Before the publication of Newton’s Principia Mathematica Philosophiae Naturalis (1689), Aristotle’s Physics was the most widely read and influential book of natural philosophy (طبیعیات).
After 1250, it constituted the core text of the discipline of natural philosophy and was, together with Aristotle’s other “natural books,” routinely studied at all European universities. Change and motion were the central topics in Aristotle’s “natural books,” and, as a consequence, came to be pivotal in medieval natural philosophy.
For Aristotle, and the medievals in his wake, motion was not merely a starting point from everyday experience, but a phenomenon whose nature needed closer investigation. This article will pay attention to medieval studies of the nature of motion, but also to medieval dynamics and kinematics (gravity, accelerated free fall, projectile motion, and qualitative changes, such as heating). The medieval discussions raised interesting ontological, semantic, and mathematical issues. The focus will be on developments at the universities of Oxford and Paris.
Introduction
Nowadays, philosophy and science are two distinct domains, separated by dividing lines that were nonexistent in the past. We often associate philosophy with the examination of unanswerable questions, whereas the core activity of science seems to be the collecting of empirical data and the creation of explanatory mathematical models.
In antiquity and in the Middle Ages, however, philosophy and science were one discipline with one shared history. Medieval natural philosophy did not originate out of nothing; it did not stand on its own shoulders but on those of ancient philosophy. The famous British philosopher Bernard Williams once remarked that “the legacy of Greece to western philosophy is western philosophy.”
Before the publication of Newton’s Principia mathematica philosophiae naturalis (1689), Aristotle’s Physics was the most widely read and influential book of natural philosophy. How could Aristotle become such an important authority? The answer to this question is linked to the site of philosophy during the Middle Ages.
After 1200, philosophical and scientific culture primarily flourished at the universities, instead of, for instance, in laboratories or learned societies, cathedral schools, monasteries, or courts. At all universities in medieval Europe, Aristotle’s works came to be compulsory reading for all students. The works of Plato or of other Greek thinkers were hardly known. This historical circumstance shaped medieval (natural) philosophy in a unique way.
In medieval universities, the curriculum of the Faculty of Arts included the seven liberal arts (grammar, rhetoric, logic, arithmetic, geometry, music, and astronomy) and what were called the three philosophies: natural philosophy, moral philosophy (or ethics), and first philosophy (or metaphysics). Aside from astronomy, then, most of the disciplines that were later called “science” fell under the heading of “natural philosophy” (philosophia naturalis). Other terms that designate the domain of natural philosophy are natural science (scientia naturalis), physics (physica), and, in the post-medieval period, physiology (physiologia).
In addition, medieval scholars were familiar with the term “science” (scientia). It was applied, however, in different ways than we are used to today. In the Middle Ages, the term “science” referred to the mental condition of possessing certain knowledge of something. It was a type of knowledge that was produced by a logical demonstration. In this respect, “science” distinguishes itself from “opinion” or “view” in that the latter are not based on any particular method and are not universally valid and certain.
In a derivative sense, “science” can also refer to a discipline in the Middle Ages, a discipline with its own domain of inquiry, principles, and methodology. In this sense, “science” is a collection of propositions about a particular topic, organized in a coherent set of arguments and proofs.
Thus, “physics” is a science, but so are metaphysics, medicine, and theology. Overall, scientia, the medieval term for science, was not exclusively reserved for physics, but was applied to any field that contained certain and valid statements.
The basic content of philosophy was constituted in medieval universities by the works of Aristotle, known as “the Philosopher,” often accompanied by the commentaries of Averroes, known as “the Commentator.” To fulfill their requirements in natural philosophy, selections were made of Aristotle’s works in natural philosophy. Natural philosophers of the Middle Ages and the Renaissance believed that Aristotle had raised and discussed crucial questions about the natural world and had explained important definitions, general principles, and concepts of physical science.
In addition to the Physics, Aristotle had provided this explanation in other “natural books” (libri naturales), such as On the Heavens (De caelo), On the Soul (De anima), On Generation and Corruption (De generatione et corruptione), “Meteorology” (Meteorologica), and The Short Natural Treatises (Parva naturalia). These books were arranged around the Physics as treatises that discussed particular aspects of natural objects.
In sum, the medieval and Renaissance scholars found their natural philosophy in the books of Aristotle rather than in the Book of Nature, written in the language of mathematics, as Galileo expressed two centuries later in a very powerful metaphor.
