The Leibniz catenary and approximation of e — an analysis of his unpublished calculations

Abstract

Leibniz published his Euclidean construction of a catenary in Acta Eruditorum of June 1691, but he was silent about the methods used to discover it. He explained how he used his differential calculus only in a private letter to Rudolph Christian von Bodenhausen and specified a number that was key to his construction, 2.7182818, with no clue about how he calculated it. Apparently, the calculations were never divulged to anyone but were discovered later among his personal papers. They may be the earliest record of an accurate approximation of the number we label e and a demonstration of its role as the base of the natural logarithm and exponential function.This, at that time, was a remarkably precise estimate for e, accomplished more than 22 years before Roger Cotes published e to 12 significant digits, and some 57 years before Euler's treatment of the logarithm in his Introductio in Analysin Infinitorum. The Leibniz construction reveals a hyperbolic cosine built on an exponential curve based on his estimated value, which implies that he understood the number as the base of his logarithmic curve. The sheets of arithmetic used by Leibniz preserved at the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, confirm this.Those sheets show how Leibniz calculated e and applied it to his catenary construction. The data actually yield e to 12 significant figures: 2.71828182845, missed by Leibniz because of a misplaced decimal point. We summarize the construction and examine the worksheets. The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts.