Archimedes

The main proposition, that two magnitudes balance at distances reciprocally proportional to the magnitudes, is proved first for commensurable and then for incommensurable magnitudes. Preliminary propositions have dealt with equal magnitudes disposed at equal distances on a straight line and odd or even in number, and have shown where the centre of gravity of the whole system lies. Take first the case of commensurable magnitudes. If A, B be the weights acting at E, D on the straight line ED respectively, and ED be divided at C so that A : B = DC : CE, Archimedes has to prove that the system is in equilibrium about C. He produces ED to K, so that DK = EC, and DE to L so that EL = CD; LK is then a straight line bisected at C. Again, let H be taken on LK such that LH = 2LE or 2CD, and it follows that the remainder HK = 2DK or 2EC. Since A, B are commensurable, so are EC, CD. Let x be a common measure of EC, CD. Take a weight w such that w is the same part of A that x is of LH. It follows that w is the same part