Aristotle as well as the medieval and Renaissance scholars who followed in his footsteps, were utterly convinced of the intelligibility of nature. Nature possesses an order that is accessible to the human mind, an order that initially discloses itself in the way we speak of nature. Investigation of the natural world begins with phenomena (phainomena) and it moves from “that what is better known to us” to “that what is more knowable by nature,” that is, the “objective” principles and causes that are concealed and intrinsic to the phenomena (Aristotle, Physica, 184a17–22 and Metaphysica, 1029b3–12).
Aristotle speaks about “phenomena” in a much broader sense than we are used to. For Aristotle, “empirical data” not only refer to the observable facts themselves, but also apply to common opinion or to scholarly opinion about these facts. It is Aristotle’s traditional method to investigate how, for instance, one speaks of a specific topic. He reflects on commonly held views, signals all kinds of problems, and subsequently offers us a more profound analysis of a “phenomenon.”
One of the essential research themes in Aristotle’s “natural books” is the phenomenon of change. What is most remarkable about the natural world is that it is susceptible to all kinds of change. Natural objects originate and perish; they are altered; they move (they change location); and they grow. From this perspective, natural philosophy includes the study of celestial bodies, of meteorological phenomena, and of important concepts that are fundamentally intertwined with change, such as “place,” “space,” and “time” and related notions such as “continuity” and “infinity”. But it also includes the study of material objects, which have the characteristics of living beings, such as human beings and animals. For the first time in the history of western thought, a systematic attempt was made to give a conceptual analysis of change and motion, including their temporal and spatial aspects.
From Athens to Western Europe
How did medieval natural philosophers become aware of the theories of a Greek thinker from the fourth century BCE? Did they read the papyrus scrolls, which Aristotle had scribbled down in Greek 1600 years earlier? The ideas and views of Greek philosophers, strolling and chatting in the local marketplace or at a gymnasium, traveled a long way before they had their impact at the universities of Oxford, Paris, or elsewhere in Europe.
Roger Bacon (1214/1220–1292) once claimed that proficiency in languages “offers the first door to wisdom, which certainly applies to the Latins, who only possess philosophical and theological texts written in a foreign language” (Roger Bacon, Opus tertium, Cap. XXVIII).
Bacon had a point: Latin wisdom had mainly been imported from Greece. If one wonders how western philosophy had become the legacy of Greece to western philosophy, then the answer is clear: through translation. From the end of the twelfth century onward, a basically alien philosophical and scientific culture was reintroduced into the Latin West that meanwhile had become Christian.
In the time span before the twelfth century, Latin natural philosophy was based on a very small number of sources. Due to political developments within the Roman Empire, the intellectual ties between the Latin West and the Greek East were gradually severed. The days when members of the intellectual elite in Rome were able to speak and write in Greek had definitely passed.
Western scholars read Aristotle in Latin, even long after 1470, when the first Greek editions of Aristotle’s works had appeared in the West. The Latin translations continued to be published for reasons of user convenience. From the Middle Ages until far into the seventeenth century, Latin simply was the language in which scholars conversed and wrote, no matter their geographical origin. Galileo, Descartes, Spinoza, and Newton, for instance, consulted Latin translations of Aristotle. The position of Latin as the universal language for scholars explains why Aristotle and other Greek thinkers were translated into Latin, rather than into Europe’s different vernacular languages.
In certain respects, the transmission of Greek science and philosophy into Latin is comparable to an infectious disease. Both pass from one community to another through contact. Whenever an “outbreak” is diagnosed, we ask ourselves “Where did it first originate?” Can all outbreaks be traced back to one primary source, or have there been several independent starting points?
How did the first Latin translations of Aristotle’s works originate? The question might seem obvious, but the answer is rather complex, at least when one wants to delve deeper than the mere enumeration of data, names, places, and titles of works. The translators came from all parts of Europe. Some worked individually, while others were part of a systematic translation movement.
The activities of the Latin translators were part of a process that can best be characterized as the appropriation and assimilation of Greek knowledge in the Latin West. The choice of phrasing implies that much more was at stake than the mere continuation and reception of Aristotle’s works.
The Greek thinkers had not come to the West as uninvited guests; they were not thrust upon western culture. On the contrary, translators went to seek out Greek erudition. Extraordinary historical circumstances offered western scholars the unique opportunity to revert to the Greek originals, as well as to their Arabic translations.
Texts written in Arabic played an important role in western philosophy and science. They were an Islamic legacy to the West. The authors came from an area stretching from North-East Africa and the South of Spain to eastern Asia, and wrote in Arabic. The Arabic translations of Aristotle’s works were only a small segment of this Islamic legacy. It further consisted of Arabic translations of other Greek authors, such as Hippocrates, Galen, Euclid, and Ptolemy and, obviously, of independent works of Islamic scientists and philosophers.
