Exhaustion (Lat. exhaurire, to draw out), a method of the ancient geometry, applied with success by Archimedes and Euclid, by which the value of an incommensurable quantity was sought by obtaining approximations alternately greater and less than the truth, until two approximations differed so little from each other that either might be taken as the exact statement. Thus the length of a circumference was sought by calculating the length of inscribed and circumscribed polygons, and increasing the number of sides until the lengths of the outer and inner polygon were sensibly the same, when that of the circumference could not differ sensibly from either. By this method the space between the polygons and the curve was exhausted, as it were, and hence the term. Exhaustion is now interesting chiefly because it was one of the methods which led, in the 17th century, to the invention of the differential calculus.