## 1 Introduction

The main purpose of this paper is to show that one of the most famous theories of motion, formulated in 1328 by Thomas Bradwardine as most of the historians of medieval science believe, was not his original achievement. As I have argued in my paper *Works by Richard Kilvington*, it was Richard Kilvington who offered weighty arguments, later used by Bradwardine to reinterpret Aristotle’s theory.^{1} That article was the result of my extensive research on the Oxford Calculators in general and on Richard Kilvington in particular.^{2} I began my study with Kilvington’s questions on motion and at the outset I noticed their dependence, as I thought, on Thomas Bradwardine’s *Tractatus de proportionibus velocitatum in motibus*. Thus, my first step was a careful study of Norman and Barbara Kretzmann’s works, especially a critical edition, English translations, and detailed study of Kilvington’s *Sophismata*.^{3} The Kretzmanns accentuate the obvious need for further study of the historical and philosophical relationships between these two men.^{4} Therefore I transcribed and attentively read all of Kilvington’s works, most of them still available only in manuscripts.^{5} My in-depth studies of all Kilvington’s works confirmed Kilvington’s probable biography and the dates of his works, as suggested by the Kretzmanns.^{6} Still, ongoing historical and doctrinal research of the 1320s brought into light new discoveries and revealed unknown facts which are significant with regard to the main subject of this paper. Thus it seems worth rediscussing the new interpretation of Aristotle’s theory of motion traditionally ascribed to Thomas Bradwardine in the light of those findings.

## 2 Richard Kilvington and Thomas Bradwardine: The Founders of the Calculators’ School

Before discussing Kilvington’s and Bradwardine’s new theory of motion I will briefly outline their relationship as seen through the eyes of the historians of medieval science. Anneliese Maier described Kilvington as not only Bradwardine’s friend but also his student.^{7} Norman Kretzmann and William Courtenay rejected the teacher-student relationship between Bradwardine and Kilvington.^{8}

There is no doubt that the roads of Bradwardine and Kilvington continued to cross. Thomas Bradwardine (ca. 1295–1349) was a Bachelor of Arts at Balliol College, Oxford, in 1321. In 1323 he became a fellow of Merton College, Oxford, where he probably remained for the next twelve years.^{9} In the same year, he became a Master of Arts, in 1340 a Doctor of theology.^{10} Richard Kilvington (ca. 1302–1361) began his study in the Arts at Oxford around 1317. Most likely, he first entered Balliol College, where he met Bradwardine. He studied theology at Oriel College, Oxford.^{11} In 1321, he became a Bachelor of Arts, in 1324 or 1325 a Master of Arts, then a Doctor of Theology in ca. 1335.^{12}

Both Bradwardine and Kilvington belonged to the circle of friends and courtiers of Richard de Bury, Bishop of Durham, who, starting in the year 1334, helped them to advance in an ecclesiastical and political career. They both were members of de Bury household in 1334–1335.^{13} Richard de Bury also introduced them to the royal court of Edward III. Bradwardine and Kilvington were members of a diplomatic mission in 1338–1340.^{14} Their academic career was followed by an ecclesiastical one. In 1333 Bradwardine was made Canon at Lincoln Cathedral, and his career was crowned with his election, in 1349, as Archbishop of Canterbury. As chancellor of St. Paul’s Cathedral in London, he was made royal chaplain in 1337 and, probably, the king’s confessor. He accompanied Edward III on his travels to Flanders and France during the campaign of 1346. Immediately after his episcopal consecration, which was held in Avignon, Bradwardine returned to England to assume his position, yet he died one month later, on the 26th of August 1349, as a victim of the first wave of the Black Death.^{15} Kilvington’s ecclesiastical career culminated in his appointment as Dean of St. Paul’s Cathedral in 1354. Along with Richard FitzRalph, Kilvington was involved in the battle against the mendicant friars, a dispute that continued almost until his death in 1361.^{16}

Thomas Bradwardine authored many significant works, which cover a number of scholarly domains. His insight and intellectual inquisitiveness earned him the title of *Doctor profundus* and a mention in Chaucer’s *Canterbury Tales*. Works by Bradwardine that have been preserved to our time are the following: two treatises in mathematics, *Arithmetica speculativa* and *Geometria speculativa*;^{17} a number of logical treatises (all written before 1328); a famous work on motion, *Tractatus de proportionibus velocitatum in motibus*, written in 1328;^{18} a *Tractatus de continuo*;^{19} two theological works, a commentary on the *Sentences*, which includes a question on future contingents, edited by Jean-François Genest as a separate treatise *De futuris contingentibus*,^{20} and *De causa Dei contra Pelagium et de virtute causarum ad suos Mertonenses*,^{21} composed in 1344. Bradwardine also authored *De memoria artificiali adquirenda*.^{22}

While most of Bradwardine’s philosophical works seem to have been composed as student textbooks or ‘guides’ for his colleagues (Mertonians), all of Kilvington’s known works, except for a few sermons, stem from his teaching at Oxford and often reflect lively discussions in the classroom.^{23} None of his works is written in the usual commentary format, following the order of books in the respective works of Aristotle. Kilvington reduced the number of topics discussed to certain central issues, which were fully developed with no more than ten questions constituting a commentary. The reduction in the range of topics is counterbalanced by a deeper analysis in the questions chosen for treatment.^{24} Some of Kilvington’s questions cover twenty folios, which in a modern edition yield about 150 pages. Only his logical treatise was not written as a commentary, but rather as ‘a guide’ for students showing how to solve sophisms. In the preface to his *Sophismata* Kilvington writes:

When we are able to call both sides into question, we will readily discern what is true and what is false, as Aristotle says in Book One of his

Topics. Therefore, in order that we may more readily discern what is true and what is false, in the present work, which consists ofsophismatato be thoroughly investigated, I intend, to the best of my ability, both to demolish the two sides of the contradiction and also to support them by means of clear reasoning. I am led to do this by the request of certain young men who have been pressing their case very hard. And so, wishing to give them something I have often heard them ask for, I have undertaken to make an attempt in that direction.^{25}

The chronology of Kilvington’s works is well established. The *Sophismata* and *Quaestiones super De generatione et corruptione*, composed before 1325, came from his lectures as a bachelor of arts; the *Quaestiones super Physicam* were composed at the latest in 1326; the *Quaestiones super libros Ethicorum* (before 1332) date from his time as a Master of Arts; after he advanced to the Faculty of Theology, he produced eight questions on Peter Lombard’s *Sentences* (1333 or 1334).^{26} As there have been some recent discoveries, I will briefly outline these findings to reassert the view that the commentary on the *Sentences* was composed in these years. In a recent paper, Chris Schabel convincingly argued that in Oxford, like in Paris, it was common practice to lecture on the *Sentences* in one year only.^{27} William Courtenay suggests Kilvington lectured on the *Sentences* between 1331 and 1335, perhaps around 1333–1334, since he is familiar with Wodeham’s and Bradwardine’s theological opinions.^{28} Bradwardine lectured in the year 1331–1332 or 1332–1333.^{29} Rega Wood is of the opinion that Wodeham lectured at Oxford after the *Norwich Lecture* (*Lectura secunda*), i.e., after 1332.^{30} Paul Streveler and Katherine Tachau provide us with the date 1332–1333 for Wodeham’s first year as a bachelor lecturing on book I of the *Sentences*, and 1333–1334 for his second year lecturing on books 2–4.^{31} According to Schabel, Wodeham’s Oxford lectures took place in 1331–1332.^{32} Thus, the *terminus ante quem* for Kilvington’s *Sentences* commentary is 1333, that is after Wodeham’s and Bradwardine’s lectures. We do not have any evidence that Kilvington lectured on the Bible, so we can safely assume that he lectured only on the *Sentences*, and following Schabel, we can safely place his lectures on the *Sentences* for one academic year only in 1333–1334, and less likely in 1334–1335.

Kilvington’s *Sentences* commentary consists of a set of eight questions,^{33} his *Ethics* form a set of eight questions,^{34} his *Physics* commentary contains eight questions,^{35} his commentary on the *De generatione et corruptione* consists of nine questions.^{36} As it appears, Kilvington’s works are of an intricate manuscript tradition. Most problematic, with regard to both the manuscript tradition and the structure, and at the same time most interesting in terms of originality, are his questions on the *Physics*. The complete set consisting of eight questions is not preserved in any codex: we find an *expositio* of the *Physics* and one question in a Vatican manuscript only, two questions are to be found only in a Sevilla manuscript, the other three only in Venice, and the next two in different libraries in Europe.

With regard to the main subject of this paper, the most interesting is the question titled *Utrum in omni motu potentia motoris excedit potentiam rei motae*, because here Kilvington offers a new interpretation of Aristotle’s rules of motion. This question was well known by the anonymous author of the commentary on the *Sentences*, to be found in ms Vatican library Vat. Lat. 968, where he discusses the new rules of motion. In her *Verschollene Aristoteleskommentare des 14. Jahrhunderts*, Maier gives the following argument to justify the date of Kilvington’s commentary on the *Physics*:

Richard Killington is the author of a now-lost commentary on the

Physics. We learn this from the anonymousSentencescommentary … which is in Vat. lat. 986, the author of which appears to be a student of Bradwardine. Killington is quoted here several times, once with these words:hoc probat Kilmiton in questionibus suis super librum Physicorum. The proof quoted is based on a famous rule that Bradwardine had established in hisTractatus proportionum: Killington’s commentary on thePhysicsmust therefore follow this, i.e., must have been after 1328.^{37}

I verified Maier’s statement and came to the conclusion that the anonymous author of the *Sentences* commentary, when debating the problem of a proportion between an active power and resistance, quoted the opinions of both Kilvington and Bradwardine and made a clear distinction between them. He also argued separately against both claims.^{38} Maier rightly recognized that Kilvington’s proof is related to Bradwardine’s function. She, however, misintepreted that Kilvington used Bradwardine’s rule for his own purpose.

Accordingly to Norman Kretzmann:

… Kilvington, unlike Bradwardine and Swineshead, for instance, did not contribute significantly to the mathematical or strictly calculatorial side of the Calculator’s work. Even when he is dealing with material we would think of as belonging to physics or mathematics. Kilvington, at least in his

Sophismata, employs only the detailed argumentation and the analysis of concepts and of language that have part of the method of philosophy in almost every period of its history.^{39}

My research confirmed Kretzmann’s opinion that Kilvington, as a logician, was not interested in calculations. He, however, made a broad use of mathematics in all philosophical inquires as well as in theology.^{40}

My claim that Kilvington’s use of the new calculus of proportions in the description of motion enabled Bradwardine to formulate the New Rule of Motion, and that Kilvington was the inventor of the new theory of motion is, nevertheless, still disputed by Edith Sylla. On many occasions she has repeated her claim, formulated in her paper in 2008:

Jung comes to the conclusion that Kilvington’s arts lectures likely occurred before 1327, thus before Bradwardine’s

On the ratios of velocities in motionsof 1328. As evidence against Weisheipl’s opinion it is only necessary to note, however, that, as Jung herself reports, Thomas Bardwardine is supposed to have read theSentencesat Oxford around 1332–1333, while we know that hisOn the ratiosis dated 1328, less than seven years earlier. It seems more reasonable to assume at most that Bradwardine and Kilvington may have been working in parallel and that each knew the work of the other before completing the edited versions of his own work.^{41}

I would readily accept Sylla’s conclusion were it not for the fact that most of the arguments justifying the ‘New Rule of Motion’, known also as ‘Bradwardine’s Rule’, are *verbatim* quotes from Kilvington’s works. This very fact, in my opinion, cannot be explained by claiming either that Kilvington and Bradwardine had been working in parallel or that Kilvington’s question was the result of that ‘he heard some ideas aurally within the Oxford context in 1320s.’^{42} It rather testifies that one of them had the work of the other put on his desk.

At the beginning of his *De proportionibus*, Bradwardine declares that his work is the first dealing with the problem of motion in a strictly mathematical way. He writes:

Since each successive motion is proportionable to another with respect to speed, natural philosophy, which studies motion, ought not to ignore the proportion of motions and their speeds, and, because an understanding of this is both necessary and extremely difficult, nor has as yet been treated fully in any branch of philosophy, we have accordingly composed the following work on the subject.

^{43}

This paragraph can be read either as a declaration that Bradwardine considers himself to be the inventor of the new theory, or that he considers himself to be the first who organized the material on this subject in a coherent, systematic way with an introduction presenting the mathematical calculus of proportions, and the next three chapters using these calculi. If the first reading were correct, Bradwardine’s claim would not be true, as will be shown below, since there were others who reinterpreted Aristotle’s theory of motion by using the new calculus of proportions; following the second reading, Bradwardine should have mentioned the others who had already argued against the false theories of motion.

The information found in the anonymous treatise *De sex inconvenientibus*, at the beginning of the fourth question on local motion (*Utrum in motu locali sit certa servanda velocitas*) is also significant.^{44} The anonymous author writes:

Firstly I argue that this is not possible, since, as the first theory states, it would follow that such speed is determined by the excess of acting potencies over resistances, or, as the second theory states, by proportions of excesses of acting potencies to resistances … The first two theories are demonstratively disproved by many, and most precisely by two: magister Thomas Bradwardine in his treatise

De proportionibusand by Adam of Pippewell, who argues subtly.^{45}

The treatise was written by someone who was associated with the Oxford Calculators. Although the exact date of its composition is unknown, it can be narrowed down considerably to 1335–1339.^{46} We only know that Adam of Pipewell, mentioned in the anonymous treatise, was a fellow of Balliol College in 1321, then a fellow at Merton College by 1325, and still present there in 1327.^{47} We can assume that Adam of Pipewell produced the work the anonymous author refers to, which was well known at Merton, before Bradwardine’s treatise. However, we have no evidence (neither the work nor any mention in the other works, except the *De sex inconvenientibus*) confirming this hypothesis, but we have at our disposal Kilvington’s commentary on the *Physics* (ca. 550 pages), composed before 1326.