Which role did Islamic science play as an intermediary in the transmission of Aristotle’s natural philosophy to the West? The picture often given is that the West came to know Aristotle’s works through translations from Arabic. These translations supposedly had given the impulse toward the western revival of philosophy and science: the light came from the East.
However, around the same time when the Latin translations from Arabic were made, Aristotle was also translated from Greek into Latin. The manuscript evidence demonstrates that, in the West, Aristotle’s works were mainly read in Latin translations that were made from the Greek original, certainly since the first half of the thirteenth century. An exception is De animalibus, one of Aristotle’s treatises on biology, which was read throughout the Middle Ages in a Latin translation made from Arabic.
More important Arabic sources, however, from a western point of view, are the Latin translations of paraphrases and commentaries by al-Fārābī (d. 950–951), Ibn Sīnā/Avicenna (980–1037), and Ibn Rushd/Averroes (1126–1198). They belonged to the philosophical movement known by the Arabic loanword falsafa (philosophia in Greek). Their works would come to play an important role in the Latin reflection on the writings of Aristotle.
The origins of the translations and the details about scholars who traveled wide distances in search for texts and who spent most of their lives translating them, are told elsewhere in this Encyclopedia (i.e. Springer Encyclopedia of Medieval Philosophy, 2020). Important translators of Aristotle’s “natural books” from Greek are James of Venice, Burgund of Pisa, and William of Moerbeke. They helped Aristotle to become the companion of any late-medieval (natural) philosopher in the West.
The Universities of Paris and Oxford
After 1250, every teacher in the arts-faculty at any university in Europe was required to teach using Aristotle’s natural books. The teaching resulted in commentaries on Aristotle’s works, which contained contemporary discussions about natural philosophy provoked by Aristotle’s text. For this reason, the commentary literature on Aristotle is a rich source of information about medieval theories.
In addition, issues in natural philosophy were discussed in the context of theological works, notably commentaries on the Sentences, and in separate treatises. The favorite format, however, was the commentary on Aristotle’s natural books. Literally hundreds of commentaries were written on Aristotle’s natural books during the Middle Ages and the Renaissance.
For reasons of convenience and historiographical tradition, this survey will devote special attention to the universities of Oxford and Paris. They were the sites of the two most prominent schools in natural philosophy, that is, the network of scholars linked to John Buridan (d. 1361) at Paris and the group linked to Thomas Bradwardine (d. 1349) at Oxford.
In recent research, it has been argued that there was no such thing as a Buridan school. More accurately, John Buridan and his alleged pupils Albert of Saxony (d. 1390) and Nicholas Oresme (fl. 1345–1360) were contemporary scholars at Paris, engaged in the discussions of their time. Even so, they remain the key figures in Parisian natural philosophy and its aftermath.
Oxford’s main protagonists were Richard Kilvington (d. 1361), Richard Swineshead (fl. 1340–1355), William Heytesbury (d. 1372/1373), and John Dumbleton (fl. 1338–1348). Richard Swineshead (not to be confounded with his contemporaneous namesakes John and Roger) came to be designated by later authors as “the Calculator” (Calculator), whereas the Oxford group in its entirety has been called “Calculators” (calculatores).
Change and Motion
As has been mentioned above, change and motion were the central topics in Aristotle’s “natural books,” and, as a consequence, came to be central in medieval natural philosophy. The Physics was only one of Aristotle’s works on natural philosophy, but from the medieval perspective, it was the most important one. It was understood to provide a characterization of the most general principles and properties of the “things that are by nature” (Aristotle, Physics, 192b9-193a30). It covered the most general truths of natural philosophy, true for all bodies.
Examples of natural things are animals and their parts, plants, and the four basic elements: earth, air, fire, and water. They are natural in a way that other objects are not, such as artifacts and things that are due to chance. But why are plants natural objects, and beds not? According to Aristotle, “things that are by nature” are distinguished from non natural things in virtue of having an inner source of moving and being at rest. In the case of natural objects, as in contrast to those artificially made by man, their specific nature disposes them to certain kinds of behavior, notably to all kinds of natural change. Fire, for instance, naturally heats other bodies. Acorns naturally develop into oak trees. Artifacts lack such an inner source of motion (although they too contain such an inner principle insofar as they are made out of natural things).
A coat, for instance, considered as a coat, does not have an inner impulse to change. As a consequence, “we must therefore see what motion is; for if it were unknown, nature too would be unknown” (Aristotle, Physics, 200 b 10–15).
