THE IMPOSSIBILITY OF SQUARING THE CIRCLE James Gregory and the limits of Cartesian geometry (přednáší Davide Crippa, Filosofický ústav AV ČR)

With the emergence of the algebraic movement in XVIth and XVIIth century geometry, the ideal that all mathematical problems should and could be solved by the most adequate means was fostered by outstanding mathematicians (Viète, Descartes). Yet it was a matter of dispute whether certain well-known problems, like the quadrature of the circle, could be solved by geometrically acceptable methods. My talk will explore this issue, considering a controversy occurred in 1668 between the Scottish mathematician James Gregory and the Dutch mathematician Christiaan Huygens, about the possibility of solving the quadrature of central conics (which included the circle) by algebraic means. Whereas the former held it was impossible, the latter believed that the circle could be squared algebraically. This controversy is significant because it hinged upon methodological or foundational questions: which were the bounds of Cartesian geometry? Are the five algebraic operations sufficient in order to express and solve all problems concerning the objects of Euclid's geometry?