The Method of Mechanical Theorems - Volume of A Sphere

Volume of A Sphere

Again, to illuminate the mechanical method, it is convenient to use a little bit of coordinate geometry. If a sphere of radius 1 is placed at x = 1, the cross sectional radius at any x between 0 and 2 is given by the following formula:

The mass of this cross section, for purposes of balancing on a lever, is proportional to the area:

Archimedes then considered rotating the region between y = 0 and y = x on the x-y plane around the x-axis, to form a cone. The cross section of this cone is a circle of radius

and the area of this cross section is

So if slices of the cone and the sphere both are to be weighed together, the combined cross-sectional area is:

If the two slices are placed together at distance 1 from the fulcrum, their total weight would be exactly balanced by a circle of area at a distance x from the fulcrum on the other side. This means that the cone and the sphere together will balance a cylinder on the other side.

In order for the slices to balance in this argument, each slice of the sphere and the cone should be hung at a distance 1 from the fulcrum, so that the torque will be just proportional to the area. But the corresponding slice of the cylinder should be hung at position x on the other side. As x ranges from 0 to 2, the cylinder will have a center of gravity a distance 1 from the fulcrum, so all the weight of the cylinder can be considered to be at position 1. The condition of balance ensures that the volume of the cone plus the volume of the sphere is equal to the volume of the cylinder.

The volume of the cylinder is the cross section area, times the height, which is 2, or . Archimedes could also find the volume of the cone using the mechanical method, since, in modern terms, the integral involved is exactly the same as the one for area of the parabola. The volume of the cone is 1/3 its base area times the height. The base of the cone is a circle of radius 2, with area, while the height is 2, so the area is . Subtracting the volume of the cone from the volume of the cylinder gives the volume of the sphere:

The dependence of the volume of the sphere on the radius is obvious from scaling, although that also was not trivial to make rigorous back then. The method then gives the familiar formula for the volume of a sphere. By scaling the dimensions linearly Archimedes easily extended the volume result to spheroids.

Archimedes argument is nearly identical to the argument above, but his cylinder had a bigger radius, so that the cone and the cylinder hung at a greater distance from the fulcrum. He considered this argument to be his greatest achievement, requesting that the accompanying figure of the balanced sphere, cone, and cylinder be engraved upon his tombstone.