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Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Loci: Convergence
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Archimedes' Method for Computing Areas and Volumes
Proposition 5 of The Method
We can use Archimedes' method to determine the center of gravity of the paraboloid. By symmetry, we know that the center of gravity lies at some point X on its axis (not indicated on the sketch). Archimedes' Proposition 5, illustrated below, shows a paraboloid and a cone inscribed inside a cylinder. Again, the dynamic version of this proposition can be accessed by clicking on Proposition 5.
The sketch illustrates that the paraboloid, left where it is, balances the cone, moved to H, where |AH| = |AD|. So by the law of the lever,
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Since the paraboloid is half the cylinder (by the previous exercise), and the cone is 1/3 of the cylinder, this implies that (1/2)|AX| = (1/3)|AH|. Hence |AX| = (2/3)|AH|. So the center of gravity of the paraboloid lies on its axis, twice as far from the vertex as from the base.
Next page >> Exercise on Proposition 6 of The Method
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