## Richard Swineshead

**-**

## Swineshead (Swyneshed, Suicet, etc.), Richard

# SWINESHEAD (SWYNESHED, SUICET, ETC.), RICHARD

(*fl. ca.* 1340–1355).

*natural philosophy*

The name Richard Swineshead is best known to the modern historian of science as that of the author of the *Liber calcualationum*, a work composed probably about 1340–1350 and famous later for its extensive use of mathematics within physics. Very little is known about this Richard Swineshead, and furthermore it appears almost certain that the little biographical data that are available about any fourteenth-century Swineshead cannot all be apportioned to one man, but that there were at least two or three men named Swineshead who may have left works in manuscript. In the early twentieth century, Pierre Duhem settled this confusion of Swinesheads to his own satisfaction by asserting that there was a John Swineshead who wrote some extant logical works, *De insolubilibus* and *De obligationibus*, and a Roger Swineshead who wrote a work on physics (found in MS Paris, Bibliothèque Nationale lat. 16621) – variously titled *De motibus naturalibus, Descriptiones motuum*, and *De primo motore* (the latter by Duhem). Duhem concluded that the famous“Calculator,”as the author of the *Liber calculationum* was often called, was not really named Swineshead at all, but was rather one Richard of“Ghlymi Eshedi,”as he is called in the explicit of the *Liber calculationum* in MS Paris, Bibliothèque Nationale lat. 6558, f. 70v. Since Duhem’s time, historians have rejected the supposed name“Ghlymi Eshedi”as a scribal error and have restored the *Liber calculationum* to Swineshead. They have not, however, completely unraveled the problem of the existence of two or three Swinesheads as authors of several logical and natural philosophical works.

The most satisfactory theory so far proposed would seem to be that of James Weisheipl, according to whom there were three fourteenth-century Swinesheads of note. One, named John Swineshead, was a fellow of Merton College from at least 1343 and pursued a career in law; he died in 1372, leaving no extant works. A second, named Roger Swineshead, was also at Oxford, but there is no record of his having been at Merton College. This Roger Swineshead wrote the logical works *De insolubilibus* and *De obligationibus* and the physical work *De motibus naturalibus*. He may have been a Benedictine monk and a master in sacred theology and may have died about 1365. The third Swineshead, Richard Swineshead, was, like John, associated with Merton College in the 1340’s and was the author of the famous *Liber calculationum*, and possibly also of two extant *opuscula, De motu* and *De motu locali*, and of at least a partial *De caelo* commentary.

Given the uncertainty of the biographical data, it seems proper that all the extant physical works ascribed to any Swineshead should be included in this article. This includes, most importantly, the *Liber calculationum*, but also the *opuscula* ascribed by Weisheipl to Richard Swineshead and the *De motibus naturalibus* ascribed by Weisheipl to Roger Swineshead. All of these works can be said to fall within the“Oxford calculatory tradition,”if not with the works of the so-called Merton school. The *De motibus naturalibus*, as the earliest work, will be described first (with folio references to MS Erfurt, Amplonian F 135), followed by the *Liber calculationum* (with folio references to the 1520 Venice edition), to be described in much greater detail, and finally by the fragmentary *opuscula* (with folio references to MS Cambridge, Gonville and Caius 499/268), which, although they most probably were written before the *Liber calculationum,* can more easily be described after that work. Weisheipl’s hypothesis will be followed as to the correct names of the authors of these works.

## De motibus naturalibus

The *De motibus naturalibus* was written at Oxford after the *De proportionibus* of Thomas Bradwardine and at about the same time (ca. 1335) as the *Regule solvendi sophismata* of William Heytesbury. In the material covered, it is similar to the latter work, and, in fact, both works treat topics that were to become standard in treatises *de motu* in the mid- and late fourteenth century. The *De motibus naturalibus* has eight parts, called *differentiae*: I. Introduction ; II. Definitions of Motion and Time; III, Generation; IV. Alteration; V. Augmentation; VI. Local Motion; VII. Causes of Motion; and VIII. Maxima and Minima. In contrast to Heytesbury’s work and to later treatises *de motu*, however, the *De motibus naturalibus* includes large sections of traditional natural philosophy as well as the logicomathematical natural philosophy typical of the later treatises. It contains many more facts about the natural world (climates, burning mirrors, tides, comets, milk, apples, frogs, worms, etc.) and lacks the strong sophismata character of some of the later works. It represents, therefore, to some extent, a stage halfway between thirteenth-century cosmological and fourteenth-century logicomathematical natural philosophy.

This position, halfway between two traditions, is represented quite strongly in the organization of the work: the treatise is fairly clearly divided into metaphysical-physical discussions and logicomathemathical treatments. Thus, for example, the three parts discussing motion in the categories of quality, quantity, and place (parts IV, V, VI) each contain two parts, a first dealing with the physics of the situation and a second dealing with the quantification of motion in that category. Although the logicomathematical topics that Roger discusses are generally those discussed by the later authors *de motu,* the order of topics in his work still reflects an Aristotelian or medical base. Whereas later authors, especially Parisian-trained authors such as Albert of Saxony, generally discussed the measures of motion with respect to cause first (*penes quid attenditur motus tanquam penes causam*) and then discussed the measures of motion with respect to effect (*tanquam penes effectum*), Roger begins with the effects of motion (as, indeed, does the Calculator after him). Furthermore, among effects, he begins with the effects of alteration rather than with the effects of local motion. In accordance with this order of treatment. Roger’s basic notions of the measurement of motion come from the category of quality rather than from causes or from locomotion, as was to be the case in seventeenth-century physics. In line with the earlier medical theory of the temperate, Roger places emphasis on mean degrees, and he considers intension at the same time as remission. When he then goes on to talk about possible mean degrees of local motion, and motions being just as fast as they are slow, and so forth, there seems to be no reasonable explanation except that he has taken“measures”which fit with the then current notions of quality and alteration and has applied them by analogy to local motion, even though the result has no apparent basis in the then current notions of local motion. Finally, to all of this, must be added the fact that Roger’s basic theoretical terms for his measurements, namely “latitude,”“degree,”and the like, are earlier found most prominently in medical theory.

From among all the material in the *De motibus naturalibus* concerned with the “measurement”of motion and therefore most closely related to the work of the Calculator perhaps the two points of greatest interest have to do with two idiosyncratic positions which Roger takes, the first having to do with the function relating forces, resistances, and velocities in motion, the second with the relation of latitudes and degrees.

First, in parts VI and VII, Roger rejects the Aristotelian position that velocity is proportional to force and inversely proportional to resistance. Thus in the first chapter of part VI Roger states five conclusions concerning natural local motion which are all aimed at showing that resistance is not required for natural motion (41va-41vb). In part VII, Roger again repeats this view (43vb-44vb). In fact, he says, the equality or inequlity of velocities is caused by the equality or inequality of the proportion of proportions of the mover to the moved, where the moved need not resist. Although Roger then accepts the mathematical preliminaries (for example, definitions of the types of proportionalities) that Bradwardine had set down as requisite for investigating velocities, forces, and resistances, he rejects Bradwardine’s function relating these variables (see the article on Bradwardine for a description of his logarithmic-type function). Where there is no resistance, Roger asserts, the proportion of velocities is the same as the proportion of moving powers. Where there are resistances, then the proportion of velocities is the same as the proportion of latitudes of resistance between the degrees of resistance equal to the motive powers and the degrees of the media (this conclusion is equivalent in modern terms to stating that velocity is proportional to the difference between the force and resistance). Concerning cases where one motion is resisted and the other is not, Roger says that the proportion of velocities follows no other proportion, or, in modern terms, that he can find no function relating the velocities to forces and resistances. Although there are some obvious resemblances between Roger’s position and the position of Ibn Bājja (Avempace) and the young Galileo, it is not a position at all common in the early fourteenth century.

Second, Roger’s combinations of latitudes and degrees for measuring motion are also unique to him, so far as is known. Like the latitudes of earlier authors, Roger’s latitudes are ranges within which a given quality, motion, or whatever, may be supposed to vary. Thus in part IV he posits the existence of three latitudes for measuring alteration, each distinguishable by reason into two other latitudes (39ra-39rb). In modern terms the first of these latitudes expresses the range within which the intensities of a quality may vary, the second expresses the range within which velocities of alteration may vary, and the third the range within which accelerations and decelerations of alteration may vary. Similarly, in part VI Roger posits five latitudes for measuring locomotion, all of them distinct from one another only in reason (43ra). In modern terms the first three of these latitudes are the ranges within which velocity or speed may vary and the last two are ranges within which accelerations and decelerations, respectively, may vary. All of these are similar to latitudes posited by the other Oxford calculators, although later there was a tendency to dispense with the latitudes of remissness and tardity that Roger posited (see below).

What is different about Roger’s system is his postulation of so-called uniform degrees. Thus Roger defines two types of degrees of heat or any other quality (38rb). One type, the “uniformly difform degree,” is a component, divisible part of a latitude of quality and is like the degrees hypothesized by later authors including the Calculator . In calling these degrees “uniformly difform,” Roger imagines that each such degree will contain within itself a linearly increasing series of degrees above some minimum and below some maximum, as indeed any segment of a latittude would contain. The other type, the “uniform degree,”is not a component part of a latitude, but rather is equally intense throughout, whereas in any part of a latitude the intensity varies. Among the Oxford calculators, only Roger makes such a distinction. Uniform degrees appear again when Roger goes on to discuss the measurement of the velocity of alteration. In the motion of intension of a quality, he says, two velocities of intension can have no ratio to one another if one subject gains a single uniform degree more than the other (39ra). Similarly Roger concludes that some local motions are incomparable to others, and that one latitude of local motion can differ from another by a single uniform degree (43va).

Roger’s postulation of uniform degrees having no proportion to latitudes does not seem to be the result of intentional atomism. Rather, the case seems to be that as a pioneer in the effort to find mathematical descriptions and comparisons of concrete distributions of qualities and velocities, he could not devise measures applicable to all cases. Earlier authors had made little attempt to deal with nonuniform distributions of qualities. Roger does try to deal with them, but he has one measure for uniform distributions (the uniform degree), and another for uniformly difform (linearly varying) distributions (the latitude), and none at all for difformly difform (nonlinearly varying) distributions. Rather than stating that he is unable to compare motions or distributions of quality that fall into different categories (which would be, from a modern point of view, the justifiable statement), he says that the motions or distributions themselves have no proportion. It seems very likely that it was exactly the kind of effort to “measure”motion represented by the *De motibus naturalibus* that motivated the Calculator to try to straighten things out in his mathematically much more sophisticated work.

## Liber calculationum

The *Liber calculationum* is by far the most famous work associated with the name Swines-head. As it appears in the 1520 Venice edition the *Liber calculationum* contains sixteen parts or *tractatus*. Some of these treatises may have been composed later than others, since they are lacking from some of the extant manuscripts. The emphasis in the *Liber calculationum* is on logicomathematical techniques rather than on physical theory. What it provides are techniques for calculating the values of physical variables and their changes, or for solving problems or sophisms about physical changes. Thus, the order of the treatise is one of increasing complexity in the application of techniques rather than an order determined by categories of subject matter, and the criteria for choosing between competing positions on various topics are often logicomathematical criteria. Thus, it is considered important that theory be complete–that it be able to handle all conceivable cases. Similarly, it is considered important that the mathematical measurements of a given physical variable be continuous, so that, for instance, the mathematical measurements of a given physical variable be continuous, so that, for instance, the mathematical measure of an intensity should not jump suddenly from zero to four degrees (unless there is reason to believe that an instantaneous change occurs physically).

As stated above, as late as the beginning of the fourteenth century natural philosophers dealing with the qualities of subjects (for instance, Walter Burley in his treatises on the intension and remission of forms) assumed tacitly that the individual subjects they dealt with were uniformly qualified. Thus, as in the pharmaceutical tradition, they could talk of a subject hot in the second degree or cold in the third degree (and perhaps about what the result of their combination would be) without questioning whether the individual subjects had qualitative variations within themselves. Roger Swineshead in the *De motibus naturalibus* attempted to deal with variations in distribution, but managed only to establish criteria for uniform and for uniformly varying (*uniformiter difformis*) distributions. Richard Swineshead in his *opuscula De motu* and *De motu locali* declared that difform distributions are too diverse to deal with theoretically (212ra, 213rb). In the *Liber calculationum*, however, he manages to deal with a good number of more complicated (*difformiter difformis*) distributions.

The overall outline of the *Liber calculationum* is as follows. It begins with four treatises dealing with the qualitative degrees of simple and mixed subjects insofar as the degrees of the subjects depend on the degrees in their various parts. Treatise I considers measures of intensity (and, conversely, of remissness, that is, of privations of intensity) per se. Treatise II, on difform qualities and difformly qualified bodies, considers the effects of variations in two dimensions–intensity and extension–on the intensity of a subject taken as a whole. Treatise III again considers two variables in examining how the intensities of two qualities, for example, hotness and dryness, are to be combined in determining the intensity of an elemental subject (this, of course, being related to the Aristotelian theory that each of the four terrestrial elements-earth, air, fire, and water-is qualified in some degree by a combination of two of the four basic elemental qualities-hotness, coldness, wetness, and dryness). Treatise IV then combines the types of variation involved in treatises II and III to consider how both the intensity and extension of two qualities are to be combined in determining the intensity of a compound (mixed) subject. Treatises I-IV, then, steadily increase in mathematical complexity.

In treatises V and VI the Calculator introduces a new dimension, that of density and rarity, and determines how density, rarity, and augmentation are to be measured. Density and rarity are mathematically somewhat more complex than qualitative intensity because, even in the simplest cases, they depend on two variables, amount of matter and quantity, rather than on one. Treatises VII and VIII, then, which consider whether reaction is possible and, in order to answer that question, discuss how powers and resistances are to the variables introduced in the preceding six parts. Treatise IX, on the difficulty of action, and treatise X, on maxima and minima, complete the discussion of the measurement of powers by determining that the difficulty of an action is proportional to the power acting and by considering how the limits of a power are determined with respect to the media it can traverse in a limited or unlimited time. Treatise IX is apparently intended to apply to all types of motion, although the examples discussed nearly all have to do with local motion. In treatise X the preoccupation with local motion becomes complete. This direction of attention to local motion is continued in treatise XI, on the place of the elements, where the contributions of the parts of a body to its natural motion are discussed. Up through treatise XI, then, the three usual categories of motion according to the medieval and Aristotelian view-alteration, augmentation, and local motion-are discussed. It is significant and typical of medieval Aristotelianism that alteration is discussed first as, so to speak, the fundamental type of motion. In treatises XII and XIII the field of attention is extended to include light, treatise XII considering the measure of power of a light source and treatise XIII considering the distribution of illumination in media.

Beginning at about treatise X the tone of the *Liber calculationum* seems to change. Whereas in the first six treatises and again in the ninth several positions are compared, in treatises X through XVI, on the whole (except perhaps for treatise XIII, which is in question form), a single view is expounded. Beginning with treatise XII and continuing at an accelerating pace to the end of the work, the parts consist mostly of long strings of conclusions concerning all the variations on the basic functions of action that can be elicited by Swineshead’s mathematical techniques. Treatise XIV consists of conclusions concerning local motion and how its velocity varies depending on the variations of forces and resistances. Treatise XV concerns what will happen if the resistance of the medium varies as the mobile is moving, or if, in a medium with uniformly increasing (*uniformiter difformis*) resistance, an increasing power begins to move. Treatise XVI concerns the various rates at which the maximum degree of a quality will be introduced into a subject depending on its initial state and the varying rate of its alteration, or on the rarefaction of the subject. Why the later treatises of the *Liber calculationum* should differ in tone from the earlier ones is, of course, not explained. It may be simply that the greater complexity involved in the later treatises prevented their being presented in the more usual scholastic question form. But another hypothesis might be that the earlier treatises bear the traces of having been used in university teaching, whereas the later treatises, although in a sense prepared for a similar purpose, never saw actual classroom use. At least the form in which we have them does not seem to reflect that use.

