University of Virginia Library

II

The more valuable scientific contributions of the
thirteenth century were in most instances those of
isolated individuals, who reformulated the science of
antiquity and made new beginnings in both experi-
mentation and mathematical analysis. The fourteenth
century saw a fuller development along these same
lines, culminating in important schools at both Oxford
and Paris whose members are commonly regarded as
the “precursors of Galileo.”

1. Theory and Experiment. These precursors worked
primarily in the area of mechanics, concentrating on
logical and mathematical analyses that led to somewhat
abstract formulations, only much later put to experi-
mental test. They never reached the stage of active
interchange between theory and experiment that char-
acterizes twentieth-century science, and that could
only be begun in earnest with the mechanical investi-
gations of Galileo and Newton. In another area of
study, however, a beginning was made even in this type
of methodology; the area, predictably enough, was
optics, which from antiquity had been emerging, along
with mechanics, as an independent branch of physics.

The reasons for the privileged position enjoyed by
optics in the late thirteenth and early fourteenth cen-
turies are many. One was the eminence it earlier had
come to enjoy among the Greeks and the Arabs. An-
other was its easy assimilation within the theological
context of “Let there be light” (Fiat lux) and the philo-
sophical context of the “metaphysics of light” already
alluded to. Yet other reasons can be traced in the
striking appearances of spectra, rainbows, halos, and
other optical phenomena in the upper atmosphere, in
the perplexity aroused by optical delusions or by an
awareness of their possibility, and above all in the
applicability of a simple geometry toward the solution
of optical problems.

Whatever the reasons, the fact is that considerable
progress had already been made in both catoptrics, the
study of reflected light, and dioptrics, the study of
refraction. In the former, the works of Euclid, Ptolemy,
and Alhazen (d. 1038) had shown that the angles of
incidence and reflection from plane surfaces are equal;
they also explained how images are formed in plane
mirrors and, in the case of Alhazen, gave exhaustive
and accurate analyses of reflection from spherical and
parabolic mirrors. Similarly in dioptrics Ptolemy and
Alhazen had measured angles of incidence and refrac-
tion, and knew in a qualitative way the difference
between refraction away from, and refraction toward,
the normal, depending on the media through which
the light ray passed. Grosseteste even attempted a
quantitative description of the phenomenon, proposing
that the angle of refraction equals half the angle of
incidence, which is, of course, erroneous. In this way,
however, the stage was gradually set for more substan-
tial advances in optics by Witelo and Dietrich von
Freiberg. Perhaps the most remarkable was Dietrich's
work on the rainbow (De iride), composed shortly after
1304, wherein he explained the production of the bow
through the refraction and reflection of light rays.

Dietrich's treatise is lengthy and shows considerable
expertise in both experimentation and theory, as well
as the ability to relate the two. On the experimental
side Dietrich passed light rays through a wide variety
of prisms and crystalline spheres to study the produc-
tion of spectra. He traced their paths through flasks
filled with water, using opaque surfaces to block out
unwanted rays, and obtained knowledge of angles of
refraction at the various surfaces on which the rays
in which he was interested were incident, as well as
the mechanics of their internal reflection within the
flask. Using such techniques he worked out the first
essentially correct explanation of the formation of the
primary and secondary rainbows (Figures 1 and 2). The
theoretical insight that lay behind this work, and that
had escaped all of his predecessors, was that a globe
of water could be thought of—not as a diminutive

watery cloud, as others viewed it—but as a magnified
raindrop. This, plus the recognition that the bow is
actually the cumulative effect of radiation from many
drops, provided the principles basic to his solution.
Dietrich's experimental genius enabled him to utilize
these principles in a striking way: the first to im-
mobilize the raindrop, in magnified form, in what
would later be called a “laboratory” situation, he was
able to examine leisurely and at length the various
components involved in the rainbow's production.

Dietrich proposed the foregoing methodology as an
application of Aristotle's Posterior Analytics wherein
he identified the causes of the bow and demonstrated
its properties using a process of resolution and com-
position. In attempting to explain the origin and order-
ing of the bow's colors, however, he engaged in a far
more hypothetical type of reasoning, and coupled this
with experiments designed to verify and falsify his
alternative hypotheses. This work, while closer meth-
odologically to that of modern science, was not suc-
cessful. There were errors too in his geometry, and in
some of his measurements; these were corrected in
succeeding centuries, mainly by Descartes and Newton.
Dietrich's contribution, withal, was truly monumental,
and represents the best interplay between theory and
experiment known in the high Middle Ages.

