University of Virginia Library

Search this document 
Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

expand sectionV. 
expand sectionIV. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionV. 
expand sectionV. 
expand sectionII. 
expand sectionIV. 
expand sectionIV. 
expand sectionI. 
expand sectionI. 
expand sectionI. 
expand sectionVI. 
expand sectionV. 
expand sectionV. 
expand sectionVI. 
expand sectionVI. 
expand sectionIII. 
expand sectionI. 
expand sectionVI. 
collapse sectionI. 
expand sectionIII. 
expand sectionVI. 
expand sectionIII. 
expand sectionIV. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionIV. 
expand sectionVII. 
expand sectionV. 
expand sectionI. 
expand sectionIII. 
expand sectionIII. 
expand sectionIII. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionIII. 
expand sectionVI. 
expand sectionIII. 
expand sectionI. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionIV. 
expand sectionIV. 
expand sectionIV. 
expand sectionVI. 
expand sectionIV. 
expand sectionIII. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionV. 
expand sectionVI. 
expand sectionIII. 
expand sectionII. 
expand sectionI. 
expand sectionII. 
expand sectionVII. 
expand sectionI. 
expand sectionI. 
expand sectionIII. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionVII. 
expand sectionV. 
expand sectionV. 
expand sectionV. 


The scientific revolution of the seventeenth century
had its remote antecedents in Greek and early medieval
thought. In the period from the thirteenth to the six-
teenth centuries, this heritage gradually took shape in
a series of methods and ideas that formed the back-
ground for the emergence of modern science. The
methods adumbrated were mainly those of experi-
mentation and mathematical analysis, while the con-
cepts were primarily, though not exclusively, those of
the developing science of mechanics. The history of
their evolution may be divided conveniently on the
basis of centuries: (1) the thirteenth, a period of begin-
nings and reformulation; (2) the fourteenth, a period
of development and culmination; and (3) the fifteenth
and sixteenth, a period of dissemination and transition.
By the onset of the seventeenth century considerable
material was at hand for a new synthesis of methods
and ideas, namely that of classical science.


Experimental science owes its beginnings in Western
Europe to the influx of treatises from the Near East,
by way of translations from Greek and Arabic, which
gradually acquainted the Schoolmen with the entire
Aristotelian corpus and with the computational tech-
niques of antiquity. The new knowledge merged with
an Augustinian tradition prevalent in the universities,
notably at Oxford and at Paris, deriving from the
Church Fathers; this tradition owed much to Platonism
and Neo-Platonism, and already was favorably disposed
toward a mathematical view of reality. The empirical
orientation and systematization of Aristotle were wel-
comed for their value in organizing the natural history
and observational data that had survived the Dark Ages
through the efforts of encyclopedists, while the new
methods of calculation found a ready reception among
those with mathematical interests. The result was the
appearance of works, first at Oxford and then at Paris,
which heralded the beginnings of modern science in
the Middle Ages.

1. Origins at Oxford. Aristotle's science and his
methodology could not be appreciated until his Physics
and Posterior Analytics had been read and understood
in the universities. Among the earliest Latin commen-
tators to make the works of Aristotle thus available
was Robert Grosseteste, who composed the first full-
length exposition of the Posterior Analytics shortly after
1200. This work, plus a briefer commentary on the
Physics and the series of opuscula on such topics as
light and the rainbow, served as the stimulus for other
scientific writings at Oxford. Taken collectively, their
authors formed a school whose philosophical orienta-
tion has been characterized as the “metaphysics of
light,” but which did not preclude their doing pioneer
work in experimental methodology.

The basis for the theory of science that developed
in the Oxford school under Grosseteste's inspiration was
Aristotle's distinction between knowledge of the fact
(quia) and knowledge of the reason for the fact (propter
). In attempting to make the passage from the one
to the other type of knowledge, these writers, implic-
itly at least, touched on three methodological tech-
niques that have come to typify modern science,
namely inductive, experimental, and mathematical.

