Mixed Mathematics 155is not entirely satisfi ed if it does not grasp, or at least investigate, the prior
causes from which these effects fi nally result.^18Baliani’s subsequent causal account of natural acceleration, because it
concerned “not effects, but the natures of things,” could not be affi rmed
as certain, being a matter of physics, not demonstrative mathematics.^19
PHYSICO- MATHEMATICS
Just a few decades after the term “mixed mathematics” appeared, some-
what mysteriously, around the beginning of the seventeenth century,
another term of similar signifi cation came into general use: “physico-
mathematics,” together with its cognates. But while “mixed mathematics”
had carried little greater specifi city (or extension) than its predecessors
such as “middle sciences,” “physico- mathematics” tended sometimes to
trade on a new cognitive claim. While “mixed mathematics” continued to
be very widely used, and with essentially the same meaning related to the
old quadrivium as well as to Aristotle’s defi nition of subordinate sciences,
“physico- mathematics” could also serve to make stronger methodological
and cognitive claims for the mathematical sciences. Just as Bacon’s use of
“mixed mathematics” heralded a widespread, though evidently indepen-
dent, use of that term, so the Dutch schoolmaster and philosopher Isaac
Beeckman’s term “physico- mathematics” in a private journal entry in 1618
was thereafter widely used to mean very different things by writers many of
whom apparently owed nothing to Beeckman’s private notation. The mat-
ter is obscure, however, since the term found its way into the work of René
Descartes and Marin Mersenne, each of whom knew and corresponded
with Beeckman, and whose own work and correspondence was in turn
widely distributed among the learned in the 1620s, ’30s, and ’40s. In fact,
Beeckman’s fi rst known use of “physico- mathematics” occurs in a reference
to the young Descartes, whom he had recently met in the Netherlands:
This Descartes has been educated with many Jesuits and other studious
people and learned men. He says however that he has never come across any-
one anywhere, apart from me, who uses accurately this way of studying that I
advocate, with mathematics connected to physics. Neither, furthermore, have
I told anyone apart from him of this kind of study.^20The novel terminology appears in the form of Beeckman’s marginal note
to this paragraph: “Very few physico- mathematicians.”^21