To justify certain steps of the computation developed in his Sand-Reckoner, Archimedes cites (without proof) the following inequalities relative to the sides of right triangles:
if of two right-angled triangles, (one each of) the sides about the right angle are equal (to each other), while the other sides are unequal, the greater angle of those toward [sc. next to] the unequal sides has to the lesser (angle) a greater ratio than the greater line of those subtending the right angle to the lesser, but a lesser (ratio) than the greater line of those about the right angle to the lesser.
That is, with reference to the two right triangles ABG, DEZ (Fig. 1), where AG equals DZ and the angle at B is greater than that at E, ZE:GB < angle B:angle E < DE:AB.