Recherches sur les courbes a double courbure

CLAIRAUT, Alexis Claude

Published by Nyon, Didot and Quillau, Paris, 1731

About this Item

"THE FIRST TREATISE ON SOLID ANALYTIC GEOMETRY" (BOYER). First edition, very rare, of Clairaut's first publication, the first extended treatment of the geometry of curves in three dimensions, written when he was only sixteen. In it Clairaut demonstrates a claim made by Newton that all cubic curves in three dimensions are projections of a cubic curve of a special form. This publication led to Clairaut's election to the Académie des Sciences at the age of eighteen, the rules of admission being suspended to accommodate the mathematical prodigy. "Alexis Claude Clairaut was one of the most precocious of mathematicians, outdoing even Blaise Pascal in this respect. At the age of ten he was reading the textbooks of L'Hospital on conics and the calculus, when he was thirteen he read a paper on geometry to the Académie des Sciences . In the year of his election Clairaut published a celebrated treatise, Recherches sur les courbesà double courbure, the substance of which had been presented to the Académie two years earlier . The treatise of Clairaut carried out for space curves the program that Descartes had suggested almost a century before - their study through projections on two coordinate planes. It was, in fact, this method that suggested the name given by Clairaut to gauche or twisted curves inasmuch as their curvature is determined by the curvatures of the two projections. In the Recherches numerous space curves are determined through intersections of various surfaces, distance formulas for two and three dimensions are given, an intercept form of the plane is included, and tangent lines to space curves are found. This book by the teenage Clairaut constitutes the first treatise on solid analytic geometry" (Boyer, History of mathematics, p. 494). "Although the expression 'double-curvature curve' (gauche curve) is attributed to Henri Pitot (1724), Clairaut's treatise on this type of figure is no less original, representing the first serious analytical study of it. The curve is determined by two equations among the three orthogonal coordinates of its 'point courant' (locus). Assimilation of the infinitesimal arc to a segment of a straight line permits determination of the tangent and the perpendiculars. It also permits rectification of the curve. This work also includes quadratures and the generation of some special gauche curves" (DSB). "In his 1731 work Recherches sur les courbes à double courbure, Clairaut studied tangents to curves of double curvature. He saw that a space curve can have an infinity of normals located in a plane perpendicular to the tangent. The expressions for the arc length of a space curve and the quadrature of certain areas on surfaces are also due to him" (Kline, Mathematical Thought from Ancient to Modern Times, p. 557). Provenance: The Earls of Macclesfield (South Library bookplate on front paste-down, small pressure stamp to title). Analytic geometry in two dimensions, i.e., the study of curves in the plane, dates from the work of Descartes, Fermat, Ghetaldi and others in the first half of the seventeenth century. Phillipe de la Hire understood, later in the century, that an equation in three unknowns represents a surface, but he did not study surfaces by means of their equations. John Wallis and Christopher Wren discussed the lines on the single-sheeted hyperboloid in the 1660s, but they did not use coordinates. Indeed, curves and surfaces in three dimensions were not studied in any detail until the eighteenth century. The first explicit analytic treatment of a curved surface was given by Antoine Parent in 1700, "an awkward treatment of the surface of a sphere, but it shows a full knowledge of space coordinates" (Boyer, History of Analytic Geometry, pp. 156-157). However, this work seems to have made little impression upon his contemporaries. Jean Bernoulli, in correspondence with Leibniz in 1715, also showed familiarity with space coordinates, but this too long went unpublished and unnoticed. It was Euler, Seller Inventory # 5797

Bibliographic Details

Title: Recherches sur les courbes a double courbure

Publisher: Nyon, Didot and Quillau, Paris

Publication Date: 1731

Edition: First edition.