With this sketch of the overall structure of the *Liber calculationum* in hand, a more detailed look at the individual treatises is now in order. Although the *Liber calculationum* is fairly well known to historians of science by title, its contents are to date only very sketchily known, evidently (*a*) because the work is quite difficult and technical and (*b*) because it is not a work known to have influenced Galileo or other figures of the scientific revolution very significantly.

** Treatise I: On Intension and Remission** . In its structure, treatise I has three basic parts. First, it discusses three positions about the measures of intensity and remissness of qualities; second, it discusses whether and in what way degrees of intensity and remissness of a quality are comparable to each other; and third, it raises and replies to three doubts about rates of variation of quality considered, for instance, as loss of intensity versus increase of remissness or as gain of intensity versus decrease of remissness.

Why should these have been topics of primary interest to Swineshead? Although historians have yet to reveal very much about the connections of the *Liber calculationum* to previous tradition, it is hardly questionable that Richard Swineshead’s mathematical inquiries here, like those of Roger Swineshead before him, take place against the background of Aristotelian and medical discussions of qualitative changes, especially changes in hotness, coldness, wetness, and dryness. As in the case of Roger Swineshead, Aristotelian and medical backgrounds may explain why Richard Swineshead starts from the assumption of double measures of quality in terms of intensity and remissness, as related, for instance, to hot and cold, rather than beginning simply from one scale of degrees. And again the medical theory of the temperate, representing health, and of departures from it leading to illness, may similarly explain Richard Swineshead’s attention in treatise I to middle or mean degrees of whole latitudes and to degrees which might be said to be “just as intense as they are remiss.”

But, furthermore, in the more immediate background of Richard Swineshead’s inquiries may have been precisely Roger Swineshead’s mathematization of the intensities of qualities, on the one hand, and Richard’s own reasoning about the defects of some of Roger’s conclusions on the other. Thus, the positions concerning the measures of intensity and remissness that Richard Swineshead considers are (1) that the intensity of any quality depends upon its nearness to the maximum degree of that quality and that remissness depends on distance from that maximum degree; (2) that intensity depends upon the distance from zero degree of a quality and remissness on distance from the maximum degree; and (3) that intensity depends upon the distance from zero degree and remissness upon the nearness to zero degree (2ra-vb). In fact, in the *De motibus naturalibus*, Roger Swineshead had held the second of these positions, and this had led him to various, sometimes peculiar, conclusions comparing the intensity and remissness of degrees (for example, 38va). Thus, Richard Swineshead may well have questioned the wisdom of a position which led to such conclusions and have looked for a better position. Beyond the earlier Aristotelian and medical theories, mathematics might have led him to refer to zero degree and to some small unit as the proper basis for a measurement of intensity. Metaphysics, however, might have led him to refer to the maximum degree of a quality, because any species may be supposed to be defined by its maximum or most perfect exemplar. Mostly on the basis of mathematical considerations, Richard concludes that both intensity and remissness ought to be measured with respect to zero degree (that is, he chooses the third position). (*Tertia positio dicit quod intensio attenditur penes distantiam a non gradu et remissio penes appropinquationem ad non gradum* [2ra].)

A result of Richard’s conclusion is that intensity and remissness are no longer symmetrical concepts. Thus, although there can be remission *in infinitum* before zero degree of a quality is reached, there cannot be intension *in infinitum* before the maximum degree of the latitude is reached. This follows as in the case of a finite line (lines often appear in the medieval manuscripts as representations of latitudes), where one can get closer to one extreme *in infinitum* (one can get halfway there, three-fourths of the way there, seven-eighths of the way there, continually halving the distance left), but one cannot get farther and farther from the same end *in infinitum* because one reaches the other end of the line. Consequently, if intensity is measured by distance from zero degree, the maximum degree of a quality must be remiss, which Richard admits (2vb).

In further sections, Swineshead elaborates the concept that remission is a privation with respect to intensity (4rb-4va), and then discusses in more detail the correlations between the latitudes of intensity and remissness and motions of intension and remission. Since remissness is measured by closeness to zero degree, the scale of remissness is an inverse scale (values of remissness are proportional to the inverse of the distance from zero degree) and smaller and smaller distances on the latitude close to zero degree correspond to greater and greater differences in degree of remissness. Swineshead apparently decides in this connection that the easiest solution to the problem of remissness is to label degrees of remissness by the same numbers as the degrees of intensity and merely to say that a degree of two corresponds to twice the remissness of the degree four (5ra). Again, since remission is a privation, if one allowed in imagination intensities beyond the maximum natural degree of the given quality, there would be an infinite latitude between any degree of remission and zero remissness, even though there is only a finite latitude of intensity between any degree of remissness and infinite remissness.

Thus, although it is not stated this way, the net effect of treatise I is to dispense with the need for talking about remissness at all: one can deal with all cases of interest while only considering intensity, and furthermore the intensities one is dealing with will be additive. In this way Richard Swineshead removes what now seems the needless complexity of double measures of quality and at the same time ends up with an additive measure. The treatise by no means provides a complete basis for actual measurements of qualities, but it does help to move in that direction by emphasizing measures of quality that are additive. For the Oxford calculators qualities were not, to use the modern terminology, “intensive magnitudes,”but were, even in their intensity, “extensive magnitudes,” again to use the modern terminology. In fact, it may be somewhat startling to the modern historian to realize that fourteenth-century authors developed their concept of “dimension” or of additive magnitude in the abstract more often through the discussion of qualitative latitudes than through the discussion of qualitative latitudes than through the discussion of spatial extension, as was to be the case in later science.

It should be noted that treatise I, in addition to determining the proper measures of intensity of qualities, also introduces many of the basic technical terms of the rest of the work. Like the *De motibus naturalibus*, it assumes that any physical variable has a continuous range, called a “latitude,” within which it can vary. In the case of qualities, this latitude starts from zero degree (*non gradus*), zero being considered as an exclusive terminus, and goes up to some determinate maximum degree, the exact number of which is usually left vague, but which is commonly assumed to be eight or ten degrees (this number arising out of the previous tradition in which there were, for instance, four degrees of coldness and four degrees of hotness, the two perhaps separated by a mean or temperate mid-degree). Within any latitude there are assumed to be a number of “degrees,” these degrees being, so to speak, parts of the latitude rather than indivisibles. Swineshead also makes distinctions between the intensities versus the extensions of qualities. In his use of the terminology of latitudes and degrees, Swineshead was by no means an innovator: he was adopting a familiar set of terms. Among others, Roger Swineshead had talked systematically of the relations of degrees and latitudes before him (although Roger had had his idiosyncratic system of “uniform degrees”). In the thoroughness of his discussion of the pros and cons of various conventions for the measurement of intensity and remissness, Richard Swineshead was, however, outstanding.

** Treatises II-IV: On Difformly Qualified Subjects; On the Intension of an Element Possessing Two, Unequally Intense, Qualities; On the Intension and Remission of Mixed Subjects.** Directly following the determination of the most appropriate “scale of measure” for intension and remission, in general, in treatise I, treatises

**II**-

**IV**form a single, interrelated whole dealing with the intensity and remissness of simple and mixed subjects insofar as the degrees or“overall measures”of these subjects depend upon the degrees had by their various parts.

Thus, treatise II treats the effects of varying intensities of a single quality as these intensities are distributed over a given subject (and hence considers the two dimensions of intensity and extension) as they bear upon the overall measure of the intensity of the whole, although it only does so for the special cases in which the variation in question is either uniformly difform over the total subject or in which the subject has halves of different, but uniform, intensities. In such cases, Swineshead is in effect asking what measure of intensity is to be assigned the whole. There are, he tells us, two ways (*opiniones or positiones*) in which this particular question can be answered: (1) the measure–or as he often calls it, the denomination–of the whole corresponds to the mean degree of the qualified subject (that is, the degree that is equidistant from the initial and final degrees of a uniformly difformly distributed quality or –to take into account the second special case at hand–from the two degrees had by the uniform, but unequally intense, halves of the subject); or (2) the subject should be considered to be just as intense as any of its parts (that is, its overall measure is equivalent to the maximum degree had by the subject [5rb; 6ra]).

Similarly, in treatise III, Swineshead considers how the intensities of two qualities–for example, hotness and dryness–in a given elemental subject (now leaving aside their extension or distribution in this subject) are to be combined in determining the intensity of the whole. We here have to do with three positions: (1) the elemental subject is as intense as the degree equidistant from the degrees of its two qualities, (2) it is as intense as its more remiss quality, or (3) it is as intense as the mean proportional degree between its qualities (9rb).

In treatise IV Swineshead combines the types of variation involved in treatises II and III to consider how both the intensity and the extension of two qualities are to be combined in determining the intensity of a *mixtum* (that is, a compound subject). Four views concerning the measure of such more“complicated”subjects are presented: (1) the intensity of a *mixtum* follows the proportion of the dominant elementary quality to the subdominant elementary quality in it, (2) every *mixtum* is as intense as its dominant elementary quality, (3) every *mistum* has an intensity in the dominant quality equal to half the difference between its two qualities, (4) the fourth position is presented in two versions: (*a*) The *mixtum* is as intense as the excess of the degree of the dominant quality over the subdominant quality, no account being taken of just what parts of the *mixtum* these qualities are distributed over, or (*b*) it is as intense as the excess between (i) what the dominant quality as extended over such and such a part contributes to the denomination or measure of the whole and (ii) what the subdominant quality as extended over its part of the subject contributes to the denomination of the whole (12va).

However, simply to tabulate the various *positiones* relative to the proper measure of intensity and remissness of the variously qualified simple and mixed subjects that Swineshead presents falls short of representing the substance of treatises II – IV. To begin with, to regard Swineshead’s major concern as the unambiguous determination of just which *positio* or theory is the correct one relative to the particular question of measure posed by each treatise is to misrepresent his real interests. At times, Swineshead appears to leave any decision as to the “best theory” an open question. Moreover, even when he does express a preference for a given *positio*, it is seldom without qualifications, and the objections he brings against the opposing,“nonpreferred”*positiones* do not necessarily imply that his primary goal was the“once-and-for-all”rejection of these other *positiones*. His primary concern was rather to show that such and such results follow from this or that *positio*, it being of secondary importance whether these results are, for one reason or another, acceptable or unacceptable (even though they are from time to time so specified); of greater significance was the exhibition of the fact that these results do follow and the explanation of how they follow.

Thus, for example, in treatise III, although Swineshead indicates his preference for the third position (*sustinenda est tertia positio*), it is nonetheless true that the objections or conclusions brought against the second,“less-preferred”position are also relevant to this third position (*sequentur igitur contra istam* [*teriam*] *positionem inconvenientia sicut contra alias*), with the difference that these same conclusions for the most part are in this instance conceded (11va–12rb). What is more, when we examine Swineshead’s procedure in presenting the objections to the presumably rejected positions, there emerges a more accurate picture of his objectives. Hence, for example, from the view that the overall degree of an element corresponds to the degree midway between the degrees of its two constituent qualities, Swineshead states that there follows the conclusion that there would occur continuously operating infinite velocities of action. This result should be rejected because then the agent in question would suddenly corrupt the patient upon which it acts. Inadmissible as this consequent of the conclusion might be, far more interesting to Swineshead (and hence more deserving of attention) is the fact that this conclusion does indeed follow (*quod tamen ista conclusio sequatur* . . .) from the *positio* under investigation (9va). If all of this is taken into account, one obtains a much better idea of what these treatises of the *Liber calculationum* are all about and is at the same time less puzzled or surprised at Swineshead’s lack of emphasis upon the definitive determination of a single, exclusively correct theory or position.

Something more of the general character of this part of the *Liber calculationum* can be derived from a slightly more detailed example drawn from treatise II. The treatise begins by examining the view that the proper measure of the kind of difformly qualified subjects in question corresponds to the mean degree of the subject. Now one of the proofs supporting this view is that, if we take a subject that is, say, either uniformly difformly hot throughout or difformly hot with each half uniformly hot, and remit the more intense half down to the mean degree while equally rapidly intending the more remiss half up to the mean degree, then, since for every part of the subject that is intended there will be a corresponding part remitted equally and no net gain or loss in intensity, it follows that at the beginning the subject contained an intensity equivalent to the mean degree.

However, this proof of the first position or view will not do according to Swineshead, since when combined with the physical assumption that heating rarefies while cooling condenses, the subject will unavoidably be rarefied in one part and condensed in another when the process of equalizing the halves of the subject is carried out; but the moment this equalization commences, the cooler, more remiss, half will, because of the rarefaction caused by heating, become greater than the more intense half, which means that throughout the whole process intension will be occurring over a greater part than is remission; therefore, Swineshead concludes, at the beginning the whole subject must be more remiss than the mean degree. An objection to this procedure is raised, but it is disposed of through a number of replies establishing that the subject must indeed initially have an overall intensity less then the mean degree (5rb–5vb).

All of this would seem to imply that Swineshead definitely rejected the first“mean degree measure”position, especially if we combine this with the fact that in the following paragraphs he appears to regard the second opposing position (that the subjects in question are just as intense as any of their parts) as acceptable (6ra–6va). However, such a judgment would be premature. Swineshead has argued not directly against the first position as such, but rather against a proof given of it. Furthermore, in the remaining (and one should note, larger and more impressive) part of treatise II, Swineshead returns to this first“mean degree”position and allows its application to difformly qualified subjects each half of which is uniform and, more generally, to“stair-step qualities”in which the intensities differ, but are uniform, over certain determinate parts of the qualified subjects. This applicability is grounded upon the fact that in a difform subject with uniform halves, a quality extended through a half “denominates the whole only half as much as it denominates the half through which it is extended.” Swineshead then generalizes this“new rule”and states that if a quality is“extended in a proportionally smaller part of the whole, it denominates the whole with a correspondingly more remiss degree than it does the part through which it is extended”(6va), thus opening the possibility of considering“stair-step”distributions.

After giving proofs for the special and general cases of his new“rule of denomination,”Swineshead raises an objection against it:“If the first proportional part of something be intense in such and such a degree, and the second [proportional part] were twice as intense, the third three times, and so on *in infinitum*, then the whole would be just as intense as the second proportional part. However, this does not appear to be true. For it is apparent that the quality is infinite and thus, if it exists without a contrary, it will infinitely denominate its subject”(6va).

Swineshead shows that this latter inference to infinite denomination does not follow and that it arises because one has ignored the proper denomination criterion he has just set forth (6vb-7ra). As a preliminary, he devotes considerable space to the important task of establishing that a subject with a quality distribution as specified by the objection is indeed just as intense as its second proportional part, and he presents in detail just how this is so (6va-6vb). The proportional parts in question are to be taken“according to a double proportion”(that is, the succeeding proportional parts of the subject are its half, fourth, eighth, etc.). Now following the arithmetic increase in intensity over the succeeding proportional parts as stipulated by the objection, it follows that the whole will have the intensity of the second proportional part of the subject. Swineshead proves this by taking two subjects–*A* and *B*–and dividing them both according to the required proportional parts. Now take *B* and“let it be assumed that during the first proportional part of an hour the first [proportional] part of *B* is intended to its double, and similarly in the second proportional part of the hour the second proportional part of it is intended to its double, and so on *in infinitum* in such a way that at the end [of the hour] *B* will be uniform in a degree double the degree it now has.”Turning then to *A*, Swineshead asks us to assume that“during the first proportional part of the hour the whole of *A* except its first proportional part grows more intense by acquiring just as much latitude as the first proportional part of *B* acquires during that period, while in the second proportional part of the same hour all of *A* except its first and second proportional parts grows more intense by acquiring just as much latitude as the second proportional part of *B* then acquires … and so on *in infinitum*,” Clearly, then, since the whole of *A* except its first proportional part is equal in extent to its first proportional part, and since the whole of *A* except its first and second proportional parts is equal to its second proportional part … and so on *in infinitum*, it follows that *A* acquires just as much, and only as much, as *B* does throughout the hour; therefore, it is overall just as intense as *B* is at the end of the hour, which is to say that it is doubly intense or has an intensity equivalent to that of its second proportional part [*Q.E.D.*].

