History of Analytic Geometry by Carl B. Boyer

Author:Carl B. Boyer [Boyer, Carl B.]
Language: eng
Format: epub, azw3
ISBN: 9780486154510
Publisher: INscribe Digital
Published: 2012-10-11T04:00:00+00:00

Fig. 33

The contribution of Clairaut in this connection should not, however, be exaggerated. The distance formulae are, after all, obvious analytical expressions of an ancient theorem named for Pythagoras but known to the Babylonians of some four thousand years ago. There can be little doubt but that their equivalents were known to the earliest analytic geometers, including Descartes and Fermat. The equations of circles and spheres, given long before Clairaut, are tantamount to distance formulae; and the rectification of curves, known since 1659, is dependent upon some such equivalent. The formulae for distance in infinitesimal analysis—ds = and ds = —appear frequently in the Recherches, but these are not to be ascribed to him. It would not be surprising, in fact, if further research would reveal explicit, as well as implicit, anticipations of Clairaut’s distance formulae, not only for the calculus, but also for analytic geometry. Nevertheless, it is important to note that, in formalization, infinitesimal analysis had at that time far outstripped Cartesian geometry, even though the invention of the latter preceded that of the former by about half a century. Formulae had been a natural outgrowth of the algorithms of Newton and Leibniz, but the coordinate geometry of Descartes and Fermat still leaned heavily upon auxiliary diagrams. Consequently, the distance formulae did not appear systematically until the time of Lagrange.

In 1731, the year of his Recherches, Clairaut published in the Mémoires of the Académie a paper relating his solid analytic geometry to the theory of higher plane curves. In this work, “Sur les courbes que l’on forme en coupant un surface courbe quelconque, par un plan donné de position,”252 he proved Newton’s well-known theorem on the projective transformation of cubic curves. Using the equation of the conical surface xyy = ax3 + bxxz + exzz + dz3, whose traces in the planes x = k are the divergent parabolas, Clairaut showed that the plane sections of this surface have as equations the various species of cubics in the Enumeratio. An interesting case of the simultaneity of ideas is found in the fact that Nicole presented a similar analytic proof of the theorem in the very same volume of the Mémoires.253 The corresponding theorem for the conics was proved, also in this volume, by Charles-Marie de la Condamine (1701–1774). He showed that the conic sections are derivable as plane sections of the cone nnxx = yy + zz, apparently the first instance in which solid analytic geometry was applied to this ancient theorem.

The study of the plane and of other surfaces in space was resumed in the following year by Jacob Hermann (1678–1733). In a paper, “De superficiebus ad aequationes locales revocatis, variisque earum affectionibus,” published in the Petersburg Commentarii for 1732—1733, he said that up to that time the geometry of surfaces other than planes and surfaces of revolution had scarcely been considered.254 This would indicate that he did not know of the Recherches of Clairaut, and that he was unaware of the work of his own associate, Euler.

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