Precisely this endeavor was undertaken in the Physics. Aristotle’s account of nature and natural objects is couched in the terminology that was primarily reserved for local motion (kinēsis, motus). But how does it relate to change in general? In an influential passage in Book 3 of the Physics, Aristotle had maintained that motion does not constitute a separate category of its own over and above the things that are moving, but is placed in several categories of entities that are capable of change: substance, quantity, quality, and place (Physics, 200b32–201a10).
Thus, “motion,” in this broad Aristotelian sense, includes (1) change of quantity (growth and decline); (2) change of quality (alteration, such as white into nonwhite); (3) locomotion; and (4) substantial change or generation and corruption.
In the first three types of change, the substance remains the same and its properties change, whereas in the latter, the substance itself changes. For this reason, medieval thinkers sometimes classified generation and corruption not as a type of motion (motus), but as a mutatio, that is, an instantaneous change. Motion, by contrast, was gradual and successive.
What Is Motion?
For Aristotle, and the medievals in his wake, motion was not merely a starting point from everyday experience, but a phenomenon whose nature needed closer investigation. The question about the nature of motion not only concerned the adequacy of Aristotle’s definition of motion, the quid nominis so to speak, but also, more interestingly, the question of what motion really is, that is, the quid rei or ontological status of motion.
In response to certain conceptual puzzles that the Eleatic philosophers Parmenides (fl. 480 BCE) and Zeno (fl. 450 BCE) had raised, Aristotle had introduced his form–matter theory of motion. If we may believe Aristotle’s account, Parmenides and Zeno had claimed that none of the things that exist come into being or pass away, or, in other words, that change is only apparent. They had argued that what comes to be must either do so from what already is, in which case it is no veritable coming-to-be, or from nothing at all (ex nihilo). The latter option, however, was considered absurd. On these logical grounds, they denied that change was possible.
Aristotle starts from the common sense assumption that perceived change is real. By his doctrine of form and matter, Aristotle tries to solve the logical impasse created by Parmenides and Zeno. He considers the objects in the world as composites of underlying matter and imposed form. From the perspective of matter, change involves continuation. The underlying substrate does not change. From the perspective of form, however, change involves real change, because it consists of the successive replacement of one form by another.
For example, if a black object turns white, the matter or substrate remains, whereas the form of blackness is replaced by that of whiteness. In similar fashion, if an acorn becomes an oak tree, the change can be described in terms of the continuation of a substrate on which a new form is imposed.
In Aristotle’s view, the replacement of one form by another is not a transition from nonbeing to being. He supposed that any change was a transition from potentiality to actuality. The black, or rather, the not-white, is potentially white. By becoming white, it becomes actually what it was already potentially. Similarly, an acorn is potentially an oak.
In sum, Aristotle’s invocation of matter and form, and of potentiality and actuality solved the logical puzzles raised by the Eleatic philosophers. Change does not involve a passage from nonbeing to being, which both Aristotle and Parmenides considered impossible, but rather a passage from potential being to actual being. As a consequence, matter and form, and potentiality and actuality became the most fundamental explanatory principles of medieval physics. Form is the principle that bears the essential properties of a substance, that is, of any object in reality. It determines what the thing is, that is, what its specific nature is. Matter is the passive recipient of the form.
Medieval attempts to define motion and discuss its ontological status were later ridiculed by Descartes. In the Rules for the Direction of the Mind (Regulae ad directionem ingenii, Rule 12 AT X, 427–27), Descartes pokes fun at the Aristotelian definition of motion. “Who doesn’t know what motion is?,” he asks rhetorically. In The World (Le Monde, AT XI 39), started around the same time, Descartes even claims that he finds the scholastic definition of motion so obscure that he is forced to leave it in “their language,” that is, “motion is the actuality of a thing in potentiality insofar as it is in potentiality” (motus est actus entis in potentia prout in potentia est).
In his works, Aristotle had made contradictory statements concerning the ontological status of motion. In Physics, book 3 (200b32–201a10) he maintained that motion is not something over and above the things in motion. In other words, motion does not constitute a separate category, but belongs to the same category that is gained by motion, that is, the category of Place in the case of local motion. In the Categories (11b1–8), however, Aristotle had claimed that motion fell into the category of Affection (passio).
Averroes tried to reconcile these incompatible statements by pointing out that in the Physics, Aristotle had set forth the more correct view, whereas in the Categories, he had maintained the more common view.
Averroes’ explanation of Aristotle’s view hinges on an analysis of motion from two different perspectives. Motion, if considered from the terminus toward which it tends, only differs from it in its degree of “more or less,” that is, in its degree of perfection. If, however, motion is considered as a process (via) toward perfection or actuality, and, as a consequence, is different from the perfection it attains, it belongs to a category of its own. When seen as a road toward actuality, motion cannot coincide with that actuality.
