Some early Greek attempts to square the circle

Abstract

THE quadrature of the circle, one of the most famous mathematical problems of antiquity no less than of more modern times, appears comparatively early in the history of Greek mathematics. In the gth century it had an established position as one of the three traditional problems, viz. quadrature or rectification of the circle, duplication of the cube and trisection of a given angle. Thinkers as different from each other as Hippocrates of ChiosI Anaxagoras the philosopher 2 and Antiphon the Sophist worked on it; and by 4I4 B.C. the endeavour square the circle had become the recognized sign of the crank.4 In what follows I shall (a) examine the evidence that we have for some of the reported attempts to square the circle, and (b) attempt to reconstruct the historical relation that subsists between these attempts. Aristotle mentions an attempt by Antiphon the Sophist to square the circle.5 -