Mixed Mathematics 163mixed mathematics, as John Heilbron’s chapter in this volume notes. Un-
der “Philosophie,” we read:
It is time to pass to the second point of this article, which is concerned with
establishing the meaning of the term “philosophy” and providing it with a
good defi nition. To philosophize, is to give the reason of things, or at least
to search for it, for to the extent that one restricts oneself to seeing and
reporting what one sees, one is only an historian. When one calculates and
measures the proportions of things, their sizes, their values, one is a math-
ematician; but he who stays to discover the reason for things, and for their
being thus rather than some other way, that is the philosopher properly
so- called.^46Mathematics simply could not constitute part of natural philoso-
phy; quantity was not physical. D’Alembert to some degree maintained
this view himself, in contrast to the assumptions of the English Newto-
nians, by distinguishing categorically between mixed mathematics and
physico- mathematics. Speaking of quantity as the “object of mathemat-
ics,” he recognized three divisions of its study: independent of real and ab-
stract things (pure mathematics); considered in “real and abstract beings”
(mixed mathematics, which includes statics); or “considered in their [the
beings’] effects investigated according to real or supposed causes”—this
last, causal variety being labeled as “physico- mathematics.”^47
D’Alembert’s scrupulous taxonomy therefore strove to maintain a dis-
tinction between physical, causal knowledge and other kinds associated
with noncausal mathematics; physico- mathematics still left him with a
place to locate his own work in Newtonian- style rational mechanics as a
kind of natural philosophy. Buffon, in the 1740s, had already expressed
similar reservations, lumping together natural history and mathematics,
much like the Encyclopédie’s discussion of philosophy, as exemplars of non-
philosophy. And Buffon was not afraid to make it clear that an absence of
true philosophy was a serious shortcoming in the study of nature.
In this century itself, where the Sciences seem to be carefully cultivated, I
believe that it is easy to perceive that Philosophy is neglected, and perhaps
more so than in any other century. The arts that people are pleased to call
scientifi c have taken its place; the methods of calculus and geometry, those
of botany and natural history, in a word formulas and dictionaries preoccupy
almost everyone. People imagine that they know more because of having
increased the number of symbolic expressions and learned phrases, and pay