Paradís, Jaume (Author)
Pla, Josep (Author)
Viader, Pelegrí (Author)
The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, ? p x+m /ndx , or under a higher hyperbola, ? p x-m /ndx with the appropriate limits of integration in each casehas a second part which was mostly unnoticed by Fermat's contemporaries. This second part of the Treatise is obscure and difficult to read. In it Fermat reduced the quadrature of a great number of algebraic curves in implicit form 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 squarequite easilyas well-known curves as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi. Keywords : History of mathematics, quadratures, integration methods
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