The same twofold analysis of motion recurs in Averroes’ commentary on Physics V. There it is couched in the terminology of change “according to matter” and “according to form.” According to matter, change and its terminus belong to the same category; according to its form, one must view change as a transmutation that takes place in time and constitutes a category of its own, namely that of Affection (passio).
In the thirteenth and fourteenth centuries, these alternative analyses of motion came to be captured under the formulas forma fluens and fluxus formae, a distinction that medieval authors usually attributed to Albert the Great. According to the forma fluens theory, change is nothing but the forms successively gained by the changeable body.
In the case of local motion, the forma fluens is the place successively attained by the mobile body. In other words, motion is the same as the perfection or form it acquires, but it represents that form in a state of flux. It is important to note that the flowing character of the form is not posited in the form itself, but results from the degree of actualization of the form in the subject. Thus, the view of motion as forma fluens did not contradict the common medieval view that forms are unchangeable.
The fluxus formae theory, on the other hand, maintained that change is not the form acquired but is “the flux” of that form – the flow, the process, or the road toward an actuality or perfection. These distinctions were, at least implicitly, in the background of fourteenth- century discussions of Aristotle’s statement that there is no change over and above real things.
In the fourteenth century, the two main positions in the debate over the nature of motion were clear: some claimed that motion is a flux, which is distinct from the mobile object and the place, whereas others advocated that motion requires nothing more than mobile body and place. However, the debate was complicated by other elements that were woven into the discussion, such as the correct interpretation of Aristotle’s and Averroes’ views and a discussion of whether local motion should be treated in the same way as alteration, that is, change of quality.
On the basis of arguments that invoke God’s omnipotence in rotating the whole cosmos, John Buridan had argued that local motion cannot be treated in the same way as alteration. Albert of Saxony found Buridan’s arguments concerning God’s omnipotence compelling and gave up the forma fluens theory for local motion, which in his view represented the theory genuinely advocated by Aristotle and Averroes.
John Buridan, Nicholas Oresme, and Albert of Saxony describe local motion as fluxus and as “being continuously in another way than before” (aliter et aliter se habere quam prius). But what is fluxus? Buridan and Albert of Saxony agree that the fluxus character of motion should be interpreted as an inherent quality or disposition (dispositio) of the mobile object, as a property of a mobile being, but of such a nature that it is purely successive.
Oresme, on the other hand, rejects the idea that the fluxus is an inherent quality, such as a form. He disqualifies this interpretation of fluxus as “the worst possible view.” How then should the fluxus be understood? Oresme introduces a new ontological entity, the modus rei or way of being, to explain the phenomenon of motion. Motion is nothing but the mode or condition of the mobile object, its condition of traversing spaces in succession. The successiveness of the mobile body, however, should not be taken in the sense of a successive thing (res successiva) that is distinct from it. Thus, Oresme’s interpretation of fluxus almost turns it into a forma fluens theory.
The Causes of Motion
Other significant problem areas to which medieval thinkers addressed themselves are the dynamic and kinematic aspects of motion, that is, motion’s relations to its causes and to distance and time, respectively. In medieval terminology, these aspects concerned the study of motion “with respect to cause” (penes causam) and “with respect to effect” (penes effectum). In the former case, some consideration was given also to the forces acting on bodies to produce motions.
Phenomena which fourteenth-century thinkers at Paris and Oxford discussed under these headings, and to which they often took a semantic and quantitative approach, were gravity, accelerated free fall, projectile motion, and also qualitative changes in a given subject, such as heating. This section discusses the approach of motion with respect to its causes, which roughly corresponds to the dynamic study of motion (as in contrast to kinematics).
Aristotle distinguished two kinds of local motion: natural and nonnatural or violent. Natural motion is the motion toward the natural place of the mobile body, which is determined by the proportion of the four elements – water, earth, air, and fire – in it. Bodies in which the elements earth or water predominate are heavy and consequently move downward, whereas bodies in which the elements air and fire predominate are light and move upward. Violent motions are motions in any direction deviating from the moving body’s natural place, for instance upward for bodies mainly consisting of the element earth.
Aristotle explained the local motion of inanimate bodies, that is of bodies that do not move themselves, by invoking the principle that motion is never spontaneous but that everything that is in motion “is moved by something” (Physics, 241 b 34; 259 a 29–31). In medieval terminology, this proposition was rendered as “everything that is moved is moved by another” (omne quod movetur ab alio movetur). Moreover, Aristotle had argued that there is no action at a spatial distance: the cause of motion and the thing moved need to be in contact (Physics, 243 a 3–4).
