William Heytesbury

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Heytesbury, William

(fl. Oxford, England, ca. 1335)

logic, kinematics.

Heytesbury was one of several scholars at Merton College, Oxford, during the second quarter of the fourteenth century whose writings formed the basis of the late medieval tradition of calculationes, the discussion of various modes of quantitative variation of qualities, motions, and powers in space and time. Other leading authors of the Merton group were Thomas Bradwardine, Richard Swineshead, and John of Dumbleton. The tradition they founded spread to the Continent in the second half of the fourteenth century and enjoyed a vogue in Italian universities during the fifteenth century and again at Paris and in the Spanish universities during at least the first third of the sixteenth century. Thereafter, however, it lost impetus with the shift of interests consequent upon the humanist movement. A question still under debate among historians of science is the precise extent of the later influence of Merton kinematics, and particularly of the Merton “mean-speed theorem,” which can be used to prove that in uniformly accelerated motion starting from rest, the distances are in the duplicate ratio of the times. Other phases of the Mertonian discussions involving the mathematical concepts of limit, infinite aggregate, and the continuum as a dense set of points, as well as distinctions now treated in quantificational logic, seem to have fallen into oblivion after the sexteenth century but are anticipatory of nineteenth-century work in these areas.

Biographical information about Heytesbury, as about the other Mertonian scholars, is meager. His name, variously spelled, appears in the records of Merton College for 1330 and 1338–1339; he may have been the William Heightilbury who with other Mertonians was appointed fellow of Queen’s College at its founding in 1340; in 1348, however, he was still—or once more—a fellow of Merton and by that year was a doctor of theology; finally, a William Heighterbury or Hetisbury was chancellor of the university in 1371.

Heytesbury’s two best-known and most influential works—the only known ones of some length, the others being short discussions of particular questions—are his Sophismata and Regule solvendi sophismata. According to the explicit of an Erfurt manuscript (Wissenschaftliche Bibliothek, Amplon. F. 135, 17r), the Regule was “datus Oxonie a Wilhelmo de Hytthisbyri” in 1335; and it is probable that the Sophismata stems from about the same time, since the two works are closely related in content, one providing rules for the resolution of different classes of real or apparent logical fallacies and the other dealing intensively with thirty-two particular sophisms. The medieval discussion of sophisms grew out of Aristotle’s Sophistical Refutations; but as we encounter it in Heytesbury, it has developed beyond the Aristotelian treatment in two directions. First, Heytesbury employs the logica moderna, a set of distinctions and work-order devices developed at the University of Paris during the thirteenth century. Second, he devotes much attention to cases and problems involving modes of purely quantitative variation in space and time.

The key innovation of the logica moderna was the theory of supposition, an analysis of the various ways in which a term is interpretable within a given proposition for some individual or individuals. For instance, in “That man disputes,” the term “that man” is said to have discrete supposition, as referring to a single, definite individual, a suppositum to which one could point. In “Some man disputes” or “Every man disputes,” on the other hand, the supposition of the subject term in either case is not discrete but common, although not in the same way. Thus, from “Some man disputes” it is permissible to descend to individual cases through an alternation: “This man disputes or this man disputes or...,” there being no existent man who is not referred to in one of the members of the alternation. But from the statement “Every man disputes” it is permissible to descend to individual cases included under the term “man” only through a conjunction: “This man disputes, and this man disputes, and...,” and so on. Finally, there are cases in which the descent is not possible through either an alternation or a conjunction, and in these cases the supposition is said to be confused only.

The kind of supposition of a term in any particular proposition is determined partly by the meaning of the predicate or subject term with which it is conjoined and partly by the “syncategorematic” terms included in the proposition—terms incapable of serving as subject or predicate but nevertheless influencing the supposition of the subject or predicate. Examples of syncategorematic terms are “any,” “all,” “some,” “necessarily,” “always,” and “immediately”.

This theory appeared, fully developed, in the works of William of Shyreswood in the middle of the thirteenth century. It seems to have derived in part from the analyses of grammarians (the first known use of the verb “to supposit” in the sense required by the theory occurs in the Doctrinale of the twelfth-century grammarian Alexander of Villa Dei) and in part from the Abelardian explication of universals: for Abelard a universal word gives rise only to a common and confused conception of many individuals and can come to determine a particular thing or particular things only in the context of a statement. Whatever its origins, the extensional analysis of the use of terms in discourse was much in vogue by Heytesbury’s time and was used by him to reveal distinctions of structure that, in modern mathematical logic, are exhibited by means of the cross-references of quantifiers and variables.

