Archimedes

Last of all, and most important for our purpose, is his use of the famous method of exhaustion for the measurement of the areas of curves and the volumes of solids.

The example of this method which will be most familiar to the reader is the proof in Euclid XII., 2, of the theorem that the areas of circles are to one another as the squares on their diameters. The proof in this and in all cases depends on a lemma which forms Prop. 1 of Euclid’s Book X. to the effect that, if there are two unequal magnitudes of the same kind and

from the greater you subtract not less than its half, then from the remainder not less than its half, and so on continually, you will at length have remaining a magnitude less than the lesser of the two magnitudes set out, however small it is.

Archimedes says that the theorem of Euclid XII., 2, was proved by means of a certain lemma to the effect that, if we have two unequal magnitudes (i.e. lines, surfaces, or solids respectively), the greater exceeds the lesser by such a magnitude as is capable, if added continually to itself, of exceeding any magnitude of the same kind as the original magnitudes. This assumption is known as the Axiom or Postulate of Archimedes, though, as he states, it was assumed before his time by those who used the method of exhaustion. It is in reality used in Euclid’s lemma (Eucl. X., 1) on which Euclid XII., 2, depends, and only differs in statement from Def. 4 of Euclid, Book V., which is no doubt due to Eudoxus.

The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon, the sophist, of Athens, a contemporary of Socrates (470-399 B.C.), who took the first step. He inscribed a square (or, according to another account, an equilateral triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and maintained that,