Philosophie naturelle et géométrie au XVIIe siècle (review)

The use of mathematics to resolve questions in natural philosophy did not begin in the seventeenth century. Geometrical diagrams and calculations based on them figure prominently in arguments on motion in fourteenth-century texts. Astronomy and optics had been geometrical from the start. Nevertheless, among many fundamental changes in early modern natural philosophy, one was a decisive reconfiguration of the role of mathematics, especially geometry, in the demonstration of general propositions.

In Philosophie natural et géométrie, a collection of papers recast into book form, Vincent Jullien holds that many of the changes evidenced in the works of Galileo, Descartes, Roberval, and Newton among others are best understood from within. Descartes would likely not have encountered Isaac Beeckman had he not been travelling with the army of Maurice of Nassau in 1619. But the content of the physico-mathematics jointly devised by them owes nothing to that circumstance. Similarly, Descartes’s unfavorable opinion of Galileo contains “not the slightest allusion” to the political relations, then fraught, of France and Italy (24).

This is worth mentioning only because, as Jullien notes, some authors have treated the arguments of natural philosophers as mere window-dressing for political disagreements. But it is difficult to see that the content of Descartes’s Géométrie has much to do with his social position; its primary determinant is the work of others on the same subject. The social element consists, for example, in Descartes’s having encountered the problem of Pappus in 1631 by way of Jacobus Golius, a professor at Leiden where Descartes had enrolled in 1630. If you want to know why that problem, that is part of the answer. But given that Descartes decided to tackle it, his solution and his methods are to be explained primarily by reference to earlier geometers. Jullien’s work, accordingly, proceeds in traditional fashion by the examination of texts and their relations. Even in a chapter entitled “Light, from the School to the Laboratory,” the “laboratory” is no concrete particular place but rather the practice of submitting questions about the nature of light to experimenta produced in laboratories.

I mention only some highlights of the work. One chapter examines the treatment of the law of refraction in Newton and Leibniz. Both reject Descartes’s proof; but Newton bases his optics on analogies that effectively subsume it under mechanics; Leibniz’s demonstration of the law places it “under the jurisdiction of metaphysics,” invoking final causes to justify his appeal to a principle according to which light travels on the “most determinate” path.

Descartes and his sometime rival Gilles Personne de Roberval are the heroes of several episodes. Roberval’s reputation has suffered from being viewed mostly through Cartesian lenses, which in this as in other cases are not notable for accuracy. That they should disagree was owing, no doubt, in part to their being in competition for certain social goods — the attention of Mersenne, for example. But again the content of their disagreements is best understood from within natural philosophy and mathematics. Descartes took the aim of natural philosophy to be the discovery of essences and fundamental laws; Roberval, like Mersenne and Gassendi, was more circumspect. Unlike Descartes, who took seriously the hypotheses he put forward in Le Monde and the Principles, Roberval in his Aristarchus (1644) treated his starting points as hypotheses only.

Three chapters take up the innovations that from the 1640s onward led to the new infinitesimal calculus of Newton and Leibniz. A chapter on “frontiers” in Descartes’s mathematics concludes that the physics of the period required the consideration of problems (e.g., the rectification of curves) that lay outside the boundaries set for mathematics by Descartes’s distinction of “geometric” from “mechanical” curves. The application of algebraic methods to geometric problems preserved in deduction the certainty of the intuitions by which the foundations...