For illustration, consider the distinction that Heytesbury makes in the Regule between the statement “Always some man will be” and the statement “Some man will be always.” In the first statement the term “man” is preceded by the syncategorematic term “always,” which according to Heytesbury has a “force of confounding” (vim confundendi) and thus confuses the supposition of the term that follows it. Hence the supposition of the term “some man” in the statement “Always some man will be” is confused only, and it is not permissible to descend either disjunctively or conjunctively to individual supposita. In the statement “Some man will be always,” on the contrary, the term “some man” is not preceded by the term “always” and its supposition therefore remains determinate, so that some particular although unspecified individual is referred to. The first statement asserts the immortality of the race of mankind; the second asserts the immortality of some particular man. In the symbols of present-day mathematical logic, the first statement becomes.

(Read: “For all x there is a y such that, if x is a time and y is a man, then y occurs in x.”) The second statement, on the other hand, becomes

(Read: “There is a y such that for all x, if x is a time and y is a man, then y occurs in x.”) Notationally, the distinction is one of the order of the universal and existential quantifiers, (x) and (Ey).

Because this distinction is crucial for the understanding of the modern definition of mathematical limit, it is of interest to find Heytesbury applying it to cases involving a mathematically conceived continuum. Thus, he distinguishes between the statement “Immediately after the present instant some instant will be” and the statement “Some instant will be immediately after the present instant.” Once again, the distinction turns on the fact that the syncategorematic term “immediately” confounds the supposition of the term following it. Thus in the first statement the term “some instant,” being preceded by the term “immediately,” has confused supposition only; and it is not permissible to descend disjunctively or conjunctively to particular instants. In the second statement the term “some instant” is not preceded by “immediately” and thus has determinate supposition; the statement therefore means that, of the infinitely many instants following the present instant, there is a determinate one that will be first. Heytesbury concludes that this second statement is false, whereas the first statement is true if expounded as meaning that, whatever instant after the present instant be taken, between that instant and the present instant there is some instant. In modern symbols, with the range of the variables restricted to instants of time,

(Read: “For all instants i, there is an instant j such that, if i is after the present instant i0, then j lies between i0 and i”) The false statement would reverse the order of the quantifiers. In effect, Heytesbury is insisting that instants in a time interval, like points on a line segment, form a dense set.

It is particularly in the two chapters of the Regule entitled “De incipit et desinit” and “De maximo et minimo” that Heytesbury’s logical sophistication in dealing with limits and extrinsically or intrinsically bounded continua comes into play. In the first of these chapters he analyzes cases in which any thing or process or state may be said to begin or to cease to be. For instance, posing the case that Plato starts to move from rest with a constant acceleration, while at the same instant Socrates starts to move from rest with an acceleration that is initially zero but increases uniformly with time, Heytesbury concludes that “both Socrates and Plato infinitely more slowly begins to be moved than Plato.” As his explication shows, what is happening here is in effect a comparison of two infinitesimals of different order: if vs is Socrates’ velocity and vp is Plato’s, then

The “De maximo et minimo” deals with the setting of boundaries to powers—for example, Socrates’ power to lift weight or to see distant objects, or the power of a moving body to traverse a medium the resistance of which varies in some specified manner. Aristotle had flatly asserted that the boundary of a power or potency is a limiting maximum. The commentator Ibn Rushd emphasized that the incapacity of a power is bounded by a minimum quod non; later Schoolmen such as John of Jandun were thus faced with the question of the relation between the maximum quod sic and the minimum quod non. One thought that comes to play a role in the discussion is that no action or motion can proceed from a ration of equality between power and resistance; this thought necessitates the assignment of the negative or extrinsic boundary, so that, for instance, Socrates’ power to lift weights is to be bounded by the minimum weight that he is unable to lift. Heytesbury was not the first to consider the assignment of such extrinsic boundaries; but his formulation of rules and analysis of cases, compared with earlier discussions, shows a more exclusive concern with the mathematical and logical aspects of the problem.

An important last chapter of the Regule entitled “De tribus predicamentis” deals with the quantitative description of motion or change in the three Aristotelian categories of place, quantity and quality. The principal aim of each of the three subschapters (“De motu locali,” “De augmentatione,” “De alteratione”) is to establish the proper measure of velocity in the given category. In the case of augmentation, Heytesbury adopts a measure involving the exponential function, which had already played a role in Bradwardine’s Tractatus de proportionibus (1328). All three subchapters exhibit the almost exclusive concern of the Mertonian calcualtores with quantitative description of hypothetical cases.