Grosseteste, for example, treated induction as a dis-
covery of causes from the study of effects, which are
presented to the senses as particular physical facts. The
inductive process became, for him, one of resolving
the composite objects of sense perception into their
principles, or elements, or causes—essentially an ab-
stractive process. A scientific explanation would result
from this when one could recompose the abstracted
factors to show their causal connection with the ob-
served facts. The complete process was referred to as
“resolution and composition,” a methodological expres-
sion that was to be employed in schools such as Padua
until the time of Galileo.

Grosseteste further was aware that one might not
be able to follow such an orderly procedure and then
would have to resort to intuition or conjecture to
provide a scientific explanation. This gave rise to the
problem of how to discern a true from a false theory.
It was in this context that the Oxford school worked
out primitive experiments, particularly in optics, de-
signed to falsify theories. They also employed observa-
tional procedures for verification and falsification when
treating of comets and heavenly phenomena that could
not be subjected to human control.

The mathematical component of this school's meth-
odology was inspired by its metaphysics of light. Con-
vinced that light (lux) was the first form that came to
primary matter at creation, and that the entire struc-
ture of the universe resulted from the propagation of
luminous species according to geometrical laws, they
sought propter quid explanations for physical phe-
nomena in mathematics, and mainly in classical geom-
etry. Thus they focused interest on mathematics as well
as on experimentation, although they themselves con-
tributed little to the development of new methods of

2. Science on the Continent. The mathematicist
orientation of the Oxford school foreshadowed in some
ways the Neo-Pythagoreanism and rationalism of the
seventeenth century. This aspect of their thought was
generally rejected, however, by their contemporaries


at the University of Paris, especially Albertus Magnus
and Thomas Aquinas. Both of the latter likewise com-
posed lengthy commentaries on the Posterior Analytics
and on the physical works of Aristotle, primarily to
put the Stagirite's thought at the service of Christian
theology, but also to aid their students in uncovering
nature's secrets. Not convinced of an underlying math-
ematical structure of reality, they placed more stress
on the empirical component of their scientific method-
ology than on the mathematical.

Albertus Magnus is particularly noteworthy for his
skill at observation and systematic classification. He
was an assiduous student of nature, intent on ascertain-
ing the facts, and not infrequently certifying observa-
tions with his Fui et vidi experiri (“I was there and
saw it for myself”). He recognized the difficulty of
accurate observation and experimentation, and urged
repetition under a variety of conditions to ensure ac-
curacy. He was painfully aware of and remonstrated
against the common failing of the Schoolmen, i.e., their
uncritical reliance on authority, including that of Aris-
totle. Among his own contributions were experiments
on the thermal effects of sunlight, which A. C. Crombie
has noted employed the method of agreement and
difference later to be formulated by J. S. Mill; the
classification of some hundred minerals, with notes on
the properties of each; a detailed comparative study
of plants, with digressions that show a remarkable sense
of morphology and ecology; and studies in embryology
and reproduction, which show that he experimented
with insects and the lower animals (Crombie, 1953).
Albert also had theoretical and mathematical interests,
stimulating later thinkers such as William of Ockham
and Walter Burley with his analysis of motion, and
doing much to advance the Ptolemaic conception of
the structure of the universe over the more orthodox
Aristotelian views of his contemporaries.

The best experimental contribution of this period,
however, was that of Peter Peregrinus of Maricourt,
whose Epistola de magnete (1269) reveals a sound
empirical knowledge of magnetic phenomena. Peter
explained how to differentiate the magnet's north pole
from its south, stated the rule for the attraction and
repulsion of poles, knew the fundamentals of magnetic
induction, and discussed the possibility of breaking
magnets into smaller pieces that would become mag-
nets in turn. He understood the workings of the mag-
netic compass, viewing magnetism as a cosmic force
somewhat as Kepler was later to do. His work seems
to be the basis for Roger Bacon's extolling the experi-
mental method, and it was praised by William Gilbert
(1540-1603) as “a pretty erudite book considering the

3. Use of Calculation. Mathematical analysis was
not entirely lacking from scientific investigation in the
thirteenth century. One unexpected source came at the
end of the century in the work of Arnald of Villanova,
who combined alchemical pursuits with those of phar-
macy and medicine. Arnald was interested in quanti-
fying the qualitative effects of compound medicines,
and refined and clarified a proposal of the Arabian
philosopher Alkindi (ninth century) that linked a geo-
metric increase in the number of parts of a quality
to an arithmetic increase in its sensed effect. The ex-
ponential function this implies has been seen by some
as a precursor of the function later used by Thomas
Bradwardine (d. 1349) in his dynamic analysis of local
motion (McVaugh, 1967).

