# University of Virginia Library

3. Merton College and Kinematics. One of the most
significant contributors to the mathematical prepara-
tion for the modern science of mechanics was Thomas
Bradwardine, fellow of Merton College and theologian
of sufficient renown to be mentioned by Chaucer in
his Nun's Priest's Tale. While at Oxford Bradwardine

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composed treatises on speculative arithmetic and ge-
ometry wherein he not only summarized the works of
Boethius and Euclid, but expanded their treatments
of ratios (proportiones) and proportions (propor-
tionalitates
) to include new materials from the Arabs
Thâbit and Ahmad ibn Yusuf. He then applied this
teaching to a problem in dynamics in his Treatise on
the ratios of velocities in motions
(Tractatus de propor-
tionibus velocitatum in motibus
) composed in 1328. By
this time various Arab and Latin writers had been
interpreting Aristotle's statements (mostly in Books 4
and 7 of the Physics) relating to the comparability of
motions to mean that the velocity V of a motion is
directly proportional to the weight or force F causing
it and inversely proportional to the resistance R of the
medium impeding it. This posed a problem when taken
in conjunction with another Aristotelian statement to
the effect that no motion should result when an applied
force F is equal to or less than the resistance R encoun-
tered. In modern notation, V should equal 0 when
F = ⩽ R, and this is clearly not the case if VF/R, since
V becomes finite for all cases except F = 0 and R = ∞

In an ingenious attempt to formulate a mathematical
relationship that would remove this inconsistency,
Bradwardine equivalently proposed an exponential law
of motion that may be written

Referred to as the “ratio of ratios” (proportio propor-
tionum
), Bradwardine's law came to be widely ac-
cepted among Schoolmen up to the sixteenth century.
It never was put to experimental test, although it is
easily shown to be false from Newtonian dynamics. Its
significance lies in its representing, in a moderately
complex function, instantaneous changes rather than
completed changes (as hitherto had been done), thereby
preparing the way for the concepts of the infinitesimal
calculus.

Bradwardine composed also a treatise on the con-
tinuum (Tractatus de continuo) which contains a de-
tailed discussion of geometrical refutations of mathe-
matical atomism. Again, in a theological work he
analyzed the concept of infinity, using a type of one-
to-one correspondence to show that a part of an infinite
set is itself infinite; the context of this analysis is a proof
showing that the world cannot be eternal. In such ways
theology, and stimulated later thinkers to make similar
applications.

Although occasioned by a problem in dynamics,
Bradwardine's treatise on ratios actually resulted in
more substantial contributions to kinematics by other
Oxonians, many of whom were fellows of Merton Col-
lege in the generation after him. Principal among these
were William of Heytesbury, John of Dumbleton, and
Richard Swineshead. All writing towards the middle
of the fourteenth century, they presupposed the valid-
ity of Bradwardine's dynamic function and turned their
attention to a fuller examination of the comparability
of all types of motions, or changes, in its light. They
did this in the context of discussions on the “intension
and remission of forms” or the “latitude of forms,”
conceiving all changes (qualitative as well as quanti-
tative) as traversing a distance or “latitude” which is
readily quantifiable. They generally employed a “let-
ter-calculus” wherein letters of the alphabet repre-
sented ideas (not magnitudes), which lent itself to subtle
logical arguments referred to as “calculatory soph-
isms.” These were later decried by humanists and more
traditional Scholastics, who found the arguments in-
comprehensible, partly, at least, because of their
mathematical complexity.

One problem to which these Mertonians addressed
themselves was how to “denominate” or reckon the
degree of heat of a body whose parts are heated not
uniformly but to varying degrees. Swineshead devoted
a section of his Book of Calculations (Liber calcula-
tionum
) to solve this problem for a body A which has
greater and greater heat, increasing arithmetically by
units to infinity, in its decreasing proportional parts (Figure 3).
He was able to show that A should be
denominated as having the same heat as another body
B which is heated to two degrees throughout its entire
length, thus equivalently demonstrating that the sum
of the series 1 + 1/2 + 1/4 + 1/8... converges to the
wardine's analysis relating to instantaneous velocity
and other concepts necessary for the calculus; signifi-
cantly his work was known to Leibniz, who wished
to have it republished.

Motion was regarded by these thinkers as merely
another quality whose latitude or mean degree could
be calculated. This type of consideration led Heytes-
bury to formulate one of the most important kinemati-
cal rules to come out of the fourteenth century, a rule
that has since come to be known as the Mertonian
“mean-speed theorem.” The theorem states that a
uniformly accelerated motion is equivalent, so far as
the space traversed in a given time is concerned, to
a uniform motion whose velocity is equal throughout
to the instantaneous velocity of the uniformly acceler-
ating body at the middle instant of the period of its
acceleration. The theorem was formulated during the
early 1330's, and at least four attempts to prove it
arithmetically were detailed at Oxford before 1350. As
in the previous case of Bradwardine's function, no

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attempt was made at an experimental proof, nor was
it seen (so far as is known) that the rule could be
applied to the case of falling bodies. The “Calcula-
tores,” as these writers are called, restricted their at-
tention to imaginative cases conceived in abstract
terms: they spoke of magnitudes and moving points,
and various types of resistive media, but usually in a
mathematical way and without reference to nature or
the physical universe. When they discussed falling
bodies, as did Swineshead (fl. 1350) in his chapter “On
the Place of an Element” (De loco elementi), it was
primarily to show that mathematical techniques are
inapplicable to natural motions of this type (Hoskin
and Molland, 1966).

A final development among the Mertonians that is
worthy of mention for its later importance is their
attempts at clarifying the expression “quantity of mat-
ter” (quantitas materiae), which seems to be genetically
related to the Newtonian concept of mass. Swineshead
took up the question of the “latitude” of rarity and
density, and in so doing answered implicitly how one
could go about determining the meaning of “amount
of matter” or “quantity of matter.” His definition of
quantitas materiae, it has been argued, is not signifi
cantly different from Newton's “the measure of the
same arising from its density and magnitude conjointly”
(Weisheipl, 1963).