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Mathematical Treasure: Leibniz's Reply to Fatio

Sidney J. Kolpas (Delaware County Community College)

Although the virulence of the calculus priority controversy in some ways continues to reverberate today, it took a long time to develop and included provocations from some seemingly unlikely characters. Nicolas Fatio de Duillier (1664–1753) was a mathematician, astronomer, and inventor from Switzerland who was elected to the Royal Society of London in 1689 and soon befriended Isaac Newton. Fatio saw a manuscript treatise Newton wrote around 1691 to describe his fluxional method for the calculus, “De quadratura curvarum.” Even after he moved to The Netherlands, Fatio urged Newton to publish the treatise. The friendship ran its course, in part because both men were ill in the 1690s. By 1699 Fatio recovered enough to publish a short book, Lineæ brevissimæ descensus investigatio geometrica duplex, cui addita est investigatio geometrica solidi rotundi in quo minima fiat resistentia (A two-fold geometrical investigation of the line of briefest descent, to which is added a geometric investigation of the solid of revolution that produces the minimum resistance), in which he proposed a solution to the brachistochrone problem and departed from the widely-held assumption that Newton and Leibniz had independently discovered the calculus by accusing Leibniz of plagiarism.

The Mathematical Treasure profiled here contains one of Leibniz’s responses to this attack. In the volume of Acta Eruditorum printed in 1700, he pointed out that he and Newton had both acknowledged that they carried out their work separately. My copy of this volume is depicted below; the full 11-page article (in Latin) may be read in the Internet Archive.

The author's copy of the 1700 volume of Acta Eruditorum.

Title page of 1700 volume of Acta Eruditorum.

First page of Leibniz's response to Fatio.

Since the brachistochrone problem involves finding the curve of fastest descent, which turns out to be a portion of a cycloid, and the cycloid is a topic of second-semester calculus (when introducing the analysis of parametric equations), I found that topic to be an opportune time to enrich the class with history. I included an overview of the Leibniz-Newton Controversy, a brief discussion of Fatio, and a marble race from point A to point B, with one marble on a linear track and one on a cycloidal track. My students really enjoyed the history and the marble race. Regarding the calculus controversy, I told my class that Newton invented calculus first, but Leibniz independently invented calculus with better notation—the notation now universally used. English pride kept their mathematicians using Newton’s notation, which is difficult to work with (for example, his derivative used a dot notation). Eventually, with the urging of Charles Babbage and others in the early 19th century, England switched to Leibniz’s notation. Babbage’s famous quotation refers to the end of England’s mathematical dotage (senility). Gabriela Sanchis provided a classroom lesson on the brachistochrone problem in her Convergence article, “Historical Activities for Calculus.”


Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time. New York: Thunder’s Mouth Press, 2006. (Note, though, the critical review by Brian E. Blank in Notices of the AMS 56(5)(May 2009): 602–610.)

Child, J. M. The Early Mathematical Manuscripts of Leibniz. New York and London: Open Court Publishing Company, 1920.

Leibniz–Newton calculus controversy. Wikipedia.–Newton_calculus_controversy.

Westfall, Richard S. Never at Rest: A Biography of Isaac Newton. Cambridge, UK: Cambridge University Press, 1980.

Index to Mathematical Treasures

Sidney J. Kolpas (Delaware County Community College), "Mathematical Treasure: Leibniz's Reply to Fatio," Convergence (June 2022)