A more noteworthy mathematical contribution was
found, however, in earlier work on mechanics, partic-
ularly in statics and kinematics, that definitely came
to fruition in the fourteenth century. Jordanus Nemo-
rarius and his school took up and developed (though
not from original sources) the mechanical teachings of
antiquity, exemplified by Aristotle's justification of the
lever principle, by Archimedes' axiomatic treatment
of the lever and the center of gravity, and by Hero's
study of simple machines. They formulated the concept
of “positional gravity” (gravitas secundum situm), with
its implied component forces, and used a principle
analogous to that of virtual displacements or of virtual
work to prove the law of the lever. Gerard of Brussels
was similarly heir to the kinematics of antiquity. In
his De motu he attempted to reduce various possible
curvilinear velocities of lines, surfaces, and solids to
the uniform rectilinear velocity of a moving point. In
the process he anticipated the “mean-speed theorem”
later used by the Mertonians, successfully equating the
varying rotational motion of a circle's radius with a
uniform translational motion of its midpoint.

Other conceptual work in the study of motive pow-
ers and resistances, made in the context of Aristotle's
rules for the comparison of motions, laid the ground-
work for the gradual substitution of the notion of force
(as exemplified by vis insita and vis impressa) for that
of cause, thereby preparing for later more sophisticated
analyses of gravitational and projectile motion.


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-
) 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-
), 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

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

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-
) 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

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-

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

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.


The fourteenth century marked the high point in
optical experimentation and in the conceptual devel-
opment of mechanics during the late Middle Ages. The
fifteenth and sixteenth centuries served mainly as pe-
riods of transition, where the underlying ideas were
diffused throughout Europe, entered into combination
with those of other cultures, and provided the proxi-
mate setting for the emergence of classical science.


Much of this interplay took place in Italy, although
France and Spain also figured in it to a limited extent.

1. Italy and Renaissance Influences. The tradition
perhaps most opposed to Scholasticism was that of
humanism, with its interest in classical antiquity, its
emphasis on the arts, and its general preference for
Plato over Aristotle. Writers such as Marsilio Ficino
and Erasmus ridiculed, respectively, the Paduan
Schoolmen and the “calculatory sophisms” of their
Parisian counterparts. Their overriding interest in
philology, moreover, led humanists to make much of
original texts, and, even in the case of Aristotle, to
confer unprecedented force on arguments from the
authority of the classical author. Yet they did make
available, in Greek and in accurate translation, the
mathematical and mechanical treatises of Euclid,
Archimedes, Apollonius, Pappus, Diophantus, and
Ptolemy—works that perforce had a salutary effect in
preparing for the new scientific mentality.

The writings of particular authors also contributed
in different ways to the coming revolution. Nicholas
of Cusa is important for his use of mathematical ideas
in elaborating his metaphysics, which prepared for the
transition, in Koyré's apt expression, “from the closed
world to the infinite universe.” He also placed great
emphasis on measurement, and preserved elements of
the medieval experimental tradition in his treatise on
“Experiments with Scales” (De staticis experi-
)—this despite the fact that most of his experi-
ments are purely fictitious and not one mentions a
numerical result. Leonardo da Vinci is perhaps over-
rated for his contributions to science, since his was
more the mentality of the engineer; his notebooks are
neither systematic nor lucid expositions of physical
concepts. Yet he too supplied an important ingredient,
wrestling as he did with practical problems of me-
chanics with great genius and technical ability. He
brought alive again the tradition of Jordanus Nemo-
rarius and Albert of Saxony, and his speculations on
kinematics and dynamics, if inconclusive, reveal how
difficult and elusive were the conceptual foundations
of mechanics for its early practitioners. Giordano
Bruno may also be mentioned as a supporter and suc-
cessor of Nicholas of Cusa; his works abound in Neo-
Platonism and mysticism, and show a heavy reliance
on Renaissance magic and the Hermetic-Cabalist tra-
dition. Of little importance for mechanics, his ideas
are significant mainly for the support they gave to
Copernicanism and to the concept of an infinite uni-

