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. (shrink)