Although, in my opinion, Richard Kilvington was the real inventor of the new theory of motion, it seems that he was unaware of the significance of his discovery. It was Thomas Bradwardine, the greatest beneficiary of Kilvington’s work, who used his arguments with great skill in formulating the New Rule of Motion that continued to be supported by Aristotelians until the early sixteenth century.^{48} In the following sections of this article, I will provide arguments to support my opinion.

## 3 Richard Kilvington and Thomas Bradwardine on Local Motion

Before discussing Kilvington’s and Bradwardine’s arguments for the new rule of motion, and their new interpretation of Aristotle’s ‘laws of dynamics’, I would like to present the most important topics raised by Kilvington in his question *Utrum in omni motu potentia motoris excedit potentiam rei motae*,^{49} composed no later than 1326, to show the parallels between this work and Bradwardine’s *De proportionibus*, written in 1328.

Kilvington’s question is divided into two main parts between which one finds a counterargument (*contra*) based on the authorities of Aristotle and Averroes, followed by a *determinatio quaestionis*. In Part I, to justify his point of view, Kilvington presents *pro* and *contra* arguments for the discussed opinions, while in Part II, he concentrates on some particularly relevant objections to the arguments previously presented. For that reason, the reader is forced to seek Kilvington’s opinions in every single argument in both parts of the question.

The four articles contained in Part I concentrate on various problems concerning the conditions of motion, respectively: an excess of an acting potency over a passive one; the limits of the acting potency of the mover; the limits of the passive potency of a body moved; and finally, when the conditions are established, the results of their interaction, i.e., the speed of motion. In the *determinatio quaestionis* Kilvington declares that, in accord with the definitions of terms presented in Part I, the answer to the question whether in every motion the acting potency (i.e., the power of a mover [F]) exceeds the passive potency (i.e., the resistance [R]) is affirmative.

Bradwardine’s *De proportionibus* consists of four chapters. The first recapitulates the theories about proportionality found in Boethius’ *Arithmetic* and Campanus of Novara’s *Commentarium super quartum librum Elementorum Euclidis*. In the second chapter, Bradwardine criticizes four theories interpreting Aristotle’s theory that speed is proportional to the acting and passive potencies involved. In the third chapter, Bradwardine introduces his own solution to the problem and ‘he commences his exegesis by quoting Aristotle and Averroes in general support of his view, after which he launches directly into his twelve theorems concerning velocity.’^{50} Finally, chapter four deals with circular motion.

Both Kilvington and Bradwardine declare that Aristotle and Averroes maintain that the potency of the mover exceeds the potency of the thing moved, and that motion must necessarily take place according to some proportion.^{51} For both ‘Calculators’ Aristotle and Averroes are the most important authorities. They refer to Aristotle’s *Physica*, *De caelo*, *De anima*, *Metaphysica*, *De generatione et corruptione* and *De motu animalium*, and to Averroes’ commentaries on these works. But the Oxford way of teaching they adopted while studying at the university is also evident in their frequent and constructive application of the treatises *De ponderibus* by Euclid, Archimedes, and Jordanus de Nemore.^{52}

## 4 Condition *sine qua non* for Motion: An Excess of the Acting Potency of the Mover over the Passive Potency of the Thing Moved

At the beginning of the first article of his question, Kilvington quotes four different opinions: (1) as some say a minimum excess for motion may be assigned (*minimum quod sic*); (2) others claim that a maximum excess which is not sufficient for motion may be assigned (*maximum quod non*); (3) still others argue that there is an excess which is sufficient to continue motion but not to initiate it; (4) finally others maintain that the same excess which is sufficient for the continuation of motion is sufficient to initiate it.^{53} While the first two opinions state that it is possible to assign an intrinsic or an extrinsic limit of the sequences of the force [F] to resistance [R] proportions, the last two views differ in deciding whether it is necessary to initiate a motion with a greater excess of an active power over resistance than the one that sustains the motion.

Kilvington refutes the first three opinions on the basis of imaginable, theoretical cases and by means of an observation of the facts.^{54} He grounds his criticism on Averroes’ statement according to which ‘any element, in its natural motion caused by its essential form, reaches its natural place and at the same time its essential form.’^{55} Kilvington first describes the following situation: ‘Suppose that a fire initiates its acting on a piece of earth for which the natural place is the centre of the Earth. Acting on this piece of earth, the fire adds levity to it and, as a consequence, causes its desire for a superior place.’^{56}

One of the arguments based on this scenario is particularly interesting here. Kilvington maintains that in the above presented case a pure heavy element, such as a piece of earth, having been warmed by fire, when it will be equally heavy and light, would be fixed half way between the inner surface of the Moon and the centre of the Earth, i.e., much above the outer surface of the air and much above a large part of the fire. This Kilvington proves as follows. Take four elements as sign: earth (E) by 1, water (W) by 10, air (A) by 100, and fire (F) by 1000.^{57} Thus, according to Aristotle, Kilvington argues, the sphere of fire does not exceed all the other elements 1000 times but more than a proportion of 9 to 1.^{58} In fact, his proof is limited to the assumption and conclusion while the most interesting is the calculation which leads to the conclusion. Since all the four elements are continuously proportional (W : E = A : W = F : A = 10 : 1), the proportion between the last and the first (F : E) is equal to (10 : 1)^{3}, which is why Kilvington claims that the fire exceeds the earth 1000 times, i.e., in a proportion (10 : 1)^{3}. However, this is not a proper calculation, because it takes into account only two elements (the first and the last), while Kilvington talks about the distance of the sphere of the three other elements to the sphere of fire. The sum of the three other elements is 111 = 100(E) + 10(W) + 1(F) and it is equal to the sphere composed of these three elements. Now, if the fourth element (F) is divided by this sum, the quotient is 9 + 1/111. The fourth element, therefore, contains the other elements taken together 9 times and a fraction, and this is what should be demonstrated.

A similar calculation, but much more elaborated and much better proven, is found in chapter IV of Bradwardine’s *De proportionibus*. Bradwardine begins his argumentation as follows:

Since, on the basis of the foregoing, and by assuming only a few more propositions to be true, it is possible to find with ease the proportion of the elements to each other, and because the discovery of this is most appropriate to natural philosophy, and has, up to this day, remained unknown, therefore (although this is admittedly not too pertinent to the present undertaking) we will, nevertheless, uncover the secret of it.

^{59}

This artful declaration opens a detailed mathematical proof based on the works of Alpharganus, Ṯābit ibn Qurra, and Ptolemy. Bradwardine’s calculus goes as follows:

The proportion between the sphere of fire and that composed of the three remaining elements is greater than the proportion of 31 to 1; for each element contains the one immediately lesser more than 32 times. Proof: Corresponding, therefore, to the four elements, take these four terms, continuously proportional in the aforementioned proportion: 1, 32, 1024, 32768, the sum of whose first three terms is 1057 and represents the sphere composed of the lower elements. If the fourth term (which represents the sphere of fire) is divided by this sum, the quotient is 31 + 1/1057. The fourth term, therefore, contains the above-mentioned sum 31 times and a fraction, and this is what we were seeking for demonstrate.

^{60}

The comparison of Kilvington’s and Bradwardine’s way of reasoning allows one to formulate the following conclusions: (1) both authors are thoroughly familiar with the new calculus of ratios; (2) Kilvington relies on the authority of Aristotle and Averroes; Bradwardine uses the theories of Alpharganus, Ṯābit ibn Qurra, and Ptolemy; (3) Bradwardine guides his readers by showing each step of his proceeding and makes a broad use of his previously proved (in Part I) axioms and theorems; Kilvington, on the other hand, forces us to reconstruct his calculations. However, it is well known that Bradwardine wrote a treatise that was used as a textbook of mathematical physics. Kilvington, on the other hand, was the author of the question *Utrum in omni motu potentia motoris excedit potentiam rei motae* that was the result of his lecturing on the *Physics*, which allowed him to ignore certain levels of reasoning visible to his students and fellow medieval readers.

Article I of Kilvington’s question ends with the acceptance of the fourth opinion above mentioned: ‘the same excess which is sufficient for the continuation of motion is sufficient to initiate it’ (*quicumque excessus sufficeret ad incohandum motum sicut ad continuandum*). Kilvington’s arguments in favour of this opinion are based on the authorities of Aristotle, Averroes, Euclid, Archimedes, and Jordanus de Nemore. In justifying his position, Kilvington first gives an example that was later repeated by Bradwardine but used for a different purpose. Bradwardine uses this argument in section three of chapter II where he disproves ‘the third erroneous theory, which claims that: (with the moving power remaining constant) the proportion of the speeds of motions varies in accordance with the proportion of resistances, and (with the resistance remaining constant) that it varies in accordance with the proportion of moving powers.’^{61}

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

… nec potest dici, quod in aliquo casu aliquis sit maximus excessus non sufficiens ad motum, quia sit excessus iste A, adhuc minor excessus A sufficit ad motum, ergo A excessus sufficit ad motum. Argumentum probo, quia in circumvolutione spherae vel rotae provenit motus alicuius partis propinquioris centro ex minori excessu quam sit A excessus, quia quanto partes spherae vel rotae centro fuerint propinquiores tanto tardius moventur, ut patet VI |
Nec potest dici quod tarditas motus non potest in infinitum duplari: quia si sic, sit A tarditas mobilis quae duplari non potest. Volvatur igitur sphera, seu corpus columnare, super axem quiescentem. Tunc in aliqua parte iuxta polum spherae, seu axem corporis columnaris, est tarditas dupla ad A ut est satis notum et facile demonstrare. Tunc cum ista parte colligetur corda fortis et longa, in cuius extremo alligetur aliquod ponderosum, quod sit B. Tunc tarditas motus B est dupla ad A tarditatem; et hoc est quod volumus demonstrare. Nec potest dicere cavillator quod motus B est motus per accidens, et in potentia tantum, ideo non facit ad propositum, quia iste motus habet motorem in actu et motum seu mobile in actu, terminus a quo et terminus ad quem in actu, tempus in actu, et spatium seu locum pertransitum in actu. Igitur est motor in actu. |

According to Kilvington, the above presented argument proves that any excess is sufficient for motion on the *sine qua non* condition that F : R > 1; if F = R, the excess is zero and motion stops. Bradwardine uses this example to undercut Aristotle’s theory:

The theory is, on the one hand, to be refuted on the ground of falsity, for the reason that a given motive power can move a given mobile with a given degree of slowness and can also cause a motion of twice that slowness. According to this theory, therefore, it can move double the mobile. And, since it can move with four times the slowness, it can move four times the mobile, and so on

ad inifnitum. Therefore, any motive power would be of infinite capacity.^{62}

Kilvington puts forward the following three arguments in support of this opinion. First, it is impossible to observe the exact moment when motion begins and when it lasts continually, and hence, it is impossible to assign a minimum excess *quod sic* or a maximum excess *quod non*. Second, an observation of different weights placed on a balance with equal arms shows that even a minimally heavier body lifts the other one. Third, in accordance with Aristotle and Averroes, motion is initiated and sustained in every case when the active power is greater than the resistance.^{63} As a consequence, motion occurs whenever the ratio of F to R is a ‘ratio of *maioris inaequalitatis*’, i.e., when F : R is greater than 1. Thus, Kilvington affirms that any proportion of force to resistance greater than 1 can produce motion.^{64}

This theory has an unexpected consequence: the Earth can move rectilinearly. The motion of the Earth is caused by its inclination to bring the geometric centre of the universe into coincidence with its centre of gravity, and because the Earth as a whole is unequally heavy and dense, and geological changes perpetually alter it, its centre of gravity shifts. If the world were eternal, this slight rectilinear motion would be infinite.^{65}

## 5 The Limits for Active and Passive Potencies

Having established a major condition for motion, Kilvington concentrates, in the next two articles, on setting boundaries to the range of causes of motion, i.e., active and passive potencies. An active potency is bounded by the patient, e.g., the weight that can be lifted or the distance that can be traversed. A passive potency is bounded by the agent by which it can be affected, as sight is bounded by the smallest object that can be seen. Commenting on Aristotle and Averroes, Kilvington raises some important queries, which open a new perspective on the solution of the problem. He is interested in answering the following questions: how is the power to be bounded if it is active or passive; if it is subject to weakening or not; if it is mutable or immutable? How to assign the boundaries of active potencies if a body moves in a medium that is not uniformly resistant? Most of the cases considered are posed *secundum imaginationem* which, however, does not make empirical verification irrelevant. Kilvington’s mathematical interest is to be observed, at the outset, in his classification of all potencies as active or passive ones, even in cases in which it is hardly possible to define them as such in accordance with Aristotelian terms. In the Oxford Calculators’ School this tradition was followed by William Heytesbury.^{66}

While discussing the problem of the limits for active potency, Kilvington presents four opinions popular at that time (he uses the wording *dicitur a quibusdam*), in the case of passive potency he mentions five opinions. His view is that an active potency is determined by an upper limit, a *maximum quod sic*, and by a lower limit, a *minimum quod non*.^{67} In order to determine Socrates’ power we can observe that he lifts 5 pounds (and not sensibly or notably more) or traverses 10 miles (and not notably more). The most proper way, however, to describe a capacity of an active potency is to determine a *minimum quod non*-limit, since every excess of an active potency over a passive one is sufficient for motion. With this conclusion Kilvington can be placed among all those masters, who, according to the anonymous author of the *Treatise on maxima et minima*,

claiming that every excess does suffice to motion, must grant and would grant, that an active capacity is bounded by a minimum upon which it cannot act. And this is the resistance equal to the active capacity, because the active capacity cannot act upon that resistance since action does not occur through a proportion of equality.