Of more direct influence, on the other hand, was
work done at the University of Padua under Averroist
and terminist influences. Aristotelianism flourished
there long after it had gone into eclipse at Oxford and
Paris, not so much in subordination to theology as it
was among Thomists, but rather under the patronage
of the Arab Averroës or of Alexander of Aphrodisias,
a Greek commentator on Aristotle. The Averroists
were Neo-Platonic in their interpretation of Aristotle,
whereas the Alexandrists placed emphasis instead on
his original text. Again, at Padua the arts faculty was
complemented not by the theology faculty but by the
inedical faculty; in this more secularized atmosphere
the scientific writings of Aristotle could be studied
closely in relation to medical problems and with much
aid from Arab commentators.

The result was the formation of a new body of ideas
within the Aristotelian framework that fostered, rather
than impeded, the scientific revival soon to be pio-
neered by the Paduan professor, Galileo Galilei.
Among these ideas some were methodological. They


derived from extended discussions of what Galileo
would refer to as the “method of analysis” (metodo
) and the “method of synthesis” (metodo com-
). Writers such as Jacopo Zabarella systema-
tized these results, showing how they could be applied
to detailed problems in physical science, thereby
bringing to perfection the methodology outlined by
Grosseteste, which has already been discussed.

More than a century before Zabarella, Paul of Venice
(Paolo Nicoletti), who had studied at Oxford in the late
fourteenth century, returned to Padua and propagated
Mertonian ideas among his students. A number of these
wrote commentaries on Heytesbury that were pub-
lished and widely disseminated throughout Europe.
Noteworthy is the commentary of Gaetano da Thiene,
who illustrated much of Heytesbury's abstract reason-
ing on uniform and difform motions with examples
drawn from nature and from artifacts that might be
constructed from materials close at hand. As far as is
known this fifteenth-century group performed no ex-
periments or measurements, but they took a step closer
to their realization by showing how “calculatory”
techniques were relevant in physical and medical in-

2. Paris and the Spanish Universities. The Paduan
school exerted considerable influence throughout
northern Italy; it also stimulated a renewed interest
in Mertonian ideas at the University of Paris at the
beginning of the sixteenth century. The group in which
this renewal took place centered around John Major
(or Jean Mair), the Scottish nominalist, who numbered
among his students John Dullaert of Ghent, Alvaro
Thomaz, and Juan de Celaya. Dullaert edited many
of the works of Paul of Venice, while he and the others
were generally familiar with the “calculatory” writings
of Paul's students. Major's group was eclectic in its
philosophy, and saw no inconsistency in making a
fusion of nominalist and realist currents, the former
embracing Oxonian and Parisian terminist thought and
the latter including Thomist and Scotist as well as
Averroist views. The Spaniard Gaspar Lax and the
Portuguese Alvaro Thomaz supplied the mathematical
expertise necessary to understand Bradwardine's,
Swineshead's, and Oresme's more technical writings.
Several good physics texts came out of this group;
especially noteworthy is that of Juan de Celaya, who
inserted lengthy excerpts from the Mertonians and
Paduans, seemingly as organized and systematized by
Thomaz, into his exposition of Aristotle's Physics
(1517). Celaya treated both dynamical and kinematical
questions, as by then had become the custom, and thus
transmitted much of the late medieval development
in mechanics (statics excluded) to sixteenth-century

Celaya was but one of many Spanish professors at
Paris in this period; these attracted large numbers of
Spanish students, who later returned to Spain and were
influential in modeling Spanish universities such as
Alcalá and Salamanca after the University of Paris. An
edition of Swinehead's Liber calculationum was edited
by Juan Martinez Silíceo and published at Salamanca
in 1520; this was followed by a number of texts written
(some poorly) in the “calculatory” tradition. Theolo-
gians who were attempting to build their lectures
around Thomist, Scotist, and nominalist concepts soon
complained over their students' lack of adequate prep-
aration in logic and natural philosophy. It was such
a situation that led Domingo de Soto, a Dominican
theologian and political theorist who had studied under
Celaya at Paris as a layman, to prepare a series of
textbooks for use at the University of Salamanca.
Among these were a commentary and a “questionary”
on Aristotle's Physics; the latter, appearing in its first
complete edition in 1551, was a much simplified and
abridged version of the type of physics text that was
used at Paris in the first decades of the sixteenth cen-
tury. It reflected the same concern for both realist and
“calculatory” interests, but with changes of emphasis
dictated by Soto's pedagogical aims.

