Natural Knowledge in the Latin Middle Ages 97powerful explanatory boost, a thesis that Roger Bacon would also defend
passionately.^63 If one was interested in causal explanations, there was
nothing inferior about the “subalternate” mathematical sciences.
There were many schemas of subalternation.^64 Grosseteste’s analysis of
Aristotelian subalternation effectively split up the old quadrivium between
mathematics (arithmetic and geometry) and the subalternate sciences of
astronomy and musical theory wedged between mathematics and natu-
ral philosophy. Aquinas (d. 1274) called this category the “intermediate
sciences” (scientiae mediae), those that “apply mathematical principles to
natural things.”^65
Were these sciences more mathematical or physical? There was no
clear consensus. In one passage about astronomy, Bacon cast his lot with
the naturales mathematici (physical mathematicians) against the puri math-
ematici (pure mathematicians) who neglected the physical. While some
loosely lumped the intermediate sciences with natural philosophy, others
called them “intermediate mathematics” (mathematica media).^66
THE EXPANSION OF THE “INTERMEDIATE SCIENCES”
Following the translations, older classifi cation schemes were cracking. The
Roman quadrivium had been oblivious to optics, which was already sophis-
ticated after Euclid and Ptolemy and became vastly more so after Ibn al-
Haytham. A composite thirteenth- century list of “intermediate sciences”
would include newer disciplines like the “science of machines” (scientia de
ingeniis) or the “science of weights” (scientia de ponderibus).^67 Aquinas had
specifi cally excluded the motion of corruptible bodies from mathematical
treatment (and the intermediate sciences) while conceding exceptional sta-
tus to astronomy, whose incorruptible entities move but are treated math-
ematically.^68 This was not the fi nal word, however. Recent successes in the
classic intermediate sciences were hard to ignore. Theodoric of Freiberg’s
mathematical explanation of refractions and refl ection in raindrops beau-
tifully illustrated the power of geometry to explain the rainbow, which
had been a natural philosophical problem for Aristotle. Successes such as
these stimulated some fourteenth- century natural philosophers to theo-
rize about problems of motion in mathematical terms. The approaches
of these calculatores associated with early- fourteenth- century Merton Col-
lege, Oxford, spread to mid- fourteenth- century Paris, then to the late-
fourteenth- century German universities, and to fi fteenth- century Italy.^69
The two most famous of these approaches deserve mention. The “lati-
tude of forms” quantifi ed changes in qualities (for example, degrees of