This tendency to quantitative description had roots in earlier discussions of kinematics (as, for example, in the thirteenth-century De motu of Gerard of Brussels) and of what was known as “the intension and remission of forms,” the variation in intensity of a quality or essence. Discussions of the latter topic prior to the fourteenth century had dealt primarily with the ontological nature of such variation; but by Heytesbury’s time the Scotian assumption that intension is an additive increase had been generally accepted, and Schoolmen turned their attention to a logical or semantic question: how to denominate a subject in which the intensity of a quality varies from one point to another, or—a question treated as analogous—how to denominate or measure a motion in which the velocity varies from instant to instant of time or from point to point of the moved body. This question merges into the mathematical problems of describing different possible modes of spatial or temporal variation of intensity and of finding rules of equivalence between one distribution of intensities and another. Thus in Heytesbury’s Regule, as in later fourteenth-century writings, any particular configuration or mode of variation of intensity in space or time is called a “latitude”; and latitudes are categorized as uniform (of constant intensity), uniformly nonuniform (the intensity varying linearly with spatial extension or time), and nonuniformly nonuniform (the intensity varying nonlinearly with spatial extension or time).

Heytesbury’s Regule is the oldest datable writing in which we find the famous Merton rule: Every latitude uniformly nonuniform corresponds to its mean degree. Thus, if the whiteness or hotness of a body varies uniformly from an intensity of two degrees at one end of the body to an intensity of four degrees at the other end, then according to Heytesbury this latitude of whiteness or hotness is equivalent to a uniform latitude of three degrees extended over the same length. This assertion rests on the presupposition—unjustified for Heytesbury and his contemporaries by any empirical measurability—that intensities of a quality are intensities of some additive quantity. In application to local motion, since intensity of motion is measured in terms of distance traversed per unit time, and distance is an additive quantity, the Merton rule leads to testable empirical consequences. Heytesbury states the rule for local motion as follows:

For whether it commences from zero degree or from some [finite] degree, every latitude [of velocity], provided that it is terminated at some finite degree, and is acquired or lost uniformly, will correspond to its mean degree. Thus the moving body, acquiring or losing this latitude uniformly during some given period of time, will traverse a distance exactly equal to what it would traverse in an equal period of time if it were moved continuously at its mean degree. For of every such latitude commencing from rest and terminating at some [finite] degree [of velocity], the mean degree is one-half the terminal degree of that same latitude (Regule [Venice, 1494], fol. 39).

The proposition implies, as Heytesbury notes, that in a uniformly accelerated motion starting from rest, the distance traversed in the second half of the time is three times that traversed in the first half—a consequence admitting of application in experimental tests. The first known assertion that the Merton theorem is applicable to free fall was made by Domingo de Soto, a Spanish Schoolman, in 1555; but it was not coupled with any attempt at empirical verification. The first experimental work on the assumption that free fall is uniformly accelerated with respect to time may have been that of Thomas Harriot, who within a few years before or after 1600 was finding the acceleration of free fall to be between 21 and 32.5 feet per second squared (for Harriot’s theory of ballistics and his researches on free fall, see British Museum MS. Add. 6789, 19r-86v); in his discussion of projectile motion in the same manuscript Harriot explicitly refers to the 1494 volume that contains Heytesbury’s works and commentaries thereon, so that a direct influence of the medieval treatises is here indicated.

In the case of Galileo, the evidence for direct medieval influence in his work on free fall is less clear and is still under debate. The Juvenilia, which may be the youthful Galileo’s notes on lectures at the University of Pisa, contains references to Heytesbury and Calculator (Swineshead) and to such Mertonian distinctions as that between a maximum quod sic and a minimum quod non, and that between a uniformly nonuniform and a nonuniformly nonuniform variation in intensity (see Le opere di Galileo Galilei, A. Favaro, ed., I, 120, 136, 139 ff., 172). But from Galileo’s letter of 1604 to Sarpi, it appears improbable that his thought on the mathematical characterization of naturally accelerated motion took its start from the Merton mean-speed theorem. According to Stillman Drake, “Galileo may have known the mean-speed rule and rejected it as inapplicable to the analysis of unbounded accelerated motion” (British Journal for the History of Science, 5 [1970], 42). It is at least a plausible suggestion, however, that a passing acquaintance with medieval discussions and calculationes involving instantaneous velocities, punctiform intensities, and different modes of variation of velocity or intensity in space or time, and also with the graphical representation of such variation that had been introduced by Oresme and was incorporated in the 1494 edition of Heytesbury’s works, may have served as general preparation for the thinking that Galileo would have to do in founding his science of motion.