In thus determining just how intense *A* is at the end of its specified intensification, Swineshead has correctly seen that in our terms the infinite geometrical series involved is convergent (if we assume the intensity of the whole of *A* at the outset to be 1, then ). But such an interpretation is misleading. Swineshead gives absolutely no consideration to anything becoming arbitrarily small or tending to zero as we move indefinitely over the specified proportional parts. Swineshead knows where he is going to end up before he even starts; he has merely redistributed what he already knows to be a given finite increase in the intensity of one subject over another subject, something that is found to be true in most instances of the occurrence of“convergent infinite series” in the late Middle Ages. Yet however Swineshead’s accomplishment is interpreted, one should note that his major concern was to show that a subject whose quality was distributed in such a manner *in infinitum* over its parts was in fact consistent with his denomination criterion and did not lead to paradox. It is also notable that, in so increasing the intensity of *A*, he could have specified that the quality in question was heat, arguing, as he had previously argued against the proof of the first position, that on grounds of the physical assumption that heating causes rarefaction, it followed that *A* would not be just as intense as its second proportional part. The fact that he did not do so lends further credence to the view that his major interest was in seeing how many“results”could be drawn out of a given position or assumption, the more complicated and surprising the results the better. One such set of results could be derived by applying a physical assumption to the proof of the first position; another, as in the present instance, by ignoring it.

This interest of Swineshead can be even better illustrated if the present example from treatise II is carried yet one step further. Immediately after answering the objection treated above, another objection is put forth claiming that“from this it follows that *A* is now only finitely intense, yet by means of a merely finite rarefaction will suddenly be made infinitely intense.”In Swineshead’s reply to the objector’s complaint that this is an absurd state of affairs and must be rejected, the important point is again Swineshead’s demonstration that this presumably absurd situation can and does obtain (7ra). We can see how this can be so if, Swineshead tells us, we take only every 2^{nth} proportional part of our previously so intensified *A* and then rarefy the second proportional part of *A* by any amount howsoever small, while rarefying each of the succeeding proportional parts twice as slowly as the preceding one. Again, in our terms we have to do with a“divergent series,”so the conclusion that *A* is“suddenly made infinitely intense”is a correct one. But to set down the general term of this“series”would be anachronistic and would credit Swineshead with something that was quite outside his thinking. What he should be credited with is ingenious, but much more straightforward. He realized that in selecting only the 2^{nth} proportional parts of *A* he had chosen parts whose intensities were successively double one another. Therefore, in deliberately specifying that the rarefaction over these parts should be successively“twice as slow,”it automatically followed that, considering both the extension and intensity of that amount added to each part by rarefaction, the resulting contribution (no matter how small) to the denomination of the whole would be the same in each instance. And since there were an infinite number of such“added parts,”the denomination of the whole immediately became infinite. Once again, in our terms, what Swineshead has done amounts to the adding of a constant amount to each term of an“infinite series,”It is more profitable, however, to view Swineshead’s concern with the infinite in another, much less modern, way. In the two“objections”that have just been cited, Swineshead has first shown that, astonishing as it might seem, a subject whose quality increases *in infinitum* as distributed over its parts is as a matter of fact only finitely intense overall. One can next take this same finitely intense subject, change it by a finite amount as small as you wish, and it immediately becomes infinitely intense. The switch from infinite to finite and then back to infinite again seems more than incidental. Swineshead was partaking of something that was characteristic of the logical–and by then physical–tradition of solving sophisms. In point of fact, at the end of treatise II he even refers to the conclusions he is dealing with (there are fifteen in all) as *sophismata* (9rb).

The foregoing fairly lengthy discussions of the first four treatises of the *Liber calculationum* should give a good impression of the character of the whole work, not forgetting that the later treatises appear to be slightly more expository in form. The descriptions of some of the special features of the treatises that follow assume the continuation of this same basic character without repeatedly asserting it.

** Treatises V-VI: On Rarity and Density; On the Velocity of Augmentation** . The fifth and sixth treatises again form a logical unit, this time concerning the quantity or rarity and density of subjects and motions with respect to quantity. The relatively long treatise

**V**(16vb–22rb) has three basic parts. It first addresses directly the question of the proper measures of rarity and density, rejecting the position (1) that rarity depends on the proportion of the quantity of the subject to its matter while density depends on the proportion of matter to quantity, and accepting the position (2) that rarity depends on quantity assuming that the amount of matter remains the same (

*raritas attenditur penes quantitatem non simpliciter sed in materia proportionata vel in comparatione ad materiam. Et ponit quod proportionabiliter sicut tota quantitas sit maior manente materia eadem, ita raritas est maior*[17ra]).

It is of interest to the modern historian to realize why Swineshead considered the first position to be significantly different from the second position; it may seem, indeed, to be nothing but an improved and more general version of the second position. The explanation of this point turns out to shed important light on the status of Bradwardine’s geometric function relating forces, resistances, and velocities, which had been propounded in his *De proportionibus* in 1328. In fact, for Swineshead, if rarity depended upon the proportion (ratio) of quantity to matter, this would have meant that, for instance, when the proportion of quantity to matter was“doubled”(*dupletur* in his terms, but“squared”in modern terms), then the rarity would be doubled, and the resulting function would have been in modern terms logarithmic or exponential in exactly the same way that Bradwardine’s function was logarithmic or exponential. It was their understanding of the meaning of the compounding or“addition” and “substraction” of proportions (equivalent to multiplying and dividing ratios in the modern sense) that essentially forced fourteenth-century thinkers to this function. Finding it difficult, therefore, to propose the dependence of rarity on the ratio of quantity to matter as this would be understood in the simple modern sense, Swineshead was therefore led to his less elegant second position as a substitute emphasizing quantity and assuming the constancy of matter as a subsidiary consideration in order to avoid the intrusion of a proportion per se.

But having proposed the dependence of rarity on quantity and assuming the matter constant, Swineshead enters in the second part of the treatise into a long consideration of how rarity should depend upon quantity. This consideration is subsumed under the question whether both rarity and density are positive entities or whether only one of them is positive, and, if so, which (17ra). Here he can rely in an important way upon his discussion in treatise I of positive and privative entities and their interrelations with regard to intensity and remissness. After an involved discussion, he concludes that density is the positive quality (and rarity privative) and what when a subject is rarefied uniformly for a given period of time it acquires quantity difformly, greater and greater quantities corresponding to equal increments of rarity as the subject becomes more rarefied (*densitas se habet positive et ex uniformi rarefactione alicuius per tempus secundum se totum difformiter acquiritur quantitas et si densius et rarius equalis quantitatis equevelociter rarefierent, rarius maiorem quantitatem acquireret quam densius* [18rb]). Thus the mathematical characteristics of the measures of rarity become similar to those of the measures of remissness in treatise I, and similiar conclusions can be reached. It follows, for instance, that the latitude of rarity between any degree of rarity and zero rarity is infinite (18vb), just as a similar conclusion had followed for remissness.

The third and last section of treatise V raises and replies to doubts, many of which are parallel to earlier considerations concerning quality. Swineshead concludes that a uniformly difformly dense body or a body with unequal degrees of uniform density in its two halves is as dense as its mean degree (18vb–20vb). Similarly, he says that bodies are as rare as their mean degrees provided that it is understood that the latitudes of rarity and density are really the same (19vb), and he disposes of a whole series of doubts about how density and rarity are to be compared by saying that the situation is the same in this case as it is in the case of intensity and remissness (*ad que omnia possunt consimiliter argui et responderi sicut arguebatur ubi tanguntur illa de intensione et remissione, mutatis illis terminis intensio et remissio in istis terminis raritas et densitas* [20vb]). Finally, he replies to a doubt about whether, if there were an infinite quantity with a part which was infinitely dense, the whole would be infinitely dense (21rb-22rb) by saying that just as in similar cases concerning qualities, so here a density extended through only a finite part of an infinite subject would not contribute anything to the denomination of the whole subject (21vb-22ra).

When to the above description of treatise V is coupled the observation that nowhere in treatise V does Swineshead directly inquire into the physical significance to be properly correlated with the concepts density and rarity, it should be clear that Swineshead’s real interest here must have been in the mathematical functions involved in the various positions, and in the consequences, whether more or less startling, that could be shown to be consistent with these functions. And again, as in earlier treatises, he concludes by saying that many more sophisms could be developed concerning this material, all of which can easily be solved if the material he has presented is well understood (22rb).

In treatise VI, Swineshead then turns from rarity and density as such to motions of augmentation, where augmentation is considered to be the same as increase of rarity. Like Roger Swineshead before him, he begins by rejecting the position espoused in Heytesbury’s *Regule solvendi sophismata* that motion of augmentation is to be measured by the proportion of the new quantity to the old quantity (22va-24va). Second, he turns to the position on augmentation held by Roger Swineshead, namely that augmentation is to be measured by quantity acquired irrespective of the quantity doing the acquiring (24va-vb), but he also rejects this position because it does not adequately handle cases in which a quantity is lost at the same time as one is added. Swineshead then replaces Roger Swineshead’s position with an improved version which he accepts, saying that augmentation and diminution should be measured by the net change of quantity of the subject (24vb). In reply to an objection concerning what happens according to this position to the concept of uniform augmentation, Swineshead admits in effect that the concepts of uniform velocities of alteration and motion will not then have an easy parallel in the case of the motion of augmentation, although one could speak of equal parts of a subject gaining equal qualities.

The most striking thing about treatise VI is perhaps the fact that the largest section of the treatise is devoted to the refutation of Heytesbury’s view concerning the proper measure of augmentation. And here there are two points to be noted about the arguments provided. First of all, a significantly large proportion of them involve what are in effect augmentations from zero quantity. Since the first position is obviously not applicable to augmentations from zero (since this would put a zero into the denominator of the proportion of quantities that it proposes as the proper measure of augmentation), one might argue that these supposed refutations of the position are misguided. And secondly, Swineshead himself eventually concedes several of the refuting arguments although he did consider the first position to be refuted. Yet these arguments are left to stand as if they were strikes against the first position. Thus, Swineshead says that the inferences which can be drawn from the first position are amazing and contrary to one’s idea of what the proper measure of a motion should be (23va)–some of these conclusions being ones that involve the unfair augmentation from zero quantity–but he then also concedes that some of these conclusions are simply true no matter how the velocity of augmentation is measured (23va). So again one might fairly draw the conclusion that Swineshead’s major concern is not really the choice between rival measures, but rather the exhibition of mathematical techniques that one might reasonably use in the discussion of any of the positions.

** Treatises VII-VIII: On Reaction; On the Powers of Things** . As in the works of the other Oxford calculators where reaction is taken up at a fairly early stage, here too the problem of reaction is really the entire problem of how two qualified bodies act on each other and involves all of the variables discussed in the preceding treatises. For those who, like Richard Swineshead, held the so-called addition of part to part theory of qualitative change, the problem was particularly acute. There were numerous well-known cases (

*experimenta*) in which reaction seemed to occur (25va). Furthermore, under the addition theory it seemed that the parts of a quality present should be able to act and react with the other qualitative parts nearby. Yet the previously accepted Aristotelian and medical theory of qualitative change had assumed that the qualities of a given subject could be represented by the single degrees of hot, cold, wet, or dry of the whole, so that if two bodies were brought close together such that one could act on the other, only that body with the higher degree, say of heat, would act as the agent or force causing change, and only that body with the lower degree would act as patient and be changed. Clearly the calculators were in a position in which they had to improve upon previous theory by taking account of distributions of quality, and yet the theoretical situation was so complex and mathematically difficult that they faced an almost impossible task.

After preliminary arguments, Swineshead takes, as the foundation of his solution to the problem, the position that the power of a subject is determined by the multitude of form (*multitudo forme*) in it, where multitude of form is determined not only by the intensity of the form and the extension, but also by what might be called the density of form (26vb). He takes density as the most important factor and asserts that if a foot length of fire were condensed to half a foot, it would still contain the same multitude of form even if the intensity were the same as before. Swineshead next turns to the question of whether the whole patient resists the agent or only the part acted on. After considering various positions, he concludes that although the whole patient does not necessarily resist the agent, the whole part of the patient directly opposite the agent resists, and that all parts do not resist equally, parts further from the agent resisting less than those closer. Unfortunately, arguments can be raised to show that no simple proportionality obtains between distance from the agent and lesser resistance. Swineshead asserts, however, that it will be more clearly understood when he deals with illuminations how resistance decreases with distance (27rb–28rb).

On the basis of these fundamentals Swineshead concludes that reaction cannot occur between uniform bodies such that the reaction is according to the quality contrary to that of the action (28va-28vb). (By a uniform body Swineshead means uniform not only according to intensity of quality but also according to the amount of form existing in equal parts of the body.) It is possible for the patient to react according to another quality–so that while the agent heats the patient the patient in turn humidifies the agent. Between difform bodies, on the other hand, there can be action with reaction in the contrary quality in another part. Where such reaction occurs the whole agent and whole patient act and resist according to their power insofar as it is applied in the given situation. Thus, Swineshead appears content in his reply to leave the unspoken and rather improbable implication that in all the observed cases of reaction according to the same quality, the qualities of the two bodies must have been difform.

The rest of treatise VII consists of the solution of three *dubia*. The first, also dealt with by John Dumbleton in his *Summa logicae et philosophiae naturalis*, concerns whether an agent will act more slowly if patients are applied to either side of it than it would if it acted on only one of the patients (29va-30ra). Swineshead is sure that if two actions concur at the same point the action will be faster, but he is not sure of the solution of the doubt if the two patients are far enough apart so that they do not act on each other. In the latter case, he says, the reader may decide for himself whether the patients will assist each other in resisting, since, although some say they do, it is hard to understand how this could be so (30ra). To a second doubt Swineshead concludes that two difform bodies which are similar in those parts nearest each other can nevertheless still act on each other (30ra-30va), and to a third doubt he concludes that bodies having maximum degrees of contraries can act on each other (30va-30vb).

It should be clear that in all of treatise VII one of Swineshead’s main questions concerns the additivity or summability of the forces and resistances he is dealing with. Indeed, elsewhere additivity was one of the major concerns of the Oxford calculators in their efforts at quantification. With respect to single dimensions such as that of intensity, the calculators were adamant that the measure of intensity should be an additive measure. Following the addition theory of the intension and remission of forms, they assumed that an intensity was equivalent to the sum of its parts. In dealing with actions and reactions of bodies, however, the basis for such additivity was not so easily found. Here, and again in treatise XI concerning local motion, Swineshead appears to concede, perhaps to his own disappointment, that difficulties appear to ensue if one attempts to treat subjects as the sums of their parts in any simple fashion. The difficulties of considering not only the forces and resistances of the parts, but also the varying distances of the parts from each other and the possible interactions of the parts on each other, made a detailed part–by–part quantitative treatment practically impossible. We may admire Swineshead’s ingenuity in the face of such odds while agreeing that it is unfortunate that the slant of Aristotelian physics towards alteration rather than local motion caused Swineshead to concentrate on such a difficult problem.

Treatise VIII again takes up the question dealt with in treatise VII concerning how the powers of things are to be measured, and it appears probable that parts of treatises VII and VIII represent Swineshead’s successive reworkings of the same basic problem. Despite nine arguments against the view that power is to be measured by multitude of form, Swineshead reaffirms his earlier conclusion that it is. The only thing that he adds here is the remark that the amount of form induced in a subject will depend upon the amount of matter present (31rb). He concedes the nine arguments or conclusions *de imaginatione*, their supposed difficulty being based on the view that, for a given form, intensity and extension are inversely related, which, he says, is only accidentally so (31rb-31va).

