EXHAUSTIONS

, or the Method of EXHAUSTIONS, a method of demonstration founded upon a kind of Exhausting a quantity by continually taking away certain parts of it.

The method of Exhaustions was of frequent use among the ancient mathematicians; as Euclid, Archimedes, &c. It is founded on what Euclid says in the 10th book of his Elements; viz, that those quantities are equal, whose difference is less than any assignable quantity. Or thus, two quantities A and B are equal, when, if to or from one of them as A, any other quantity as d be subtracted, however small it be, then the sum or difference is respectively greater or less than the other quantity B: viz, d being an indefinitely small quantity, if A + d be greater than B, and A - d less than B, then is A equal to B.

This principle is used in the 1st prop. of the 10th book, which imports, that if from the greater of two quantities be taken more than its half, and from the remainder more than its half, and so on; there will at length remain a quantity less than either of those proposed. On this foundation it is demonstrated, that if a regular polygon of infinite sides be inscribed in a circle, or circumscribed about it; then the space, which is the difference between the circle and the polygon, will by degrees be quite exhausted, and the circle become ultimately equal to the polygon. And in this way it is that Archimedes demonstrates, that a circle is equal to a right-angled triangle, whose two sides about the right angle, are equal, the one to the semidiameter, and the other to the perimeter of the circle. Prop. 1 De Dimensione Circuli.

Upon the Method of Exhaustions depends the Method of Indivisibles introduced by Cavalerius, which is but a shorter way of expressing the method of Exhaustions; as also Wallis's Arithmetic of Insinites, which is a farther improvement of the Method of Indivisibles; and hence also the Methods of Increments, Differentials, Fluxions, | and Infinite Series. See some account of the Method of Exhaustions in Wallis's Algebra, chap. 73, and in Ronayne's Algebra, part 3, pa. 395.