Fermat’s Treatise on Quadrature : a New Reading

The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, ∫ x dx, or under a higher hyperbola, ∫ x dx— with the appropriate limits of integration in each case—, has a second part which was not understood by Fermat’s contemporaries. This second part of the Treatise is obscure and difficult to read and even the great Huygens described it as “published with many mistakes and it is so obscure (with proofs redolent of error) that I have been unable to make any sense of it”. Far from the confusion that Huygens attributes to it, in this paper we try to prove that Fermat, in writing the Treatise, had a very clear goal in mind and he managed to attain it by means of a simple and original method. Fermat reduced the quadrature of a great number of algebraic curves to the quadrature of known curves: the higher parabolas and hyperbolas of the first part of the paper. Others, he reduced to the quadrature of the circle. We shall see how the clever use of two procedures, quite novel at the time: the change of variables and a particular case of the formula of integration by parts, provide Fermat with the necessary tools to square very easily curves as well-known as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.