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Dictionary of the History of Ideas | ||

*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

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

tionalitates

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(

the ratios of velocities in motions

*Tractatus de propor-*

tionibus velocitatum in motibus) composed in 1328. By

tionibus velocitatum in motibus

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

*V*∝

*F*/

*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

Bradwardine made use of mathematics in physics and

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

value 2. Swineshead considerably advanced Brad-

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

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).

Dictionary of the History of Ideas | ||