# Mathematical Treasure: Leibniz's Papers on Calculus

Author(s):
Frank J. Swetz (The Pennsylvania State University)

Gottfried Wilhelm Leibniz (1646-1716) was a true polymath recognized for his excellence in many fields, particularly philosophy, theology, mathematics, and logic. He is considered a cofounder, along with Isaac Newton, of the Calculus. In 1682, Leibniz, together with a fellow German philosopher and scientist, Otto Mencke (1644-1703), founded a scholarly journal, Acta Eruditorum [Reports of Scholars], in Leipzig. The journal was intended for the German-speaking regions of Europe, despite being written almost entirely in Latin. Acta Eruditorum, a monthly journal, would become the vehicle for much of the mathematical publication of Leibniz and the Bernoullis and would eventually be the forum through which Leibniz defended his priority in the development of calculus.

According to J. J. O’Connor’s and E. F. Robertson’s biography in the MacTutor History of Mathematics Archive, Leibniz “developed the basic features of his version of the calculus” while living in Paris during the 1670s:

In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the $\int f(x)\,dx$ notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar $d(x^n)=nx^{n-1}dx$ for both integral and fractional $n.$

Leibniz began publishing his calculus results during the 1680s. We present on the following pages three famous articles on the Calculus, published by Leibniz in Acta Eruditorum in 1684, 1686, and 1693:

The portrait of Leibniz above is from the Convergence Portrait Gallery. It appears there courtesy of the Dibner Library of Science and Technology, The Smithsonian Institution Libraries, and its usage must conform to the Library’s rules and standards. The images in the following three pages of this article are used through the courtesy of the Lilly Library, Indiana University, Bloomington, Indiana. You may use them in your classroom; for all other purposes, please seek permission from the Lilly Library.

##### Reference

J. J. O’Connor and E. F. Robertson, “Biography of Gottfried Wilhelm von Leibniz,” MacTutor History of Mathematics Archive

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Leibniz's Papers on Calculus," Convergence (June 2015)