^{68}

Kilvington then goes on to discuss the limits of passive potencies. According to the definition Heytesbury later adopted, ‘A passive potency is one which, inasmuch as it is susceptible to less or can be affected by less, is susceptible to a greater, or can be affected by a greater, and not vice versa.’^{69} Kilvington accepts the *minimum quod sic*-limit with respect to circumstances for a passive potency, since—as he puts it—it happens that we cannot see not only a small thing, but also a big one, for instance a cathedral, if we stay too close to it. When we ask, however, about Socrates’ capacity of good vision, we point out the smallest thing he can scarcely see. And this is also Aristotle’s point of view. What happens, however, is that a weak active power, such as a drop of water, overcomes the resistance of a rock. Also a small amount of fire can overcome the resistance of a large amount of water. This happens because every passive potency can be affected in part.^{70}

To sum up, Kilvington indicated most of the issues concerning the problem of setting boundaries to potencies involved in an action-passion process. Although he did not explicitly formulate general rules concerning different types of division, his debates reveal that he approved the following conditions for the existence of limits. First, there must be a range in which the potency can act or be acted upon, and another range in which it cannot act or be acted upon. Second, the potency should be capable of taking on a continuous range of values between zero potency and the value that serves as a boundary, and no other values.^{71}

## 6 The Measurement of the Speed of Motion

The fourth and final article of Kilvington’s question contains an extensive debate on how to correctly ‘measure’ the speed of motion (v). This time, Kilvington does not mention any other opinions of his contemporaries. He begins the debate with the following statement:

If the question is true, the speed of motion varies either as the difference whereby the power of the mover exceeds the resistance offered by the thing moved or it varies with the proportion of the active power of the mover to the passive power of the thing moved.

^{72}

The two theories mentioned in this passage state respectively that the speed of motion is proportional to the arithmetical difference between an active potency and a passive one (v ~ F – R) or that it is proportional to the proportion of F to R (v ~ F : R).

Presented and reviewed are only these two theories. Bradwardine refutes four theories: the first one claims that the speed of motion is proportional to the arithmetical difference between an active potency and a passive one; the second one states that speed is proportional to F – R/R (as it seems, this is originally Bradwardine’s idea); the third theory states that the speed is proportional to the proportion of F to R; ‘the fourth erroneous theory takes … a non-mathematical view of the relation of velocity to forces, claiming that velocities vary neither as a proportion of forces nor as an arithmetical excess of motive force over resistive force,’ but vary, instead, by a ‘natural relation of mover to moved.’^{73} Since I am of the opinion that Kilvington composed his questions on the *Physics* in 1326, at the latest, and Bradwardine composed his treatise on proportions in 1328, I am presenting Kilvington’s arguments first and then Bradwardine’s ones.

In refuting the first opinion (v ~ F – R), both Kilvington and Bradwardine begin with quotes from Averroes’ commentary on *Physics* IV, commentum 71 (*omnis motus est secundum excessum potentiae moventis super rem motam*)^{74} and *Physics* VII, commentum 39 (*secundum excessum potentiae alterantis super potentiam alterati erit velocitas motus alterationis in quantitate temporis*).^{75} Bradwardine adds a quote from Aristotle’s *De caelo et mundo* I and from *Physics* IV, commentum 35.^{76} Then they both make a similar statement. Kilvington: ‘But this theory may be refuted in many ways.’ Bradwardine: ‘The present theory, may, however, be torn down in several ways.’^{77} Kilvington presents the following arguments against this theory.

(1) Undoubtedly, this theory challenges Aristotle’s and Averroes’ rule of motion, which holds that two movers would move two *mobilia* taken together with the speed equal to the speed with which one of them would move one *mobile*. Suppose that F_{1} = F_{2} = 2 and R_{1} = R_{2} = 1, then F_{1} + F_{2} = 4, R_{1} + R_{2} = 2 and, in accordance with the rule: v_{2} ~ (F_{1} + F_{2}) – (R_{1} + R_{2}) = 4 – 2 = 2 while in the case of the motion caused by a separate acting mover v_{1} ~ F_{1} – R_{1} = 2 – 1 = 1, thus v_{1} ≠ v_{2}.

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

… si haec esset vera, tunc falsificaretur regula Aristotelis et Commentatoris VII |
Secundum istum modum sequitur quod, aliquo motore movente aliquod mobile per aliquod spatium in aliquo tempore, medietas motoris [movet] medietatem moti per medietatem excessus; sicut quaternarius excedit binarium per binarium, et medietas eius (scilicet binarius) excedit medietatem eius (scilicet unitatem) per unitatem tantum, que est medietas prioris excessus. Et falsitas consequentis patet per Aristotelem, septimo |

(2) This theory is also contrary to the following rule of Aristotle and Averroes: ‘half of a given mobile is moved by a half of a given mover in the same time and on the same distance as a whole mobile moved by a whole mover.’^{78} The given numerical example is the same: suppose that F = 4, R = 2, then F – R = 2 and the whole mover exceeds the whole mobile by 2, while a half of F = 2 exceeds half of a mobile R = 1 by 1 (½F – ½R = 2 – 1 = 1).

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Item, ex ista opinione sequitur quod alia regula Aristotelis sit falsa, quae ponitur in textu commenti 36 in VII |
Nec potest dici quod Aristoteles et Averroes intelligent, in locis praedictis, per proportionem seu analogiam, proportionalitatem arithmeticam seu aequalitatem excessuum, ut dicunt quidam, quia septimo |

(3) Now to demonstrate the flaw in Averroes’ theory, Kilvington gives a series of examples from sense experience:

(a) ‘If to a single man who is moving some weight which he can scarcely manage with a very slow motion, a second man joins himself, the two together can move it much more than twice as fast.’^{79}

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Item, probo quod velocitas motus non semper sequitur excessum, quia videmus quod, si aliquis unus homo moveat aliquod ponderosum motu valde tardo, quod ponderosum potest vix ille homo movere, si alius homo sibi coniungitur, illi duo homines movebunt illud ponderosum multo plus quam in duplo velocius; et tamen excessus non duplicatur nisi ⟨duplo⟩ praecise; ergo velocitas non sequitur excessum. |
Tertio ex ista positio super mendacio arguenda, quoniam experimentum sensibile docet huius positionis contrarium. Videmus enim quod, uno homine movente aliquod ponderosum (quod potest vix solus movere motu valde tardo), si aliquid sibi adiungatur, illi duo movent illud multo plus quam in duplo velocius. |

(b) The same principle is quite manifest in the case of weight suspended from a revolving axle, which it moves insensibly during the course of its own downward motion (as is the case with clocks): ‘If an equal clock weight is added to the first, the whole descends and the axle, or wheel, turns much more than twice as rapidlym (as is sufficiently evident to sight).’^{80}

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Istud confirmo: si in horilogio per certum tempus moveretur rota certa velocitate, movente addito, licet non dupletur pondus, dupla velocitate movebitur, ut experimento notum est. Illud apparet manifeste de pondere suspenso ad aliquod circumvolubile, quod per suum descensum movet insensibiliter et movet⟨ur⟩ rota quodam motu insensibili. Sic accidit in horilogio: ad pondus si suspendatur iterum tantum pondus, totum descendet cum impetu et circumvolvet rotam multo plus quam in duplo velocius, ut patet manifeste sensationi. |
Et patet manifeste de pondere suspenso ad axem circumvolubilem, quod per suum descensum movet insensibiliter et volvit axem seu rotam motu insensibili (sicut accidit in horologio) ad quod si suspendatur tantum pondus, totum descendet et circumvolvet axem seu rotam plus quam in duplo velocius (ut sensui sufficienter constat). |

Bradwardine uses these two examples to disprove the second false theory: ‘with the moving power remaining constant, the proportion of the speeds of motions varies in accordance with the proportion of resistances. And, with the resistance remaining constant, that it varies in accordance with the proportion of moving powers’.^{81}

(c) The next example Kilvington gives is as follows: ‘suppose that one man carries a weight on some distance, running as fast as he can. Then if another man, who runs as fast as the first one, joins him, they both would carry the weight and move with the same speed as before.’^{82} This means that, despite the active power being doubled, the speed of motion would remain the same. As a consequence, speed does not vary in accordance with the arithmetical proportion between an active power and a mobile.

(d) Sense experience shows that ‘if three men haul a ship with ineffective power, they can only turn it around and not move it forward; however, if a fourth joins them, they can tow the ship in a straight line over a great distance.’^{83}

(e) ‘A man, carrying a load with which he can move very slowly, being burdened with an additional, much smaller load, he will move twice as slowly and with great difficulty.’^{84} Bradwardine gives a similar example of a fly, a boy, and a strong man who, when burdened with an extra load, do not move much slower than without that additional load.^{85}

(4) Finally, the presented opinion has to be rejected because a geometric proportion (i.e., a similarity of proportions) of movers to the *mobilia* would not produce equal speeds, since it does not represent an equality of excesses. For, although the proportion of 4 to 2 and 2 to 1 are the same, the excess of one term over the other is 2 in the first case, and 1 in the second. The conclusion is contrary to Aristotle’s and Averroes’ statement that equal proportions between movers and mobiles result in equal speed.

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Item, si velocitas motus sequeretur excessum, tunc ex similitudine motus ad mota et ex proportione istorum geometrica non sequitur velocitas aequalis, quia non sequitur ex aequalitate proportionum aequalitas excessuum, ut manifeste patet. Nam aequalis est proportio quattuor ad duo et duorum ad unum, et tamen excessus sunt inaequales sicut binarius et unitas. Et per consequens ad quod deducitur est contra Aristotelem et Commentatorem VII |
Tertio sic: Tunc ex proportione geometrica, scilicet similitudine proportionum motorum ad sua mota, non sequitur aequalis velocitas motuum, quia nec excessuum; quoniam eadem est proportio duorum ad unum et sex ad tres; excessus tamen unius est unitas, alius autem ternarius. Consequens autem ad quod deducitur est falsum et contra Aristotelem septimo |

Some of these arguments, as it was shown above, are to be found in the first part of Bradwardine’s chapter II, where he criticizes the first false theory: ‘the proportion between the speeds with which motions take place varies as the difference whereby the power of the mover exceeds the resistance offered by the thing moved.’^{86}

According to Kilvington, the above mentioned arguments *contra* justify the need to adopt the second theory, that speed of motion is proportional to the proportion of F : R. In support of his opinion, he gives the last argument quoted below, which Bradwardine also uses to close the discussion at the end of the first part of Chapter II. To reinforce his view, Kilvington refers to authorities. Bradwardine uses the expression *ex prioribus*, i.e., the arguments taken from chapter I. So again he leaves the impression that he is the first one who criticizes this false theory.

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Pro istis et consimilibus dicitur secunda opinio quod velocitas motuum sequitur proportionem moventis ad motum et non excessum, quia si sequeretur excessum, tunc sequeretur quod, si aliquod mobile simplex quod in dupla proportione excedit, non posset moveri in aliquo medio subtiliori mundi ( |
Tunc sequeretur quod, si terra pura movetur in aliquo medio quod in dupla proportione excederet vel maiori, non potest moveri in duplo velocius in aliquo medio alio. Non enim posset excedere aliquod medium per duplum excessum, quoniam tunc totum esset excessus. Et tunc, manente eodem motore, non in infinitum per subtiliationem medii posset velocitas motus generari; |

Chapter III, in which Bradwardine presents his own theory, begins with the following grandiose ‘opening’:

Now that these fogs of ignorance, these winds of demonstration, have been put to flight, it remains for the light of knowledge and of truth to shine forth. For true knowledge proposes a fifth theory which states that the proportion of the speeds of motions varies in accordance with the proportion of the power of the mover to the power of the thing moved.

^{87}

Kilvington begins article IV of his question, where he presents the new calculus of proportions and the new interpretation of Aristotle’s rules of motion, saying: ‘from what has been said the second theory follows,’^{88} i.e., speed is proportional to the proportion of F to R (v ~ F : R). To support their views, both Kilvington and Bradwardine quote the same paragraphs from Averroes.^{89}

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

… dicit Commentator sic uniformiter: ‘cum ergo sic fuerint duo motores et duo mota, et proportio unius motoris ad suum motum fuerat sicut proportio reliqui motoris ad suum reliquum motum, tunc illi motus erunt aeque veloces, et cum diversabitur proportio diversabitur et motus.’ Et in fine in eodem dicit sic: ‘diversitas motuum in velocitate et tarditate est secundum proportionem hanc, quae est inter duas potentias, motivam scilicet et resistivam.’ Et II |
Et hoc est quod vult Averroes, super quarto |

In the first part of his questions, Kilvington formulates seven theorems (*conclusiones*)—as Crosby translates this term—against this theory,^{90} and in part II he addresses these arguments, some of which he totally accepts, others he modifies.^{91} Some of these arguments are also found in chapters II and III of Bradwardine’s treatise. Since Kilvington’s discussion spreads out over 16 pages, I will present only the main theorems.

