One area of what would become physics that saw serious developments during the European Middle Ages was motion, both fall and projectile motion. I’ve dealt with various aspects of these developments in earlier blog posts but I’m going to bring the strands together in one episode here.
The first thing to note is that the theories of motion of both Aristotle and John Philoponus re-entered European intellectual intercourse during the Scientific Renaissance in the twelfth century. This was the beginning of the Aristotelian philosophy that became the core of the syllabus on the scholastic medieval university following its elucidation by Albertus Magnus (c. 1200–1280) and his student Thomas Aquinas (c. 1225–1274). However, as I am very fond of repeating, Edward Grant points out that medieval Aristotelian philosophy is not Aristotle’s philosophy. The medieval university scholars changed and developed Aristotle’s ideas and this is very true of his concepts of fall.
Perhaps the most significant development in the study of fall was made by the so-called Oxford Calculatores, a group of scholars at Merton College Oxford, in the first half of the fourteenth century. However, the Calculatores didn’t start from scratch but were influenced by a couple of earlier medieval scholars, who we will briefly look at before turning our attention to Merton College.
Gerard of Brussels (French: Gérard de Bruxelles, Latin: Gerardus Bruxellensis) is a somewhat obscure figure from the early thirteenth century, who is only really known through his book Liber de motu (On Motion), which was composed sometime between 1187 and 1260. In this book, he, to paraphrase the title of a Marshall Clagett essay on his work, reduces curvilinear velocities to uniform rectilinear velocities.
The Liner de motu contains thirteen propositions, in three books. In these propositions the varying curvilinear velocities of the points and parts of geometrical figures in rotation are reduced to uniform rectilinear velocities of translation. The four propositions of the first book relate to lines in rotation, the five of the second to areas in rotation, and the four of the third to solids in rotation, Gerard’s proofs are particularly noteworthy for their ingenious use of an Archimedean-type reductio demonstration, in which the comparison of figures is accomplished by the comparison of their line elements. (Marshall Clagett, DSB)
The second influence comes from Walter Burley (or Burleigh c. 1275–1344/45), who had been a fellow of Merton College until about 1310, following which he spent time in Paris on the Sorbonne before spending seventeen years as a clerical courtier in England then in Avignon. Burley wrote extensive commentaries on the works of Aristotle but perhaps most importantly he was one of the first medieval scholars to recognise the priority of propositional logic over the term or syllogistic logic of Aristotle.
The principal figures of the Calculatores were Thomas Bradwardine (c. 1300–1349), William of Heytesbury (c. 1313–1372/73), Richard Swineshead,(fl. c. 1340–1354) and John of Dumbleton (d. c. 1349).
Thomas Bradwardine devoted much of his life’s work to theology but it is his work in mathematics and physics that interests us here. He wrote two introductory works on mathematics Geometria speculativa (Speculative Geometry) and Arithmetica speculative (Speculative Arithmetic), which are elementary text books on the subjects.
His application of logical and mathematical analysis to Aristotle’s theories of motion, however, were totally groundbreaking. The emphasis of Bradwardine’s work is on kinetics– The branch of mechanics concerned withmotion of objects, as well as the reason i.e. the forces acting on such bodies (Wiktionary). In his Tractatus de proportionibus velocitatum in motibus, which handles the ratio of speeds in motion, he derived the mean speed theorem– a uniformly accelerated body (starting from rest, i.e. zero initial velocity) travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body (Wikipedia)–which is the key element in the laws of fall and is often falsely attributed to Galileo.
