102.51 Archimedes' quadrature of the parabola

摘要

Every segment bounded by a parabola and a chord is equal to four thirds of the area of the triangle with the same base as the segment and equal height. Archimedes proved this by the method of exhaustion, a strategy attributable to Eudoxus. We know that Archimedes often found his mensuration results initially by means of mechanics, especially the law of the lever, and then proved them, if necessary, by rigorous geometrical reasoning. The following looks at this beautiful result using algebraic techniques.