THE CONTROVERSY BETWEEN GREGORY AND HUYGENS ON THE QUADRATURE OF THE CIRCLE
Davide CRIPPA (Max Planck Institute for the History of Science)
With the emergence of the algebraic movement in 16th and 17th 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, not only because it is
one of the first episodes in which the impossibility of solving an open problem is at the centre of
a debate, but also because this debate 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?
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