Mixed Mathematics 151work when he devised the fi rst version of his new heliocentric system of
the universe.^4
The terms being used in the sixteenth century to designate those parts
of the quadrivium that went beyond arithmetic and geometry were es-
pecially owing to Aristotle’s discussions of them in the Posterior Analyt-
ics and the Metaphysics. Aristotle had argued that a science ought prop-
erly to be built up deductively from its fundamental principles, and that
the conclusions of that science should only employ terms homogeneous
with those of its principles. This doctrine, called metabasis in Greek, thus
envisaged multiple sciences, each logically self- contained. In the case of
the mathematical sciences of nature, however, Aristotle saw that this argu-
ment could not apply: a conclusion in geometrical optics about light rays
would be rooted in the principles of geometry, not statements about light
rays themselves, and hence the requisite homogeneity of terms would not
apply. Nonetheless, Aristotle recognized such arguments as legitimately
scientifi c, and hence established for them the category of “subordinate
sciences” (in later Latin usage, scientiae subalternatae). Thus the proofs of
a mathematical science of nature that employed geometrical demonstra-
tions were regarded as subordinate to those of geometry itself, that is, de-
pendent upon them, and analogously for mathematical sciences making
use of arithmetical demonstrations.^5 In the thirteenth century, Thomas
Aquinas had employed the term scientiae mediae (“middle sciences”) to
designate the same category, since such a science was located between
its superior discipline and its inferior subject matter, between pure math-
ematics and physics. In the sixteenth century, Niccolò Tartaglia, in his
Italian Euclid (1569), referred to these “impure” mathematical disciplines
as “mixed” (miste).^6 The establishment of the term “mixed mathematics”
by the beginning of the seventeenth century simply used the preexisting
category in its specifi c reference to mathematical scientiae mediae or sub-
alternatae (Aquinas recognized a few nonmathematical instances of the
latter), and applied to them the new term, in Latin, mathematicae mix-
tae, to designate the entire fi eld.^7 Thus the important Jesuit mathemati-
cian and pedagogue Christoph Clavius described astronomy and music
in the quadrivium as representing “mixed” mathematics in the widely
read “Prolegomena” to his Opera mathematica of 1612; a work on logic of
1608 by the Lutheran scholar Gregor Horst also discusses what he calls
disciplinae Mathematicae mixtae—each, and no doubt many others, doing
so evidently independently of Bacon’s use of the English term “mixed
mathematics” in The Advancement of Learning (1605).^8 The term rapidly
became standard in the works of Jesuit as well as non- Jesuit mathemati-