The historical record is quite straightforward: the matter was simply not seen as a very serious issue. In Transcendental Curves in the Leibnizian Calculus, 2017 2.3.4 Foundations of infinitesimal methodsĪnother point on which the preconceived opinions of modern authors lead to a distorted view of history is that of the foundations of infinitesimal methods.
![stochastic calculus with infinitesimals stochastic calculus with infinitesimals](https://i.stack.imgur.com/RFDwf.png)
Although Euler had already noticed that the coordinates of a point on a surface could be expressed as functions of two independent variables, it was Gauss who first made a systematic use of such a parametric representation, thereby initiating the concept of “local chart” which underlies differential geometry. Until Gauss’ fundamental article Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) published in Latin in 1827 (of which one can find a partial translation to English in Spivak (1979)), surfaces embedded in R 3 were either described by an equation, W( x, y, z) = 0, or by expressing one variable in terms of the others. The study of differential properties of curves and surfaces resulted from a combination of the coordinate method (or analytic geometry) developed by Descartes and Fermat during the first half of the seventeenth century and infinitesimal calculus developed by Leibniz and Newton during the second half of the seventeenth and beginning of the eighteenth century.ĭifferential geometry appeared later in the eighteenth century with the works of Euler Recherches sur la courbure des surfaces (1760) (Investigations on the curvature of surfaces) and Monge Une application de l’analyse à la géométrie (1795) (An application of analysis to geometry).
![stochastic calculus with infinitesimals stochastic calculus with infinitesimals](https://image.slidesharecdn.com/stochasticcalculus-rlk-130113061431-phpapp01/95/stochastic-calculus-53-638.jpg)
Paycha, in Encyclopedia of Mathematical Physics, 2006 Curves and Surfaces