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 the number of sides, nearly exhausted the spaces between the polygons and circumferences. From the theorem that similar polygons inscribed in circles are to each other as the squares on their diameters, geometers may have divined the theorem attributed to Hippocrates of Chios that the circles, which differ but little from the last drawn polygons, must be to each other as the squares on their diameters. But in order to exclude all vagueness and possibility of doubt, later Greek geometers applied reasoning like that in Euclid, XII. 2, as follows: Let C and c, D and d be respectively the circles and diameters in question. Then if the proportion $$\scriptstyle{D^2:d^2=C:c}$$ is not true, suppose that $$\scriptstyle{D^2:d^2=C:c'}$$. If $$\scriptstyle{c' \; < \; c}$$, then a polygon p can be inscribed in the circle c which comes nearer to it in area than does c'. If P be the corresponding polygon in C, then $$\scriptstyle{P:p=D^2:d^2=C:c'}$$, and $$\scriptstyle{P:C=p:c'}$$. Since $$\scriptstyle{p \; > \; c'}$$, we have $$\scriptstyle{P \; > \; C}$$, which is absurd. Next they proved by this same method of reductio ad absurdum the falsity of the supposition that $$\scriptstyle{c' \; > \; c}$$. Since c' can be neither larger nor smaller than c, it must be equal to it, Hankel refers this Method of Exhaustion back to Hippocrates of Chios, but the reasons for assigning it to this early writer, rather than to Eudoxus, seem insufficient.

Though progress in geometry at this period is traceable only at Athens, yet Ionia, Sicily, Abdera in Thrace, and Cyrene produced mathematicians who made creditable contributions to the science. We can mention here only Democritus of Abdera (about 460–370 B.C.), a pupil of Anaxagoras, a friend of Philolaus, and an admirer of the Pythagoreans. He visited Egypt and perhaps even Persia. He was a successful geometer and wrote on incommensurable lines, on geometry, on numbers, and on perspective. None of these works are extant. He used to boast that in the construction of plane figures with proof no one had yet surpassed him, not even