If in the work of Oresme, the concept of the“quantity of quality”was to become fundamental, where quantity of quality was the product of intensity times extension, here we see Swineshead’s effort to deal with objections based on a similar concept. Whatever objections there may be to Oresme’s concept of“quantity of quality”from a modern point of view, students of medieval science have in recent years become so familiar with the concept that there is a tendency to assume the use of a similar concept in other late medieval authors attempting the quantification of qualities or forms. It deserves emphasizing, therefore, that Richard Swineshead and the other Oxford calculators of 1330–1350 were familiar with concepts like Oresme’s quantity of quality, but rejected them in favor of quantifications in terms of intensities alone or in terms of something like Swineshead’s multitude of form. This rejection of “quantity of quality”helps explain, among other things, Swineshead’s less than total happiness in treatise II with the“mean degree measure”of difform qualities, a measure that he might otherwise have been expected to favor because of its mathematical attractiveness. Swineshead does not always assume, as Oresme’s quantity of quality concept implies, that when the extension of a subject is decreased, the form remaining unchanged, the intensity will increase, and he even goes so far as to assert that a form could be condensed to a point without its intensity increasing (3lrb).

** Treatises IX-X: On the Difficulty of Action; On Maxima and Minima.** The ninth and tenth treatises in a sense carry further Swineshead’s treatment of action and the forces causing it. Treatise X has a clear precedent in earlier discussions of maxima and minima, in particular in the discussion found in Heytesbury’s

*Regule solvendi sophismata*, and treatise IX probably is related, although in a non-obvious way, to Bradwardine’s

*De proportionibus*. The problem covered in treatise IX (and also in Dumbleton’s

*Summa*, part VI) seems to have arisen because of the Calculator’s acceptance of Bradwardine’s function for measuring velocities. For the standard Aristotelian position concerning the relation of forces, resistances, and velocities, there was a simple relationship between forces and the velocities produced with a given resistance, such that each equal part of the force could be interpreted as contributing an equal part of the velocity (and here again arose the question of additivity). For the Bradwardinian position, on the other hand, multiples of a force, with the resistance remaining constant, did not produce equal multiples of the velocity. As a result, one needed some other measure of what a force could do. On this subject, Swineshead first rejects two positions: (1) that the action or difficulty produced depends upon the proportion of greater inequality with which the agent acts, so that an agent acting from a greater proportion produces a greater difficulty (3lva); and (2) that the action or difficulty produced depends on a proportion of lesser inequality, because an agent closer in power to the strength of the resistance tires more in acting (3lva).

The position that Swineshead adopts is the same as that adopted by Dumbleton, namely that the difficulty produced is proportional to the power acting to its ultimate (3lvb). In a given uniform resistance there will be an action of maximum difficulty that cannot be produced in that medium, namely the difficulty equal to the power of the resistance (31vb-32ra). When the power of the agent is doubled, the difficulty it can produce is also doubled. The latitude of difficulty or range of all possible difficulties is infinite (32ra).

The remainder of treatise IX consists of the raising of twelve arguments against Swineshead’s preferred position and of his replies to them. In these arguments the connection with Bradwardine’s function manifests itself. A common assumption behind the objections is that difficulty (or action) and motion (or velocity) ought to be proportional to each other, so that to move something twice as fast is to produce a double action. As stated above, in the commonly assumed Aristotelian“function,”a double force, a double velocity, and presumably a double action or difficulty all seem to be correlated with each other. This is not so in the Bradwardinian function, where a doubling of force does not usually correlate with a doubling of velocity. Swineshead simply asserts here, therefore, that difficulty can be correlated with force or power, and that velocity and difficulty produced do not necessarily correspond to each other (32vb-33rb). The main effect of treatise IX, therefore, is to clear away objections that might be raised about Bradwardine’s function by those still thinking in an Aristotelian framework.

Treatise X concerns maxima and minima only with respect to the traversal of space in local motion, and the ground it covers is quite standard. Having stated some familiar definitions and suppositions concerning maxima and minima of active and passive powers, Swineshead states two rules: (1) that both debilitatable and nondebilitatable powers have a minimum uniform resistance that they cannot traverse (in familiar scholastic terminology, a *minimum quod non*); and (2) that with respect to media there is a maximum power that cannot traverse a given medium, namely the power equal to the resistance of the medium (a *maximum quod non* [34rb]). Thus, for a power, say, equal to 3, there will be a minimum resistance it cannot traverse, that is, the resistance 3, and, conversely, for the resistance equal to 3, the power equal to 3 will be the maximum power that cannot traverse it.

The rest of treatise X consists of rules for assigning maxima and minima under a variety of possible conditions, that is, when the medium is uniform and when it is difform, when there is a time limit and when there is not, when the power is constant and when it weakens in acting, and when the medium is infinite and when it is finite. Although they may take a while to decipher, these conclusions are mostly the simple results of the assumption that the force must be greater than the resistance for motion to occur. Swineshead himself seems to feel that he is traversing familiar ground, and the treatise is therefore quite short.

** Treatise XI: On the Place of an Element** . Swineshead’s concern in treatise XI is a single problem relating to the motion of a heavy body in the vicinity of its natural place at the center of the universe: whether in free fall, assuming a void or nonresistant medium, the heavy body will ever reach the center of the universe in the sense that the center of the body will eventually coincide with the center of the universe. If we regard the heavy body in question as a thin rod (

*simplex columnare*), the variables that Swineshead has to deal with in resolving the problem become evident: as soon as any part of the rod passes the center of the universe, that part may be considered as acting as a resistance against its continued motion. Now one position that can be taken in resolving the problem is that the body acts as the sum of its part and that therefore the parts of the rod“beyond the center”actually do resist its motion. Assuming this, lengths or segments of the rod will function both as distances traversed as the rod approaches the center and as the forces and resistances involved in determining such a traversal. If we also assume, with Swineshead, Bradwardine’s“function”relating velocities with the forces and resistances determining them, then the task to be carried out is to discover a way to apply this“function”to the“distance-determined”forces and resistances acting upon the falling rod in order to calculate the relevant changes in velocity and thus ascertain whether or not the center of the rod ever will reach the center of the universe.

In what is mathematically perhaps the most complicated and sophisticated section of the *Liber calculationum*, Swineshead accomplishes this task and replies that, on the assumption of the rod acting as the sum of its parts, the two centers will never come to coincide. He presents his argument axiomatically, beginning from a number of strictly mathematical *suppositiones* and *regulae* and then moving to their application to the problem at hand. It will be easier to indicate something of the nature of his accomplishment if the order is, at least in part, reversed. Thus, Swineshead clearly realizes and emphasizes the fact that the distance remaining

between the center of the rod and the center of the universe will always be equal to half the difference between that (greater) part of the rod which is still on this side of the center of the universe and that (lesser) part which is beyond (37ra) (in terms of Figure 1, *CD* = (*F*_{1} – *R*_{1})/2). This obtains no matter what space intervals we consider in the rod’s progressive motion toward the center. When this is added to the fact that, with any given motion of the rod, whatever is subtracted from the segment this side of the center of the universe is added to the segment beyond the center, thus determining the“new”forces and resistances obtaining after that motion, Swineshead then has a way to apply Bradwardine’s“function”to the whole problem. Divide, for example, the remaining distance (*CD*) into proportional part (according to a double proportion); we know the relation of the distance between the centers to the difference between the relevant forces and resistances: CD= (*F*_{1} – *R*_{1})/2, but successively following the particular division specified of the remaining distance it is also true that this same half-difference between *F*_{1} and *R*_{1} is equivalent to the excess of *F*_{1} less one-fourth of (*F*_{1} – *R*_{1}) over *R*_{1} plus one-fourth of (*F*_{1} – *R*_{1}). But in a strictly mathematical “suppositio” it is stated (and then proved) that the proportion of the thusly decreased *F*_{1} to the thusly increased *R*_{1} will be less than the “sub-double” of the proportion between the original, unaltered *F*_{1} and *R*_{1}. (*Si inter aliqua sit proportio maioris inequalitatis, et quarta pars excessus maioris supra minus auferatur a maiori et addatur minori, tunc inter illa in fine erit proportio minor quam subdupla ad proportionem existentem inter ista duo in principio* [35vb].) In modern symbols:

At the same time, the decreased *F*_{1} and the increased *R*_{1} have given us the *F*_{2} and *R*_{2} operative after the rod has moved over the first proportional part of the distance between centers, assuming that its speed is uniform throughout this motion, and, moving over the second proportional part, we can similarly derive *F*_{3} and *R*_{3} and relate them to *F*_{2} and *R*_{2} by means of the same “mathematical supposition,” and so on, over succeeding proportional parts and *F*’s and *R*’s. However, the ordering of the force-resistance proportions that is the burden of this mathematical supposition is precisely what is at stake in Bradwardine’s“function”claiming that increases and (in the particular problem at hand) decreases in velocity correspond to increases and decreases in the proportion between force and resistance. Bradwardine can therefore be directly applied, yielding resultant velocities over succeeding proportional parts each of which is“more remiss”than half the preceding one (*V*_{1} > 2*V*_{2} > 4*V*_{3} > . . .). Since succeeding proportional parts of the distance decrease by exactly half (*CE*_{1} = 2*E*_{1}*E*_{2} = 4*E*_{2}*E*_{3} = . . .), it follows that the time intervals for each increment of distance must increase *in infinitum*, which means that the that the center of the rod will never reach the center of the universe (*maius tempus requireretur ad pertransitionem secunde partis proportionalis quam ad pertranstitionem prime … et sic in infinitum. Ergo in nullo tempore finito transiret C totam illam distantiam* [37rb]).

Swineshead has reached this result by applying a particular, proportional part division to the distance remaining between the centers, but he provides for the generalization of this division by specifying (and proving) his crucial mathematical supposition in a general form (36va); then we may presumably take any succeeding proportional parts whatsoever in determining the fall of the rod. But his own use of this more general supposition occurs in a second, different proof of the conclusion that the rod will never reach the center of the universe. In our terms, the first proof summarized above assumes a constant velocity–and hence constant force-resistance proportions–over the relevant distance intervals, thus employing a discontinuous, step function in resolving the problem. In a more compact, and more difficult, proof Swineshead comes more directly to grip with his variables as exhibiting a continuous function. Less tractable, and hence more difficult to represent adequately in modern terms, than the first proof, its substance is tied to the proportional comparison decrease or losses with what we would term rates of decrease or loss (for example, *motus velocius proportionabiliter remittetur quam excessus; ergo excessus tardius et tardius proportionabiliter remittetur*[37rb]).

In both proofs Swineshead has in effect assumed that the rod or heavy body in question is a *grave simplex*, a limitation that he addresses himself to by considering, in reply to several objections, the body as a *mixtum* (37va). Far more important, however, is another objection. It claims, in effect, that the assumption behind Swineshead’s whole procedure up to that point-namely, that the rod does act as the sum of its parts, must be false because it implies that there would exist natural inclinations that would be totally without purpose and vain (*appetitus. . . omnino otiosus. . . vanus*), an inadmissible consequent (37rb).

Swineshead therefore sets forth a second, alternative position, one in which the heavy body in question acts as a whole, where its parts contribute to the natural inclination or desire (*appetitus*) of the whole where in a manner that is not given precise mathematical determination (37vb - 38ra). As might be expected, Swineshead spends far less time treating, and seems much less interested in, this second position, in spite of the fact that it is apparently the true one. This brevity fits well with the whole tenor of the *Liber calculationum* and with what has been noted above of the greater interest in deriving results than in just what the results are. The treatment based on the first,“false”*positio* of the whole body as the sum of its parts also fits well with much rest of the rest of the *Liber calculationum*, where the mathematical and logical determination of the contribution of parts to wholes is so often a central issue.

** Treatises XII-XIII: On Light; On the Action of Light.** These two treatises are concerned with light, first with respect to the power of the light source and second with respect to the illumination produced.

The power of a light source, Swineshead states, is measured in the same way as the power of other agents, namely by the multitude of form (38ra). Equal light sources, then, will be those that are not only equal in intensity but also equal in multitude of from (38rb). Thus, if sources with equal multitudes of form are intended by equal latitudes of intensity, they will gain equally in power, but if sources with unequal multitudes of form are intended by equal latitudes of intensity, the one with more form will increase more in power than the other.

On the basis of these presuppositions, Swineshead then draws a number of conclusions or rules treating what happens when either the quantity of light source is varied (by adding or subtracting matter so that the multitude of form is changed) or the intensity of these sources is varied. He concludes, for instance, that if there are two light sources of different intensity, which at the outset are either equal or unequal in quantity, but which then diminish equally in quantity, then proportionally as one is more intense than the other it will diminish in power more rapidly (38rb-38va). Swineshead does not believe that changing the quantity (extension) of a light source without adding or subtracting matter will change its power, but he says that his conclusions can be proved even better by those who hold such a view (38vb).

It should be noted that in this treatise Swineshead does assume that there is a correlation between intensity and multitude of form, something which he felt it necessary to state as an explicit hypothesis when he dealt with similar problems in treatise II (for example, 8va) and something which he, in effect, ignored in treatise VIII. Had his interests been in determining the one correct physical theory, it is hard to believe that he would not have brought these contexts together and somewhere stated what he felt to be the true physical situation with regard to the connections between the intensity, extension, and“density”of a form. With his attention falling as entirely on the quantitative side as it does, he lets apparent inconsistencies slide, covered by the remark that the connection between the intensities and extensions of a given form are only accidental (31rb).

Treatise *XIII* consists of the solution of two major doubts and a long string of conclusions. In reply to the first doubt, Swineshead concludes that every light source produces its entire latitude, from its maximum degree down to zero degree, in every medium in which it suffices to act, but that a source will cast its light to a greater distance in a rarer medium and to a lesser distance in a denser medium (39vb). Light is remitted (is less intense) at more distant points because of the indisposition caused by the medium between the source and the distant point, so since there is no medium between the source and the point next to it there is no remission at that point.

In reply to the second doubt Swineshead concludes that a light source casts a uniformly difform illumination in a uniform medium (40rb-va). Considering the medium between the source and a distant point as an impediment subtracting from the intensity of illumination, Swineshead comes up with a physically reasonable relation leading to a uniformly difform distribution, avoiding the trap of making the intensity inversely proportional to the distance from the light source. This no doubt is the explanation Swineshead had in mind in treatise VII as being helpful in understanding how distant parts combine their actions and resistances (28rb).

These two replies are then followed by a series of fourteen conclusions (numbered 13–26 in sequence with the conclusions of treatise XII) intended to make clearer what has preceded. They appear quite complex, but we may suppose that he arrived at them by a simple visualization of the situation with few if any mathematical calculations. Thus, Swineshead concludes first that if a light source acts in a uniform medium and if a part of the medium next to the agent is made more dense without changing its quantity (extension), then at every point of the rest of the medium farther from the agent the illumination will be remitted with the same velocity as the illumination at the extreme point of the part made denser is remitted (40vb-41ra). Imagining a graph of the original uniformly

difform illumination, we see that this conclusion amounts to saying that when part of the medium is condensed, the slope of decrease of intensity will become steeper in that part, but in the remaining part the slope of decrease will remain the same, being shifted down parallel to itself to connect with the new, more remiss degree at the extreme point of the condensed part (see Figure 2).

Like this first conclusion, Swineshead’s other conclusions are easy to understand on the basis of graphs, although he makes no reference to visualizations of the conclusions. In our terms, the variables that he has to work with in the conclusions are that the quantity of the light source determines the rate of decrease of intensity or slope of the distribution of intensities in a given medium, whereas the intensity of the source determines the degree from which that decrease starts. The density of the medium, on the other hand, also determines the slope of the distribution of intensities when the source remains the same. Swineshead can then partially offset changes in the quantity of the light source by changes in the density of the medium, or vice versa, as it suits his purpose (cf. conclusion 19 [41vb]).