Theorem I: ‘A body twice as heavy as another body would not move in the same medium with a double speed.’^{92} The following two numerical examples confirm this claim:

(1) ‘Suppose that a body moves downward with its natural motion and that the resistance of a medium is exceeded by the moving potency (gravity) like 3 to 1; then if we double the gravity of the body (i.e., if we double its weight), the potency would exceed the resistance as 6 to 1, though it does not mean that the speed of motion would be doubled.’^{93} In accordance with Euclid’s definition of a double proportion, a double proportion is a proportion multiplied by itself. Thus, a ‘correct’ double proportion of 3 to 1 is 9 to 1, and not a proportion of 6 to 1, since 9 : 1 = (9 : 3)(3 : 1) = (3 : 1)(3 : 1).^{94} The proportion of 6 to 1 is less than 9 to 1, and if we take for granted the theorem of Averroes and Aristotle that speed is proportional to the proportion of F to R, it turns out that the speed would be less than twice the previous one; so a body twice as heavy in the same medium would not move twice as fast: v_{2} < 2v_{1} and not v_{2} = 2v_{1}.^{95}

Bradwardine’s reasoning is the same. However, he does not provide any numerical example, but instead he formulates the following general conclusion (his Theorem IV): ‘If the proportion of the power of the mover to that of its mobile is greater than two to one, when the motive power is doubled the motion will never attain twice the speed.’^{96}

(2) We can also prove, Kilvington argues, that in the same medium, the speed of a body twice as heavy might be greater than twice the previous one. ‘Suppose that an active power exceeds a resistance in a proportion of 6 to 4; hence, in accordance with Averroes’ theorem, a doubling of the active power gives the proportion of 12 to 4, which is greater than the proportion of 6 to 4.’^{97} The correct calculus, however, gives a proportion of 9 to 4, since: (9 : 4) = (9 : 6)(6 : 4) = (3 : 2)(3 : 2) which is a proportion double of 6 to 4, while 12 to 4 is not.^{98} Since the proportion of 12 to 4 is greater than 9 to 4, the speed of motion would be greater than the double of the previous one, so that a body twice as heavy in the same medium would not move twice as fast: v_{2} > 2v_{1} and not v_{2} = 2v_{1}.

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Ut potest demonstrari per definitionem proportionis duplicate positam V Euclidis … si fuerint tria continue proportionabilia proportione inequalitatis, tunc proportio tertii ad primum est proportio secundi ad primum duplicata. |
Si fuerit proportio maioris inequalitatis primi ad secundum ut secundi ad tertium, erit proportio primi ad tertium dupla ad proportionem primi ad secundum et secundi ad tertium. |

However, it can be noticed, at first glance, that such calculation of proportions forces changes in the interpretation of Aristotle and Averroes, Archimedes, Euclid, and Jordanus de Nemore, who state that speed is proportional to the proportion of the moving power to the resistance of the body moved.^{99} While noticing the contradiction, Kilvington confirms that the value of speed depends on the proportion of active power to resistance, and only in a case when F : R is 2 : 1 would the doubling of F result in double the speed. Bradwardine says the same in his Theorem III.

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Et numquam movebitur duplum grave praecise in duplo velocius in eodem medio nisi quando grave subduplum excedat resistentiam medii in proportione dupla. |
Si potentiae moventis ad potentiam sui moti sit dupla proportio, eadem potentia movebit medietatem eiusdem moti velocitate praecise duplata. |

Theorem II: A mobile, e.g., a piece of earth, which moves in water will not move twice as fast in air twice as rare. ‘This conclusion follows from what has been said above,’ Kilvington writes, since if the proportion of the heaviness of a simple body in downward motion (F) to the resistance of a medium (R) is greater than 2 : 1, then in a medium doubly rare, e.g. in air, the body would move slower than twice. If F : R_{w} > 2 : 1, and R_{a} = 2R_{w,} then F : R_{a} < 2v.^{100} The same point is made by Bradwardine in his Theorem IV quoted above.^{101}

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Et si illud grave simplex excedat resistentiam aquae in proportione maiori quam in dupla, tunc excedet resistentiam duplae subtilitatis in proportione maiori quam dupla ad proportionem primam, sicut demonstrari potest ut prius per definitionem proportionis duplicatae. Ergo tunc movebitur tale grave in aere duplae subtilitatis praecise plus quam in duplo velocius quam movebatur in aqua. |
Si potentiae moventis ad potentiam sui moti sit maior quam dupla proportio, potentia motiva geminata motus eiusdem duplam velocitatem nequaquam attinget. |

Theorem III: The following rules given by Aristotle and Averroes are false: (1) If a given power moves a given mobile through a given distance in a given time, the same power will move twice the same mobile through half the distance in an equal time, and through the same distance in twice the time. (2) If a given power moves a given mobile through a given distance in a given time, double the power will move that mobile through double the distance in an equal time.^{102} If F_{1} = F_{2} and R_{1} = 2R_{2}, then t_{1} = t_{2} (t-time) and s_{1} = 2s_{2} (s-distance), or then s_{1} = s_{2} and t_{1} = 2t_{2}. If F_{1} = 2F_{2} and R_{1} = R_{2,} then t_{1} = t_{2}, and s_{1} = 2s_{2}. Bradwardine quotes these rules in Chapter II, part 3, in support of, as he says, the third wrong theory: ‘(with the moving power remaining constant) the proportion of the speeds of motions varies in accordance with the proportion of resistances, and (with the resistance remaining constant) that it varies in accordance with the proportion of moving powers.’^{103} In his opinion, ‘the theory is, however, refutable on two grounds: first, on that of insufficiency, second, because it yields false consequences.’^{104}

In Kilvington’s opinion, the above mentioned rules are false, since when the proportion of F_{1} : R_{1} = 3 : 1 , and when F would be doubled, such a moving power would not move a body with a speed v_{2} = 2v_{1}, but with v_{2} < 2v_{1}. If F_{1} : R_{1} = 3 : 1, then double F_{1} (read multiplied by 2) results in the proportion F_{2} : R_{2} = 6 : 1, if R_{1} = R_{2}, and 6 : 1, as was said above, does not double the proportion of 3 : 1, but 9 : 1 doubles 3 : 1, and 6 : 1 < 9 : 1. On the other hand, when F : R = 3 : 2, then: v_{2} > 2v_{1}, because if F_{2} = 2F_{1}, then F_{2} : R_{2} = 6 : 2, which is not a correct calculus, since 9 : 4 doubles 3 : 2, and 9 : 4 < 6 : 2, so v_{2} > 2v_{1}. 3 : 2, and 9 : 4 < 6 : 2, so v_{2} > 2v_{1}.

Kilvington’s calculation appears to be confusing. What he seems to be doing here is applying the new calculus of proportions for Aristotle’s old theory: speed is proportional to a proportion of F : R. Thus in both cases, either when F_{1} : R_{1} > 2 : 1 or when F_{1} : R_{1} < 2 : 1, v_{2} is not equal to 2v_{1}. In the first case, speed is proportional to a proportion v_{2} ~ 2F_{1} : R_{1} = 6 : 1 and this does not double 3 : 1, and a proportion 6 : 1 is less than 9 : 1. So that is the reason why he states that speed v_{2} < 2v_{1}. In the second case, when speed–v_{1} is proportional to 3 : 2, double the proportion would result in the speed v_{2} > 2v_{1}.

In his comment to the above-mentioned theorem III, Kilvington, followed later on by Bradwardine, explains how the double proportion should be understood:

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Et per idem ad tertiam conclusionem dico quod non oportet quod, ‘si aliquis motor possit movere aliquod mobile per aliquod spatium in aliquo tempore, quod possit movere medietatem illius mobilis per duplum spatium in equali tempore.’ Immo dico quod Philosophus intelligit per ‘medietatem mobilis’ talem partem mobilis quae habet se ad motorem in proportione subdupla ad proportionem totius mobilis ad eundem motorem. Et in aliis regulis intelligit per ‘duplum agens’ habens proportionem ⟨duplam⟩ ad eandem resistentiam. Et sic intelligendo verificentur omnes istae regulae, si cetera fuerunt paria. Et sic loquitur Philosophus in locis preallegatis ‘quod grave duplum movebitur praecise in duplo velocius in eodem medio,’ id est, grave habens praecise proportionem duplam ad resistentiam illius medii; et ⟨haec est⟩ regula proportionis inter grave subduplum et illud medium. |
Conclusio autem allegata ex septimo |

Kilvington believes that the above definition of a double proportion allows explaining Archimedes’, Jordanus de Nemore’s and Euclid’s theorems, which were quoted above to support the contrary theory. Likewise Bradwardine claims that Jordanus’ theorem should be read in this way.^{105}

Theorem IV: Two heavy simple bodies, one of which moves in water one foot deep and the other in subdouble dense air two feet deep, would not move downward with the same speed in the same time, since even though in both cases the proportion of the active power to the resistance is the same, the resistances of air and water are not the same with regard to their intensity, and thus the speed would not be the same.^{106} In Kilvington’s opinion this is so, because there are two different resistances: one is intrinsic (*intensive*), and depends on the density of the medium, the other is extrinsic (*extensive*) and depends on the distance to be traversed.^{107}

Theorem V: ‘An object may fall in the same dense medium faster than another.’^{108} To prove this conclusion, Kilvington presents two arguments based on the new calculus of proportions and on Aristotle’s and Averroes’ statement that a medium can be rarefied *ad infinitum*. He also gives evidence for contradictory statements: (1) ‘a heavy mixed body may not fall in another medium twice as fast as it moves now,’ (2) ‘a heavy mixed body may fall in another medium twice as fast as it moves now.’^{109} Bradwardine presents the above conclusions in a shorter version in his Theorem XI: ‘An object may fall in the same medium both faster, slower, and equally with some other object that is lighter than itself’.^{110} As Crosby explains:

An important idea embodied in Bradwardine’s development of this theorem is that, in motions involving mixed bodies, the internal resistance is to be added to any external resistance which is present, i.e., if a mixed body may be represented by f/r, then the velocity of such a body through a medium possessing the resistance R, is to be represented as v = f/R+r, and not v = (f/r)/R. Therefore, though velocity remains the function of a proportion of forces, coacting forces are to be added to each other rather than made proportionate; the resultant motive force is proportionate to the resultant resistive force.

^{111}

The meaning of Kilvington’s proofs can be described in the same way.

Theorem VI: ‘A body will move at equal speed in a medium and in a vacuum.’^{112} Kilvington offers a proof of this statement by using the new calculus of proportions.^{113} His final conclusion, presented in Part II of his question, is the same as Bradwardine’s Theorem XII: ‘All mixed bodies of similar composition will move at equal speeds in vacuum’.^{114}

Richard Kilvington, |
Thomas Bradwardine, |
---|---|

Omnia talia mixta proportionalis compositionis aeque velociter moverentur in vacuo. |
Omnia mixta compositionis consimilis aequali velocitate in vacuo movebuntur. |

Theorem VII: ‘If the speed varies in accordance with the proportion of an active power to resistance, a heavy pure body may move infinitely fast.’^{115} To support this claim, Kilvington shows that in a downward motion the speed of motion of a pure heavy body increases infinitely, since at the beginning of the second part of a distance traversed, a body has to overcome half as great a resistance of the medium, because if at the beginning F : R = 2 : 1, in half the distance F : R = F : ½R = 4 : 1 = (2 : 1)^{2}, and thus a body moves twice as fast; at the beginning of the fourth part F : R = (2 : 1)^{3}, and so on, thus before it attains its place, it will move with an infinite speed.^{116} In Part II of his question, Kilvington affirms that we can state that a body might move with an infinite speed only if the term ‘infinite’ is understood syncategorematically, which means that a body would move twice as fast, three times, four times, and so one *in infinitum*, since there is no *maximum quod sic*-limit for the speed of motion. Moreover, since there is no proportion between the speed of accelerated motion and a uniform motion, one can say, syncategorematically, that the speed of accelerated motion is infinite in comparison to uniform motion.^{117}

## 7 Conclusion

To conclude, in his questions on the *Physics*, Kilvington made it clear that he was aware that Aristotle’s and Averroes’ rules of motion, given in *Physics* VII, were not universally applicable. He was convinced that the correct calculus of proportions was formulated by Euclid in his *Elements*. As a consequence, he noticed that the proper understanding of Euclid’s definition of operations on proportions necessitates a new interpretation of Aristotle’s and Averroes’ theory of motion. On the one hand, Euclid’s and Archimedes’ theory of operations on proportions concludes that doubling a ratio corresponds to squaring the fraction which we form from the ratio. On the other hand, Aristotle’s and Averroes’ statements clearly indicate that speed is proportional to a proportion of an active power to resistance, which is not squared but simply multiplied by two. Having noticed the contradiction of these two views, Kilvington first presented two main arguments against the Aristotelian proposition and finally concluded that, when talking about a power moving one half of a mobile, Aristotle means precisely a double ratio between F and R; when talking about a power moving a mobile twice as heavy, he means taking the square root of the ratio of F : R. The new mathematical rule describing the speed of motion is in accord with those of Aristotle only in one case: if the ratio of the power of the mover to that of its mobile is 2 : 1, the same power will move half the mobile with exactly twice the speed. It has already been shown that Kilvington’s reinterpretation of Aristotle’s and Averroes’ rules of motion reveals that the speed cannot be described by sole multiplication. Thus, as it was shown above, on the new interpretation, ‘double’ the ratio of 3 : 2 (or the ratio 3 : 2 *duplicata*) is the ratio 9 : 4, not 6 : 2, and the ratio of 3 : 1 *dupla* or *duplicata* is equal to 9 : 1, not to 6 : 1. Kilvington’s calculus provides values of the ratio of F to R greater than 1 : 1 for any speed down to zero, since any root of a ratio greater than 1 : 1 is always a ratio greater than 1 : 1. And, with his additional assumption that any excess, however small, of an active power over resistance is sufficient to initiate motion and to continue it, he may have described a very slow motion with a speed greater than 0 and less than 1 (0 > v < 1). Hence, he avoids a serious weakness of Aristotle’s theory, which cannot explain the mathematical relationship of F and R in very slow motions, when speed is lesser than 1.

Murdoch and Sylla emphasized that with Bradwardine’s treatise *On the proportions of velocities* ‘the situation changed rather dramatically in 1328.’ In their opinion, Bradwardine ‘removed the whole problem of relating velocities, forces, and resistances from the context of an exposition of Aristotle’s words and investigating it in its own right.’^{118} Although Sylla identified four questions on motion, found in a Venice manuscript (Biblioteca Nazionale Marciana, Ms. lat. VI. 72 [2810]), as authored by Richard Kilvington, she did not notice the strong similarities between Kilvington’s and Bradwardine’s works, and she repeated Maier’s views.^{119} Maier, as it was said at the beginning of this paper, noticed the contextual relations between Kilvington’s and Bradwardine’s works, but she was convinced that Kilvington used ‘Bradwardine’s rule’. My research shows that it was in fact Kilvington who initiated the study of the mathematics of motion and used the ‘correct’ technique of calculus of proportions, which allowed him to resolve the inconsistences of Aristotle’s theory, although he did not further develop the new theory in such a consistent and precise way as Bradwardine did. Kilvington seems to have taken the first step towards reinterpreting Aristotle and Averroes. He, and most likely some others, opened the new chapter in the history of medieval physics.

