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The Surface Area of a Scalene Cone as Solved by Varignon, Leibniz, and Euler

The work translated alongside this article is Euler’s contribution to the problem of the scalene cone, in which he discusses the earlier work of Varignon and Leibniz. He improves Varignon’s solution and corrects Leibniz’s more ambitious solution. Here I will discuss the problem, provide extensive notes on Euler’s paper, and lay out its relation to the other two. For ease of reference translations of the papers of Varignon and Leibniz are appended, with the Latin originals. Euler’s version in Latin is available at https://scholarlycommons.pacific.edu/euler-works/133 1 The Problem 1.1 The History of the Problem Finding the surface area of a right cone, a cone with circular base and vertex directly over the center of the circle, was known from antiquity, certainly by Archimedes. The problem of finding the surface area of a scalene (or oblique) cone, one with circular base but vertex not necessarily directly over the center of the circle, seems to have been solved first by Pierre Varignon. In 1727 he published his result in the Miscellanea Berolinensia [3]. He called his paper a schediasma, a Latin version of a word that in Greek means a caprice, or whimsy. Varignon probably meant a sketch or rough draft. In fact, the mathematics in his paper is fully formed, but the layout and referencing are not carefully done. He solved the problem by giving a curve whose arc length at certain points gives the surface area of the corresponding part of the cone. In the same volume, in fact in the article immediately following Varignon’s, Gottfried Leibniz [4] gave his suggested improvements. Varignon’s curve was transcendental, implicitly using trigonometric functions. Leibniz gave an algebraic curve to solve the problem and outlined how to extend his solution to base curves other than the circle. The two papers share a single diagram, printed in between them. 10 Curtin: The Surface Area of a Scalene Cone as Solved by Varignon, Leibniz Published by Scholarly Commons, 2021 In his introductory paragraph, Leibniz asserted that Gilles de Roberval told Leibniz he had a solution, but never published it, but Leibniz noted that he could not find such work among Roberval’s papers, after his death. A bit later, Euler took up the problem, publishing his solution in 1750 [2]. In his introduction, he referred to the two earlier papers by Varignon and Leibniz. He remarked that Leibniz’s solution was excellent, but involved a slight error. Euler used a Greek word in Latin form, sphalma meaning a misstep or stumble, perhaps to soften the criticism of a mathematician he greatly admired. Euler gave his own version of Varignon’s solution, then extended it in several ways. He explained where Leibniz went wrong, then corrected the misstep. 1.2 The Differential Approach A cone is defined by starting with a circle in the plane and a vertex V that is above the plane but not necessarily located directly over the center of the circle. A point on the circle A is selected and line segment V A is connected. Then the cone is defined by following points M around the circle as VM sweeps out any section AVM of the cone that is desired. This dynamic definition allows infinitesimal methods to be used. For all three authors, a solution to the problem is a curve whose arc length gives the areas of the sectors of the cone. For either the arc length problem or the surface area problem, the integrals that result are rarely integrable in simple terms, since they both involve the element of the arc length ds = √ dx2 + dy2 or something similar.