Oxford Calculators

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Richard Swineshead, Calculator, 1520

The Oxford Calculators were a group of 14th-century thinkers, almost all associated with Merton College, Oxford; for this reason they were dubbed "The Merton School". These men took a strikingly logico-mathematical approach to philosophical problems. The key "calculators", writing in the second quarter of the 14th century, were Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbleton. These men built on the slightly earlier work of Walter Burley and Gerard of Brussels.


The advances these men made were initially purely mathematical but later became relevant to mechanics. They used Aristotelian logic and physics. They also studied and attempted to quantify every physical and observable characteristic, like heat, force, color, density, and light. Aristotle believed that only length and motion were able to be quantified. But they used his philosophy and proved it untrue by being able to calculate things such as temperature and power.[1] They developed Al-Battani's work on trigonometry and their most famous work was the development of the mean speed theorem, (though it was later credited to Galileo) which is known as "The Law of Falling Bodies".[2] Although they attempted to quantify these observable characteristics, their interests lay more in the philosophical and logical aspects than in natural world. They used numbers to philosophically disagree and prove the reasoning of "why" something worked the way it did and not only "how" something functioned the way that it did.[3]

The Oxford Calculators distinguished kinematics from dynamics, emphasizing kinematics, and investigating instantaneous velocity. They first formulated the mean speed theorem: a body moving with constant velocity travels the same distance as an accelerated body in the same time if its velocity is half the final speed of the accelerated body.

The mathematical physicist and historian of science Clifford Truesdell, wrote:[4]

The now published sources prove to us, beyond contention, that the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics texts, were discovered and proved by scholars of Merton college.... In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe. Almost immediately, Giovanni di Casale and Nicole Oresme found how to represent the results by geometrical graphs, introducing the connection between geometry and the physical world that became a second characteristic habit of Western thought ...

In Tractatus de proportionibus (1328), Bradwardine extended the theory of proportions of Eudoxus to anticipate the concept of exponential growth, later developed by the Bernoulli and Euler, with compound interest as a special case. Arguments for the mean speed theorem (above) require the modern concept of limit, so Bradwardine had to use arguments of his day. Mathematician and mathematical historian Carl Benjamin Boyer writes, "Bradwardine developed the Boethian theory of double or triple or, more generally, what we would call 'n-tuple' proportion".[5]

Boyer also writes that "the works of Bradwardine had contained some fundamentals of trigonometry". Yet "Bradwardine and his Oxford colleagues did not quite make the breakthrough to modern science."[6] The most essential missing tool was algebra.

Thomas Bradwardine[edit]

Thomas Bradwardine was born in 1290 in Sussex, England. An attending student educated at Balliol College, Oxford, he earned various degrees. He was a secular cleric, a scholar, a theologist, a mathematician, and a physicist. He became chancellor of the diocese of London and Dean of St Paul's, as well as chaplain and confessor to Edward III. During his time at Oxford, he authored many books including: De Geometria Speculativa (printed in Paris, 1530), De Arithmetica Practica (printed in Paris, 1502), and De Proportionibus Velocitatum in Motibus (printed in Paris in 1495).

Aristotle suggested that velocity was proportional to force and inversely proportional to resistance, doubling the force would double the velocity but doubling the resistance would halve the velocity (VαF/R). Bradwardine objected saying that this is not observed because the velocity does not equal zero when the resistance exceeds the force. Instead, he proposed a new theory that, in modern terms, would be written as (Vαlog F/R), which was widely accepted until the late sixteenth century.[7]

William Heytesbury[edit]

William Heytesbury was a bursar at Merton until the late 1330s and he administered the college properties in Northumberland. Later in his life he was a chancellor of Oxford. He was the first to discover the mean-speed theorem, later "The Law of Falling Bodies". Unlike Bradwardine's theory, the theorem, also known as "The Merton Rule" is a probable truth.[7] His most noted work was Regulae Solvendi Sophismata (Rules for Solving Sophisms). Sophisma is a statement which one can argue to be both true and false. The resolution of these arguments and determination of the real state of affairs forces one to deal with logical matters such as the analysis of the meaning of the statement in question, and the application of logical rules to specific cases. An example would be the statement, "The compound H2O is both a solid and a liquid". When the temperature is low enough this statement is true. But it may be argued and proven false at a higher temperature. In his time, this work was logically advanced. He was a second generation calculator. He built on Richard Klivingston's "Sophistimata and Bradwardine's "Insolubilia". Later, his work went on to influence Peter of Mantura and Paul of Venice.[8]

Richard Swineshead[edit]

Richard Swineshead was also an English mathematician, logician, and natural philosopher. The sixteenth-century polymath Girolamo Cardano placed him in the top-ten intellects of all time, alongside Archimedes, Aristotle, and Euclid.[7] He became a member of the Oxford calculators in 1344. His main work was a series of treatises written in 1350. This work earned him the title of "The Calculator". His treatises were named Liber Calculationum, which means "Book of Calculations". His book dealt in exhaustive detail with quantitative physics and he had over fifty variations of Bradwardine's law.

John Dumbleton[edit]

John Dumbleton became a member of the calculators in 1338-39. After becoming a member, he left the calculators for a brief period of time to study theology in Paris in 1345-47. After his study there he returned to his work with the calculators in 1347-48. One of his main pieces of work, Summa logicae et philosophiae naturalis, focused on explaining the natural world in a coherent and realistic manner, unlike some of his colleagues, claiming that they were making light of serious endeavors.[9]

See also[edit]


  1. ^ Agutter, Paul S.; Wheatley, Denys N. (2008) "Thinking About Life"
  2. ^ Gavroglu, Kostas; Renn, Jurgen (2007) "Positioning the History of Science"
  3. ^ Paul S. Agutter, and Denys N. Wheatley (ed.). Thinking About Life. Springer. ISBN 978-1-4020-8865-0. 
  4. ^ Clifford Truesdell, Essays in The History of Mechanics, (Springer-Verlag, New York, 1968)
  5. ^ Carl B. Boyer, Uta C. Merzbach. A History of Mathematics. 
  6. ^ Norman F. Cantor (2001). In the Wake of the Plague: The Black Death and the World it Made. p. 122. 
  7. ^ a b c Mark Thakkar (2007). "The Oxford Calculators". Oxford Today. 
  8. ^ Longeway, John. "William Heytesbury". Stanford Encyclopedia of Philosophy. 
  9. ^ Molland, George (23 September 2004). "Dumbleton, John". Oxford Dictionary of National Biography. 


  • Sylla, Edith (1999) "Oxford Calculators", in The Cambridge Dictionary of Philosophy.
  • Gavroglu, Kostas; Renn, Jurgen (2007) "Positioning the History of Science".
  • Agutter, Paul S.; Wheatley, Denys N. (2008) "Thinking About Life"

Further reading[edit]

  • Carl B. Boyer (1949), The History of Calculus and Its Conceptual Development, New York: Hafner, reprinted in 1959, New York: Dover.
  • John Longeway, (2003), "William Heytesbury", in The Stanford Encyclopedia of Philosophy. Accessed 2012 January 3.
  • Uta C. Merzbach and Carl B. Boyer (2011), A History of Mathematics", Third Edition, Hoboken, NJ: Wiley.
  • Edith Sylla (1982), "The Oxford Calculators",in Norman Kretzmann, Anthony Kenny, and Jan Pinborg, edd. The Cambridge History of Later Medieval Philosophy: From the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, New York: Cambridge.
  • Boccaletti, Dino (2016). Galileo and the Equations of Motion. Heidelberg, New York: Springer. ISBN 978-3-319-20134-4.