One innovation in Soto's work has claimed the at-
tention of historians of science. In furnishing examples
of motions that are “uniformly difform” (i.e., uniformly
accelerated) with respect to time, Soto explicitly men-
tions that freely falling bodies accelerate uniformly as
they fall and that projectiles (presumably thrown up-
ward) undergo a uniform deceleration; thus he saw the
distance in both cases to be a function of the time of
travel. He includes numerical examples that show he
applied the Mertonian “mean-speed theorem” to the
case of free fall, and on this basis, at the present state
of knowledge, he is the first to have adumbrated the
correct law of falling bodies. As far as is known, Soto
performed no measurements, although he did discuss
what later thinkers have called “thought experiments,”
particularly relating to the vacuum. An extensive sur-
vey of all physics books known to be in use in France
and Spain at the time has failed to uncover similar
instance of this type, and one can only speculate as
to the source of Soto's examples.

3. Italy Again: Galileo. With Soto, the conceptual
development of medieval mechanics reached its term.
What was needed was an explicit concern with meas-
urement and experimentation to complement the
mathematical reasoning that had been developed along
“calculatory” and Archimedean lines. This final devel-
opment took place in northern Italy, again mainly at
Padua, while Galileo was teaching there. The stage was
set by works of considerable mathematical sophis-


tication, under the inspiration of Archimedes, by six-
teenth-century authors such as Geronimo Cardano,
Nicolo Tartaglia, and Giovanni Battista Benedetti. Also
the technical arts had gradually been perfected, and
materials were at hand from which instruments and
experimental apparatus could be constructed.

The person of Galileo provided the catalyst and the
genius to coordinate these elements and educe from
them a new kind of synthesis that would reach perfec-
tion with Isaac Newton. Galileo received his early
university training at Pisa around 1584, where his
student notebooks (Juvenilia) reveal an acquaintance
with many Schoolmen, including Soto, an edition of
whose Physics appeared at Venice in 1582. Galileo used
their terminology in an early treatise On Motion (De
), and only gradually departed from it. His teacher
at Pisa, Francesco Buonamici, himself a classical Aris-
totelian, seemingly gave a muddled account of the
medieval tradition, and it is difficult to know how well
Galileo understood what was presented. Actually this
matters little; what is important is that the ideas that
contributed to the developing science of mechanics
were at hand for himself or another to use. Classical
science did not spring perfect and complete, as Athena
from the head of Zeus, from the mind of Galileo or
any of his contemporaries. When it did arrive, it was
a revolution, and no one can deny this, but it was a
revolution preceded by a strenuous effort of thought.
The genesis of that thought makes an absorbing, if little
known, chapter in the history of ideas.


The principal sources and bibliography are given in M.
Clagett, The Science of Mechanics in the Middle Ages (Madi-
son, 1959); also E. A. Moody and M. Clagett, eds., The
Medieval Science of Weights
(Madison, 1952). See too A. C.
Crombie, Robert Grosseteste and the Origins of Experi-
mental Science, 1100-1700
(Oxford, 1953), p. 195; idem,
Medieval and Early Modern Science, 2nd rev. ed., 2 vols.
(Cambridge, Mass., 1961); E. J. Dijksterhuis, The Mecha-
nization of the World Picture,
trans. C. Dikshoorn (Oxford,
1961); J. A. Weisheipl, The Development of Physical Theory
in the Middle Ages
(New York, 1959); P. Duhem, Études
sur Léonard de Vinci,
3 vols. (Paris, 1906-13; reprint 1955),
a pioneer work of great scope, but requires revision in light
of the researches of A. Maier in her Studien zur Naturphi-
losophie der Spätscholastik,
5 vols. (Rome, 1949-58), espe-
cially Vol. I, DieVorläufer Galileis im 14 Jahrhundert (Rome,
1949) and Vol. V, Zwischen Philosophie und Mechanik
(Rome, 1958). See also: P. Duhem le Système du monde,
10 vols. (Paris, 1913-59); idem, To Save the Phenomena, trans
E. Doland and C. Maschler (Chicago, 1969; original in
French, 1908).

