In November 1618, Descartes, then twenty-two, met and worked for two months with Isaac Beeckman, a Dutch scholar seven years his senior. Beeckman was one of the first supporters of a corpuscular-mechanical approach to natural philosophy. However, it was not simply corpuscular mechanism that Beeckman advocated to Descartes. He also interested Descartes in what they called “physico-mathematics.” In late 1618, Beeckman (1939–53, I:244) wrote, “There are very few physico-mathematicians,” adding, “(Descartes) says he has never met anyone other than me who pursues enquiry the way I do, combining Physics and Mathematics in an exact way; and I have never spoken with anyone other than him who does the same.” They were partly right. While there were not many physico-mathematicians, there were of course others, such as Kepler, Galileo, and certain leading Jesuit mathematicians, who were trying to merge mathematics and natural philosophy (Dear 1995, 168–79).

Physico-mathematics, in Descartes’ view, deals with the way the traditional mixed mathematical disciplines, such as hydrostatics, statics, geometrical optics, geometrical astronomy, and harmonics, were conceived to relate to the discipline of natural philosophy. In Aristotelianism, the mixed mathematical sciences were interpreted as intermediate between natural philosophy and mathematics and subordinate to them. Natural philosophical explanations were couched in terms of matterand cause, something mathematics could not offer, according to most Aristotelians. In the mixed mathematical sciences, mathematics was used not in an explanatory way, but instrumentally for problem solving and practical aims. For example, in geometrical optics, one represented light as light rays. This might be useful but does not facilitate answering the underlying natural philosophical questions: “the physical nature of light” and “the causes of optical phenomena.” In contrast, physico-mathematics involved a commitment to revising radically the Aristotelian view of the mixed mathematical sciences, which were to become more intimately related to natural philosophical issues of matter and cause. Paradoxically, the issue was not mathematization. The mixed mathematical sciences, which were already mathematical, were to become more “physicalized,” more closely integrated into whichever brand of natural philosophy an aspiring physico-mathematician endorsed.

Three of Descartes’ exercises in physico-mathematics survive. The most important is his attempt, at Beeckman's urging, to supply a corpuscular-mechanical explanation for the hydrostatic paradox, which had been rigorously derived in mixed mathematical fashion by Simon Stevin (AT X 67–74, 228; Gaukroger and Schuster 2002).