The Method of Archimedes
2. The ``Method'' in action
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Archimedes determined the ratio of the volume of a sphere to the volume of the circumscribed cylinder. The actual construction involves the cylinder concentric to the circumscribed cylinder but with double the diameter (and consequently four times the volume). Another essential ingredient is the cone with the same base as the large cylinder, and with the same height. According to Euclid, XII, 10 (this link is to the invaluable Perseus Project at Tufts) the volume of that cone is 1/3 the volume of the cylinder. What Archimedes gets from his Method is the equation:
Vol(Sphere) + Vol(Cone) = (1/2)Vol(Large Cylinder).
So the volume of the sphere is 1/2 - 1/3 = 1/6 the volume of the large cylinder, using Euclid's result. It follows that the volume of the sphere is 4/6 = 2/3 of the volume of the circumscribed cylinder. This is the equation Archimedes wanted engraved on his tombstone.
The balancing argument runs as follows. We imagine the sphere (red) the cone (blue) and the large cylinder (mauve) to be set up horizontally, as shown here,
We choose an arbitrary slice through these three solids, perpendicular to their common axis. That slice cuts out three circles, as shown. Archimedes shows by elementary geometry that if the two smaller circles are slid along the axis to a point at twice height of the cylinder, and the large circle is left where it is, their areas (thought of as masses) will exactly balance about a fulcrum (balance-point) at the center of the figure.
If we imagine doing this for all the slices together, we will have balanced, on the right, the entire sphere and the entire cone, their masses concentrated at the end-point of the axis, and on the left the cylinder in its original position, since none of its slices had to be moved. The balancing force of the cylinder is the same if all of its mass is concentrated at its center of mass, which is halfway down the axis to the left. The equation for balancing masses m and M at distances d and D on opposite sides of the fulcrum is m d = M D. Here m is the mass of the cylinder, d is half the height of the cylinder, M is the sum of the masses of sphere and cone, and D is the height of the cylinder. It follows that M = (1/2)m, the equation we needed.