Special studies include C. B. Boyer, The Rainbow: From
Myth to Mathematics
(New York, 1959); Jean Buridan,
Quaestiones super libros quattuor de caelo et mundo, ed.
E. A. Moody (Cambridge, Mass., 1942); M. Clagett, Archi-
medes in the Middle ages, Vol. I. The Arabo-Latin Tradition

(Madison, 1964); idem, Nicole Oresme and the Medieval
Geometry of Qualities and Motions
(Madison, 1968); H. L.
Crosby, Jr., ed. and trans., Thomas of Bradwardine, His
Tractatus de Proportionibus,Its Significance for the De-
velopment of Mathematical Physics
(Madison, 1955); S.
Drake and I. E. Drabkin, eds. and trans., Mechanics in
Sixteenth Century Italy
(Madison, 1969); H. Élie, “Quelques
Maîtres de l'université de Paris vers l'an 1500,” Archives
d'histoire doctrinale et littéraire du moyen âge,
18 (1950-51),
193-243; N. W. Gilbert, Renaissance Concepts of Method
(New York, 1960); E. Grant, ed. and trans., Nicole Oresme,
De proportionibus proportionum and ad pauca respicientes
(Madison, 1966); M. A. Hoskin and A. G. Molland, “Swines-
head on Falling Bodies: An Example of Fourteenth-Century
Physics,” The British Journal for the History of Science, 3
(1966), 150-82; A. Koyré, From the Closed World to the
Infinite Universe
(Baltimore, 1957), and idem, Études
3 vols. (Paris, 1939), much of which is sum-
marized in idem, Metaphysics and Measurement: Essays in
the Scientific Revolution
(Cambridge, Mass., 1968); E.
McMullin, ed., Galileo: Man of Science (New York, 1967);
M. McVaugh, “Arnald of Villanova and Bradwardine's
Isis, 58 (1967), 56-64; A. D. Menut and A. J. Denomy,
eds., Eng. trans. by Menut, Nicole Oresme, Le Livre du ciel
et du monde
(Madison, 1968); E. A. Moody, “Galileo and
His Precursors,” Galileo Reappraised, ed. C. L. Golino
(Berkeley, 1966), pp. 23-43; idem, “Galileo and Avempace:
The Dynamics of the Leaning Tower Experiment,” Journal
of the History of Ideas,
12 (1951), 163-93, 375-422. J. E.
Murdoch, Rationes Mathematice: Un Aspect du rapport
des mathématiques et de la philosophie au moyen âge
1962); J. H. Randall, Jr., The School of Padua and the
Emergence of Modern Science
(Padua, 1961); C. B. Schmitt,
“Experimental Evidence for and against a Void: The Six-
teenth-Century Arguments,” Isis, 58 (1967), 352-66; W. A.
Wallace, The Scientific Methodology of Theodoric of Freiberg
(Fribourg, 1959); idem, “The Enigma of Domingo de Soto:
Uniformiter Difformis and Falling Bodies in Late Medieval
Physics,” Isis, 59 (1968), 384-401; idem, “The 'Calculatores'
in Early Sixteenth-Century Physics,” The British Journal for
the History of Science,
4 (1969), 231-32; idem, “Mechanics
from Bradwardine to Galileo,” Journal of the History of
32 (1971), 15-28; J. A. Weisheipl, “The Concept of
Matter in Fourteenth Century Science,” in The Concept of
ed. E. McMullin (Notre Dame, 1963), pp. 319-41;
C. Wilson, William Heytesbury: Medieval Logic and the
Rise of Mathematical Physics
(Madison, 1960).


[See also Abstraction; Alchemy; Astrology; Authority; Cau-
sation; Continuity; Islamic Conception; Neo-Platonism;
Optics; Renaissance Humanism.]