volume can be derived from this rela­tion­ship.

Let's use a formula for a cylinder volume — product of the cylinder base area by its height. The base area is $\pi \cdot R^2$, cylinder height is $2 \cdot R$, where $R$ is the ball radius. Thus, the cylinder volume is $(\pi \cdot R^2) \cdot (2 \cdot R) = 2 \cdot \pi \cdot R^2$.

Multi­plying by the coef­fi­cient of $2/3$, we get the formula for the ball volume — $4/3 \cdot \pi \cdot R^3$.

It is worth consid­ering the possi­bility of dividing the cylinder into two parts equal in height. If we balance the ball with the cylinder having the height equal to the ball radius, the scaleвЂ™s arms rela­tion will be exactly $4:3$

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 (106—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.