The myth of Leibniz's proof of the fundamental theorem of calculus (Q2818542)

According to common historiography Leibniz gave a first, geometrical version of the fundamental theorem of calculus in a paper from 1693, \textit{Supplementum geometriae dimensoriae}. In that paper, Leibniz stated: ``I shall now show that the general problem of quadratures can be reduced to the finding of a curve that has a given law of tangency.'' Stripped off from its geometrical vest, this claim is usually taken to be a version of the fundamental theorem of the calculus, that is: \(\int_{a}^{b} f(x)dx = F(b)-F(a)\), where \(F'(x) = f(x)\). The paper under review attacks this thesis, arguing that Leibniz's principal goal in his 1693 article was very different from that of proving the inverse relation between the operations of integration and differentiation. The proper context to understand Leibniz's paper is the construction-based geometry of Euclid and Descartes. Descartes gave a geometrical definition of ``algebraic curves'', as those curves constructible by certain tracing machines which, as complex as they might be, nevertheless possess in common with the ruler and the compass the fundamental requirement of unicity and continuity of the tracing motions. Descartes' programme greatly extended the boundaries of geometry, but still left out from it transcendental curves, in particular the quadratrix or the Archimedean spiral, on the ground that their construction involved synchronizing two motions according to a transcendental ratio (i.e., \(\pi\)). Leibniz's 1693 paper tackles precisely this issue. Just like Descartes did for algebraic curves, Leibniz considered devices which could trace curves on the basis of certain properties of their tangents. The simplest example is the tractrix, a curve traced by dragging a heavy body over a horizontal plane by a cord or a rod, which is fixed to the body, while the other end is moved along a straight line. In the 1693 paper, Leibniz implemented tractional motion by a device that allowed one to construct any curve satisfying a differential equation of the form: \(\frac{dy}{dx} = f(x)\), with \(f(x)\) being a given function. A corollary of the use of such machine is the possibility of solving quadrature problems by reducing them to rectifications: this is where Leibniz's version of the fundamental theorem appears. However, in Leibniz's account its role is far from fundamental, the fundamental issues at stake being instead that of legitimating transcendental geometry by the very same guiding criterion used by Descartes for algebraic curves, and that of constructing integrals such as: \(\int \sqrt{1+x^4}dx\) geometrically. These integrals cannot be computed using the inverse relation between differentiation and integration. This paper is a welcome contribution to the literature on Leibniz's mathematics, so far mostly restricted to the issue of infinitesimals, since it casts new light onto Leibniz's foundational interest for the legitimation of transcendental geometry. I notice that the manuscripts written between 1673 and 1676 (in particular [\textit{G. W. Leibniz}, Sämtliche Schriften und Briefe. Reihe 7. Mathematische Schriften. Band 3. 1672--1676. Differenzen, Folgen, Reihen (Latin). Berlin: Akademie Verlag (2003; Zbl 1038.01515); Sämtliche Schriften und Briefe. Reihe 7. Mathematische Schriften. Band 5. 1674--1676. Infinitesimalmathematik (Latin). Berlin: Akademie Verlag (2008; Zbl 1155.01006)]) confirm that Leibniz became interested in the topic already in Paris, i.e., before the invention of calculus. Leibniz's idea was, back then, to extend the range of permissible constructions by allowing the possibility of bending strings, so as to generate certain transcendental curves (like the cycloid) by one and continuous motion. It is possible that Leibniz came to see the generality of tractional motion only later on, and built on the latter his project of legitimating the realm of transcendental curves.