Descartes and the infinitesimal (Q2866541)

The author's purpose in this article is to link Descartes' conceptions of curves, his dealings (in the plural) with the infinitely (or indefinitely) small, and his epistemology. According to the latter, only that can be an object of demonstration which corresponds to a ``clear and distinct'' idea in the understanding (l'entendement), even though the imagination (``image-forming'') faculty may assist -- but this faculty fails when faced with the infinite or the infinitely small.NEWLINENEWLINEThe infinitely large belongs to Descartes' metaphysics only. Solely the infinitely small presents itself in his mathematics, and Descartes is shown to have two distinct strategies for dealing with it. One is in the \textit{Géométrie} from 1637, which deals with curves that can be represented by algebraic equations. For these, Descartes finds the normal in a given point (which then yields the tangent) as the radius of a circle that ``touches the curve [in the point in question] but does not cut it'' -- which means that the single equation resulting from the equation system possesses a double root. Here, the infinitely small only intervenes (surreptitiously) if we do not accept the notion of ``touching without cutting'' as a primary concept but interpret it through concepts of infinitesimal analysis. Since Descartes does not see things in this way, he can challenge Fermat's method for finding tangents, which relies on a notion of infinitesimal approach to zero.NEWLINENEWLINEWhen taking up once again mechanical curves (a topic he had been interested in long before 1637), Descartes has to use a different strategy for finding tangents. This is examined on the example of his analysis of the cycloid from the following year. Here, the rolling circle is understood as the limit of a rolling ``polygon with an infinity of sides''. As the author observes, Descartes here ``combines a strong geometric intuition with a kind of infinitesimal reasoning'' similar to that which had been ``applied by Archimedes and the Arabic infinitesimalists'' without explicit infinitesimal notions -- and Descartes can still reject Fermat's use of infinitesimals tending toward zero. And while the algebraic method of the \textit{Géométrie} can be characterized as ``clear and distinct'', that applied to mechanical curves is only ``clear, allowing degrees and imperfections'' and therefore also admitting the in(de)finitely small.