In his Tractatus de proportionibus (1328) Bradwardine also produced important advances in Eudoxus’ theory of proportion anticipating the concept of exponential growth. He also criticised Aristotle’s concepts of the relationship between powers and speeds:
In Book VII of Physics, Aristotle had treated in general the relation between powers, moved bodies, distance, and time, but his suggestions there were sufficiently ambiguous to give rise to considerable discussion and disagreement among his medieval commentators. The most successful theory, as well as the most mathematically sophisticated, was proposed by Thomas Bradwardine in his Treatise on the Ratios of Speeds in Motions. In this tour de force of medieval natural philosophy, Bradwardine devised a single simple rule to govern the relationship between moving and resisting powers and speeds that was both a brilliant application of mathematics to motion and also a tolerable interpretation of Aristotle’s text. (Wikipedia, attributed to Lindberg and Shank without a reference but I think is a paraphrase from Laird, see footnote 1)
In his work Bradwardine referenced Gerard of Brussels Liber de motu and his use of philosophy and logic is obviously influenced by the writings of Walter Burley.
William of Heytesbury, the second member of the group was at heart a logician, whose main work was Regulae solvendi sophismata (Rules for Solving Sophisms). Part IV of this work is devoted to problems concerning the three species of change: local motion or change of place, augmentation or change of size, and alteration or change of quality. In the section on local motion, he distinguished uniform motion, which he defined as motion in which equal distances are transversed at equal speeds from nonuniform motion or difform motion.[1]
On local motion he writes:
For whether it commences from zero degree or from some [finite] degree, every latitude [of velocity], provided that it is terminated at some finite degree, and is acquired or lost uniformly, will correspond to its mean degree. Thus, the moving body, acquiring or losing this latitude uniformly during some given period of time, will traverse a distance exactly equal to what it would traverse in an equal period of time if it were moved continuously at its mean degree. For of every such latitude commencing from rest and terminating at some [finite] degree [of velocity], the mean degree is one-half the terminal degree of that same latitude (Regule [Venice, 1494], fol. 39). (Curtis A. Wilson, DSB)
The third member of the group Richard Swineshead is regarded as the best mathematician of the group and therefore was known as The Calculator. His magnum opus was a series of treatises known as the Liber calculationum (Book of Calculations) written around 1350. His reputation as a mathematician carried down to the Renaissance and the Early Modern Period, with Julius Caesar Scaliger (1484–1558), Gerolamo Cardano (1501–1576) and Gottlieb Leibniz (1646–1716) all expressing admiration for it.
The emphasis in the Liber calculationum is on logicomathematical techniques rather than on physical theory. What it provides are techniques for calculating the values of physical variables and their changes, or for solving problems or sophisms about physical changes. (John E. Murdoch & Edith Dudley Sylla, DSB)
The fourth member John of Dumbleton, who wrote a sort of general encyclopaedia, Summa logicae et philosophiae naturalis, which amongst other things includes the work of the other three.
His proof of the “Merton mean speed theorem” is interesting and in some ways reminiscent of the geometric method of exhaustion. He also considered how mathematical techniques could be applied to motions other than local motion. (A. G. Molland, DSB)
What we see in general by the Oxford Calculatores is the systematic application of logical and, above all, mathematical analysis to problems that Aristotle, who basically rejected mathematics in his physics, only dealt with philosophically. This approach is something that is widely and falsely believed first took place in the seventeenth century during the so-called scientific revolution. One thing that they did not do, however, is to back up their purely theoretical application of mathematical analysis with any form of experimentation.
The work of the Merton scholars did not take place in a bubble but quickly spread throughout the European academic community and it found people who took it up and propagated in at the university in Paris amongst a group who have become known as the Paris Physicists.
The first Parisian scholar we will look at is Nicole Oresme (1325–1382), a polymath who wrote on a very wide range of topics, became Bishop of Lisieux and a counsellor of King Charles V (1338–1380). Oresme was the thinker who introduced the concept of comparing the celestial motions to a mechanical clock, a concept that would go on to have a long and varied history all the way down to the nineteenth century. Otherwise, his cosmology remained fairly traditional.
On the mathematical side he wrote a treatise on the theory of proportions, Proportiones propotionum (The Ratio of Ratios), the starting point for which was Bradwardine’s Tractatus de proportionibus (1328). It is essentially a treatment of fractional exponents conceived as ratios of ratios.
