Proof
Inscribe a pyramid on a polygonal base in both cone and cylinder;
XII.7 makes the pyramid one-third the prism; apply X.1 to refine the
inscribed polygon to fill the circle (XII.2); the limit gives the
cone-to-cylinder ratio.
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Depends on (3)
- X.1Proposition X.1Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and…
- XII.2Proposition XII.2Circles are to one another as the squares on the diameters.
- XII.7Proposition XII.7Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
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