(i.e., atoms) and geometric theorems. Under pressure from the radicals, Aris-
totle was now being creatively used by the conservative camp.
A group centered on Balliol and Merton colleges carried these ingredients
to more refined conclusions: William of Heytesbury (192 in Figure 9.6), Rich-
ard Swineshead (193), Richard Billingham (195), Richard Kilvington (189),
John Dumbleton (196), and others. Many of them were nominalists in the
strict sense, working out technical advances based on Ockham’s logic; they
also generalized Bradwardine’s mathematical functions for uniformly acceler-
ated motion and expanded his physics. More specialized than the generation
of Bradwardine, Burley, and Ockham, they stayed away from the controversies
of theology. Bradwardine in his later years did the opposite, perhaps because
of his visits to Paris with the king, where he would have encountered the
radicalism of Autrecourt’s circle. Bradwardine attacked the Ockhamists as
“Pelagians” for their emphasis on free will, while he himself fell back on realist
doctrines of cause to assert God’s deterministic qualities. Oxford was no closed
school of nominalists but a network in creative tension.^14 The Merton calcu-
lators were friends of “new thinkers” in other realms as well, such as the poet
Chaucer. The most famous product of the Balliol-Merton axis, at the end of
the 1300s as the calculators had faded, was the theologian Wyclif, a Scotist
and extreme realist, theological follower of Bradwardine, and inspiration for
the proto-Protestant heresy of John Hus.
The Balliol-Merton calculators exemplify the concentration on the topics
of the arts curriculum which was the key to the creativity of this period. At
the same time, this was one of the limitations on their influence. Much of
their work was meant for disputations of advanced undergraduates, rather
than for the purpose of making discoveries in natural philosophy (Sylla, in
CHLMP, 1982: 540–563). They did not make theoretical physics independent
of academic logic or philosophy, but emphasized imaginary cases as aids in
preparing for scholastic arguments. If there was no connection to empirical
research, neither was the connection to mathematics taken very far. Burley and
Kilvington rarely calculated; Bradwardine and Heytesbury typically surveyed
mathematical possibilities but gave only a general indication of results. This
work culminated in Swineshead’s Liber Calculationum (ca. 1350), which treats
physics problems through verbal reasoning about relationships among vari-
ables but contains little actual numerical analysis (DSB, 1981: 13:184–213).
(Similarly among the Parisian group, Oresme’s “Cartesian coordinates” were
described but apparently not actually used for mathematical constructions.)
This work was later admired by mathematicians such as Cardano and Leibniz
for its achievements beyond the level of the Greeks; yet the Humanists of the
next century saw in it nothing but involuted reasoning on scholastic exercises
and dismissed it as barbaric. The Oxford calculators could be recognized
Academic Expansion: Medieval Christendom^ •^493