In this treatment Oresme made a new and apparently original distinction between irrational ratios of which the fractional exponents are rational, for example, and those of which the exponents are themselves irrational. In making this distinction Oresme introduced new significations for the terms pars, partes, commensurabilis, and incommensurabilis. Thus pars was used to stand for the exponential part that one ratio is of another.
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It should also be noted that Oresme composed an independent tract, the Algorism of Ratios, in which he elucidated in an original way the rules for manipulating ratios. (Marshall Clagett, DSB)
Oresme also tackled the problem of falling bodies:
In discussing the motion of individual objects on the surface of the earth, Oresme seems to suggest (against the prevailing opinion) that the speed of the fall of bodies is directly proportional to the time of fall, rather than to the distance of fall, implying as he does that the acceleration of falling bodies is of the type in which equal increments of velocity are acquired in equal periods of time. He did not, however, apply the Merton rule of the measure of uniform acceleration of velocity by its mean speed, discovered at Oxford in the 1330’s, to the problem of free fall, as did Galileo almost three hundred years later. Oresme knew the Merton theorem, to be sure, and in fact gave the first geometric proof of it in another work, but as applied to uniform acceleration in the abstract rather than directly to the natural acceleration of falling bodies.
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The mention of Oresme’s geometrical proof of the Merton mean speed theorem brings us to a work of unusual scope and inventiveness, the Tractatus de configurationibus qualitatum et motuum composed in the 1350’s while Oresme was at the College of Navarre. This work applies two-dimensional figures to hypothetical uniform and nonuniform distributions of the intensity of qualities in a subject and to equally hypothetical uniform and nonuniform velocities in time. (Marshall Clagett, DSB)
Oresme’s use of two-dimensional geometrical figures to represent change comes very close to the concept of using a rectangular coordinate system to plot change. For what we would call the vertical axis he used the term latitudo and for the horizontal axis longitudo, terms used in his time primarily for astronomical coordinate systems.
He shows that his method of figuring the latitude of forms is applicable to the movement of a point, on condition that the time is taken as longitude and the speed as latitude; quantity is, then, the space covered in a given time. In virtue of this transposition, the theorem of the latitudo uniformiter difformis became the law of the space traversed in case of uniformly varied motion; thus Oresme published what was taught over two centuries prior to Galileo’s making it famous. (Wikipedia, paraphrasing Pierre Duhem & Marshall Clagett)
Our second fourteenth century Parisian is Oresme’s contemporary Jean Buridan (c. 1301–c. 1359/62). Buridan is a rare example of a scholar, who spent his whole life teaching at the university but remained in the arts faculty, never moving up into one of the three higher faculties.
With Buridan we move away from the Oxford Calculatores and Aristotle’s concepts of fall to his concepts of projectile motion. Buridan rejected Aristotle’s thoughts on the topic and accepted instead the theory of John Philoponus (c. 490– c. 570) via the Islamic scholar Abu’l-Barakāt Hibat Allah ibn Malkā al-Baghdādī (c. 1080–1164/5). In this theory the mover imparts something to the projectile and it is this that keeps it in motion. Abu’l-Barakāt called this mayl qasri (a violent inclination), Buridan gave it the name still used today, impetus.
…after leaving the arm of the thrower, the projectile would be moved by an impetus given to it by the thrower and would continue to be moved as long as the impetus remained stronger than the resistance, and would be of infinite duration were it not diminished and corrupted by a contrary force resisting it or by something inclining it to a contrary motion (Questions on Aristotle’s Metaphysics XII.9: 73ra).