It is so natural to us to visualize these conclusions that it is hard to imagine that Swineshead did not do so also. Dumbleton, in the corresponding part of his *Summa*, did make an explicit geometrical analogy, and the tradition of using triangles or cones to represent the dispersal of light in optics would also have prompted mental images of triangles. It is very probable then that the mathematics behind treatise XIII was a simple “visualized geometry”.

** Treatise XIV: On Local Motion** . One of the most exhaustively developed sections of the

*Liber calculationum*, treatise XIV begins by stipulating that its contents will be formulated under the assumption “that motion is measured in terms of geometric proportion” (

*motum attendi penes proportionem geometricam*[43va]). As immediately becomes obvious, this is Swineshead’s elliptical way of informing his readers that he will be accepting Bradwardine’s view that variations in velocities correspond directly to variations in the force-resistance proportions determining those velocities (in modern terminology, that arithmetic changes in velocity correspond to geometric changes in the relevant force-resistance ratios). With this as base, what Swineshead accomplishes in setting forth the forty-nine

*regulae*that constitute treatise

**XIV**is to give a relatively complete “catalog” of just which

*kinds*of changes in velocity correspond to which

*kinds*of changes in force and resistance and vice versa.

He does this in strict axiomatic fashion, the first three rules presenting what can be regarded as the basic mathematics of proportion which will serve, together with his assumption of Bradwardine’s “function”, as the key to all that follows. As indicated above, there is no doubt that the medieval tradition of compounding proportions was at the root of Bradwardine’s logarithmic-type function, and it is precisely this that Swineshead makes explicit in his initial rules. Thus, the first rule tells us that “whenever a force (*potentia*) increases with respect to a constant resistance (*resistentiae non variatae*), then it will acquire as much proportionally relative to that resistance as it will itself be rendered greater” (43va). That is to say, if some force *F*_{1} acting on a constant resistance *R* increases to *F*_{2}, then the proportional increase in *F* (the proportion *F*_{2} : *F*_{1}) is equal to the increase of *F*_{2} : *R* over *F*_{1} : *R*. As Swineshead makes clear in his proof of this rule, this amounts to a compounding of the proportions involved, that is, when *F*_{2} : *F*_{1} is “added to” *F*_{1} : *R* the result is *F*_{2} : *R*. (Note that following the medieval convention proportions are added to one another [just as are numbers or line segments], when we would say they are multiplied.) In the second and third rules Swineshead establishes corresponding relations for the cases of a decrease in force and an increase or decrease in resistance (the force than being held constant [43va]).

That these rules provide the basis of what Swineshead was attempting to do in treatise **XIV** can be seen as soon as one introduces the motion or velocities that correspond to or “result from” the force-resistance proportions whose “mathematics of change” he has just established. Thus, to return to the first rule as an example, if we assume that a velocity *V*_{1} corresponds to the proportion *F*_{1} : *R* and a velocity *V*_{2} to the proportion *F*_{2} : *R*, then, because *F*_{2} : *R* is greater than *F*_{1} : *R* by the proportion *F*_{2} : *F*_{1} (which is what the compounding of these proportions asserts), it follows that the velocity added is precisely the velocity that results from the proportion *F*_{2} : *F*_{1} (and which would result from a proportion *F* : *R* equal to *F*_{2} : *F*_{1} standing alone). Given this, and the corresponding relations when a resistance is allowed to vary while the force is held constant, Swineshead can determine all that he wishes concerning changes in velocity by paying attention only to the relevant force-force or resistance-resistance proportions representing the changes. When, and only when, these proportions are equal will the corresponding positive or negative increments of velocity be equal.

In deducing his succeeding *regulae* on such a basis, Swineshead does not “calculate” velocity increments from given force-force or resistance-resistance proportions, nor velocities from given force-resistance proportions. In this he was strictly medieval. Rather, he always compares (at least) *pairs* of *F* : *F* or *R* : *R* proportions. Thus, whenever unequal forces increase or decrease with equal swiftness (equevelociter)–which means that *F*_{2} – *F*_{1} = *F*_{4} – *F*_{3} – then the resultant *F*_{2} : *F*_{1}, *F*_{4} : *F*_{3} proportions will be unequal, whence it follows that the corresponding velocity increments will be unequal. If, on the other hand, the forces increase or decrease proportionally (*eque proportionabiliter*)– which means that *F*_{2} : *F*_{1} = *F*_{4} : *F*_{3}–then the corresponding velocity increments will be equal. And the same thing holds for increasing or decreasing resistances, the forces being held constant.

Beginning, then, with two rules (4 and 5 [43va]) that apply a change in a *single* force or resistance to, correspondingly, *two* constant resistances or forces (whence the relevant single *F*_{2} : *F*_{1} or *R*_{2} : *R*_{1} proportions function as pairs since they are applied to pairs of *R* or *F*), Swineshead sets forth the implications of his mathematics of force-resistance changes to changes in velocity. In rules 4 and 5, inasmuch as one has a single *F*_{2} : *F*_{1} or *R*_{2} : *R*_{1} proportion doing double duty, the corresponding velocity increments are naturally the same. (For example, rule 4 reads: “Whenever a force increases or decreases with respect to two equal or unequal, but constant, resistances, it will intend or remit motion with respect to each [of these resistances] with equal swiftness.”) Note should be taken of how, throughout treatise XIV, Swineshead handles what we would consider as positive versus negative increments of velocity. Since all determining force and resistance proportions are always of greater inequality (e.g., *F*_{2} : *F*_{1} where *F*_{2} > *F*_{1} when it is a question of increase in force, *F*_{1} : *F*_{2} where *F*_{1} > *F*_{2} when decrease is involved), velocity increments are always added, a procedure that follows directly from Swineshead’s basic technique of compounding proportions. This means that when it is a question of the remission of motion arising from decreasing forces or increasing resistances, then increments are added to the motion at the end of the change in question, the “sum” of these increments plus the final motion or velocity giving the motion or velocity at the beginning of the change. When the motion is intended, or the velocity increments are positive, the addition is naturally made to the motion obtaining at the beginning of the change.

In rules 6 through 15 (43va–43vb) Swineshead applies the two kinds of force or resistance change, that is, equally swift (*equevelociter*) or equally proportional (*eque proportionabiliter*) increases or decreases, to *paris* of changing forces or resistances and infers the corresponding changes in velocity. Thus far, the force and resistance changes can be considered as discrete. However, beginning with rule 19 [43vb] Swineshead faces the case of the uniform and the continuous change of a single force or resistance. Now a single uniformly increasing force acting on a constant resistance, for example, will “generate” pairs of proportions *F*_{2} : *F*_{1}, *F*_{3} : *F*_{2}, *F*_{4} : *F*_{3}, etc., where the succeeding proportions have “common terms” because the increase in force is continuous and where *F*_{2} – *F*_{1} = *F*_{3} – *F*_{2} = *F*_{4} – *F*_{3} = . . . because the increase is uniform. This latter fact entails that *F*_{2} : *F*_{1} > *F*_{3} : *F*_{2} > *F*_{4} : *F*_{3} >. . . , which in turn implies that the increments of velocity will become successively smaller and smaller. Thus, the first half of rule 19 reads: “If a force increases uniformly with respect to a constant resistance, it will intend motion more and more slowly”.

It is important, however, to be able to deal with at least certain kinds of nonuniform or difform changes in force and resistance. Hence, in rules 21 and 22 (43vb–44ra), arguing by a *locus a maiori*, Swineshead shows that if (as he has just established) a uniform gain in force or resistance entails a, respectively, slower and slower intension or remission of motion, then a nonuniform, slower and slower gain in force or resistance will necessarily also entail slower and slower intension and remission. Similarly, if a uniform loss in force or resistance entails a, respectively, faster and faster remission or intension of motion, then a nonuniform faster and faster loss in force or resistance also entails faster and faster remission and intension. The difform changes involved in a faster and faster gain, or a slower and slower loss, of force or resistance are not treated, since in such cases no inferences can be made about the resultant *F* : *F* and *R* : *R* proportions and, hence, about the resultant velocity increments.

However, the types of difform change in force or resistance that Swineshead can and does treat are precisely those needed in much of the remainder of treatise **XIV** . Up to this point changes in resistance have been independently given. Beginning with rule 23 (44ra) these changes are ascribed to the medium through which a mobile (represented by the force acting upon it) moves. Furthermore, all increase or decrease of resistance has hitherto been considered merely *relative to time* (whether it be *equevelociter* or *eque proportionabiliter*, no matter). But to ascribe variations in resistance to a medium is to speak of increase or decrease relative to *space*. Consequently, the problem facing Swineshead is to connect increments of resistance with respect to space to increments of resistance with respect to time, which is exactly what rule 23 does. Thus, a uniformly difform medium is one in which equal increments of resistance occur over equal spaces or distances. We also know that any body moving through such a medium in the direction of increasing resistance will move continuously more and more slowly (that is, the spaces *S*_{1}, *S*_{2}, *S*_{3},. . . traversed in equal times successively decrease). But these two factors imply that, in equal times, the mobile will encounter smaller and smaller increments of resistance. Hence, equal increments of resistance (Swineshead calls them “latitudes of resistance”) over space have been connected to decreasing increments of resistance over time. Accordingly, rule 23 reads: “If some force begins to move from the more remiss extreme of a uniformly difform medium and remains constant in strength, then the resistance with respect to it will increase more and more slowly”. However, this slower and slower increase in resistance over time is precisely one of those kinds of difform change Swineshead was able to deal with in rules 21 and 22. This allows him to infer in rule 24 (44rb) that the motion of a mobile under a constant force through a uniformly difform medium in the direction of increasing resistance entails that the motion in question will undergo continuously slower and slower remission.

There are media, however, in which the distribution of resistance over equal spaces is difform, but not uniformly difform. What can be said of them? We know what equal changes of velocity are associated with a resistance that changes uniformly proportionally over time (rules 11 and 27 [43va, 44va–44vb]). Thus, if we imagine a medium of uniform resistance which increases in resistance equiproportionally over time as a constant force mobile moves through it, the mobile will remit its motion uniformly, that is, will undergo equal negative increments of velocity in equal times (rule 28 [44vb–45va]). With this rule in hand, Swineshead then imagines another medium with a resistance constant in time but difform with respect to space and having at each point the resistance which was at the corresponding point of the first medium when the mobile was at that point. The mobile will then have the same motion in the second medium as it had in the first, that is, a uniformly difform motion. This means that (rule 29 [45va–;45vb]) there can be a medium with resistance distributed difformly over space in such a way as to cause a mobile moving in it under a constant force to remit its motion uniformly (even though Swineshead could not describe this distribution).

Nevertheless, what Swineshead has established in rule 29 is of considerable importance for much of the remainder of treatise XIV. Rules 30–43 [45vb–;48rb] all have to do with what will, or will not, occur when other constant or changing forces move through a medium in which (again rule 29) a given constant force uniformly remitted its motion. Thus, in rule 30 [45vb–46rb], Swineshead proves that two unequal constant forces cannot both uniformly remit their motion in the same medium. It is worthy of note that to prove this rule, Swineshead has to determine where the mobile that does remit its motion uniformly is at the middle instant of its motion. To do this he uses the ratio of space traversed in the first half of the time to the space traversed in the second half of the time and to find this ratio he uses the famous “Merton mean speed theorem”, which he proves for the occasion (45vb–46ra). Apart from the fact that Swineshead gives four different proofs of the theorem, it here appears as a fairly routine lemma. He does not assign it any special importance, and does not even give it the honor of labeling it as a separate rule or conclusion.

Holding in mind the constant force specified in rule 29 as able to cause the uniform remission of motion in a given medium, Swineshead concludes treatise XIV by considering what will transpire when constant forces greater or lesser than that constant force are brought into play and when greater or lesser forces that are undergoing continuous intensification or remission are involved (rules 31–43). The last of these rules points out that a constant force greater than that specified in rule 29, but acting in the same medium, will give rise to a faster and faster remission of motion, that is, to a difformly difform motion. This leads to the final rules (44–49 [48rb–48vb]) of the treatise, which together function as a kind of appendix stipulating various facts and relations concerning difformly difform motions. As a whole, treatise XIV is an extremely impressive exhibition of just which cases of the different kinds of variation in force, resistance, and velocity that can be drawn out of Bradwardine’s “function” are amenable to determination and treatment. As in the case of many of the other treatises of the *Liber calcalationum*, there is a substantial increase in complexity from the beginning to the end of treatise XIV. But perhaps one of Swineshead’s most signal accomplishments is his success in the latter part of the treatise in connecting variations in resistance over time with variations in resistance over time with variations in resistance over space. For to the medieval supporter of Bradwardine’s function (or of “Aristotle’s function” as well for relating forces, resistances, and velocities), motion in a medium that was nonuniform was exceedingly problematic. As soon as the resistance in a medium was allowed to vary, one had to face the difficulty that the degree of resistance of the medium determined the velocity of the motion, while at the same time the velocity determined where in the medium the mobile would be and hence the resistance it would encounter. One seemed caught in a situation involving a double dependency of the relevant variables on each other. But Swineshead’s “translation” of spatial increments of resistance into temporal ones automatically rendered the resistance of the medium time dependent and thus circumvented the troublesome double dependency.

** Treatises XV–XVI: On a Nonresisting Medium or on the Increase of Power and Resistance; On the Induction of the Highest Degree.** Treatises XV and XVI are continuations of treatise XIV and add ever more complications. In treatise XV Swineshead again considers the local motions of constant or changing powers in extended media, but this time he allows the resistance of the medium to vary while the mobile is moving through it or (we would say) takes the increase of power as an independent variable. In the key rule 29 of treatise XIV Swineshead had considered the motion of a mobile through a uniform medium with resistance changing over time, but this was a tool to allow him to deal with spatially difform resistances, the distributions of which he could not otherwise describe. Treatise XV, however, begins by dealing with temporally changing resistances in their own right.

The first conclusion of treatise XV is an example of Swineshead’s mathematical ingeniousness, not in that he does complex mathematics, but in that he sees how to avoid complex mathematics. The conclusion concerns a nonresisting medium (or, in modern terms, a fixed space or vacuum) in which a resistance begins to be generated. The resistance first appears at one end of the medium and moves progressively across the medium in such a way that the resistance increases uniformly from that end up to the point where the resistance ends. In modern terms, then, we might represent the resistance graphically by a straight line that rotates around the origin, starting in a vertical position, rotating at a decreasing rate (so that any point of the line has a constant horizontal velocity), and increasing in length so that the maximum height of the end point is a constant (see Figure 3). If, Swineshead concludes, a mobile begins to move from the same extreme of the medium at which the resistance begins to be generated, then it will move with a constant velocity (always keeping pace with the progress of a given degree of resistance), provided that the maximum resistance moves away from the mobile faster than the mobile could move with that resistance (48vb). Swineshead proves this conclusion first by showing that there could not have been any initial period of time during which the mobile increased or decreased its velocity, and second by showing that the mobile could not later begin to move faster or more slowly than its given resistance. In the later proof he argues, for instance, that if the mobile were supposed to increase its velocity, then it would immediately begin to encounter greater resistances, implying a decrease rather than an increase in velocity and thus a contradiction; and if the mobile were supposed to decrease its velocity, then conversely it would immediately begin to encounter lesser resistances implying an increase rather than a decrease in velocity and thus another contradiction. So, therefore, it must continue with a constant velocity. As stated above, from a modern point of view, any position that connects resistance with velocity as Bradwardine’s function does would seem to be very problematic when applied to difform resistances, given that position (and therefore resistance) would determine the velocity of the

mobile, and yet velocity (and initial position) would also determine position, involving a double dependency. We see Swineshead here, however, not only coping with the problems of such a double dependency, but even playing with it and in a sense making sport to come up with ever more intriguing and complicated conclusions. Following the first conclusion, Swineshead proceeds to show that if the motion of the latitude of resistance is accelerated or decelerated the motion of the mobile will accelerate or decelerate also (49ra–50ra), and he goes on from there to prove nine other conclusions concerned with the generation of a latitude of resistance in a nonresisting medium (50vb–51rb).