I. Original Works. MSS giving the Regule solvendi sophismata in whole or in part are Biblioteca Antoniana, Padua, Scaff. XIX, MS.407, fols. 28–32, 53–56; library of the University of Padua, MS.1123, 14c, fols. 50–65; MS.1434, 15c, fols. 1–26; and MS.1570, 15c, fols. 131–137; Bodleian Canon. Misc. MS.221, fols. 60–82; MS.376, 15c, fols. 30–32; MS.409, A.D. 1386, fols. 1–18; and MS.456, A.D. 1467, fols. 1–43; Bruges, Stadsbibliotheek, 497, 14c, fols. 46–59; and 500, 14c, fols. 33–71; Bibliotheca Marciana, Zanetti Latin MS.310, fols. 1–3; and VIII. 38 (XI, 14), a. 1391, fols. 40–54; Erfurt, Amplonian MS.135, fols. 1–17; and Vat. Lat. MS.2136, 14c, fols. 1–32; and MS.2138, 14c, fols. 89–109. It was published at Pavia in 1481 and at Venice in 1491 (fols. 4–21) and 1494 (fols. 7–52).

The Sophismata exists in the following MSS: Bibliotheca S. Johannis Baptistae, Oxford, MS.198, 14c, fols. 1–175; library of the University of Padua, MS.842, fols. 1–149; and MS.1123, 14c, fols. 97–172; Bodleian, Canon. Misc. MS.409, A.D. 1386, fols. 29–98; Bibliotheca Marciana, Zanetti Latin MS.310, fols. 54–79; Paris, Bibliotheque Nationale, Latin MS.16134, 14c, fols. 81–146; and Vat. Lat. MS.2137, 14c, fol. 1 et seq.; and MS.2138, 14c, fols. 1–86. It was Published at Pavia in 1481 and at Venice in 1491 (fols. 29–99) and 1494 (fols. 77–170).

De sensu composito et diviso is in following MSS: Biblioteca Nazionale, Florence, Cl. V, MS.43, 15c, fols. 38–44; library of the University of Padua, MS.1434, 15c, fols. 26–27; Bodleian, Canon. Misc. MS.219, A.D. 1395, fols. 4–6; Bologna University, MS.289.II.2, fols. 1–4; Bibliotheca Marciana, Zanetti Latin MS.310, fols. 49–53; and Vat. Lat. MS.2136, 14c, fols. 32–36; MS.3030, fols. 55–58; MS.3038, 14c, fols. 15–22; and MS.3065, 15c, fols. 140–143. It was published at Venice in 1491 (fols. 2–4), 1494 (fols. 2–4), and 1500 (fols. 1–23).

De veritate et falsitate propositionis was published at Venice in 1494 (fols. 183–188).

“Casus obligationis” is in MS: Bodleian, Canon. Latin MS.278, 14c, fol. 70; Bibliotheca Marciana, Zanetti Latin MS.310, fol. 96; and Vat. Lat. MS.3038, 14c, fols. 37–39.

“Tractatus de eventu futurorum” is in the MS Bibliotheca Marciana, MS.fa.300 (X, 207), 14c, fols. 78–79.

“Tractatus de propositionum multiplicium significatione” is available as Bibliotheca Marciana, Latin MS. VI, 160 (X, 220), a. 1443, fols. 252–253.

The following are doubtful works: Consequentie, in MS as Corpus Christi College, Oxford, MS.293, fol. 337 et seq., and published at Bologna; Probationes conclusionum, in MS as Vat. Lat. MS.2189, fols. 13–38, where the work is given the title “Anonymi conclusiones,” and published at Venice in 1494 (fols. 188–203); “Regulae quaedam grammaticales,” in MS as British Museum, Harleian MS.179; and “Sophismata asinina,” available as: Biblioteca Nazionale, Florence, C1.V, MS.43, 15c, fols. 45–46; library of the University of Padua, MS.1123, 14c, fols. 18–22; and MS.1570, 15c, fols. 113–130; Bodleian, Canon. Latin MS.278, 14c, fols. 83–87; and Bibliotheca Marciana, Zanetti Latin MS.310, fols. 122–126.

II. Secondary Literature. References to the relevant literature will be found in Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, Wis., 1959), pp. 683–698; and Nicole Oresme and the Medieval Geometry of Qualities and Motions (Madison, Wis., 1968), pp. 105–107; and Curtis Wilson, William Heytesbury: Medieval Logic and the Rise of Mathematical Physics (Madison, Wis., 1956), pp. 212–213.

