Fermat's Treatise on Quadrature: A New Reading

UPF Economics and Business Working Paper 775

31 Pages Posted: 16 Nov 2005

See all articles by Jaume Paradís

Jaume Paradís

Universitat Pompeu Fabra

Pelegrí Viader

Universitat Pompeu Fabra

Josep Pla

University of Barcelona

Date Written: February 2004


The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, R x+m/n dx, or under a higher hyperbola, R x-m/n 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.

Keywords: History of mathematics, quadratures, integration methods

JEL Classification: C00

Suggested Citation

Paradís Balaux, Jaume and Viader Canals, Pelegrí and Pla, Josep, Fermat's Treatise on Quadrature: A New Reading (February 2004). Available at SSRN: https://ssrn.com/abstract=848544 or http://dx.doi.org/10.2139/ssrn.848544

Jaume Paradís Balaux (Contact Author)

Universitat Pompeu Fabra ( email )

Ramon Trias Fargas 25-27
Barcelona, 08005
(34-3) 542 24 04 (Phone)
(34-3) 542 17 46 (Fax)

Pelegrí Viader Canals

Universitat Pompeu Fabra ( email )

Ramon Trias Fargas 25-27
Barcelona, 08005
(34-93) 542 27 33 (Phone)
(34-93) 542 17 46 (Fax)

Josep Pla

University of Barcelona ( email )

Gran Via de les Corts Catalanes, 585
Barcelona, 08007

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