It should be noted that whereas Philoponus thought that which was imparted somehow got used up in flight, Buridan thought, as noted above, that it got slowed down and eventually stopped by the air resistance and gravity. This was an important development in the theory in the direction inertia. Falsely Buridan thought that rotational motion at uniform angular velocity as due to a rotational impetus analogous to the rectilinear impetus Out of this, Buridan thought that it was impetus which was the cause of celestial motion. With no wind resistance or gravity in the celestial sphere impetus would be perpetual.
A pupil of Buridan’s was the Italian, Dominicus de Clavasio (fl. mid-fourteenth century).
He taught arts at Paris during 1349–1350, was head of the Collège de Constantinople at Paris in 1349, and was an M.A. by 1350. Dominicus received the M.D. by 1356 and was on the medical faculty at Paris during 1356–1357. He was astrologer at the court of John II and may have died between 1357 and 1362.
Dominicus is the author of a Practica geometriae written in 1346; a questio on the Sphere of Sacrobosco; a Questiones super perspectivam; a set of questiones on the first two books of the De caelo of Aristotle, written before 1357; and possibly a commentary on Aristotle’s Meteorology. He mentions in the Practica his intention to write a Tractatus de umbris et radiis. (Claudia Kren, DSB)
In his questiones on the first two books of the De caelo he adopted the impetus theory as an explanation of projectile motion as well as of acceleration in free fall.
“When something moves a stone by violence, in addition to imposing on it an actual force, it impresses in it a certain impetus. In the same way gravity not only gives motion itself to a moving body, but also gives it a motive power and an impetus, … .” (Dominicus, De caelo)
His work was clearly influenced by both Oresme and Buridan.
The Polish priest, theologian and philosopher, Joannes Cantius (Polish: Jan z Kęt or Jan Kanty English: John Cantius, 1390–1473) in his philosophical works also propagated Buridan’s impetus theory.
The most important propagator of Buridan’s impetus theory was his student Albert von Rickmersdorf (Latin: Albertus de Saxonia; English: Albert of Saxony c. 1316–1390). Following studies in Prague and Paris, Albert taught as a professor in Paris from 1351 to 1362. In 1353 he was rector of the Sorbonne. From 1363 he was at the court of Pope Urban V in Avignon, who appointed him first rector of the University of Vienna in 1365. In 1366 he was appointed Bishop of Halberstadt, where he remained until his death.
Albert was influenced by both Oresme and Buridan and he followed Buridan in adopting and propagating the theory of impetus. It appears that there was, at least on paper and interchange on the topic between the teacher and his student. In his Quaestiones super libros Physicorum, Albert appears to have referenced an earlier version of Buridan’s work with the same title, as in a later version of the same work Buridan references Albert’s work. One result is that Albert doesn’t use the term impetus, which Buridan only introduced in the later version of his work, but rather a virtus motiva or virtus impressa, an impressed power.
In his own contribution to the theory of impetus Albert added a third stage to the two stage theory of John Philoponus.
- Initial stage. Motion is in a straight line in direction of impetus which is dominant while gravity is insignificant
- Intermediate stage. Path begins to deviate downwards from straight line as part of a great circle as air resistance slows projectile and gravity recovers.
- Last stage. Gravity alone draws projectile downwards vertically as all impetus is spent.
Both Nicole Oresme and Jean Buridan were significant figures in the world of medieval philosophy and their modifications and development of the Aristotelian theories of motion were influential down to the beginnings of the Early Modern Period
[1] Walter Roy Laird, Change and Motion, in The Cambridge History of Science, Volume 2, Medieval Science, CUP, ppb 2015, p. 429
The Renaissance Mathematicus 
“[Burley]… was one of the first medieval scholars to recognise the priority of propositional logic over the term or syllogistic logic of Aristotle.”
I am curious as to the evidence of this. I’m not a Burley expert, but I’ve read a decent amount of his, both in Latin and in translation. He certainly worked on the theory of consequences and other forms that we would consider propositional logic, but so did Ockham and Buridan and many others. Burley, as far as I am aware, was very much a man of his time in his treatment of logical matters. But I would love to be shown otherwise.