Gregory's converging double sequence: A new look at the controversy between huygens and gregory over the “analytical” quadrature of the circle

Abstract

As is well known, upon publication of his Vera circuli et hyperbolae quadratura (Padua 1667), James Gregory became involved in a bitter controversy with Christiaan Huygens over the truth of one of his major propositions. It stated that the area of a sector of a central conic cannot be expressed “analytically” in terms of the areas of an inscribed triangle and a circumscribed quadrilateral. Huygens objected to Gregory's method of proof, and expressed doubts as to its validity. As Gregory's iterative limiting process, employing an infinite double sequence, uses a combination of geometric and harmonic means, one may apply to it methods developed by the young Gauss for dealing with a similar process based on the combination of arithmetic and geometric means. This yields both the Leibnizian series forπ/4 and the product found by Vie`te for2/π, and thus serves to illuminate the structure of Gregory's procedure and the nature of Huygens' criticism.