Britannica Sources used in this answer

How did Eudoxus’s method of exhaustion help calculate areas and volumes?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

The method of exhaustion, attributed to Eudoxus of Cnidus, was a rigorous logical technique used to determine the areas and volumes of geometric figures by approximating them with simpler shapes of known dimensions, such as polygons or prisms.123

Rather than using limits or infinitesimal quantities, Eudoxus employed a bisection principle: any given magnitude can be reduced by successive halving until it is smaller than any other assigned magnitude. To calculate an area or volume, he would "exhaust" the figure by inscribing or circumscribing it with a sequence of simpler shapes.142

For example, to find the volume of a pyramid, he demonstrated that a stack of prisms could be placed inside the pyramid to exceed any volume less than the target, while another stack could undercut any volume greater than the target. By showing that the volume could be neither greater nor less than a specific value (such as $B h / 3$ ), he proved that the volume must be exactly that amount. This method allowed Eudoxus to prove fundamental results, including that the areas of circles are proportional to the squares of their diameters and that the volumes of pyramids and cones are one-third those of prisms and cylinders with the same base and height.254