As Sylla explains with respect to Bradwardine:

Bradwardine’s function can be represented as a correlation between the latitude of proportion and of velocity, such that the proportion of equality corresponds to zero velocity and greater proportions correspond to greater velocities.

^{120}

Precisely the same conclusion applies to Kilvington.^{121} It seems that Kilvington invented the new theory of motion and applied the new calculus of proportions to Aristotle’s ‘laws of motion’, while Bradwardine employed many of Kilvington’s arguments and examples to substantiate the theory. Kilvington, however, was not the only one. Also ‘a third author’, as S. Rommevaux-Tani notices, ‘plays a role in this story, Adam of Pipewell, with whom the author of the *Tractatus de sex inconvenientibus* says he agrees. The precise connection between Kilvington and Pipewell is unknown, but Adam of Pipewell was a fellow at Oxford when Kilvington probably wrote his *Questiones*.’^{122} Regrettably, none of Pipewell’s works have survived to our time. It is worth noting that not only Kilvington’s *Quaestiones super Physicam* but also his *Quaestiones super de generatione et corruptione*, written ca. 1322/1323, testify to a lively discussion of the new calculus of ratios.^{123} At that time Kilvington was appointed at Balliol College, where both Bradwardine and Pipewell were most likely residing. Thus, it is very well possible that they, and maybe some others, were debating the mathematics of motion. It is also possible that Bradwardine lectured in arithmetic and geometry related to his work on motion at that time.

In his question *Utrum potentia motoris excedit potentiam rei motae*, which belongs to his commentary on the *Physics* written in 1326 at the latest, Richard Kilvington clearly explains the mathematical conception of compounding proportions implicit in what is generally called ‘Bradwardine’s Rule’. So here we seem to have a case of ‘Stigler’s law of eponymy’, which states that no scientific discovery is named after its original discoverer.^{124} By studying Kilvington’s work, we can trace what was going on with the Oxford Calculators before Bradwardine’s *De proportionibus*, which has previously been considered the founding document of Oxford calculatory work. In an attempt to trace the impact, spread, and decline of quantifying Aristotle, we should realize that the activity of quantifying motion had a prehistory before 1328.

## Acknowledgements

I gratefully acknowledge that this chapter is a result of project nr 2018/31/B/HS1/00472 funded by the National Science Centre, Poland. I thank Edith Sylla for her comments on my paper.

E. Jung-Palczewska, ‘Works by Richard Kilvington,’ *Archives d’ histoire doctrinale et littéraire du Moyen Âge* 67 (2000), 181–223.

I am grateful to Edith Sylla, who first drew my attention to Richard Kilvington’s questions on motion found in Venezia, Biblioteca Nazionale Marciana, Ms. lat. VI. 72 (2810).

Richard Kilvington, *Sophismata*, eds. N. Kretzmann and B.E. Kretzmann, Oxford 1990 (Auctores Britannici Medii Aevi, 12) and [Richard Kilvington], *The Sophismata of Richard Kilvington*, transl. N. Kretzmann and B.E. Kretzmann, New York 1991.

Kretzmann and Kretzmann, ‘Introduction,’ in [Richard Kilvington], *The Sophismata*, XXVIII.

A critical edition by Monika Michałowska of Kilvington’s questions on Aristotle’s *Ethics* has been published recently: Richard Kilvington, *Quaestiones super libros Ethicorum*, ed. M. Michałowska, Leiden 2016 (Studien und Texte zur Geistesgeschichte des Mittelalters, 121). Robert Podkoński and I have been working on Kilvington’s questions on the *De generatione et corruptione*. I am also working on a critical edition of the *Questiones super Physicam*, which, most likely, will be published next year.

Kretzmann and Kretzmann, ‘Introduction,’ in [Richard Kilvington], *The Sophismata*, XXIII–XXVII.

See A. Maier, *Die Vorläufer Galileis im 14. Jahrhundert* (Studien zur Naturphilosophie der Spätscholastik, 1), 2nd ed., Rome 1966 (Storia e letteratura, 22), 174; Eadem, *An der Grenze von Scholastik und Naturwissenschaft* (Studien zur Naturphilosophie der Spätscholastik, 3), 2nd ed., Rome 1952 (Storia e letteratura, 52), 266–267; Eadem, *Ausgehendes Mittelalter*. *Gesammelte Aufsätze zur Geistesgeschichte des 14. Jahrhunderts*, 1, Rome 1964 (Storia e letteratura, 97), 75.

N. Kretzmann, ‘Richard Kilvington and the Logic of Instantaneous Speed,’ in A. Maierù and A. Paravicini Bagliani (eds.), *Studi sul XIV secolo in memoria di Anneliese Maier*, Rome 1981 (Storia e letteratura, 151), 143–178, at 144–145; W.J. Courtenay, *Schools and Scholars in Fourteenth-Century England*, Princeton, NJ, 1987, 244.

See E.B. Fryde and J.R.L. Highfield, ‘An Oxfordshire Deed of Balliol College’, *Oxoniensia* 20 (1955), 40–45.

See J.A. Weisheipl, ‘The Place of John Dumbleton in the Merton School,’ *Isis* 50 (1959), 439–454, at 446, n. 45.

See Kretzmann and Kretzmann, ‘Introduction,’ in [Richard Kilvington], *The Sophismata*, XXIV.

See Kretzmann and Kretzmann, ‘Introduction,’ in [Richard Kilvington], *The Sophismata*, XXVII; Jung-Palczewska, ‘Works,’ 222.

See G. Leff, *Bradwardine and the Pelagians*, Cambridge 1957 (Cambridge Studies in Medieval Life and Thought, Series 2, 5), 3.

See Kretzmann and Kretzmann, ‘Introduction,’ in [Richard Kilvington], *The Sophismata*, XXVII.

See E. Wilks Dolnikowski, *Thomas Bradwardine. A View of Time and a Vision of Eternity in Fourteenth-Century Thought*, Leiden 1995 (Studies in the History of Christian Thought, 65), 5.

See E. Jung, ‘Richard Kilvington,’ in E.N. Zalta (ed.), *The Stanford Encyclopedia of Philosophy* (Winter 2016 Edition), URL = *Towards the Modern Theory of Motion. Oxford Calculators and the New Interpretation of Aristotle*, Łódź 2020 (Research on Science and Natural Philosophy, 4), 13–17.

Thomas Bradwardine, *Geometria speculativa*, ed. and transl. A.G. Molland, Stuttgart 1989 (Boethius. Texte und Abhandlungen zur Geschichte der exakten Wissenschaften, 18).

Thomas Bradwardine, *Tractatus de proportionibus velocitatum in motibus*, ed. H.L. Crosby, Jr., *Thomas of Bradwardine. His* Tractatus de proportionibus. *Its Significance for the Development of Mathematical Physics*, Madison, WI, 1955 (University of Wisconsin Publications in Medieval Science, 2). The colophon of one of the copies of Bradwardine’s treatise mentions the year 1328: ‘Explicit tractatus de proportionibus editus a magistro Thoma de Bradelbardin. Anno Domini MCCC28’ (ibid., 140).

Thomas Bradwardine, *Tractatus de continuo*, ed. J.E. Murdoch, *Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine’s* Tractatus de Continuo, unpublished Ph.D. dissertation, University of Wisconsin, 1957.

On Bradwardine’s questions on the *Sentences* see, e.g., J.-F. Genest and K.H. Tachau, ‘La lecture de Thomas Bradwardine sur les Sentences,’ *Archives d’ histoire doctrinale et littéraire du Moyen Âge*, 57 (1990), 301–306; J.-F. Genest, ‘Les premiers écrits théologiques de Bradwardine: textes inédits et découvertes récentes,’ in G.R. Evans (ed.), *Mediaeval Commentaries on the* Sentences *of Peter Lombard*. 1: *Current Research*, Leiden 2002, 395–421. For a critical edition of the question *Utrum Deus habeat prescienciam omnium futurorum contingencium ad utrumlibet*, see J.-F. Genest, ‘Le *De futuris contingentibus* de Thomas Bradwardine,’ *Recherches Augustiniennes et Patristiques*, 14 (1979), 249–336. Chris Schabel and Severin Kitanov prepared a critical edition of questions 7–9 from the set of nine (for the titles, see Genest, ‘Les premiers écrits théologiques,’ 397). See S. Kitanov and C. Schabel, ‘Thomas Bradwardine’s Questions on Grace and Merit from His *Lectura* on the *Sentences* at Oxford, 1332–1333,’ *Archives d’ histoire doctrinale et littéraire du Moyen Âge* 89 (2022) (forthcoming).

Thomas Bradwardine, *De causa Dei contra Pelagium et de virtute causarum ad suos Mertonenses libri tres*, ed. H. Savile, London 1618.

Edited by M. Carruthers, ‘Thomas Bradwardine, *De memoria artificiali adquirenda*,’ *The Journal of Medieval Latin* 2 (1992), 25–43. For an English translation, see M. Carruthers, *The Book of Memory: A Study of Memory in Medieval Culture*, New York 1990, 281–228. Additionally, see also M. Carruthers and J.M. Ziolkowski (eds.), *The Medieval Craft of Memory. An Anthology of Texts and Pictures*, Philadelphia, PA, 2002, 205–214.

See E. Jung, *Arystoteles na nowo odczytany. Ryszarda Kilvingtona ‘Kwestie o ruchu*’, Łódź 2014, 107–316.

See for example Kilvington, *Quaestiones super libros Ethicorum*, 63–336.

[Richard Kilvington], *The Sophismata*, 1.

See Jung, ‘Works by Richard Kilvington,’ 222.

See C. Schabel, ‘Ockham, the *Principia* of Holcot and Wodeham, and the Myth of the Two-Year *Sentences* Lecture at Oxford,’ *Recherches de Théologie et Philosophie Médiévales*, 87 (2020), 59–102. See also W.O. Duba and C. Schabel, ‘Remigio, Scotus, Auriol and the Myth of the Two-Year *Sentences* Lecture at Paris,’ *Recherches de Théologie et Philosophie Médiévales* 84 (2017), 143–179.

See Courtenay, *Schools and Scholars*, 272.

See C. Schabel, ‘Richard FitzRalph vs. William Skelton, 1331–1332: The Attribution of the *Determinationes* in a Florence Manuscript,’ in M. Dunne and I. Nolan (eds.), *A Companion to Richard FitzRalph* (forthcoming). I am grateful to Chris Schabel for sharing his texts with me.

See Adam Wodeham, *Lectura secunda in librum primum Sententiarum*, eds. R. Wood and G. Gál, New York 1990, 1: 32*–37*.

Robert Holcot, *Questions on Future Contingents*, eds. P.A. Streveler and K.H. Tachau, *Robert Holcot. Seeing the Future Clearly. Questions on Future Contingents*, Toronto, ON, 1995 (Studies and Texts, 119), 27.

See Schabel, ‘Ockham, the *Principia* of Holcot and Wodeham,’ 71.

The commentary on the *Sentences* is found in the following manuscripts: (1) Bologna, Biblioteca comunale dell’Archiginnasio, Ms. 985, 1^{ra}–52^{va}; (2) Brugge, Stedelijke Openbare Bibliotheek, Ms. 188, 3^{ra}–54^{vb}; (3) Brugge, Stedelijke Openbare Bibliotheek, Ms. 503, 80^{ra}–105^{rb}; (4) Erfurt, Universitätsbibliothek, Dep. Erf., CA F. 105, 122^{rb}–182^{rb}; (5) London, British Library, Harley, Ms. 3243, 111^{rb}–131^{rb}; (6) Paris, Bibliothèque Nationale de France, Ms. lat. 14.576, 117^{ra}–199^{vb}; (7) Paris, Bibliothèque Nationale de France, Ms. lat. 15.561, 198^{ra}–226^{vb}; (8) Praha, Národní Knihovna České Republiky, Ms. III. B. 10, 130^{ra}–139^{vb}, 140^{va}–212^{vb}; (9) Wrocław, Biblioteka Uniwersytecka, Ms. IV. F. 198, 15^{ra}–45^{rb}; (10) Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 4353, 1^{r}–58^{r}; (11) Tortosa, Biblioteca de la Catedral y del Cabildo de la Sanctísima Iglesia Catedral, Ms. 186, 35^{ra}–66^{ra}. For a description of the manuscripts see M. Michałowska (ed.), *Richard Kilvington on the Capacity of Created Beings, Infinity, and Being Simultaneously in Rome and Paris*. Critical Edition of Question 3 from *Quaestiones super libros Sententiarum*, Leiden 2021 (Studien und Texte zur Geistesgeschichte des Mittelalters, 130), 35–49. The eight questions, from the Bologna manuscript are titled as follows: (1) *Utrum Deus sit super omnia diligendus*; (2) *Utrum per opera meritoria augeatur habitus caritatis quo Deus est super omnia diligendus*; (3) *Utrum omnis creatura sit suae naturae certis limitibus circumscripta*; (4) *Utrum quilibet actus voluntatis per se malus sit per se aliquid*; (5) *Utrum peccans mortaliter per instans solum mereatur puniri per infinita instantia interpolata*; (6) *Utrum aliquis nisi forte in poena peccati possit esse perplexus in his quae pertinent ad salutem*; (7) *Utrum omnis actus factus extra gratiam sit peccatum*; (8) *Utrum aliquis possit simul peccare venialiter et mereri vitam aeternam*. A critical edition by E. Jung and M. Michałowska of Kilvington’s question 4 will be submitted to Brill in 2022.

