The Garden of Archimedes
A Museum for Mathematics

Analytic geometry and the problem of tangents

works in the section

Pierre Fermat, Methodus ad disquirendam maximam et minimam in Oeuvres de Fermat par P. Tannery et C. Henry, tome I, Paris, Gauthier-Villars, 1891 [first edition 1679].
Réné Descartes, Discours de la Méthode pour bien conduire sa raison, & chercher la verité dans les sciences. Plus La Dioptrique, Les Meteores et La Géométrie qui sont des essais de cette méthode, Leyde, de l'imprimerie de Ian Maire, 1637 [anastatic reprint 1987].
Réné Descartes, Geometria a Renato des Cartes anno 1637 gallice edita in linguam latinam versa opera atque studio Francisci Schooten, Lugduni Batavorum, ex officina Ioannis Maire, 1649 [first edition 1637].
Gilles Personne de Roberval, Observations sur la composition des mouvements et sur le moyen de trouver les touchantes des lignes courbes, in Divers ouvrages des Mathématique et de Physique par Messieurs de l'Academie Royale des Sciences, Paris, de l'Imprimerie Royale, 1693.

At the same time of the progresses in squaring methods, the Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences was published in France by René Descartes (1596-1650), whose name is latinised as Cartesio, accompanied by three essays.
One of them is the Géométrie, a unique printed mathematical work by Descartes, often remembered as the first text written with a language and format familiar to a modern reader. The work spread among mathematicians, especially thanks to the two Latin editions that followed, which had commentaries by Franz van Schooten, the first in 1649 and the second, in two volumes, in 1659-1661.
The method exposed by Descartes combines algebra and geometry, translating one into the other reciprocally. This was a moment of profound renovation which can be identified as the birth of analytic geometry.

Moving in a similar direction, using algebra of an antiquated kind, we also find Pierre de Fermat (1601-1665), who independently reached the identification of equations and geometric loci. After the publication of the Géométrie, in a letter to Mersenne, correspondent of Descartes and of many scientists of the time, Fermat exposed his method of finding maxima and minima. Observing that the difference between a curve and its tangent has in its tangential point a minimum (or a maximum), he uses such method to determine the tangents to a curve. His results initially spread only through written correspondence. The method was first published in the fifth volume of Supplementum Cursus Mathematici (1642) written by Herigone and was printed as Methodus ad disquirendam maximam et minima only in 1679.

The problem of the construction of a tangent to a curve can be found in the Géométrie under the equivalent form of the construction of the normal. The technique used by Descartes is the one of considering a circle of variable centre on one of the axes and to impose the algebraic condition that the circle has two intersections coinciding with the curve in the tangential point.

The methods by Descartes and Fermat can obviously be applied only to polynomial equations or that can be traced back to them, as however equations of the considered curves always are, and become practically useless as the complexity of the equation increases. A different method, where the tangent is determined through kinematic considerations on the curve, is used by Gilles Personnes de Roberval (1602-1675) and made known in 1644 by Mersenne. In the same year Torricelli published his Geometric works containing very similar techniques. Through the kinematic method, tangents to parabolas of a superior order, to spirals and to cycloids can be determined. During the following decades, the analytical method gave origin to a series of rules for the calculus of tangents, as in the works by Hudde, Sluse, Gregory, Barrow, Wallis.

Panels of the exhibition (only italian available)

History of calculus...