Clark University
                        Origins of Calculus
David Joyce
Department of Mathematics and Computer Science
Clark University

September, 2013

Veritas et Falsitas
Zeno of Elea (ca. 490–ca. 430) shows Youths the Doors to Truth and False.
Fresco in the Library of El Escorial, Madrid.
Date 1588–1595, by Bartolomeo Carducci or Pellegrino Tibaldi

The School of Athens
Fresco in in the Apostolic Palace in the Vatican
Date 1509–1510, by Raphael
Includes figures for Pythagoras (ca. 570–ca. 490), Plato (427–347), Aristotle (384–322), Euclid (fl. ca. 295), Archimedes (287–212)

William of Ockham, from a stained glass window at a church in Surrey.
William of Ockham (ca. 1285–ca. 1349)

Miniature from Oresme’s Traité de l’espere, Bibliothèque Nationale, Paris
Nicole Oresme (ca. 1320–1382)

Nicole Oresme was at the University of Paris and expanded the analytic study of changing quantities. He had a graphical interpretation very similar to the modern graph y = f(x) of a function in the (x,y)-plane, but analytic geometry and coordinates were yet to be developed by Fermat and Descartes in the 1600s.

He represented time as a line, much as Aristotle had done long before, so that instants in a time interval were represented by points on a horizontal line segment AB, which he called the longitude. Given a moving object, at each instant in time E that moving object has a velocity, and he represented that velocity by a vertical line segment EF proportional to the velocity; each vertical line segment he called a latitude. These latitudes together formed a plane region ABDC, which he called a form, bounded on the bottom by the original longitude AB, on the left by the initial latitude AC representing the initial velocity, on the right by the final latitude BD representing the final velocity, and on the top by the curve CFD which he called the summit curve. He then argued that the area of that form ABDC is proportional to the distance traveled.

This is now what is called the Fundamental Theorem of Calculus. It says that the area under the graph of the velocity equals the distance travelled. In modern terminology, that says the integral of the derivative equals the the total change in the function. In modern notation, that looks like

Thus, the Fundamental Theorem of Calculus predates most of the rest of calculus by almost 300 years.

Dispute of Queen Cristina Vasa and René Descartes
Date 1884, by Nils Forsberg (1842–1934). Museo nacional de Versailles.
René Descartes (1596–1650)

y2 = x3 – x + 1        

Pierre de Fermat
Date 1886, by Alexandre Falguière (1831–1900). Musée des Augustins
Pierre de Fermat (1601–1665)

Portrait of Isaac Newton
Date 1689, by Sir Godfrey Kneller (1646–1723)
Isaac Newton (1642–1727)

  • Newton and Leibniz are often called the founders of Calculus based on their work in the late 17th century

  • But the Fundamental Theorem of Calculus, derivatives, and many other parts of calculus predated them

  • They built their calculus on the mathematics of their predecessors

British two pound coins

A portrait of Gottfried Wilhelm von Leibniz
17th century, artist unknown. Public Library of Hannover
Gottfried Wilhelm Leibniz (1646–1716)

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