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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. 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). |
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,
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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.