The second main part of treatise XV concerns the motion of powers augmenting from zero degree in uniformly difform resistances. Again, perhaps the first conclusion of this part may serve as an example of the fourteen conclusions proved. If there is a uniformly difform medium terminated at zero degree in which a power begins to move as it augments uniformly from zero degree (always moving according to the proportion of its power to the point of the medium at which it is), then this power will continually move uniformly (51rb–51va).

The further conclusions of this second part all concern temporally constant uniformly difform resistances, but the powers are allowed to vary in different ways, both uniformly and difformly. These conclusions differ from the conclusions of treatise XIV not only because the resistances involved are uniformly difform rather than difformly difform, but also because the changes of resistance and power are given as the independent variables rather than the velocities, as in treatise XIV.

In this treatise as in the last, the attempt seems to be to give a general description of the possible interrelations of power, resistance, and velocity and their changes. As in the last, too, the main variations of the independent variables considered are uniformly difform variations. This seems to result from the mathematical tractability of such situations rather than from any observational or theoretical context that made such variations likely, as was the case for illumination in treatise XIII, where there was a common belief that the distribution of illumination from a light source was uniformly difform. Within these limitations, Swineshead does, however, manage to build up more and more complex conclusions without doing much explicit mathematics. He rather simply picks those cases, however complex they may appear, about which something can be said on the basis of the general characteristics of the functions involved.

In the third and last main part of treatise XV Swineshead should then have combined his two variables to allow both resistance and power to vary at the same time and to draw conclusions. The treatise simply ends, however, with the words: “It remains to inquire how both [resistance and power] may simultaneously be acquired”. (The 1520 edition adds: “and first are posited rules, etc”.) Given the complexity of the conclusions that resulted when only resistance or power was allowed to vary, one could hardly blame Swineshead for failing to push on further.

Treatise XVI is broken down into five chapters, each considering a class of problems concerning the induction of the maximum degree. Chapter 1 considers the alterations of larger and smaller uniformly difform subjects altered either uniformly throughout or by a uniformly difform latitude of alteration. Chapter 2 considers cases where a difformly difform latitude of alteration is extended through the part remaining to be brought to maximum degree in the same way as it was extended through the whole at the start.

Chapters 1 and 2 both consider cases where the alteration is extended at the start through the whole subject. Chapter 3, then, considers cases where the alteration does not extend through the whole subject at the start, but rather begins to be generated at the more intense extreme of the subject. Chapter 4 considers how the induction of the maximum degree is to be measured when the subject is rarefied or condensed during the alteration, deciding that such induction should not be measured by the fixed space outside the subject, but rather, with certain qualifications, by the subject itself. Chapter 5 considers how it may occur, through the successive generation of alteration in a subject, that the subject remains or becomes uniformly difform. Treatise XVI may be the most complex of the *Liber calculationum*, but perhaps enough has been said about the previous treatises so that its character can be imagined without the detailed examination of any of its conclusions.

As in the previous treatises, Swineshead in treatise XVI is preoccupied with uniformly difform alterations and the like, probably because they were well defined, whereas with difformly difform alterations the situation becomes overly complex. It might be remembered that, where alterations are concerned, as in this treatise, there was the common view that all qualitative actions, like light, decrease uniformly difformly (that is, linearly) as one moves away from the agent or source. So here again, as in the treatise on illumination, there might be a physical reason for emphasizing uniformly difform distributions. Nevertheless, if there was such a reason, it is well in the background.

## Opuscula

Of the three short *opuscula* that may be assigned to Richard Swineshead, two by explicit ascription to a Swineshead as author, and one by its position between the two others, one is a partial commentary on the *De caelo* and the other two are partially repetitive treatments of motion.

** In librum de caelo.** Apparently part of a commentary, beginning from text 35 of book I of the

*De caelo*, this short fragment is in two main parts, the first dealing with Aristotle’s proofs that an infinite body cannot move locally, and the second dealing with the relation of substances to their qualities, in connection with the possibility of action and passion between infinite bodies.

In the first main part, Swineshead considers (1) the proofs that an infinite body cannot rotate, drawing paradoxes concerning the intersections of infinite lines during such rotation: (2) the proofs against translational motion of infinites: and (3) Aristotle’s arguments that there cannot be an infinite body so that, *a fortiori*, there cannot be any motion of an infinite body.

In the second main part, Swineshead considers (1) Aristotle’s discussion of the possible action and passion of infinites, both with relation to simple subjects and with relation to compounds or mixtures: (2) the question of whether substantial forms can vary within some latitude, apparently with the idea that, if elemental forms can be remitted, then a finite action of an infinite subject might be within the range of possibility; (3) a proof that elements can exist without their qualities; (4) a proof that there cannot be mixed bodies of two or more elements of degrees as remiss as desired (in this part, Swineshead mentions Dumbleton by name and refutes some arguments he makes in part IV of his *Summa*); and (5) arguments concerning the possible perpetuation and duration of compounds. The opinions expressed in this work do not seem to be in conflict with the conclusions of the *Liber calculationum*.

** De motu** . This second short work (following in Cambridge MS Gonville and Caius 499/268 directly after the

*De caelo*fragment) is not explicitly ascribed to Swineshead, but its similarity to the last short work of the manuscript, which is ascribed to Swineshead, is so great that there is every reason to ascribe it also to Swineshead. In the Seville manuscript Colombina 7–7-29 the works (the

*De motu*and the following

*De motu locali*)appear as one, but this seems hardly plausible since there is a great deal of overlap and repetition between them. It is more likely that they are successive drafts of the same work than that they are both sections of a single larger work.

The *De motu* contains an introduction concerning the material, formal, efficient, and final causes of motion, and two main sections, the first dealing with the measurement of motion with respect to cause and the second with the measurement of motion with respect to effect. Only local motion is considered. The subject matter of both sections is similar to that of treatise XIV of the *Liber calculationum*. Roughly speaking, both the *De motu* and the *De motu locali* seem to occupy an intermediate position between Heytesbury’s *Regule* on local motion and the *Calculationes*, a natural supposition being that Swineshead began from Heytesbury’s work and went on to develop his ideas from that point.

Concerning the consideration of motion with respect to cause, Swineshead begins by expounding Bradwardine’s function relating powers, resistances, and velocities (212ra). This is followed by a number of rules, some of which are the same as, and others similar to, the conclusions of treatise XIV of the *Liber calculationum* (212ra–212va). Swineshead says, for instance, that if a constant power begins to move in the more remiss extreme of a uniformly difform resistance, it will remit its motion more and more slowly (212ra–vb). (Cf. rule 24 of treatise XIV of the *Liber calculationum*).

In the second section, Swineshead begins with a number of statements concerning the measurement of motion with respect to effect–for instance, that uniform local motion is measured by the line described by the fastest moved point (212vb)–many of which have close analogues in Heytesbury’s *Regule*. He states the mean speed rule (213ra) and the rule that in a motion uniformly accelerated from zero or decelerated to zero three times as much is traversed by the more intense half of the motion as by the more remiss half of the motion (213ra), which he derives as a consequence of the mean speed rule.

The second section concludes with five conclusions and a statement concerning the measurement of difform motion. The first three conclusions have to do with the traversal of extended resistances, one of these being the thirtieth rule of treatise XIV of the *Liber calculationum*: if one constant power remits its motion uniformly to zero in a given difform resistance, no other greater or lesser constant power will uniformly intend or remit its motion traversing the same medium (213ra). Concerning velocity in difform motion, Swineshead says that it is not to be measured by the maximum line that is described, but rather by the line that would be described, if the velocity were continued for a period of time (213rb).

The *De motu*, then, consists mostly of a series of conclusions along with several stipulations as to how motion is to be measured with respect to cause and with respect to effect. Although the conclusions are so divided, there is little to distinguish them.

** De motu locali** . This short work receives its title from its explicit, and hence we retain the name despite the fact that this last fragment contains treatments of alteration as well as of local motion. The first two sections of this work correspond to the two main sections of the

*De motu*, and many of the rules or conclusions stated are the same.

The work starts with a series of conclusions concerning the effect on velocity of increasing or decreasing the power or resistance in motion (213rb-vb). Many of these conclusions, including eight of the first nine, appear also in the *De motu*, and a similar number, although not always the same ones, appear also in the *Liber calculationum,* treatise XIV. Although these conclusions presuppose Bradwardine’s function relating powers, resistances, and velocities, that function is not explicitly stated as it was in the *De motu*.

These rules are followed by a section concerning the measures of resistance, for instance that two resistances of which the most intense degrees are equal must be themselves equal (213vb); and that a motor equal to the maximum degree of a uniformly difform resistance will move in it eternally, never completely traversing it (214ra).

The second section of the *De motu locali* concerns the measures of motion with respect to effect, starting with the stipulation that the velocity of local motion is measured by the line that would be traversed by the fastest moved point (if there is one), provided that it continued its velocity uniformly for a period of time (214rb). Difformly difform motions, Swineshead says, always correspond to some degree within their range of variation (*ibid.*), and uniformly difform motions correspond to their middle degrees (214rb-va). On this basis Swineshead mentions four types of sophisms that can arise from the comparison of accelerations to velocities of which some, he says, are possible, that is, not self-contradictory, and some impossible and to be rejected (214va-vb). A mobile can never begin from an infinite part of a magnitude and traverse some part uniformly (215ra), nor can a motion be remitted uniformly from an infinite degree of velocity (*ibid.*).

If one wants to know how much is traversed by a uniformly difform motion starting and ending at a degree, all one can say in general is that more is traversed than by a mobile moving for the same time with half the maximum degree of the uniformly difform motion (214vb-215ra). All local motions are as fast as any of their parts, and all subjects moved locally are moved as fast as any of their parts, the former being true for all motions but the latter being true only for local motions and not for alterations (215ra).

The third and last section (215ra-215rb) of the *De motu locali* concerns measures of alteration. Like local motion the measure of alteration with respect to cause is the proportion of the power of the altering agent to the power of the altered patient. With respect to effect, the velocity of motion of alteration depends on the maximum latitude of quality that would be acquired by any part of the subject if the velocity were continued for some time period. Irrespective of the measure of velocity of alteration, a subject need not be altered as fast as any of its parts, but often is altered more slowly than a part of it closer to the agent. To determine how fast a subject is altered one has to consider the degree to which it corresponds, calculating the contribution of various degrees to the subject’s denomination by the proportion of the subject through which they are extended.

Many sophisms can arise from the comparison of velocities of alteration to the velocities with which subjects are altered. Something may be altered with a faster velocity of alteration and yet be more slowly altered, and hence the two separate measures of alteration must be kept separate. On this basis, it is not contradictory for an agent to alter faster than anything is altered by it. If a uniformly difform alteration is extended throughout a subject, the subject will be altered as the middle degree of that alteration. With brief references to other ways of dealing with alteration the *De motu locali* ends.

## Conclusion

As is evident from the discussions of its separate treatises, the *Liber calculationum* places what may seem to be an uncommon emphasis upon the generation of *conclusiones, regulae, objectiones*, and *sophismata*. This unceasing generation of results occurs in other works of the calculatory tradition, but it reaches a high with Richard Swineshead, so much so as to be nearly the defining characteristic of the *Liber calculationum*. Results are drawn to the very limits of manageability. Swineshead’s *opuscula, De motu* and *De motu locali*, exhibit the beginnings of this effort to educe results. We, operating with modern mathematics, could generate many more such results, but the subclass that Swineshead himself generates and treats almost completely exhausts the results he could have dealt with, given the techniques at his disposal. And this excogitation of results occurs whether Swineshead is examining two or more *positiones* or *opinions* relative to a given topic (when, as said above, only slight attention is paid to deciding definitively between *opiniones*) or only one. The major difference between these two cases appears to be that, in the former, Swineshead is more apt to label as *sophismata* the results he is generating. In any event, he makes it quite clear that *conclusiones* can be elicited, objected to, and resolved on both or all sides when a plurality of *opiniones* is at stake (*multe conclusiones possunt elici ex dicits, ad quarum tamen utramque partem probabiles possunt fieri rations, que per predicta, si bene intelligantur, satis faciliter solvuntur* “34ra”).

Furthermore, there is almost no discussion in the *Liber calculationum* of the contexts in which one might expect the situations represented by these results to occur, nor, indeed, any time spent investigating whether they can occur. Instead, the work proceeds almost entirely *secundum imaginationem*, as Heytesbury’s *Regule solvendi sophismata* had before, it. In fact, among all other “Metronian” works, the *Liber calculationum* is most like Heytesbury’s *Regule*. Some of the earlier Oxford calculators, like Walter Burley and Roger Swineshead, had fairly frequently considered natural, as well as *de imaginatione*, situations. Richard Swineshead, by contrast, quite consistently imagines situations that will illustrate and drew results out of various theories rather than taking examples from natural occurrence. The emphasis is upon developing and having a set of techniques as complete as one can make it. If we may trust an inference from the later commentary of Gaetano of Thiene on Heytesbury’s *Regule*, one of the reasons for this emphasis was simply the fact that every “calculator” ought to have a system applicable to every conceivable situation (. . . *dicit* [*sc.*, Heytesbury] *quod hoc est tamen impossibile . . . physice loquendo . . . Sed dicit ille magister bene scis hoc, sed quia non implicat contradictionem et est satis imaginabile, ideo calculators non debent fugere casum* [Venice ed., 1494, 48va]).

The “techniques” that are presented in the *Liber calculationum* may strike the modern reader as basically mathematical. But one must consider such a judgement with care. To begin with, the evidence of the extant manuscripts of the *Liber calculationum* tells us that the medievals themselves did not regard it (or any of the other “calculatory” works, for that matter) as mathematical in the sense of Euclid, Boethius’ *Arithmetica*, or Jordanus de Nemore. When the *Liber calculationum* does not take up an entire codex, it or fragments from it invariably appear with other treatises, questions, or notes no natural philosophy or logic. Nevertheless, there is no doubt that, the evidence of medical codification aside, mathematical functions and val codification pervade Swineshead’s major work. They are applied, however, not in order to understand how some phenomenon normally occurs (as was the case in medieval optics, statics, and astronomy), but in a thoroughly *secundum imagunatinem* fashion that is totally different, from the Greek-based mathematical tradition inherited by the Middle Ages. A good deal of material from the Greek tradition was, of course, utilized by Swineshead, but he utilized it in most un-Greek way. Mathematical functions are applied in order to determine all conceivable contributions that parts could make to wholes, to distinguish the discontinuous from the continuous, and to encompass situations or results involving infinite values. It is for these kinds of problems that the *Liber calculationum* contained the required techniques. To the medieval scholar with the patience and ability needed to comprehend this work, the purpose seems to have been that he should learn to operate with the techniques and rules given by Swineshead–just as one solving sophisms to which the *Liber calculationum* from time to time refers–so expertly as to be able to handle situations and *casus* in every corner of the fourteenth century realm, be it physical, logical, medical, theological, or whatever.

## Dissemination and Influence of the Liber calculationum

Although the work of Swineshead’s fellow Mertonians and the English “calculatory” tradition in general is generously represented in later fourteenth-century natural philosophy (very notably so, for example, in the work of Oresme), specific evidence of the *Liber calculationum* during this period is not especially plentiful, although shortly after mid-century the English logician Richard Ferebrich appears to employ parts of treatise XIV in his *Calculationes de motu.* Most of the extant manuscripts of Swineshead’s major work are of the fifteenth century, and the records we have of its occurrence in library catalogues for the most part date from 1400 and later. It does appear, however, in at least two “student notebooks” of the later fourteenth century: in one merely in terms of fragmentary traces (MS Bibliothèque Nationale, fonds latin 16621), in the other more substantially (MS Worcester Cathedral, F. 35).