Both propositions seemed to require that mover and moved were separate entities. When medieval commentators turned to these two basic Aristotelian principles of the explanation of motion, they concentrated their discussions on two problem areas that seemed to present counterexamples to Aristotle’s axioms: one in the field of natural motion, and one in the field of violent motion.
Natural Motion: The Explanation of Gravity and Accelerated Free Fall
In violent motion, it was not difficult to see that an external motive force was involved; but how were mover and moved body to be distinguished in natural motion? Some medieval scholars followed Thomas Aquinas’ lead and concluded that the generator (generans) gave the mobile body its form and everything that followed from that form, including its motive powers and its disposition to move, that is, its heaviness (gravitas) or lightness (levitas).
In this way, the generator acted as a kind of remote motive cause in natural motion. A body’s natural motion was conceived of as the consequent of its original generation. In case a body was impeded to move after it was generated, the cause of its natural motion was identified with whatever removed the impediment to its motion (removens impedimentum). The natural motion, for example, of a stone hanging on a rope was caused by the agent that cuts the rope.
However, the generator and the remover of the impediment could not explain the continuation of a body’s natural motion, since they were not in continuous simultaneous contact with the moving body. How did fourteenth-century authors solve this problem?
A decisive viewpoint was taken by John Duns Scotus (c. 1265–1308). He concluded that heavy and light bodies are moved by their heaviness and lightness, and in this sense, they move themselves. This was an important deviation from Aristotle, Averroes, and earlier commentators. They had emphasized that the generator was the real cause of the fall of a body, and that the substantial form, heaviness, and lightness were, at best, lower-order instrumental causes.
After Duns Scotus, however, authors such as John of Jandun, William Ockham, Walter Burley, John Buridan, Marsilius of Inghen, and Albert of Saxony embraced the position that heaviness and lightness were the proximate causes of a body’s unobstructed natural motion. In this way, they gave up Aristotle’s principle that mover and moved are separate entities. The mover was therefore no longer an extrinsic cause (such as the generator had been), but was now conceived of as an intrinsic principle, inhering in the moving body.
As a consequence, the important Aristotelian distinction between the motion of animate and inanimate bodies became blurred. None of these fourteenth-century authors, however, stressed the new turn they had taken, but, on the contrary, tried to justify Aristotle’s views by “glossing” them.
Another question raised in the context of falling bodies was acceleration: a body in natural fall moves with accelerated velocity, not with uniform speed. Discussions about “heaviness” and “lightness” ignored the phenomenon of acceleration. The causal explanation of acceleration took its point of departure from the rules concerning motion, which Aristotle had provided in Book 7 of the Physics (249 b 27–250 a 9) and book 1 of De caelo (273 b 30–275 a 20). In these books, he had discussed the notions of “quicker,” “slower,” and “of equal speed” and formulated the relations that obtain between velocities and the forces and resistances that determine them. Aristotle’s passages are often presented as his “mechanics” and certainly helped to develop late-medieval mechanics.
With respect to natural motion, in which weight acts as the motion’s internal force, and the density of the medium as the resistance, Aristotle states a number of rules. In sum, they provide two options to account for the acceleration of a falling body: it was caused by either an increase in force or a decrease in resistance. Most late medieval thinkers, however, eliminated the second option.
The most influential theory was proposed by John Buridan. He maintained that the body’s weight not only caused its downward fall, but also generated an impressed force in the falling body, which he called impetus. Since the body’s weight does not expire, but continues to act as a cause, it continues to produce new increments of impetus, which successively increase the falling body’s velocity. This explanation was basically followed by other thinkers in Paris, such as Nicholas Oresme, Albert of Saxony, and Marsilius of Inghen.
Violent Motion: The Explanation of Projectile Motion
Buridan also uses his impetus theory to solve another problem in Aristotelian physics, namely that of the explanation of projectile motion. Projectile motion seemed to contradict Aristotle’s principles about movers and mobile bodies: how could the mobile body continue its motion after it had lost contact with the moving force, say, the hand throwing a stone?
Aristotle had claimed that it was the surrounding medium, which accounted for the continuation of the projectile’s motion. Simultaneously with the projectile, the medium too received a motive force from the projector, which it continued to transmit by means of the activation of the medium, thus pushing the projectile along. In this way, Aristotle could uphold his principle that the external force is in continuous contact with the mobile body: the projector’s role had been taken over by the medium.
This account came to be rejected in the fourteenth century. The medium’s role in continuing the projectile’s motion was considered problematic. Francis of Marchia (fl. 1320) pointed out that if the original projector could produce some motive force in the medium, there was no reason why it could not do so also in the projectile itself. Indeed, this “left-behind force” (virtus derelicta) or impressed force (vis impressa) given to the projectile was the real proximate cause of the projectile’s continued motion.
