98 Shank
heat, speed, whiteness, and so forth) and analyzed them as a function of
time. A second approach, “the proportions of speeds in motions” associ-
ated with Thomas Bradwardine, criticized Aristotle’s treatment of speeds
in relation to forces and resistances and used proportion theory to develop
original solutions to this and related problems. Both brought mathemati-
cal tools to bear on natural philosophical problems.^70 In their own lists of
“intermediate sciences,” Henry of Langenstein (d. 1397) included the “lat-
itude of forms” and Regiomontanus (d. 1476) “the proportions of speeds
in motions.” At the interface between mathematics and natural philoso-
phy, the intermediate sciences were clearly a growing category.^71 These are
but two of the new technical “analytical languages”^72 that characterized
the era. Scholars created new logical and logico- mathematical techniques
that they put to use in many facets of natural philosophy and even theol-
ogy. They transformed the character of natural philosophy and theology
so deeply that one historian sees in just one of them “a new intellectual
factor... a typical change of mentalities.”^73
Not everyone appreciated these innovations or their diffusion. The
sixteenth- century Aristotelian natural philosopher Pietro Pomponazzi,
for example, criticized the Liber calculationum of Richard Swineshead, the
Oxford calculator, for its excess of mathematics in natural philosophy,
and disapproved of this new hybrid.^74 Galileo’s analyses of free fall and
projectile motion would eventually show just how pregnant such hybrids
could become.FIRST PRINCIPLES OF NATURAL KNOWLEDGE
Grosseteste’s analysis of the intermediate sciences had presupposed that
natural knowledge emerges, at least in part, from a deductive framework.
Any conclusion must properly be able to trace its roots to reliable fi rst
principles. Those who worried about the ultimate source of principles nec-
essary to ground all the sciences faced several options.
For many, especially thinkers in the Augustinian tradition, Christian
theology seemed the best repository of the highest principles. As fi rst
cause of everything and guarantor of all knowledge, God would surely
ground fi rst principles securely. But others insisted on distinguishing
Christian theology from a metaphysics or “theology” of fi rst principles
that, like Aristotle’s, was accessible to all rational thinkers, regardless of
creed. This distinction was effectively built into the structure of the uni-
versities. Masters of arts who were not studying theology could not treat
(revealed) theology, but they routinely commented on the “theology” in