If we ask for evidence of the dissemination and influence not of Mertonian ideas in general or of some particular idea like the so-called Merton mean speed theorem, but rather of parts of the *Liber calculationum* itself, the fifteenth and sixteenth centuries are far richer. The first center of interest is Italy, in the middle and toward the end of the fifteenth century, where Swineshead appears as part of the broader preoccupation with English logic and natural philosophy. He appears, moreover, in terms both *pro* and *contra.* He suffers part of the humanist criticism of the “barbari Britanni” that one finds in the likes of Coluccio Salutati; for Leonardo Bruni he is one of those “quorum etiam nomina perhorresco” and he even gave his name to the “sophisticas quisquilias et *suisetica* inania” complained about loudly by Ermolao Barbaro and others.

Yet it seems fair to claim that one of the major reasons for these humanist complaints was not Swineshead’s works themselves, but rather the fact that they and other English “calculations” had attracted the attention of a fair number of Italian scholars. Thus, his views are found among those of others treated in works *De reactione* written by Gaietano de Thienis, Giovanni Marliani, Angelus de Fossambruno, Vittore Trincavelli, and Pietro Pomponazzi. He also appears to have been very much in the center of the interest in “calculatory” matters in Padua, and especially Pavia, around the mid-fifteenth century, something that jibes extremely well with the number of times that the *Liber calculationum* was published in these cities. We also know that Nicolletto Vernia, earlier a student of both Paul of Pergola and Gaietano de Thienis, went to Pavia to study the *Calculationes Suisset,* information that fits very well indeed with other things we are able to put together about Vernia’s interests. Pomponazzi relates that he engaged in a dispute with Francesco di Nardò armed with *argumentis calculatorüs;* but we have more direct evidence of his concern with Swineshead from the fact that he refers to the “Calculator” in his (unedited) *Questio de anima intellectiva* and from the fact that Vernia owned a copy of the *Liber calculationum* (now Biblioteca Vittorio Emanuele, MS 250).

When one turns to the Italian commentaries or questions on the *Liber calculationum,* the first thing to be noted is that most of this literature is preoccupied only with treatise **I:** *De intensione et remissione formarum.* It was this part of Swineshead that occupied Pomponazzi in his own treatise of the same title written in 1514, and we have similar sixteenth-century works by Tiberio Baccilieri, Cardinal Domenico Grimani, and Hieronymus Picus. Pomponazzi is critical of Swineshead insofar as the “scale of measure” he proposed in treatise I of the *Liber calculationum* maintains the inverse proportionality of intension and remission and ignores their proper ontological status. Intension and remission should be viewed, Pomponazzi felt, as, respectively, perfection and imperfection; this done, one would not, as Swineshead, “measure” remission by nearness to zero degree, but rather, as an imperfection, in terms of its distance from the maximum degree. Pomponazzi refers to a similar disagreement with Swineshead’s “non gradum measure” in his *Super libello de substantia orbis,* where he specifically complains that it runs counter to the *via Aristotelis.* Pomponazzi’s criticism can be partially explained by the fact that in his (unedited) lectures on Aristotle’s *Physics* he felt that Swineshead and other English “calculators” put too much mathematics (*ille truffe spectant ad mathematicum*) and “geometricalia” into natural philosophy, and (as he complained in his *De intensione et remissione formarum*) constructed a *scientia* that was *media inter physicas et mathematicas.*

Italian *expositiones* or *questiones* on other parts of the *Liber calculationum* are rarer than those on treatise **I.** Bassanus Politus composed a *Tractatus proportionum* specifically claimed to be *introductorius**ad calculationes Suisset;* it sets forth in succinct fashion little more than the standard mathematics of proportion and proportionality drawn from such authors as Euclid and Boethius. More to the content of parts of the *Liber calculationum* is Marliani’s *Probatio cuiusdam sententie calculatioris de motu locali.* Its concern is with the views of Swineshead on “mean degree measure” in treatise II and his proofs for the “mean speed theorem” in treatise XIV. The two most complete Italian commentaries on the *Liber calculationum* are unedited and unstudied. One is by Christopher de Recaneto, *doctor in artibus* at Padua in 1454, and covers but treatise I and part of treatise **II.** The other is more extensive, commenting on treatises I–V and VII–VIII, but we know almost nothing of its author, Philippus Aiuta. All we know is that Marliani wrote a *difficultates* sent to Philippus and (as we learn from the incipit of the present commentary) that he was a doctor in arts and medicine. The commentary itself, apparently a compilation of Philippus’ view made in 1468 by one Magister Bernardinus Antonius de Spanochiis, relates Swineshead to any number of other mathematicians (especially Euclid) and philosophers, both contemporary, earlier medieval, and ancient. But perhaps its most intriguing aspect is the explicit tendency to render Swineshead more comprehensible by the addition of appropriate figures (*quasdam ymagines in marginibus*).

The second major center of interest in Swineshead was Paris at the beginning of the sixteenth century, where a considerable amount of work was done with the *Liber calculationum,* largely by a group of Spanish and Portugueses scholars. The earliest, and certainly most impressive, among them appears to be Alvaro Thomaz. His *Liber de triplici motu,* published in Paris in 1509, contains an extensive, two-part, preliminary treatise expounding all aspects of the mathematics of proportions, and itself treats all parts of the *Liber calculationum* that deal with *motus.* Alvaro also includes much material drawn from Nicole Oresme, not only explaining, but on occasions expanding what he finds in Swineshead and Oresme. One notable instance of such “expansion” is his treatment of the “infinite series” treated by both of these fourteenth-century authors. Following Alvaro by a few years, both John Dullaert of Ghent (in 1512) and Juan de Celaya (in 1517) include comprehensive expositions of “calculatory” material in their *Questiones* on Aristotle’s *Physics.* Celaya treats a good number of *conclusiones* drawn from parts (treatises **I, II, IV, VI, XIV, XV** ) of the *Liber calculationum* and appears to have followed Thomaz in at least the structure of much of what he includes. Another Spanish member of this Paris school was Luis Coronel, who does not, like Celaya, include a lengthy connected exposition of Swineshead material in his *Physice perscrutationes* (published in 1511), but who nevertheless does discuss a fair number of issues and passages from scattered treatises (**I, II, V, X, XI** ) of the *Liber calculationum.* At times he seems to lack a proper understanding of what he is discussing from Swineshead and even complains how *prolixissime et tediose* the reasoning is (in this instance referring to treatise XI). The same kind of complaint and lack of comprehension is probably in part behind the remarks of Diego de Astudillo in his *Questiones* on Aristotle’s *Physics* when he excuses his omission of “calculatory disputations” since he would “confound the judgments of beginners . . . ignorant of mathematics.” Indeed, if it is a proper appreciation of Swineshead’s accomplishments that one has in mind, then none of the “commentators,” Italian or Spanish, save Thomaz, qualifies.

Some Renaissance figures continued the fifteenth-century humanist criticism of English “subtilitates” and ridiculed (Luis Vives, for example) Swineshead’s work, but others, such as Julius Scaliger and Cardano, praised him as outstandingly acute and ingenious. The most famous later “appreciation” of the Calculator was that of Leibniz. Perhaps initially learning of Swineshead through Scaliger (whom he mentioned in this regard), Leibniz confessed to a certain admiration of Swineshead even before he had had the opportunity to read him. How thoroughly Leibniz finally did read the *Liber calculationum* we do not know; but we do know that he went to the trouble to have the 1520 Venice edition of it transcribed (today Hannover, Niedersächssische Landesbibliothek, MS 615). In Leibniz’ eyes, Swineshead’s primary accomplishment lay in introducing mathematics into scholastic philosophy, although we should perhaps take “mathematics” to include a certain amount of logic, since at times he coupled Swineshead with Ramon Lull in having accomplished this task. In any event, what Swineshead had done was in his eyes much in harmony with Leibniz’ own convictions about the relation of mathematics and “mathematical” logic to philosophy.

Because of the remarks of the likes of Cardano, Scaliger, and Leibniz, Swineshead found his way into eighteenth-century histories of philosophy such as that of Jacob Brucker. After that he was forgotten until Pierre Duhem rediscovered him at the beginning of the twentieth century in his rediscovery of medieval science as a whole.

## BIBLIOGRAPHY

I. Original Works. Manuscripts. * of the *Liber calculationum* include Cambridge, Gonville and Caius 499/268, 165r–203v (14c.), tr. **I–XI, XV, XII–XIII** ; *Erfurt, Stadtbibl., Amplon. 0.78, 1r–33r (14c.), contains an abbreviated version of tr. **I–II, IV–VIII** . Ascribed to one “clymiton” (presumably Killington?) by Schum in his catalogue of the Amplonian MSS, apparently drawing his information from Amplonius Ratinck’ fifteenth-century catalogue; *Padua, Bibl. Univ. 924, 51r–70r (15c.), tr. **I, VII, VIII, IX** ; *Paris, Bibl. Nat. lat. 6558, Ir–;70v (dated 1375), tr. **I–XI, XV, XII–XIII;** *Paris, Bibl. Nat. lat. 16621 (14c.), *passim*, fragments (often in altered form); e.g., 52r–v, 212v from tr. XIV; *Pavia, Bibl, Univ., Aldini 314, Ir–83r(15c.), tr. **I–XI, XV, XII–XIII, XVI, XIV** ; Perugia, Bibl. Comm. 1062, Ir–82r (15c.); *Rome, Bibl. Angelica 1963, 1r–106v (15c.), tr. **I–IX, XI–XVI** ; *Rome, Bibl. Vitt. Emanuele 250, 1r–82r (15c.), contains all tr. except III in the following order: **I, II, V, VI, XIV** (incomplete), **XV, XIV** (complete), **XVI, XII, XIII, VII, VIII, IX, IV, X, XI** : belonged to Nicoletto Vernia; *Vatican, Vat. lat. 3064, 1r–120v (15c.), tr. **I–X, XVI, XI, XV, XIV, XII–XIII, III** (again); *Vatican, Vat. lat. 3095, 1r-119v (15c.), tr. **I–X, XVI, XI, XV, XII–XIV** ; Vatican, Chigi E. IV. 120, 1r–112v (15c.); Venice, Bibl. Naz. San Marco., lat. VI, 226, 1r–98v (15c.). tr. **I–II, IV–XII, III** (according to L. Thorndike); *Worcester, Cathedral F. 35, 3r, 27r–65v, 70r–75v (14c.), contains, in following order, fragment (Reg. 1–4) of **XIV, I–IV, VI–VII, XII–XIII, XVI, V** ; Cesena, Bibl. Malatest., Plut. IX, sin cod VI.

Editions include Padua *ca*. 1477; Pavia 1498; Venice, 1520. A modern edition of treatise XI has been published in the article of Hoskin and Molland cited below.

The *Opuscula* are *In librum de caelo* (Cambridge, Gonville and Caius 499/268, 204r–211v “15c.”Worcester, Cathedral F. 35, 65v–69v “14c.”, incomplete); *De motu* (Cambridge, Gonville and Caius 499–268, 212r–213r [15c.]; Oxford, Bodl., Digby 154, 42r–44v “14c.” and Seville, Bibl. Colomb. 7–7-29, 28v–30v [15c.]; and *De motu locali* (Cambridge, Gonville and Caius 499/268, 213r–215r [15c].” Seville, Bibl. Colomb. 7–7-29, 30v–34r [15c.]).

The *De motibus naturalibus* of Roger Swineshead; Erfurt, Stadtbibl., Amplon. F. 135, 25r–47v (14c.); Paris, Bibl. nat., lat. 16621, 39r, 40v–51v, 54v–62r, 66r–84v (14c.), Fragment of part II, all of parts **III–VIII** ; Venice, Bibl. Naz. San Marco lat, VI, 62, 11 lr(15c.), definitions of part IV. For MSS of the *De obligationibus*

*Many of the **MSS** do not contain all sixteen *tractatus* of the *Liber calculationum*; the incipits and explicits of each *tractatus* have been checked by the authors for those **MSS** indicated by an asterisk.

*and De insolubilibus* ascribed to Roger, see the article by Weisheipl on Roger cited below.

The *Questiones quatuor super physicas magistri Ricardi* (in **MSS** Vatican, Vat. lat. 2148, 71r–77v; Vat. lat 4429. 64r–70r; Venice, San Marco lat. **VI** , 72, 81r–112r, 168r–169v) that have been tentatively ascribed to Swineshead by Anneliese Maier are in all probability not his, but rather most likely the work of Richard Killington (or Kilvington).

II. Secondary Literature. The two basic biographical-bibliographical sources are A. B. Emden, *A Biographical Register of the University of Oxford to A.D. 1500* (Oxford, 1957–1959), 1836–1837, which includes material on Roger and John; and James A. Weisheipl, “Roger Swyneshed, O.S.B., Logician, Natural Philosopher, and Theologian,” in *Oxford Studies Presented to Daniel Callus* (Oxford, 1964), 231–252. These both replace G .C . Brodrick, *Memorials of Merton College* (Oxford, 1885), 212–213. Emden has a long list of variant spellings of Swineshead such as Swyneshed, which he prefers, Suicet, Suincet, etc.

On Swineshead and the calculatory tradition in general, see Pierre Duhem, *Études sur Léonard de Vinci*, **III** , “Les précureseurs parisiens de Galilée” (Paris, 1913), 405–480; most of this material is reprinted with some additions, omisions, and changes in Duhem’s *Le système du monde*, VII (Paris, 1956), 601–653. See also Marshall Clagett, *The Science of Mechanics in the Middle Ages* (Madison, Wis., 1959), chs. 4–7; Anneliese Maier, *Die Vorläufer Gulileis im 14. Jahrhundert (=Studien zur Naturphilosophie der Spätscholastik*, I), 2nded. (Rome, 1966); *Zwei Grundprobleme der scholastischen Naturphilosophie (=Studien, II),* 3rd ed. (Rome, 1968); *An der Grenze von Scholastik und Naturwissen-schaft (=Studien, III)*, 2nd ed. (Rome, 1952); John Murdoch, “*Mathesis in philosophiam scholasticam introducta*. The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology,” in *Arts libéraux et philosophie au moyen âge* (=Acts du quatrième Congrès International de Philosophie Médiévale), (Montreal–Paris, 1969), 215–254; A. G. Molland, “The Geometrical Background to the ’Merton School’” in *British Journal for the History of Science*, **4** (1968), 108–125; Edith Sylla, *The Oxford Calculators and the Mathematics of Motion, 1320–1350: Physics and Measurement by Latitudes* (unpublished diss., Harvard Univ., 1970); “Medieval Quantifications of Qualities: The ’Merton School,’” in *Archive for History of Exact Sciences*, **8** (1971), 9–39; “Medieval Concepts of the Lalitude of Forms: The Oxford Calculators,” in *Archives d’histoire doctrinale et littéraire du moyen âge*, **40** (1973), 223–283, More particularly on Swineshead see James A. Weisheipl, “Ockham and Some Mertonians,” In *Mediaeval Studies*, **30** (1968), 207–213; Lynn Thorndike, *A History of Magic and Experimental Science*, **8** vols. (New York, 1923–1958), III, 370–385.

See the following on particular treatises of the *Liber calculationum* (although much information is also provided on the individual treatises in some of the more comprehensive literature above): tr. I: Marshall Clagett, “Richard Swineshead and Late Medieval Physics,” in *Osiris*, **9** (1950), 131–161; tr. II: John Murdoch, “Philosophy and the Enterprise of Science in the Later Middle Ages,” in The *Interaction Between Science and Philosophy*, Y. Elkana, ed. (Atlantic Highlands, N.J., 1974), 67–68; tr. VII: Clagett, *Giovanni Marliani and Late Medieval Physics* (New York, 1941), ch. 2; tr. X: Curtis Wilson, *William Heytesbury: Medieval Logic and the Rise of Mathematical Physics* (Madison, Wis., 1956), ch. 3; tr. XI: M. A. Hoskin and A. G. Molland, “Swineshead on Falling Bodies: An Example of Fourteenth-Century Physics,” in *British Journal for the History of Science*, **3** (1966), 150–;182; A. G. Molland, “Richard Swineshead on Continuously Varying Quantities,” in *Actes du XII ^{e} Congrès International d’Histoire des Sciences*,

**4**(Paris, 1968), 127–;130; John Murdoch, “Mathesis in philosophiam . . .,” 230–;231, 250–254; tr. XIV: John Murdoch,

*op cit*., 228–230; Marshall Clagett,

*The Science of Mechanics*. . . , 290–304, for Swineshead’s proof of the so-called mean speed theorem.