This solution had been anticipated by the Greek commentator John Philoponus (d. after 575), who had also found it implausible that the medium should serve as a motive force rather than as resistance to the projectile’s motion. His work, however, was not available in translation, and neither were the several writings of Islamic thinkers who had developed theories of impressed force (mail) in projectile motion.
John Buridan introduced the technical term “impetus” for this impressed force and developed an advanced theory about it. In Buridan’s view, impetus was a motive force transmitted from the initial mover that could act in any direction to which it had been. Buridan took a first step toward the quantification of impetus by declaring that its strength was determined by the velocity with which the initial mover had moved the projectile.
Furthermore, he considered its strength proportionate to the projectile’s weight, which in turn depended on its quantity of matter (quantitas materiae). Buridan substantiated this rule by reference to the phenomenon that a leaden projectile can be thrown further than a wooden one of the same volume and shape, because it has a greater capacity to receive impetus.
Buridan applied his impetus theory not only to the explanation of the acceleration of falling bodies, but also to that of celestial motions. He declared that at the moment of creation, God might have imprinted an impetus on the heavens. Since in the case of the heavenly motions, the impetus is not producing a violent motion away from their natural place, and therefore encounters no resistance, it will continue forever. In the sublunar region, however, the impetus is corrupted by the mobile body’s tendency to move to its natural place, and by resistances (from the medium, for instance) acting on the body. In sum, Buridan conceived of impetus as a quasi-permanent quality inherent in the mobile body, which, under terrestrial conditions, interacted with the body’s natural tendencies, and dissipated as a consequence.
Albert of Saxony adhered to the same position, even though he avoided the terminology of impetus and spoke instead of “moving force” (virtus motiva) or “moving quality” (qualitas motiva). Still other slight variations were introduced by Nicholas Oresme and Marsilius of Inghen. New medieval concepts such as impetus and weight (gravitas) helped abolish Aristotle’s principle that mover and moved are separate entities that must be in contact. Instead, they represent an internalization of Aristotle’s external motive force. The medieval impetus theory was to remain the dominant explanation of projectile motion until the seventeenth century, although it had little impact in Oxford.
The Quantification of the Causes of Motion
At Oxford, scholars took a more quantitative approach toward the causal explanation of motion. The quantitative rules for force and resistance by which Aristotle had tried to capture the interaction between mover and moved (Physics, book 7), came to be criticized at the beginning of the fourteenth century. In his Treatise on Proportions (Tractatus de proportionibus), written in 1328, Thomas Bradwardine gave a general treatment of how one can relate a change of speed of a mobile to a variation in its causes, or as he put it, “of the proportion between the speeds with which motions take place with respect to both moving and resisting powers.”
Bradwardine’s mathematical approach had a tremendous influence. At Oxford, Richard Swineshead further developed Bradwardine’s insights by applying them to two new problems. In his Book of Calculations (Liber calculationum), written about 1350, Swineshead specifies, among other things, Bradwardine’s theorem for different kinds of changes in velocity, such as uniform, difform, uniformly difform, and so on. Moreover, he applies Bradwardine’s function to the fall of a heavy body near the center of the universe.
Bradwardine’s theorem of motion was incorporated into the Parisian commentaries on the Physics. Starting with John Buridan, other fourteenth-century scholars at the University of Paris began to show familiarity with Bradwardine’s views. Moreover, Albert of Saxony and Nicholas Oresme elaborated Bradwardine’s function in their Treatise on Proportions (Tractatus proportionum) and Treatise on Ratio of Ratios (Tractatus de proportionibus proportionum), respectively. They abstracted from the physical problems that Bradwardine had discussed and, instead, concentrated on a calculus of proportions as such.
Logic and Geometry in Natural Philosophy
The quantification of motion not only occurred in medieval dynamics, but also in kinematics, which relates motion to time and space traversed. This type of approach should be seen in the light of the introduction of logic and geometry in natural philosophy.
Scholars at Paris had a predilection for providing semantic analyses of the terms in which their physical problems were formulated. Terms such as “motion,” “nature,” “change,” “alteration,” “point,” “space,” “time,” and “instant,” for example, were submitted to an analysis that employed all the logical techniques available.
Especially the theory of supposition was fundamental. It was a tool that analyzed a term’s reference within the context of a proposition and in this way determined the meaning and truth of that proposition. The supposition theory provided, for instance, different semantic analyses of the propositions “man is a species” and “man is a three-letter word.”
