The Impossibility of Squaring the Circle in the 17th Century

This book is about James Gregory's attempt to prove that the quadrature of the circle, the ellipse and the hyperbola cannot be found algebraically. Additonally, the subsequent debates that ensued between Gregory, Christiaan Huygens and G.W. Leibniz are presented and analyzed. These debates eventually culminated with the impossibility result that Leibniz appended to his unpublished treatise on the arithmetical quadrature of the circle. The author shows how the controversy around the possibility of solving the quadrature of the circle by certain means (algebraic curves) pointed to metamathematical issues, particularly to the completeness of algebra with respect to geometry. In other words, the question underlying the debate on the solvability of the circle-squaring problem may be thus phrased: can finite polynomial equations describe any geometrical quantity? As the study reveals, this question was central in the early days of calculus, when transcendental quantities and operations entered the stage. Undergraduate and graduate students in the history of science, in philosophy and in mathematics will find this book appealing as well as mathematicians and historians with broad interests in the history of mathematics.

Delivers an unprecedented perspective on this topic Gives a fresh picture of the mathematics in the 17th Century based on previously unstudied documents Conveys mathematics with minimal technical requirements and thorough explanation so that the narrative can be followed by graduate and undergraduate students in sciences and humanities

Contenu
FM.-Introduction.- James Gregory and the Impossibility of Squaring the Central Conic Sections.- Leibniz's Arithmetical Quadrature of the Circle.- Conclusion.-BM.