Abstract

To justify certain steps of the computation developed in his Sand-Reckoner, Archimedes cites the following inequalities relative to the sides of right triangles: if of two right-angled triangles, the sides about the right angle are equal, while the other sides are unequal, the greater angle of those toward [sc. next to] the unequal sides has to the lesser a greater ratio than the greater line of those subtending the right angle to the lesser, but a lesser than the greater line of those about the right angle to the lesser. That is, with reference to the two right triangles ABG, DEZ, where AG equals DZ and the angle at B is greater than that at E, ZE:GB < angle B:angle E < DE:AB.