Another much-used semantic tool was the analysis of a term’s position within a proposition, the “word-order,” so to speak. This aspect was expressed in the technical vocabulary of distinguishing between the categorematic and the syncategorematic use of a term. Since these semantic aspects significantly affected the truth-value of the propositions in which the physical problems were stated, they had a bearing on the solution of these problems.
At Oxford, the application of logic in kinematics was blended with mathematical techniques. As a matter of fact, one of the distinctive features of most of the Calculators’ treatises is that they originated out of a logical, disputational context. This is especially true for the Sophismata, collections of counterintuitive statements called “sophisms” that served as examples to illustrate semantic theories. Often, sophismata have a purely logical character, but especially at Oxford a new genre originated, that of the mathematical–physical sophisms.
This emphasis on logico-mathematical techniques, rather than directly on physical theory, is present in the treatises of Heytesbury and Swineshead and also in the Sophismata of Richard Kilvington. A typical example of the type of problems discussed there is the truth of sophism 34, “Plato can move uniformly during some time and as fast as Socrates now moves”.
The quantification of kinematic aspects of motion was introduced in the context of the so-called doctrine of the latitude of forms (latitudo formarum), a theory that was developed to deal with the different degrees that may be assigned to one and the same quality. For instance, one banana can be more yellow than another one, or one object hotter than another one.
The idea that qualities in a subject can exist in varying degrees was first expressed in Aristotle’s Categories (10 b 26 and following). It gave rise to two problem areas that were only very loosely connected. The first one, called the problem of the intension and remission of forms in medieval terminology, developed further the idea that qualities can become more or less, that is, that they can undergo intensification and remission (intensio et remissio). It focused on ontological issues such as the search for the subject of the strengthening or weakening: was it the quality itself, or the subject’s participation in the quality? And how was a quality intensified or weakened?
The two most prominent alternatives that had emerged by the fourteenth century were the succession theory and the addition theory. Discussions of the intension and remission of forms tacitly assumed that a quality was uniformly distributed in a subject and uniformly changed over time. However, precisely this assumption was further investigated in the second problem area, namely that of the latitude of forms (latitudo formarum).
This problem area studied the phenomenon of qualitative changes in a subject from the perspective of space and time. The point of departure was the idea that qualities can be non-uniformly, or difformly (difformiter) according to medieval terminology, distributed over a given subject and that they can change difformly.
The scholars at Oxford focused on the problem of measuring these qualities, especially the uniformly difform qualities, that is, that class of qualities that strengthened or weakened at a constant rate from one end of the subject to the other. By analogy, they addressed questions of measures of other types of change, especially of local motion. It was in this context of measuring qualities and motions that the concept of latitude came to play a pivotal role. It is important to note that these discussions often were hypothetical.
The philosophical analysis of the degrees of qualities helped to develop the idea of velocity as a magnitude to which can be attributed a numerical value and by which motions can be measured. The idea had been quite alien to Aristotle, who had conceived of velocity as an unquantifiable concept.
One theorem that developed out of the latitude of forms context has been characterized by at least one historian of science as “probably the most outstanding single medieval contribution to the history of mathematical physics”: the mean speed theorem (Grant 1996, p. 101). This theorem measures uniformly accelerated motions with respect to the spaces traversed by comparing them with uniform motions.
The mean speed theorem states that a mobile moving with a uniformly accelerated motion covers the same space in a given time as it would if it moved for the same time with a uniform speed equal to the speed at the middle instant of the duration of its acceleration.
The mean speed theorem has attracted much scholarly attention, because it was mentioned by Galilei in his Discourses on Two New Sciences (Day 3, theorem 1) and applied to the free fall of bodies, which is an example in nature of a uniformly accelerated (i.e., uniformly difform) motion.
The Calculators provided many proofs of the theorem, but none was as easy to visualize as Oresme’s geometrical proof, which was also employed by Galilei. Oresme noted that a right triangle is equal in area to a rectangle whose height is the mean height of the triangle. In this way he graphically compared the quantity of a uniformly difform quality to that of a uniform quality of mean intensity.
Other Parisians besides Oresme were also well acquainted with the Oxford’s Calculators’ discussions of the measure of the effects of motion, as is clear from their discussions in commentaries on the Physics. As Edith Sylla has convincingly argued, the aim of the logical and geometrical discussion of kinematics was to instruct students to think and argue clearly and exactly. Newton’s Principia mathematica philosophiae naturalis still carries the echos of medieval natural philosophy in its title, although the emphasis had now definitely shifted toward the mathematical approach.
Source: Encyclopedia of Medieval Philosophy, editor: Henrik Lagerlund, second edition pp. 1271 -1283, Springer, 2020.