2. Nominalism and Its Influence. Most historians are
agreed that some break with Aristotle was necessary
before the transition could be made from natural phi-
losophy to science in the classical sense. One step
toward such a break came with the condemnation, in
1277, by Étienne Tempier, Bishop of Paris, of 219
articles many of which were linked to an Aristotelian-
Averroist cosmology. Concerned over God's omni-
potence, the bishop effectively proclaimed that several
worlds could exist, and that the ensemble of celestial
spheres could, without contradiction, be moved (by
God) in a straight line. The general effect of his con-
demnation was to cause many who were uncritically
accepting Aristotle's conclusions as demonstrated and
necessarily true to question these. The way was thus
opened for the proposal and defense of non-Aristotelian
theses concerning the cosmos and local motion, some
with important scientific ramifications.

Another step came with the rise of nominalism or
terminism in the universities. Under the auspices of
William of Ockham and his school, this movement
developed in an Aristotelian thought context but
quickly led to distinctive views in logic and natural
philosophy. Its theory of supposition questioned the
reality of universals or “common natures,” generally
admitted by Aristotelians, and restricted the ascription
of reality to individual “absolute things” (res absolutae),
which could be only particular substances or qualities.
Quantity, in Ockham's system, became merely an ab
stract noun: it cannot exist by itself; it can increase
or decrease without affecting the substance, as is seen
in the phenomena of rarefaction and condensation; and
by God's absolute power it can even be made to disap-
pear entirely, as is known from the mystery of the
Eucharist. Thus, with Ockham, quantity became a
problem more of language than of physical science;
his followers soon were involved in all manner of
linguistic analyses relating to quantity, but not infre-
quently the physical problems involved got lost in a
maze of logical subtleties.

Ockham's treatment of motion went along similar
lines. Convinced that the term “local motion” desig-
nates only the state of a physical body that may be
negatively described as not at rest, he effectively de-
nied the reality of motion. Moreover, since motion is
not a real effect, it does not require a cause, and
hence the Aristotelian rule “whatever moves is moved
by another” (quidquid movetur ab alio movetur) is no
longer applicable to it. Some have seen in this rejection
of motor causality a foreshadowing of the law of inertia
or even the principle of relativity (Sir Edmund Whit-
taker, E. J. Dijksterhuis). Undoubtedly there are some
affinities between Ockham's analysis and those of classi-
cal and modern mechanicians, but the identification
need not be pressed. Ockham's more direct contri-
bution would seem to lie in his preparing the way for
sophisticated, if highly imaginative, calculations of
spatiotemporal relationships between motions with
various velocities. These calculations opened the path
to considerable advances in kinematics, soon to be
made at Merton College in Oxford.

Nominalism quickly spread from Oxford to the
universities on the Continent, where it merged its
thought patterns with both “orthodox” and “hetero-
dox” (from the viewpoint of the Christian faith) schools
of Aristotelianism. From this amalgam came a renewed
interest in the problems of physical science, a consid-
erably revised conceptual structure for their solution,
and a growing tolerance of skepticism and eclecticism.
Most of the fruits were borne in mechanics and astron-
omy, but some were seen in new solutions to the prob-
lems of the continuum and of infinity. Nicholas of
Autrecourt is worthy of mention for his advocacy of
atomism—at a time when Democritus' thought was
otherwise consistently rejected—and for his holding a
particulate theory of light. His skepticism generally has
led him to be styled as a “medieval Hume” and as
a forerunner of positivism.

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

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 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

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).

4. Paris and the Growth of Dynamics. As in the
thirteenth century an interest in science with emphasis
on the mathematical began at Oxford, to be followed
by a similar interest with emphasis on the physical at
Paris, so in the fourteenth century an analogous pattern
appeared. The works of the English Calculatores were
read and understood on the Continent shortly after the
mid-fourteenth century by such thinkers as John of
Holland at the University of Prague and Albert of
Saxony at the University of Paris. Under less pro-
nounced nominalist influence than the Mertonians, and
generally convinced of the reality of motion, the Con-
tinental philosophers again took up the problems of
the causes and effects of local motion. Particularly at
Paris, in a setting where both Aristotelian and terminist
views were tolerated, “calculatory” techniques were
applied to natural and violent motions and new ad-
vances were made in both terrestrial and celestial
dynamics.

The first concept of significance to emerge from this
was that of impetus, which has been seen by historians
of medieval science, such as Duhem, as a forerunner
of the modern concept of inertia. The idea of impetus
was not completely new on the fourteenth-century
scene; the term had been used in biblical and Roman
literature in the general sense of a thrust toward some
goal, and John Philoponus, a Greek commentator on
Aristotle, had written in the sixth century of an “in-
corporeal kinetic force” impressed on a projectile as
the cause of its motion. Again Arabs such as Avicenna
and Abū'l-Barakāt had used equivalent Arabic termi-
nology to express the same idea, and thirteenth-century
Scholastics took note of impetus as a possible explana-
tion (which they rejected) of violent motion. What was
new about the fourteenth-century development was the
technical significance given to the concept in contexts
that more closely approximate later discussions of
inertial and gravitational motion.