For functions of *F. R*. and *V* similar to Roger Swineshead’s, one may consult Ernest Moody, “Galileo and Avempace: The Dynamics of the Leaning Tower Experiment,” in *Journal of the History of Ideas*, **12** (1951), 163–;193, 375–;422. For the “addition” theory of qualitative change mentioned in the discussion of treatise VII, see E. Sylla, “Medieval Concepts of the Latitude of Forms....” This article also contains a more complete discussion of the ideas of Roger Swineshead.

*The dissemination and influence of the Liber calculationum*. To date, the most adequate treatment of the spread of late-medieval natural philosophy in general is Marshall Clagett’s chapter on “English and French Physics, 1350–1600,” in *The Science of Mechanics in the Middle Ages* (Madison, Wis. 1959). This chapter contains (630–631) a brief account of a segment of Richard Ferebrich’s (Feribrigge) *Calculationes de motu*.

*The Italian reception of Swineshead*. On the humanist criticism of “Suissetica” and other, equally infamous, “English subtleties,” the point of departure is Eugenio Garin, “La cultura fiorentina nella seconda metà del trecento e i ‘barbari Britanni,’” as in his *L’età nuova. Ricerche di storia della cultura dal XII al XVI secoto* (Naples, 1969), 139–177; the relevant passages from Coluccio Salutati and Leonardo Bruni Aretino, as well as many other similar sources are cited therein. To this one might add the two anonymous letters of Ermolao Barbaro, edited by V. Branca; *Epistolae* (Florence, 1943), II, 22–23 (on which see the article by Dionisotti below). The various works *De reactione* treating Swineshead’s opinion are Angelus de Fossambruno, MS Venice, Bibl. Naz. San Marco, VI, 160, ff. 248–252r: *Gaietano de Thienis. Tractatus perutilis de reactione*, Venice ed., 1491: Giovanni Marliani, *Tractatus de reactione* and *In defensionem tractatus de reactione*, both printed in his *Opera omnia*, II (Pavia, 1482) [on which see Clagett, *Giovanni Marliani and Late Medeival Physics*, (New York, 1941), ch. 2]; Victorus Trincavellus, *Questio de reactione iuxta doctrinam Aristotelis et Averrois commentatoris*, printed at the end (69r–74r) of the 1520 Venice ed. of the *Liber calculationum*; Pietro Pomponazzi, *Tractatus acutissimi … de reactione* (Venice, 1525).

The case for Pavia as the fifteenth-century center of “calculatory,” and especially Swineshead, studies is made by Carlo Dionisotti, “Ermolao Barbaro e la Fortuna di Suiseth,” in *Medioevo e Rinascimento: Studi in onore di Bruno Nardi* (Florence, 1955), 219–253. Dionisotti briefly treats the evidence for Nicoletto Vernia’s studies in Pavia on the Calculator, but see also Eugenio Garin, “Noterelle sulla filosofia del Rinascimento I: A proposito di N. Vernia,” in Rinascimento, **2** (1951), 57–62. The reference from Vernia’s *De anima intellectiva* (MS Venice, Bibl. Naz. San Marco, VI, 105, ff. 156r–160r) was furnished by Edward Mahoney, who is preparing an edition of the text. The relevant bibliography of the Italian commentaries on treatise I, *De intensione et remissione formarum*, of the *Liber calculationum* is Tiberio Baccilieri: Bruno Nardi, *Sigieri di Brabante nel pensiero del Rinascimento italiano* (Rome, 1945), 138–139; Pearl Kibre, “Cardinal Domenico Grimani, ’Questio de intensione et remissione qualitatis’: A Commentary on the Tractate of that Title by Richard Suiseth (Calculator),” in *Didascaliae: Studies in Honor of Anselm M. Albareda*, Sesto Prete, ed. (New York, 1961), 149–203; Charles Schmitt, “Hieronymus Picus, Renaissance Platonism and the Calculator,” to appear in Anneliese Maier Festschrift; Curtis Wilson, “Pomponazzi’s Criticism of Calculator,” in *Isis*, **44** (1952), 355–363. Other references to Pomponazzi on Swineshead can be found in Pietro Pomponazzi. *Corsi inediti dell’insegnamento padovano, I: Super Libello de substantia orbis, exposition et Quaestiones Quattuor* (1507); *Introduzione e testo a cura di Antonino Poppi* (Padua, 1966); and Bruno Nardi, *Saggi sull’ Aristotelismo Padovano dal secolo XIV al XVI* (Florence, 1958).

The relevant sources or literature concerning other parts of the *Liber calculationum* are Bassanus Politus, *Tractatus proportionum introductorius ad calculationes Suiset* (Venice, 1505); Giovanni Marliani, *Probatio cuiusdam sententie calculatoris*, in his *Opera omnia*, II (Pavia, 1482), ff. 19r–25r (on which see Clagett, *Giovanni Marliani and Late Medieval Physics* [New York, 1941], ch. 5); Christopher de Recaneto, *Recolecte super calculationes*, MS Venice, Bibl. Naz. San Marco, Lat. VI, 149, ff. 31r–49v. On Recaneto, see Nardi, *Saggi sull’Aristotelismo Padovano dal secolo XIV al XVI* (Florence, 1958), 117–;119, 121–122; Philippus Aiuta, “Pro declaratione Suiset calculatoris,” MS Bibl. Vaticana, Chigi E. VI, 197, ff. 132r–149r. Finally, mention should be made of a totally unexamined sixteenth-century work on the same part of the *Liber calculationum* by Raggius of Florence, MS Rome, Bibl. Casan, 1431 (B.VI.7).

*The Parisian-Spanish reception of Swineshead*. The relevant primary sources are [Alvaro Thomaz], *Liber de**triplici motu proportionibus annexis magistri Alvari Thome Ulixbonensis philosophicas Suiseth calculationes ex parte declarans* (Paris, 1509): John Dullaert of Ghent, *Questiones super octo libros phisicorum Aristotelies necnon super libros de celo et mundo* (Lyons, 1512): [Juan de Celaya], *Expositio magistri ioannis de Celaya Valentini in octo libros phisicorum Aristotelis: cum questionibus eiusdem secundum triplicem viam beati Thome, realium et nominalium* (Paris, 1517): Ludovicus Coronel, *Physice perscrutationes* (Paris, 1511): Diego de Astudillo, *Quaestiones super octo libros physicorum et super duos libros de generatione Aristotelis, una cum legitima textus expositione eorundem librorum* (Valladolid, 1532). Of the secondary literature on these–and other related–figures, the two basic articles are by William A. Wallace, “The Concept of Motion in the Sixteenth Century,” in *Proceedings of the American Catholic Philosophical Association*, **41** (1967), 184–195: “The “Calculatores’ in Early Sixteenth-Century Physics,” in *British Journal for the History of Science*, **4** (1969), 221–232. More detailed bio-bibliographical information on the school of Spanish scholars at Paris in the early sixteenth century can be found in H. Elie, “Quelques maîtres de l’université de Paris vers l’an 1500,” in *Archives d’histoire doctrinale et littéraire du moyen âge,***18** (1950–1951), 193–243: and R. Garcia Villoslada, *La universidad de Paris durante los estudios de Francisco de Vitoria, O.P., 1507–1522, Analecta Gregoriana*, XIV (Rome, 1938). On Alvaro Thomaz’ treatment of the “infinite series” in Swineshead (and in Nicole Oresme) see Marshall Clagett, *Nicole Oresme and the Medieval Geometry of Qualities and Motions* (Madison, Wis., 1968), 496–499, 514–516; and H. Wieleitner, “Zur Geschichte der unendlichen Reihen im christlichen Mittelalter,” in *Bibliotheca mathematica*, **14** (1913–1914), 150–168. Some biographical details on Alvaro can be found in J. Rey Pastor, *Los matemáticos españoles del siglo XVI* (Toledo, 1926), 82–89.

*Later sixteenth- and seventeenth-century appreciation of Swineshead*. Appropriate references to Cardano’s *De subtilitate* and Julius Scaliger’s *Exotericarum exercitationum* are given in Jacob Brucker’s *Historia critica philosophiae*, III (Leipzig, 1766), 851. This work contains (849–853) numerous quotations from other authors referring (pro and con) to Swineshead as well as a brief example of the *Liber calculationum itself*, Leibniz’references to Swineshead are too numerous (we know of at least eight of them) to cite completely, but some of the most important of them can be found in L. Couturat’s *Opuscules et fragments inédits de Leibniz* (Paris, 1903), 177, 199, 330, 340. Indication of the manuscript copy Leibniz had made of Swineshead can be found in Eduard Bodemann. *Die Handschriften der königlichen offentlichen Bibliothek zu Hannover* (Hannover, 1867), 104–;105.

John E. Murdoch

Edith Dudley Sylla

## Swineshead (Swyneshed, Suicet, Etc.), Richard

# SWINESHEAD (SWYNESHED, SUICET, ETC.), RICHARD

(*fl*. c. 1340–1355), *natural philosophy, calculations*.

For the original article on Swineshead see *DSB*, vol. 13.

In view of the thorough treatment of the works ascribed to one or another Swineshead in the original *DSB*, this postscript will be relatively brief. In the past thirty years, no new evidence has emerged that would contradict James Weisheipl’s argument that Richard Swineshead and Roger Swyneshed were distinct authors, despite being confused in manuscript attributions (recent articles on Roger’s logic tend to choose the spelling “Swyneshed,” whereas the spelling “Swineshead” is more frequently used for Richard, the Calculator, but the name is the same). The current consensus is that Richard Swineshead, fellow of Merton College, Oxford, was the author of the *Calculationes*(or *Liber Calculationum*), because of which he came to be known throughout Europe in the succeeding centuries as the “Calculator.” On the other hand, Roger Swyneshed was the author of influential works on logic, the *Obligationes* and *Insolubilia*(both of which were edited and published by Paul Spade in the later 1970s), and probably also of the work on motion known alternately as *Descriptiones motuum* or *De motibus naturalibus*, which applies logical (sophistical) and mathematical techniques to questions of Aristotelian natural philosophy. In light of this situation, there are two postscripts on “Swineshead,” this one for Richard, and another for Swyneshed (Swineshead), Roger.

The *Liber Calculationum* has been seen as a work applying mathematics to physics, but it is probably best understood, together with William Heytesbury’s *Regule Solvendi Sophismatum*(Rules for solving sophismata), as having been written to assist Oxford University arts students in preparing for their obligatory disputations on sophismata as well as on the subjects of Aristotelian natural philosophy (Sylla, 1982). Traditionally, logic formed a large part of the undergraduate curriculum, and student disputations accordingly were intended to hone students’ acuteness in logic. After the success of Thomas Bradwardine’s *De proportionibus velocitatum in motibus*(On the ratios of velocities in motions) of 1328, however, mathematical approaches using proportions were added to the stock of analytical tools that students were expected to be able to apply in disputations. Whereas some fourteenth-century Oxford arts masters were primarily interested in natural philosophy, and Roger Swyneshed’s *Descriptiones motuum* as well as John Dumbleton’s *Summa logicae et philosophiae naturalis* alternate between natural philosophical discussions and applications of the new analytical tools, Swineshead’s *Liber Calculationum* is more completely directed to assisting students with the application of the analytical tools. Nevertheless, as Stefano Caroti has shown, Swineshead’s views on natural philosophical topics such as “reaction” clearly fall within the range of English discussions current in his day. Here the issue was, for instance, how a hot and a cold body can each act on the other, the hot body heating the cold one and the cold body simultaneously cooling the hot body, given that Aristotelian physics stipulates that for action to occur, the power of the agent must be greater than the resisting power of the patient or body acted on.

In the years since the original *DSB*, considerable work has been done on the reception of the *Liber Calculationum*. As indicated in the original article, much attention was paid to Calculator’s work in Paris, various Italian universities, and elsewhere on the Continent. Whereas some scholars (and especially humanists) rejected the calculatory approach as overly technical and quibbling, a few such as Alvarus Thomas of Lisbon fully mastered Swineshead’s ideas together with those of Nicole Oresme, who built upon the ideas of both Thomas Bradwardine (q.v.) and Swineshead. Ultimately, the mathematics of Bradwardine’s rule, which was the basis of Treatise XIV of the *Liber Calculationum*, became foreign territory to philosophers as arithmetical operations on proportions were assimilated to arithmetical operations on numbers.

Ironically, Bassanus Politus’s *Tractatus proportionum introductorius ad calculations Suicet*(Venice, 1505) is among the works rejecting Bradwardine’s approach to proportions, thereby rendering itself useless in understanding key parts of Swineshead’s work, to which it purports to be an introduction. When G. W. Leibniz on many occasions praised Swineshead for having been among the first to introduce mathematics into natural philosophy, and when he commissioned a hand copy to be made of the 1520 printed edition of the *Liber Calculationum* available in the Royal Library in Paris, he was proposing to publish a series of new editions of the treasures (*cimelia*) or most valuable works in all fields to come out since the beginning of printing.

## BIBLIOGRAPHY

### SUPPLEMENTARY BIBLIOGRAPHY

Caroti, Stefano. “Da Walter Burley al *Tractatus Sex Inconvenientium*: La Tradizione Inglese della discussione medievale ‘De reactione.’” *Medioevo*21 (1995): 257–374. Deals with Swineshead’s *Calculationes*, pp. 331–357.

———. “*Reactio* in English Authors.” In *La Nouvelle Physique du XIV ^{e} Siecle*, edited by Stefano Caroti and Pierre Souffrin, 231–250. Florence, Italy: Leo S. Olschki, 1997. A shorter English version of the preceding Italian article.

Lewis, Christopher. “The Fortunes of Richard Swineshead in the

Time of Galileo.” *Annals of Science*33 (1976): 561–584.

———. *The Merton Tradition and Kinematics in Late Sixteenth and Early Seventeenth Century Italy*. Padua, Italy: Antenore, 1980.

Sylla, Edith D. “The Oxford Calculators.” In *The Cambridge History of Later Medieval Philosophy*, edited by Norman Kretzmann, Anthony Kenny, and Jan Pinborg, 540–563. Cambridge, U.K.: Cambridge University Press, 1982.

———. “The Fate of the Oxford Calculatory Tradition.” In

*L’Homme et Son Univers au Moyen Age*, edited by Christian Wenin, 692–698. Actes du septième congres international de philosophie médiévale (30 aout–4 septembre 1982 (Philosophes Mediévaux, vol. 27). Louvain-la-Neuve, Belgium: Editions de l’Institut Superieur de Philosophie, l986.

———. “Alvarus Thomas and the Role of Logic and Calculations in Sixteenth Century Natural Philosophy.” In *Studies in Medieval Natural Philosophy*, edited by Stefano Caroti, 257–298. Biblioteca di Nuncius, Studi e Testi I. Florence, Italy: Leo S. Olschki, 1989. As late as the early sixteenth century, Alvarus Thomas of Lisbon, working at Paris, was a thorough master of Swineshead’s calculations.

———. “*Calculationes de motu locali* in Richard Swineshead and

Alvarus Thomas.” In *Mathématiques et théorie du mouvement (xiv ^{e}-xvie siècles)*, edited by Joël Biard and Sabine Rommevaux. Villeneuve d’Ascq, France: Presses Universitaires du Septentrion, in press 2007.

*Edith Dudley Sylla*

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