Richard Kilvington, *Quaestiones super libros Ethicorum*, 11–26.

The questions are the following: *Expositio super primum librum Physicorum* (Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 4353, 125^{r}–129^{v}); (1) *Utrum omne scitum sciatur per causam* (Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 4353, 130^{r}–143^{v}); (2) *Utrum omne quod generetur ex contrariis generetur* (Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 4353, 124^{v}; Sevilla, Biblioteca Capitular y Colombina, Ms. 7-7-13, 40^{ra}–50^{vb}; Erfurt, Universitätsbibliothek, Dep. Erf., CA O. 74, 70^{ra}–86^{va}); (3) *Utrum in omni generatione tria principia requirantur* (Sevilla, Biblioteca Capitular y Colombina, Ms. 7-7-13, 27^{rb}–37^{ra}; Paris, Bibliothèque Nationale de France, Ms. lat. 6559, 71^{rb}–88^{rb}; Brugge, Stedelijke Openbare Bibliotheek, Ms. 503, 41^{va}–50^{vb}); (4) *Utrum omnis natura sit principium motus et quietis* (Sevilla, Biblioteca Capitular y Colombina, Ms. 7-7-13, 37^{ra}–40^{vb}); (5) *Utrum potentia motoris excedit potentiam rei motae* (Venezia, Biblioteca Nazionale Marciana, Ms. lat. VI. 72 [2810], 81^{ra}–89^{rb}; Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 2148, 76^{ra–vb}); (6) *Utrum qualitas suscipit magis et minus* (Venezia, Biblioteca Nazionale Marciana, Ms. lat. VI. 72 [2810], 89^{rb}–101^{ra}; Paris, Bibliothèque Nationale de France, Ms. lat. 16.401, 149^{v}–166^{v}; Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 2148, 71^{ra}–75^{ra}; Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 4429, 64^{r}–70^{v}; Paris, Bibliothèque Nationale de France, Ms. lat. 6559, 121^{ra}–131^{ra}; Oxford, Bodleian Library, Ms. Canon. Misc. 226, 61^{v}–65^{r}; Praha, Národní Knihovna České Republiky, Ms. III. B. 10, 140^{va}–152^{vb }; Cambridge, Peterhouse Library, Ms. 195, 70^{rb}–72^{ra}); (7) *Utrum aliquod motus simplex possit moveri aeque velociter in vacuo et in pleno* (Venezia, Biblioteca Nazionale Marciana, Ms. lat. VI. 72 [2810], 101^{ra}–107^{vb}); (8) *Utrum omne transmutatum in transmutationis initio sit in eo ad quod primitus transmutatur* (Venezia, Biblioteca Nazionale Marciana, Ms. lat. VI. 72 [2810], 107^{vb}–112^{rb}).

The questions are the following: (1) *Utrum augmentatio sit motus ad quantitatem*; (2) *Utrum numerus elementorum sit aequalis numero qualitatum primarum*; (3) *Utrum ex omnibus duobus elementis possit tertium generari*; (4) *Utrum continuum sit divisibile in infinitum*; (5) *Utrum omnis actio sit ratione contrarietatis*; (6) *Utrum omnia elementa sint adinvicem transmutabilia*; (7) *Utrum mixtio sit miscibilium alteratorum unio*; (8) *Utrum omnia contraria sint activa et passiva adinvicem*; (9) *Utrum generatio sit transmutatio distincta ab alteratione*. They are to be found, as a complete or incomplete set, in the following manuscripts: (1) Brugge, Stedelijke Openbare Bibliotheek, Ms. 503, 20^{vb}–50^{vb}; (2) Cambridge, Peterhouse Library, Ms. 195, 60^{ra}–81^{rb}; (3) Erfurt, Universitätsbibliothek, Dep. Erf., CA O. 74, 35^{ra}–86^{va}; (4) Kraków, Biblioteka Jagiellońska, Ms. 648, 40^{ra}–53^{rb}; (5) Paris, Bibliothèque Nationale de France, Ms. lat. 6559, 61^{ra}–132^{vb}; (6) Sevilla, Biblioteca Capitular y Colombina, Ms. 7-7-13, 9^{ra}–27^{ra}; (7) Firenze, Biblioteca Nationale Centrale, Cod. Conv. Soppr. B. VI. 1681, 37^{ra}–77^{vb}.

Maier, *Ausgehendes Mittelalter*, 1: 253: ‘Richard Killington ist der Verfasser einer heute verschollenen Erklärung der Physik. Wir erfahren dies aus dem in philosophischer und naturphilosophischer Beziehung hochinteressanten Anonymen Sentenzenkommentar, der sich im Vat. Lat. 986 findet und dessen Verfasser ein Schüler Bradwardines zu sein scheint. Killington wird hier mehrfach zitiert, einmal mit diesen Worten: *hoc probat Kilmiton in questionibus suis super librum Physicorum*. Der zitierte Beweis bezieht sich aus einer berühmten Regel, die Bradwardine in seinem *Tractatus proportionum* aufgestellt hatte: Killingtons *Physikkommentar* muss also nach diesem, d. h. nach 1328 entstanden sein.’

Città del Vaticano, Biblioteca Apostolica Vaticana, Ms. Vat. lat. 986, 56^{rb}: ‘Primus articulus est utrum proportio actionum sequatur proportionem potentiae activae ad passivam seu potentiam resistivam … *Primo hoc probat Kilmington in quaestionibus suis super* Physicorum, nam si sic, igitur subduplus motor non movebit in eodem tempore aeque velocitatis medietatem mobilis, quod movet duplus motor. Quod videtur falsum: cum universaliter enim videtur, qualis est proportio totius ad totum, talis est proportio partis ad partem. Probatur consequentia nam totus motor excedit totum mobile per certum excessum et medietas motoris excedit medietatem motoris per medietatem excessus et per consequens, si velocitatis sequitur excessum potentiae non erit tanta velocitas etc. *Secundo ad hoc arguit Bradwadinus in* Proportionibus *suis*, quia si sic, igitur sequitur quod si aliquis motor excedet suam resistentiam per minorem excessum quam alius suam, tardius moveret illam. Consequens falsum; igitur. Consequentia clara … *Ad primam dico* quod intelligit quod omnis motus provenit a proportione potentiae maioris aequalitatis motoris ad motum … *Ad secundam* iste est intellectus quod secundum quod proportio potentiae motoris ad potentiam rei motae est maior secundum hoc maior velocitas est, sed hoc non est contra me.’ See, Jung-Palczewska, ‘Works,’ 215.

N. Kretzmann, ‘*Tu scis hoc esse omne quod est hoc*: Richard Kilvington and the Logic of Knowledge,’ in N. Kretzmann (ed.), *Meaning and Inference in Medieval Philosophy. Studies in Memory of Jan Pinborg*, Dordrecht 1988 (Synthese Historical Library, 32), 225–245, at 225–226.

See, e.g., Jung-Palczewska, ‘Works,’ 199–203, 218–220; E. Jung and R. Podkoński, ‘Richard Kilvington on Proportions,’ in J. Biard and S. Rommevaux (eds.), *Mathématiques et théorie du mouvement, XIV ^{e}–XVI^{e} siècles*, Villeneuve d’ Ascq 2008 (Histoire des sciences), 81–101.

See E.D. Sylla, ‘The Origin and Fate of Thomas Bradwardine’s *De proportionibus velocitatum in motibus* in Relation to the History of Mathematics,’ in W.R. Laird and S. Roux (eds.), *Mechanics and Natural Philosophy before the Scientific Revolution*, Dordrecht 2008 (Boston Studies in the Philosophy of Science, 254), 67–119, at 95. See also E.D. Sylla, ‘Guide to the Text,’ in John Buridan, *Quaestiones super octo libros Physicorum Aristotelis (secundum ultimam lecturam)*, Libri III–IV, eds. M. Streijger and P.J.J.M. Bakker, Guide to the Text by E.D. Sylla, Leiden 2016 (Medieval and Early Modern Philosophy and Science, 27), XX–CCXVIII, at XXXIV, LXXXI, LXXXII, CXXIX, CLXXXI.

E.D. Sylla’s comment to my paper.

Thomas Bradwardine, *De proportionibus*, 65^{1–5}: ‘Omne motum successivum alteri in velocitate proportionari contingit; quapropter philosophia naturalis, quae de motu considerat, proportionem motuum et velocitatum in motibus ignorare non debet. Et quia cognitio illius est necessaria et multum difficilis, nec in aliqua parte philosophiae tradita est ad plenum, ideo de proportione velocitatum in motibus fecimus istud opus’ (transl. Crosby, 65).

Anonymous, *Tractatus de sex inconvenientibus*, ed. S. Rommevaux-Tani, *De sex inconvenientibus, un traité anonyme de philosophie naturelle au XIV ^{e} siècle*, édition, introduction et étude doctrinale, Paris 2022, 297–364. For a semi-critical edition of question IV see Anonymous,

*Tractatus De sex incovenientibus*, q. IV, ed. J. Papiernik, in Jung and Podkoński,

*Towards the Modern Theory of Motion*, 297–390.

The corresponding passage of the Latin text according to Rommevaux-Tani’s edition runs as follows: ‘Et arguo primo quod non, quia ex isto tunc sequitur quod talis velocitas attenderetur penes excessum potentiarum moventium ad potentias resistentes, sicut ponit una positio; aut penes proportionem excessuum potentiarum moventium ad potentias resistentes, sicut ponit secunda positio … Prime due positiones demonstrative a pluribus improbantur, precise a duobus famosis, a magistro Thoma de Bradwardyn in tractatu suo *De proportionibus* et a magistro Adam Pippewell, qui subtiliter hoc demonstravit’ (ed. Rommevaux-Tani, 295). All English translations are mine unless noted otherwise.

For the problems of dating this text see Rommevaux-Tani’s contribution in this book and her introduction in the edition previously mentioned. Additionally, see also Jung and Podkoński, *Towards the Modern Theory of Motion*, 22–29.

See G.C. Brodrick, *Memorials of Merton College with Biographical Notices of the Wardens and Fellows*, Oxford 1885, 195; A.B. Emden, *A Biographical Register of the University of Oxford to A.D. 1500*, 3: P to Z, Oxford 1959, 1484.

See, e.g., Sylla, ‘The Origin and Fate,’ 67–95; S. Rommevaux, ‘A Treatise on Proportion in the Tradition of Thomas Bradwardine: The *De proportionibus libri duo* (1528) of Jean Fernel,’ *Historia Mathematica* 40 (2013), 164–182.

For Jung’s critical edition of this question, see Jung and Podkoński, *Towards the Modern Theory of Motion*, 215–266.

Crosby (ed.), *Thomas of Bradwardine*, 38.

See Richard Kilvington, *Utrum in omni motu*, 251–252, § 91. See also Thomas Bradwardine *De proportionibus*, 108, 470–500. Quotations from Kilvington’s text come from Jung’s critical edition, mentioned above in n. 49. Quotations and English translations from Bradwardine’s text come from Crosby’s edition and translation mentioned above in n. 18.

For the Oxford *curriculum* see, e.g., Courtenay, *Schools and Scholars*, 219–250; W.J. Courtenay, ‘The Role of English Thought in the Transformation of University Education in the Late Middle Ages’, in J.M. Kittelson and P.J. Transue (eds.), *Rebirth, Reform and Resilience. Universities in Transition, 1300–1700*, Columbus, OH, 1984, 108–115; J.E. Murdoch, ‘*Mathesis in philosophiam scholasticam introducta*. The Rise and Developement of the Application of Mathematics in Fourteenth-Century Philosophy and Theology,’ in *Arts libéraux et philosophie au Moyen Âge. Actes du quatrième Congrès international de philosophie médiévale, Montréal 1967*, Montreal 1969, 216–227; J.E. Murdoch, ‘Mathematics and Sophisms in Late Medieval Natural Philosophy,’ in *Les genres littéraires dans les sources théologiques et philosophiques médiévales: définition, critique et exploitation. Actes du colloque international de Louvain-la-Neuve, U.C.L., Institut d’ études médiévales*, Louvain-la-Neuve 1982 (Textes, Études, Congrès, 5), 85–100; E.D. Sylla, *The Oxford Calculators and the Mathematics of Motion, 1320–1350. Physics and Measurement by Latitudes*, dissertation, Harvard University, 1970 (repr. with preface and errata, New York 1991), 182–304; M. Clagett, ‘The Impact of Archimedes on Medieval Science,’ *Isis* 50 (1959), 419–429, n. 162; M. Clagett and E.A. Moody, *The Medieval Science of Weights (Scientia de ponderibus): Treatises Ascribed to Euclid, Archimedes, Thabat Ibn Qurra, Jordanus de Nemore and Blasius of Parma*, Madison, WI, 1952 (University of Wisconsin Publications in Medieval Science, 1), 3–20.

Richard Kilvington, *Utrum in omni motu*, 215, § 1: ‘Et probo quod non, quia tunc vel esset dare: (1) minimum excessum sufficientem ad motum, ut ponunt quidam, (2) vel maximum excessum non sufficientem ad motum, ut ponunt alii; (3) vel quicumque excessus sufficeret ad continuandum motum non tamen ad incohandum, ut ponunt tertii, (4) vel quicumque excessus sufficeret ad incohandum motum sicut ad continuandum, ut ponunt quarti.’

Richard Kilvington, *Utrum in omni motu*, 215–219, §§ 2–11.

Averroes, *Commentum magnum super libro De celo et mundo Aristotelis* IV, comm. 22, eds. F.J. Carmody and R. Arntzen, Leuven 2003 (2 vols.) (Recherches de Théologie et Philosophie médiévales. Bibliotheca, 4), 694^{98–101}: ‘… illud quod movetur ad suum locum non movetur nisi secundum quod aliquod movetur ad suam formam; quod enim movetur ad formam est illud quod est in potentia forma.’

