Fermat’s method of quadrature

Jaume Paradís; Josep Pla; Pelegrí Viader

Revue d'histoire des mathématiques (2008)

  • Volume: 14, Issue: 1, page 5-51
  • ISSN: 1262-022X

Abstract

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The Treatise on Quadratureof Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, ∫ x + m / n d x , or under a higher hyperbola, ∫ x - m / n d x —with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of theTreatise 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 square—quite easily—as well-known curves as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.

How to cite

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Paradís, Jaume, Pla, Josep, and Viader, Pelegrí. "Fermat’s method of quadrature." Revue d'histoire des mathématiques 14.1 (2008): 5-51. <http://eudml.org/doc/274941>.

@article{Paradís2008,
abstract = {The Treatise on Quadratureof Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, $\int x^\{+m/n\}dx$, or under a higher hyperbola, $\int x^\{-m/n\}dx$—with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of theTreatise 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 square—quite easily—as well-known curves as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.},
author = {Paradís, Jaume, Pla, Josep, Viader, Pelegrí},
journal = {Revue d'histoire des mathématiques},
keywords = {history of mathematics; quadratures; integration methods},
language = {eng},
number = {1},
pages = {5-51},
publisher = {Société mathématique de France},
title = {Fermat’s method of quadrature},
url = {http://eudml.org/doc/274941},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Paradís, Jaume
AU - Pla, Josep
AU - Viader, Pelegrí
TI - Fermat’s method of quadrature
JO - Revue d'histoire des mathématiques
PY - 2008
PB - Société mathématique de France
VL - 14
IS - 1
SP - 5
EP - 51
AB - The Treatise on Quadratureof Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, $\int x^{+m/n}dx$, or under a higher hyperbola, $\int x^{-m/n}dx$—with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of theTreatise 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 square—quite easily—as well-known curves as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.
LA - eng
KW - history of mathematics; quadratures; integration methods
UR - http://eudml.org/doc/274941
ER -

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