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I wish to God these calculations had been executed by steam.
In H. Eves, In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Archimedes' Method for Computing Areas and Volumes
Exercise on Proposition 4 of The Method
Investigate Archimedes' Proposition 4 below, which concerns a paraboloid (the solid obtained by rotating the curve x = y2 about the x-axis) inscribed inside a cylinder. A static version of the balancing appears below. If you click on Proposition 4, you will get a dynamic version.
a) Describe the balancing relationship that is illustrated in the sketch.
b) Use this relationship to determine a relationship between the volume of the paraboloid and the volume of the cylinder.
c) Find a formula for the volume of the paraboloid in terms of r, the radius of the cylinder.
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