Richard Kilvington, *Utrum in omni motu*, 215, § 4^{2}–216, § 4^{6}: ‘Et consequentiam probo, quia pono quod ignis incipiat agere in terram puram, cuius centrum est centrum mundi. Tunc quacumque levitate inducta in illa terra erit terra mixta, ergo appetit locum superiorem quam prius quando fuit pura, quia illud quod minus grave est, intensive appetit locum superiorem.’ A similar reasoning is found in Kilvington’s commentary on *De generatione et corruptione*, written before his *Physics* commentary, and in the questions on the *Physics*: *Utrum in omni generatione tria principia requirantur* and *Utrum natura sit principium motus et quietis*.

This assumption is based on the text of Aristotle’s *De generatione et corruptione*, II, 333a21–27: ‘If it is meant that they are comparable in their amount, all the “comparables” must possess an identical something whereby they are measured. If, e.g. one pint of Water yield ten of Air, both are measured by the same unit; and therefore both were from the first and identical something. On the other hand, suppose they are not “comparable in their amount” in the sense that so-much of the one yields so much of the other, but comparable in “power of action” (a pint of Water, e.g. having a power of cooling equal to that of ten pints of Air); even so, they are “comparable in their amount” though not qua “amount” but qua “so-much power”.’ See Aristotle, *De generatione et corruptione (On Generation and Corruption)*, transl. H.H. Joachim, in *The Basic Works of Aristotle*, ed. R. McKeon, New York 2001, 518.

Richard Kilvington, *Utrum in omni motu*, 216–217, § 7^{3–8}: ‘Signetur ergo terra per unum, aqua per 10, aer ⟨per⟩ 100, et ignis per 1000. Tunc sequitur quod ex dicto Philosophi, quod ignis in sua sphaera excedit alia omnia tria elementa residua in maiori proportione quam in millecupla qualis est proportio 9 ad 1; ergo punctus medius inter centrum et concavum orbis lunae est supra valde magnam partem ignis.’

Thomas Bradwardine, *De proportionibus*, 132^{196–200}: ‘Quia per quaedam praedictorum, paucis aliis vero coassumptis, proportio elementorum adinvicem faciliter sciri potest, et eius scientia multum philosophiae congruit naturali, et hucusque latuit cooperta (licet praesenti negotio non multum pertineat) eius latentiam detegemus’ (transl. Crosby, 133).

Thomas Bradwardine, *De proportionibus*, 136^{270–281}: ‘Proportio spherae ignis ad spheram compositam ex tribus elementis residuis est maior proportione 31 ad 1. Nam (per proximam) quodlibet elementum maius trigesies et bis continet proximam sibi minus. Igitur, ad istas quattuor elementorum disponantur isti quattuor termini continue proportionales proportione praedicta (1, 32, 1024, 32768) quorum, si res primi congregantur in unum, 1057 constituent: quod spheram ex inferioribus elementis compositam representat. Per quod congregatum, si quartus terminus (qui ignem significat) dividatur, 31 exhibunt et remanet unitas dividenda. Quartus igitur terminus congregatus praedictum, trigesies et semel amplius continebit. Et hoc cuius demonstrationem quaesivimus’ (transl. Crosby, 137). Bradwardine also operates with the series (1, 10, 100, 1000) of four terms but in order to prove that the distance of the outer surface of the air from the center of the Earth is more than 10 and less than 11 times the radius of the Earth. See Thomas Bradwardine, *De proportionibus*, 138^{298–306}.

Thomas Bradwardine, *De proportionibus*, 94^{183–185}: ‘Sequitur de tertia opinio erronea, quae ponit proportionem velocitatum in motibus (manente eodem motore vel aequali) sequi proportionem passorum, et (manente eodem passo vel aequali) sequi proportionem motoris’ (transl. Crosby, 95).

Thomas Bradwardine, *De proportionibus*, 98^{254–259}: ‘Est autem ista ex mendacio arguenda, quia aliqua potentia motiva localiter potest movere aliquod mobile aliqua tarditate, et potest movere dupla tarditate. Ergo (per istam positionem) potest movere duplum mobile. Et potest movere quadrupla tarditate: igitur quadruplum mobile, et sic in infinitum. Igitur quaelibet potentia motiva localiter esset infinita.’ (transl. Crosby, 99).

Richard Kilvington, *Utrum in omni motu*, 252–258, §§ 93–105.

Richard Kilvington, *Utrum in omni motu*, 253, § 93^{12–13}: ‘Pro primo articulo dico, quod quilibet excessus sufficit ad motum.’

Richard Kilvington, *Utrum in omni motu*, 253, § 95^{3–21}: ‘Sed dico: omnibus aliis paribus concedi potest conclusio et iuxta sententiam Philosophi, quod terra est centrum in continuo motu. Et quando dicitur quod Philosophus vult in *De motu animalium* quod omne motum in suo motu necessario indiget fixo, ideo centrum indiget terra quiescente, dico quod intelligit quod terra quiescit a tali motu circulari quali movetur coelum. Non enim volo ponere quod terra moveatur motu circulari aeternaliter circa suum centrum, sicut posuit Plato … Sed dico, quod ceteris ⟨paribus⟩ terra movetur aeternaliter dicendo alterius motibus secundum partem graviorem, verum tamen motu valde tardo et insensibili. Et quiescit terra a motu sensibili, non obstante quod insensibiliter moveatur … Dico etiam ultra quod, si terra naturaliter debet esse sphaerica, est tam vallosa et in quibusdam partibus plana, propter convenientiorem habitudinem animalium, sicut aqua congregata in certis locis non obstante quodam terram undique circumdare, ita quod ista sunt quodammodo violenta et aliqualiter propter naturam, non tamen violenta propter peius sed propter melius, et talia violenta sunt sive possunt esse aeterna.’ See also Jung-Palczewska, ‘Works,’ 216–217; E. Grant, ‘Cosmology,’ in D.C. Lindberg (ed.), *Science in the Middle Ages*, Chicago 1978 (Chicago History of Science and Medicine), 265–302 esp. 290–291. John Buridan also presents the same conclusions. See John Buridan, *Quaestiones super libris quattuor De caelo et mundo*, ed. E.A. Moody, Cambridge, MA, 1942, 231.

See, e.g., E. Jung, ‘Richard Kilvington,’ in E.N. Zalta (ed.), *The Stanford Encyclopedia of Philosophy* (Winter 2016 Edition), URL = *The Stanford Encyclopedia of Philosophy* (Spring 2018 Edition), URL =

For a detailed discussion of the limit-decision problem see, e.g., C. Wilson, *William Heytesbury: Medieval Logic and the Rise of Mathematical Physics*, Madison, WI, 2nd ed., 1960 (University of Wisconsin Publications in Medieval Science, 3), and J. Longeway, *William Heytesbury: On Maxima and Minima. Chapter 5 of* Rules for Solving Sophismata *with an Anonymous Fourteenth-Century Discussion*, Dordrecht 1984 (Synthese Historical Library, 26).

See Longeway, *William Heytesbury*, 135.

Richard Kilvington, *Utrum in omni motu*, 259, § 114^{27–29}: ‘Et ideo verum est quod, sicut potentia passiva terminatur per minimum cum certis circumstantiis, ita potest terminari per maius cum circumstantiis proportionabilibus.’

Richard Kilvington, *Utrum in omni motu*, 259, § 114.

For a detailed discussion, see E. Jung, ‘Richard Kilvington on Local Motion,’ in P.J.J.M. Bakker (ed.) with the collaboration of E. Faye and C. Grellard, *Chemins de la pensée médiévale. Études offertes à Zénon Kaluza*, Turnhout 2002 (Textes et études du Moyen Âge, 20), 116–128.

Richard Kilvington, *Utrum in omni motu*, 233, § 48^{29–31}: ‘Item, si quaestio esset vera, tunc velocitas motus vel sequeretur excessum potentiae moventis super potentiam rei motae vel proportionem potentiae moventis super potentiam rei motae.’

Crosby (ed.), *Thomas of Bradwardine*, 38. 37.

Richard Kilvington, *Utrum in omni motu*, 234, § 49; Thomas Bradwardine, *De proportionibus*, 86^{9–11}. Averroes, *In Physicam* IV, comm. 71, 161^{rb–va}: ‘Proportio tarditatis ad tarditatem est sicut proportio impedimentis ad impediens ut dicit Avempace … Et si concesserimus quod proportio motuum, que fuerit in medio, adinvicem est sicut proportio impedimentis ad impedientem, quando idem motum movetur in vacuo, movetur motu indivisibili, et in instanti, sed sequitur necessario ut moveatur in tempore, cuius proportio ad tempus in quo movetur in medio est sicut proportio excessus potentie motoris super rem motam.’

Richard Kilvington, *Utrum in omni motu*, 234, § 49; Thomas Bradwardine, *De proportionibus*, 86^{14–15}. Averroes, *In Physicam* VII, comm. 39, 337^{va}.

Thomas Bradwardine, *De proportionibus*, 86^{5–16}.

Richard Kilvington, *Utrum in omni motu*, 234, § 50^{8}: ‘Sed contra istam opinionem potest sic argui multipliciter.’ Thomas Bradwardine, *De proportionibus*, 86^{17}: ‘Haec autem opinio destrui poterit multis modis.’

Cf. Aristotle, *Physica* VII, 250a25–28; Averroes, *In Physicam* VII, comm. 36, 335^{va}. Averroes writes: ‘Ut declaverunt Geometrae.’ Kilvington writes: ‘Sic declarant geometrae et dicit Commentator VII *Physicorum* commento 36.’ Bradwardine states: ‘Sicut universaliter demonstrant geometri,’ and supplements this statement with a mathematical argument referring to the introductory chapter of his treatise.

Transl. Crosby, 99.

Transl. Crosby, 99.

Thomas Bradwardine, *De proportionibus*, 94^{183–186}. For Latin quote see above, n. 62 (transl. Crosby, 95).

Richard Kilvington, *Utrum in omni motu*, 236, § 54^{4–9}: ‘Item, posito quod aliquis homo trahat unam fabam per unam cordam currendo ita velociter sicut potest, tunc si alius homo tantae potentiae ad currendum sibi iniungatur ad trahendum illam fabam praedictam, illi duo homines non trahent velociter quam unus illorum per se.’

Richard Kilvington, *Utrum in omni motu*, 236, § 55^{9–18}: ‘Posito quod tres homines forte nitantur trahere navem quam non possunt movere nisi girando huc illuc, tunc isti tres homines unam actionem in navem faciunt. Si addatur quartus homo aequalis potentiae alicuius praedictorum, illi quattuor homines movebunt navem directe per magnum spatium, ut experimento notum est. Ex quo sequitur, quod minus quam agens duplum facit plus quam duplam actionem, quia si primi tres homines agant in navem prius directe per medietatem tanti spatii, per quantum movent isti quattuor homines, fecissent medietatem actionis praecise quam faciant isti quattuor homines.’ See also Averroes, *In Physicam* VIII, comm. 23, 359^{ra}; Aristotle, *Physica* VIII, 253b.

Richard Kilvington, *Utrum in omni motu*, 236, § 56^{21–24}: ‘Item, exemplum potest adduci de homine portante aliquod pondus cum quo vix potest ire lento passu. Tunc minori addito quam aequali pondere movebitur in duplo tardius et vix aliquo modo vadit.’

Thomas Bradwardine, *De proportionibus*, 92^{145–150}: ‘Videmus enim quod musca portando aliquod modicum velociter multum volat, et puer aliquod modicum velociter satis movet, et homo fortis unum magnum mobile (quod vix potest movere) movet valde tarde. Et licet illi mobile apponatur maius quam Musca vel puer posset movere, movet totum non multum tardius tunc quam prius.’

Thomas Bradwardine, *De proportionibus*, 86^{5–6}: ‘Prima ponit proportionem velocitatum in motibus sequi excessum potentiae motoris ad potentiam rei motae’ (transl. Crosby, 87).

Thomas Bradwardine, *De proportionibus*, 110^{1–5}: ‘His igitur ignorantiae nebulis demonstrationum flatibus effungatis, superest ut lumen scientiae resplendeat veritatis. Scientia autem veritatis ponit quintam opinionem, dicentem quod proportio velocitatum in motibus sequitur proportionem potentiae motoris ad potentiam rei motae’ (transl. Crosby, 111).

Richard Kilvington, *Utrum in omni motu*, 237, § 59^{7}: ‘Ex quibus concluditur haec opinio secunda.’

Cf. Averroes, *In Physicam*, IV, comm. 71, 160^{vb}; VII, comm. 35, 335^{ra}; *In De coelo* II, comm. 36.

Richard Kilvington, *Utrum in omni motu*, 238–251, §§ 60–89.

Richard Kilvington, *Utrum in omni motu*, 262–266, §§ 123–135.

Richard Kilvington, *Utrum in omni motu*, 238, § 61^{9}: ‘Duplum grave praecise non movebitur praecise in duplo velocius in eodem medio quam grave subduplum.’

Richard Kilvington, *Utrum in omni motu*, 238, § 61^{11–14}: ‘Posito quod grave simplex excedat resistentiam medii sicut tria excedunt unum, tunc duplata sua gravitate se habebit ad eandem resistentiam sicut sex ad unum, quae proportio non est dupla ad primam proportionem.’