A recent study of Heytesbury’s work on the liar paradox is Alfonso Maierü, “Il problema della verità nelle opere di Guglielmo Heytesbury,” in Studi medievali, 3rd ser., 7 , fasc. 1 (Spoleto, 1966), 41–74.

Curtis A. Wilson

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(before 13131372/3)

William Heytesbury, a fellow at Merton College in Oxford from 1330, belonged to the second generation of Mertonian "Calculators." His work depends on Richard Kilvington's Sophismata (1325) and Thomas Bradwardine's Insolubilia and Tractatus de Proportionibus (1328). His technique was to analyze sophismataambiguous problematic statements whose truth or falsehood is to be assessed under specified assumptionsand apply supposition theory, a form of semantic-logical analysis, to the explication of their underlying logical grammar. He is particularly noted for his work on motion and the continuum.

Heytesbury's most popular work was the Rules for Solving Sophismata (1335), which contains six treatises: "On Insoluble Sentences (Insolubilia)," dealing with self-referential paradoxes; "On Knowing and Doubting," concerning reference in intensional contexts; "On Relative Terms," considering the reference of relative pronouns; "On Beginning and Ceasing" and "On Maxima and Minima," about continua; and "On the Three Categories," on velocity and acceleration in changes of place, quantity, and quality.

In "On Beginning and Ceasing," Heytesbury considers the sophisma "some part of an object ceases to be seen by Socrates," given that the object is not now, but will, immediately after now, be partly occluded by an object passing in front of it. This statement may assert that there is a given part of the object that will, in every moment after this one, be entirely occluded, and if so, it is false. Or it may assert that at every moment after this present moment, there will be some part of the object entirely occluded at that moment (a different part for each moment), and then it is true.

The Rules became popular, and remained important on the European continent even after the Mertonians began to be ignored in Britain. It was taught at Padua and Paris through the early sixteenth century, influencing the Paduan school, fifteenth-century Italian logicians such as Paul of Venice (d. 1429), and the school of John Major at Paris. With the rest of medieval logic, Heytesbury's work sank into obscurity after that. In addition to Rules Heytesbury wrote two collections of sophismata, in one of which the (obviously false) statement, "you are a donkey," was repeatedly derived from seemingly harmless admissions. He also wrote some shorter works; for instance, "On the Compounded and Divided Senses," which deals with scope ambiguities similar to that involved in the preceding example.

In the sixth chapter of Rules, Heytesbury states the mean-speed theorem for uniformly accelerated motion: A uniformly accelerated body will, over a given period of time, traverse a distance equal to the distance it would traverse if it moved continuously in the same period at its mean velocity (one-half the sum of the initial and final velocities). Elsewhere, he points out, in a particular case, that a uniformly accelerated body will, in the second equal time interval, traverse three times the distance it does in the first. Domingo de Soto observed the applicability of the mean-speed theorem to free fall in 1555.

See also Bradwardine, Thomas; Kilvington, Richard; Medieval Philosophy; Paul of Venice.


works by william heytesbury

Hentisberi de sensu composito et diviso, Regulae solvendi sophismata. Venice: Bonetus Locatellus, 1494.

"William of Heytesbury on the Three Categories." Selections translated by E. A. Moody. In The Science of Mechanics in the Middle Ages, edited by Marshall Claggett. Madison: University of Wisconsin Press, 1959. Reprinted in A Source Book in Medieval Science, edited by Edward Grant. Cambridge, MA: Harvard University Press, 1974.

"William of Heytesbury on the 'Insoluble' Sentences." Translated with notes by Paul Spade. Toronto: Pontifical Institute of Medieval Studies, 1979.

On Maxima and Minima: Chapter 5 of "Rules for Solving Sophismata," with an anonymous fourteenth-century discussion. Translated with introduction and study by John Longeway. Dordrecht: D. Reidel Publishing Company, 1984.

"'The Compounded and Divided Senses,' and 'The Verbs Know and Doubt.'" Translated by Norman Kretzmann and Eleonore Stump. In The Cambridge Translations of Medieval Philosophical Texts. Vol. 1: Logic and Philosophy of Language. Cambridge, U.K.: Cambridge University Press, 1988.

works about william heytesbury

Wilson, Curtis. William Heytesbury: Medieval Logic and the Rise of Modern Physics. Madison: University of Wisconsin Press, 1960.

John Longeway (2005)