The impossibility of squaring the circle in the 17th century. A debate among Gregory, Huygens and Leibniz (Q1626805)

This book is the abridged and improved version of the doctoral dissertation of the author. Therein he studies ``several attempts to prove the impossibility of solving three fundamental problems in geometry by algebraic means: the squaring of the circle, the ellipse and the hyperbola within the mathematical context of Seventeenth Century'' (p. 1). In other words, it is a matter of the quadratures of the central conic sections. The used texts belong to the period 1667 up to 1676. In 1667, James Gregory published his treatise \textit{Vera circuli et hyperbolae quadratura}. It stated the impossibility of squaring the central conic sections thus leading to a vivid debate about this impossibility result in mathematics. While the first, introductory chapter of the book under review deals especially with impossibility results in classical mathematics the second chapter discusses Gregory's handling of the impossibility of squaring the central conic sections. The third chapter investigates Leibniz's criticism of Gregory's results and explains Leibniz's own \textit{De quadratura arithmetica}. The last version of this treatise was written in 1676 and its last proposition (Proposition 51) consists of a new, improved proof of the algebraic unsolvability of the quadrature of the central conic sections. The conclusion of this charming book is dedicated to the role and goals of impossibility results in early modern geometry. These results ``became central in the work of many outstanding early modern mathematicians'' (p. 173).