On the medieval theory of proportion, see Sylla, ‘The Origin and Fate.’ As J.E. Murdoch and E.D. Sylla, ‘The Science of Motion,’ in D.C. Lindberg (ed.), *Science in the Middle Ages*, Chicago 1978 (Chicago History of Science and Medicine), 206–264, at 225–226, explain: ‘The standard medieval mathematics of proportions in general terms tells us that the only way to increase or decrease proportions is to do so by (in *our* terms) multiplying or dividing proportion by proportions. For example, to increase the proportion 3 to 1 to its double would be to multiply it by itself (yielding 9 to 1 as its double); to decrease the proportion 16 to 1 to its half would be to divide it by proportion 4 to 1 (yielding 4 to 1 as its half) or to increase the proportion 3 to 2 to the proportion of 2 to 1 would be to multiply it by the proportion 4 to 3 (yielding 12 to 6, which equals 2 to 1). One should note a difference between the first two of these examples and the third. In the first two, we have a multiplication or a division which amounts, respectively, to taking powers or roots. Thus, for the first example we have (in modern notation): 9/3=3/1, and thus multiplying 3/1 by itself gives 9/3 · 3/1=9/1. For the second example, we have: 16/4=4/1 and thus dividing 16/1 by 4/1 gives 16/1=16/4 · 4/1. However, in the third example, we do not have equal proportions, and, therefore, we do not have a problem of powers or roots. But the multiplication of 3/2 by the *unequal* proportion 4/3 is still what is involved in the “increase” of 3/2. Thus: 3/2 · 4/3=12/6.’ For more information about the standard medieval calculus of proportions, see also J.A. Weisheipl, ‘Ockham and some Mertonians,’ *Mediaeval* *Studies* 30 (1968), 163–213, repr. in J.I. Catto and T.A.R. Evans (eds.), *The History of the University of Oxford*, 2: *Late Medieval Oxford*, Oxford 1992, 607–658; M. Clagett, *The Science of Mechanics in the Middle Ages*, Madison, WI, 1959 and 1961 (University of Wisconsin Publications in Medieval Science, 4), 440–444, 465–503; A.G. Molland, ‘The Geometrical Background to the “Merton School”,’ *British Journal for the History of Science* 4 (1968–1969), 108–125; J.E. Murdoch, ‘The Medieval Language of Proportions: Elements of the Interaction with Greek Foundations and the Development of New Mathematical Techniques,’ in A.C. Crombie, *Scientific Change. Historical Studies in the Intellectual, Social and Technical Conditions for Scientific Discovery and Technical Invention, from Antiquity to the Present*. *Symposium on the History of Science, University of Oxford 9–15 July, 1961*, London 1963, 237–271.

Richard Kilvington, *Utrum in omni motu*, 238, § 61^{14}–239, § 61^{4}: ‘Nam proportio nonupla est praecise dupla ad proportionem triplam, ut potest demonstrari per definitionem proportionis duplicatae positam V Euclidis, nam sicut unum ad tria, ita tria ad novem. Ergo proportio novem ad unum est praecise dupla ad proportionem ⟨tres⟩ ad unum, per definitionem proportionis duplicatae quae est quod, si fuerint tria continue proportionabilia proportione inaequalitatis, tunc proportio tertii ad primum est proportio secundi ad primum duplicata. Ex quo sequitur, quod proportio nonupla est praecise dupla ad proportionem triplam. Et per consequens proportio sextupla est minor quam dupla ad triplam. Et sequitur quod, si grave excedit resistentiam medii in quo movetur in tripla proportione praecise, quod duplum grave non movebitur in illo medio in duplo velocius, immo minus quam in duplo velocius, cum velocitas motus sequatur proportionem, ut ponit haec opinio.’

Thomas Bradwardine, *De proportionibus*, 112^{64–66}: ‘Si potentiae moventis ad potentiam sui moti sit maior quam dupla proportio, potentia motiva geminate motus eiusdem duplam velocitatem nequaquam attinget’ (transl. Crosby, 113).

Richard Kilvington, *Utrum in omni motu*, 239, § 62^{5–11}: ‘Posito quod aliquod grave excedat resistentiam medii in quo movetur sicut sex excedunt quattuor. Tunc duplata sua gravitate se habebit ad suam resistentiam eandem sicut duodecim ad quattuor, quae est maior quam dupla proportio ad proportionem sex ad quattuor. Nam sicut sex ad quattuor, ita novem ad sex, quia utraque proportio est sexquialtera. Ergo proportio novem ad quattuor est dupla ad proportionem sex ad quattuor, quoniam sexquialtera proportio duplicata est dupla ad unam sexquialteram.’

In the fourteenth century, various formulations of this definition were known. See Euclid (Campanus), *Elementa*, ed. H.L.L. Busard, *Campanus of Novara and Euclid’s Elements*, Wiesbaden 2005 (Boethius. Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaft, 51/1), book V, def. X, 168: ‘Cum fuerint tres quantitates continue proportionales, dicetur proportio prime ad tertiam proportio prime ad secundam duplicata’; Euclid, *Elementa*, eds. H.L.L. Busard, *The Mediaeval Latin Translation of Euclid’s Elements Made Directly from the Greek*, Wiesbaden 1987 (Boethius. Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaft, 15), book V, def IX, 109: ‘Quando vero tres quantitates proportionales fuerint, prima ad tertiam duplicem proportionem habere dicitur quam ad secundam’; Euclid (Adelard of Bath), *Elementa*, ed. H.L.L. Busard, *The First Latin Translation of Euclid’s* Elements *Commonly Ascribed to Adelard of Bath*, Toronto, ON, 1983 (Studies and Texts, 64), 146: ‘Cum fuerint tres quantitates proportionales, erit proportio prime ad tertiam sicut proportio prime ad secundam repetita’; Euclid (Robert of Chester), *Elementa*, eds. H.L.L. Busard and M. Folkerts, *Robert of Chester’s (?) Redaction of Euclid’s* Elements, *the so-called Adelard II Version*, Basel 1992 (Science Networks-Historical Studies, 9), 1: 161: ‘Cum fuerint tres quantitates proportionales, dicetur proportio prime ad terciam proportio prime ad secundam duplicata’; Euclid (Johannes de Tinemue), *Elementa*, eds. H.L.L. Busard, *Johannes de Tinemue’s redaction of Euclid’s* Elements *the so-called Adelard III version*, Stuttgart 2001 (Boethius. Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaft, 45), 129: ‘Cum fuerint tres quantitates proportionales, erit proportio prime ad tertiam, prime ad secundam geminata. Verbi gratia: Proportio 9 ad 4 est quasi proportio 9 ad 6 dupla quia constat quasi totaliter ex ea que est inter 9 ad 6 et 6 ad 4 cuius equaliter sit, utrobique sit sexquialtera.’ It seems that Kilvington follows Johannes de Tinemue while Bradwardine follows Axiom 7 from Book V of Euclid’s *Elements*: ‘Si fuerit proportio maioris inequalitatis primi ad secundum ut secundi ad tertium, erit proportio primi ad tertium dupla ad proportionem primi ad secundum et secundi ad tertium’ (Thomas Bradwardine, *De proportionibus*, 78^{297–302}).

Richard Kilvington, *Utrum in omni motu*, 240, § 62. Euclid, *Liber de ponderoso et levi et comparatione corporum ad invicem*, ed. M. Clagett, in M. Clagett and E.A. Moody, *The Medieval Science of Weights (Scientia de ponderibus): Treatises Ascribed to Euclid, Archimedes, Thabat Ibn Qurra, Jordanus de Nemore and Blasius of Parma*, Madison, WI, 1952 (University of Wisconsin Publications in Medieval Science, 1), 23–32, at 28; Jordanus de Nemore, *Liber de ponderibus*, ed. E.A. Moody, in Clagett and Moody, *The Medieval Science of Weights*, P.01, 154: ‘Inter quelibet gravia est virtutis et ponderis eodem ordine sumpta proportio’; R1.01, 174: ‘Inter quelibet gravia est virtutis et ponderis eodem ordine sumpta proportio’; Jordanus de Nemore, *Elementa super demonstrationem ponderum*, ed. E.A. Moody, in Clagett and Moody, *The Medieval Science of Weight*, E.1, 128: ‘Inter quelibet gravia est velocitatis in descendendo eodem ordine sumpta proportio, descensus autem et contrarii motus proportio eadem sed permutata’; Thomas Bradwardine, *De proportionibus*, 96^{236–238}; 100^{368}–102^{343}; 102^{364}–104^{385}.

Richard Kilvington, *Utrum in omni motu*, 241, § 63^{4–18}: ‘Secundo ex ista opinione sequitur ista conclusio, quod idem grave simplex motum in aqua alicuius densitatis non movebitur praecise in duplo velocius in aere duplae subtilitatis. Haec autem conclusio sequitur per similes determinationes prioribus. Nam si illud grave excedat resistentiam aquae in proportione maiori quam dupla, tunc excedit aerem duplae subtilitatis in proportione minori quam dupla ad illam proportionem, ut patet per deductionem priorem; ergo illud grave movebitur in aere duplae subtilitatis minus quam in duplo velocius quam movebitur in ista aqua. Et si illud grave simplex excedat resistentiam aquae in proportione maiori quam in dupla, tunc excedet resistentiam duplae subtilitatis in proportione maiori quam dupla ad proportionem primam, sicut demonstrari potest ut prius per definitionem proportionis duplicatae. Ergo tunc movebitur tale grave in aere duplae subtilitatis praecise plus quam in duplo velocius quam movebatur in aqua.’

See below, p. 74. The same situation occurs when the resistance is halved. See Crosby (ed.), *Thomas of Bradwardine*, 39.

Richard Kilvington, *Utrum in omni motu*, 242, § 64^{1–11}: ‘Tertio ex ista opinione sequitur ista conclusio, quod istae regulae Philosophi in VII *Physicorum* sint falsae. Quarum una ponitur in textu commento 37 et isto commento, quae est illa: “si aliquis motor moveat mobile per aliquod spatium in aliquo tempore, eadem potentia motoris movebit medietatem illius mobilis per duplum spatium in aequali tempore et per aequale spatium in medietate temporis,” ut dicitur commento 36. Et alia regula ponitur textu commenti ultimi, quae est quod “si aliqua potentia moveat aliquod mobile per aliquod spatium in aliquo tempore, dupla potentia movebit idem mobile per duplum spatium in aequali tempore,” quod est ex praecedentibus notum esse falsum.’ See Aristotle, *Physica* VII, 250a1–4; Averroes, *In Physicam* VII, comm. 36, 335^{va}; Thomas Bradwardine, *De proportionibus*, 96^{215–219}.

For the Latin, see above, n. 61.

Thomas Bradwardine, *De proportionibus*, 96^{247–249}: ‘Ista tamen positio est dupliciter arguenda; primo super insufficientia, secundo super mendacio consequentiae’ (transl. Crosby, 97).

Richard Kilvington, *Utrum in omni motu*, 263, § 126^{8–10}: ‘Et per hanc glossam dissolvi poterunt conclusions Archimedis et Jordani *De ponderibus* et Euclidis, quae sunt in contrarium allegatae.’ Thomas Bradwardine, *De proportionibus*, 100^{338}: ‘Conclusio autem allegata *De ponderibus* similiter debet intelligi …’

Richard Kilvington, *Utrum in omni motu*, 263, § 127^{13–21}: ‘Dico quod duo motus duorum gravium aequalium, quorum unus descendit in aqua pedalis quantitatis et aliud in aere bipedalis quantitatis duplae tamen subtilitatis, non sunt aeque veloces nec aequalis est proportio motoris ad suam resistentiam utrobique. Et quando dicitur quod resistentia aeris et resistentia illius aquae sunt aequales, dico quod non, quia non sunt aequales extensive (notum est) nec intensive, quia iste aer propter eius subtilitatem non potest tantum resistere suo motori ad causandum motum retardari, sicut potest ista aqua propter eius densitatem, propter hoc quod partes eius propinquius iacent et melius poterunt applicari ad causandum motum tardum.’

For Kilvington’s discussion of this distinction see *Utrum in omni motu*, 243, § 67; 263, §§ 127–128. For the parallel in Bradwardine, see *De proportionibus*, 116–120 (transl. Crosby, 117–121).

Richard Kilvington, *Utrum in omni motu*, 244, § 686^{6–7}: ‘Aliquod corpus movetur velocius alio et hoc in medio denso.’

Richard Kilvington, *Utrum in omni motu*, 244–246, §§ 68–75.

Thomas Bradwardine, *De proportionibus*, 114^{98–99}: ‘Quantumcumque gravius in eodem medio tardius et velocius illo et aequali velocitate potest descendere’ (transl. Crosby, 115).

Crosby (ed.), *Thomas of Bradwardine*, 41.

Richard Kilvington, *Utrum in omni motu*, 246, § 74^{11–12}: ‘Aliquid movetur in pleno et in vacuo aeque velociter.’

Richard Kilvington, *Utrum in omni motu*, 246–247, §§ 74–75.

Thomas Bradwardine, *De proportionibus*, 116^{127–128} (transl. Crosby, 117).

Richard Kilvington, *Utrum in omni motu*, 247, § 76^{17–18}: ‘Aliquid movebitur infinita velocitate, si velocitas motus sequatur proportionem.’

Richard Kilvington, *Utrum in omni motu*, 247–248, §§ 76–80.

Richard Kilvington, *Utrum in omni motu*, 265–266, §§ 133–135. Cf. Thomas Bradwardine, *De proportionibus*, 124^{293–296}.

Murdoch and Sylla, ‘The Science of Motion,’ 224.

See Sylla, *The Oxford Calculators*, 436.

Sylla, *The Oxford Calculators*, 403.

Since both Kilvington’s question and Bradwardine’s treatise were put together by the authors themselves, and were not *reportationes* of classroom disputations, it seems that Bradwardine, like Ockham before, did not assume that his readers would be using his treatise in conjunction with Kilvington’s work. See E.D. Sylla, ‘Walter Burley’s *Tractatus primus*: Evidence Concerning the Relations of Disputations and Written Works,’ *Franciscan Studies* 44 (1984) (*William of Ockham [1285–1347] Commemorative Issue*), 1: 257–274, at 258.

See Rommevaux-Tani’s contribution to the present volume.

See Jung and Podkoński, ‘Richard Kilvington on Proportions.’

See S.M. Stigler, ‘Stigler’s Law of Eponymy’, *Transactions of the New York Academy of Sciences* 39 (1980), 147–158.