The first to speak of impetus in such a context seems
to have been the Italian Scotist Franciscus de Marchia.
While discussing the causality of the Sacraments in a
commentary on the Sentences (1323), Franciscus em-
ployed impetus to explain how both projectiles and
the Sacraments produced effects through a certain
power resident within them; in the former case, the
projector leaves a force in the projectile that is the
principal continuer of its motion, although it also
leaves a force in the medium that helps the motion
along. The principal mover is the “force left behind”
(virtus derelicta) in the projectile—not a permanent
quality, but something temporary (“for a time”), like

heat induced in a body by fire, and this even apart
from external retarding influences. The nature of the
movement is determined by the virtus: in one case it
can maintain an upward motion, in another a sideways
motion, and in yet another a circular motion. The last
case allowed Franciscus to explain the motion of the
celestial spheres in terms of an impetus impressed in
them by their “intelligences”—an important innova-
tion in that it bridged the Peripatetic gap between the
earthly and the heavenly, and prepared for a mechanics
that could embrace both terrestrial and celestial phe-
nomena.

A more systematic elaborator of the impetus concept
was John Buridan, rector of the University of Paris and
founder of a school there that soon rivaled in impor-
tance the school of Bradwardine at Oxford. Buridan,
perhaps independently of Franciscus de Marchia, saw
the necessity of some type of motive force within the
projectile; he regarded it as a permanent quality, how-
ever, and gave it a rudimentary quantification in terms
of the primary matter of the projectile and the velocity
imparted to it. Although he offered no formal discus-
sion of its mathematical properties, Buridan thought
that the impetus would vary directly as the velocity
imparted and as the quantity of matter put in motion;
in this respect, at least, his concept was similar to
Galileo's impeto and to Newton's “quantity of motion.”
The permanence of the impetus, in Buridan's view, was
such that it was really distinct from the motion pro-
duced and would last indefinitely (ad infinitum) if not
diminished by contrary influences. Buridan also ex-
plained the movement of the heavens by the imposition
of impetus on them by God at the time of the world's
creation. Again, and in this he was anticipated by
Abū'l-Barakāt, Buridan used his impetus concept to
explain the acceleration of falling bodies: continued
acceleration results because the gravity of the body
impresses more and more impetus.

Despite some similarities between impetus and in-
ertia, critical historians such as A. Maier have warned
against too facile an identification. Buridan's concept,
for example, was proposed as a further development
of Aristotle's theory of motion, wherein the distinction
between natural and violent (compulsory) still ob-
tained. A much greater conceptual revolution was
required before this distinction would be abandoned
and the principle of inertia, in its classical under-
standing, would become accepted among physicists.

Buridan's students, Albert of Saxony and Marsilius
of Inghen, popularized his theory and continued to
speak of impetus as an “accidental and extrinsic force,”
thereby preserving the Aristotelian notions of nature
and violence. Albert is important for his statements
regarding the free fall of bodies, wherein he speculates
that the velocity of fall could increase in direct pro-
portion to the distance of fall or to the time of fall,
without seemingly recognizing that the alternatives are
mutually exclusive. (This confusion was to continue in
later authors such as Leonardo da Vinci and the young
Galileo.) Albert himself seems to have favored distance
as the independent variable, and thus cannot be re-
garded as a precursor of the correct “law of falling
bodies.”

Perhaps the most original thinker of the Paris school
was Nicole Oresme. Examples of his novel approach
are his explanation of the motion of the heavens using
the metaphor of a mechanical clock, and his specula-
tions concerning the possible existence of a plurality
of worlds. An ardent opponent of astrology, he devel-
oped Bradwardine's doctrine on ratios to include irra-
tional fractional exponents relating pairs of whole-
number ratios, and proceeded to argue that the ratio
of any two unknown celestial ratios is probably irra-
tional. This probability, in his view, rendered all astro-
logical prediction fallacious in principle. Oresme held
that impetus is not permanent, but is self-expending
in its very production of motion; he apparently associ-
ated impetus with acceleration, moreover, and not with
sustaining a uniform velocity. In discussing falling
bodies, he seems to suggest that the speed of fall is
directly proportional to the time (and not the distance)
of fall, but he did not apply the Mertonian mean-speed
theorem to this case, although he knew the theorem
and in fact gave the first geometrical proof for it.
Further he conceived the imaginary situation of the
earth's being pierced all the way through; a falling
body would then acquire an impétuosité that would
carry it beyond the center, and thereafter would oscil-
late in gradually decreasing amplitudes until it came
to rest. A final and extremely important contribution
was Oresme's use of a two-dimensional figure to plot
a distribution of the intensity of a quality in a subject
or of velocity variation with time (Figure 4). Possibly
this method of graphical representation was antici-
pated by the Italian Franciscan Giovanni di Casali, but
Oresme perfected it considerably, and on this account
is commonly regarded as a precursor of Descartes'
analytic geometry.