Merton calculators (act. c.1300–c.1349) were a band of Oxford scholars who in the first half of the fourteenth century produced works applying mathematical or logicomathematical methods to questions of natural philosophy; the techniques they developed remained influential in the study of physics until the time of Galileo.
Thomas Bradwardine,
William Heytesbury,
John Dumbleton, and
Richard Swineshead, all fellows of Merton College, and
Richard Kilvington, who was probably a fellow of Oriel College, were prominent members of this mathematizing group. Together they became known to later generations simply as the ‘calculators’. The calculators focused on problems raised by Aristotle's
Physics, such as how to quantify the qualities of matter, especially the quality of motion.
Aristotelian physics had long held a place in the medieval university curriculum. One of the celebrated early fourteenth-century
Physics commentators was
Walter Burley, a sometime Mertonian whose
De intensione et remissione formarum explained motion and other qualities as a succession of changing forms of increasing perfection as they approached a terminal result. Burley, however, did not frame his ‘theory of succession’ with mathematical analysis. By contrast Richard Kilvington, in his
Quaestiones super physicam, took special notice of Aristotle's
Physics VII.5, which dealt with proportions between motive force and resistance. Aristotle had claimed that doubling the motive force (F) or halving the resistance (R) would double the velocity (V) of an object, implying that V was proportional to F/R. Kilvington reasonably concluded that the mathematics of proportionality and ratios should be applied to the study of motion.
That is precisely what Thomas Bradwardine did with his
Tractatus de proportionibus velocitatum in motibus, completed in 1328. He argued that the relationship between velocity and the ratio of force and resistance that Aristotle had posited was erroneous, because it allowed for motion to occur when the motive force was less than the opposing resistance. Bradwardine proposed instead that changes in velocity were proportional to powers of the ratio between force and resistance; for instance, if velocity were doubled the F/R ratio must be multiplied by itself, or squared; if the velocity were tripled the F/R ratio would be raised to the third power, and so on. A consequence of this exponential relationship was that when force and resistance were equal to one another the quantity of motion would be zero. Bradwardine's theorem was the single most influential contribution made by the calculators to natural philosophy. Even so, it was a mathematical operation
secundum imaginationem, founded on absolutely no empirical observation of real moving objects. This has led some modern commentators to suggest that Mertonian natural philosophy altogether ignored nature.
Whether it ignored nature or not, the academic context for the study of motion and other physical questions was disputational. That is to say, university students debated
sophismata or puzzle-questions on a variety of topics, not infrequently natural philosophy, as exercises in logical reasoning. Kilvington's textbook
Sophismata showed some tendency to mathematize physical puzzles, but William Heytesbury's
Regulae solvendi sophismata, of 1335, most clearly illustrates this activity, with examples of
sophismata that competent responders were expected to know how to explain with the aid of calculation. The six sections of the
Regulae include problems of defining the beginning and ending of motion, maxima and minima of qualities, and the Aristotelian categories of traversing space—local motion, augmentation, and alteration. Heytesbury appears to have been the first to articulate what J. A. Weisheipl has called the ‘Mertonian theorem of mean speed’ (Weisheipl, 637). The problem was how to determine the effective velocity of an object accelerating at a uniform rate. Heytesbury's answer was that the effective velocity could be taken at the middle instant, or mean speed, between the initial and final velocities. It is interesting that Heytesbury presented cases like that of an object accelerating to infinite speed, even though he believed them to be impossible in nature. Edith Sylla contends that for Heytesbury problems of natural philosophy were chiefly instruments of honing the student's skills in disputational logic rather than having a pedagogical value in their own right.
Writing his
Summa logicae et philosophiae naturalis some time in the 1340s, John Dumbleton ambitiously attempted to cover the entirety of Aristotelian natural philosophy. He treated the problem of motion from the mathematical and logical perspectives of Bradwardine and Heytesbury. Dumbleton distinguished between these two streams of discourse, one which concerned the causes of motion (a branch of mechanics later to be called dynamics) and the other which described its effects (kinematics). Sadly, the promise of the
Summa was not fully kept; the treatise was left unpolished and incomplete, perhaps on account of its author's death (he may have been a victim of the black death in 1349).
The achievements of Merton calculatory scholarship are expressed most profoundly in Richard Swineshead's
Liber calculationum, of about 1350. In it Swineshead applied intricate quantitative analyses to numerous topics involving local motion, the density of objects, the intensity of light, and the behaviour of objects moving near the centre of the earth. An example of the level of detail with which Swineshead scrutinized each problem is his exploration of Bradwardine's theorem in more than fifty cases of varying motive force and resistance affecting an object's calculated velocity. Swineshead also grappled with problems of a new order of theoretical complexity, as when he discussed the intensity and remission of natural qualities in a non-homogeneous extended object, assuming the ratio of hotness, coldness, wetness, and dryness at any one point within the object would not necessarily be the same as the ratio at all other points. The
Liber calculationum was a
tour de force of the analytical language of fourteenth-century logicomathematical reasoning. Many scholars admired the work of Swineshead ‘the Calculator’; few actually made it all the way through the exceedingly dense text.
Richard is not to be confused with his likely kinsman and fellow Oxonian Roger Swineshead, whose own treatise,
De motu, was not without merit. Several other contemporary Merton fellows pursued mathematical fields.
John Maudith,
Simon Bredon, and
William Rede were noteworthy astronomers who performed an enormous amount of practical calculation. Even the physician John Gaddesden tried to determine the efficacy of medicines by measuring the ratios of their active ingredients. The dynamism of the Merton calculators must have had something to do with the synergy and collegiality of their fellowship. It is known that Bredon and Heytesbury were lifelong friends, for instance. Members of the most distinguished intellectual circle in Oxford also had to deal personally with practical issues of college administration and estate management. Although their scholarship had an otherworldly cast to it, the calculators did not shut themselves up in an ivory tower. Bradwardine, who rose to become archbishop of Canterbury, and Heytesbury, who served as chancellor of the university, were certainly successful men of action.
The black death and subsequent political and theological controversies ignited by John Wyclif (who was temporarily at Merton in the 1350s) may have led to the decline of Merton calculatory activity after 1350. Bradwardine and perhaps Dumbleton succumbed to the plague, while a new generation of scholars turned to law and other subjects. Ralph Strode, who became common serjeant of the city of London as well as composing an important treatise on logic, was one of them. None the less, the works of the calculators were copied and distributed across Europe. Bradwardine's
De proportionibus inspired the great Parisian scientist Nicole Oresme (
d. 1382), as well as a host of other commentators. Heytesbury's works were popular in Italian universities and were printed in multiple editions; it is not unlikely that the young Galileo was familiar with them. Swineshead continued to impress Renaissance scholars, too, for although some complained of his trivial
quisquiliae, others, like the celebrated sixteenth-century Italian mathematician Girolamo Cardano, praised him as one of the greatest intellects the world had ever known.
Keith Snedegar