Abstract
Around 1746, Euler took up the problem of the surface area of scalene cones, cones in which the vertex does not lie over the center of the base circle. Calling earlier solutions by Varignon and Leibniz insightful but incomplete and extending his solution to conical bodies with noncircular bases, Euler published his results in 1750 (On the Surface Area of Scalene Cones and Other Conical Bodies: De superficie conorum scalenorum aliorumque corporum conicorum). He had not actually calculated any particular areas—not surprisingly, as they generally lead to elliptic integrals. Instead, he showed how to reduce the problem to calculating the arclength of certain curves, carefully elucidating the many ways these curves may be defined. Although the curves seem naturally to involve transcendental quantities, he showed how to adjust so only algebraic quantities are needed. Some details of Euler’s solution for the scalene cones are presented here.
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- 1.
These appeared in the same volume of a journal of the Royal Prussian Society in Berlin (see bibliography.) Unfortunately, I obtained copies of these papers too late to include any discussion of them here.
- 2.
This is Euler’s notation. For him, du was just part of the integrand and not required, as in our modern convention, to serve also as the right end of the integral. After this section, I will generally use modern conventions.
- 3.
These curves are interesting to graph, but I do not see that that they have any special geometric significance for the current problem.
References
Ronald S. Calinger, Leonhard Euler: Mathematical Genius in the Enlightenment, Princeton University Press 2015
Daniel J. Curtin, On the Surface Area of Scalene Cones and Other Conical Bodies: A translation of Leonhard Euler: De superficie conorum scalenorum aliorumque corporum conicorum, Euler Archives, to appear 2017
Leonhard Euler, De superficie conorum scalenorum aliorumque corporum conicorum, Novi Commentarii academiae scientiarum Petropolitanae 1, 1750, pp. 3–19
Gottfried Wilhelm Leibniz, Additio: Ostendens Explanationem superficiei conoïdalis cujuscunque; & speciatim explantionem superficiei Coni scaleni, ita ut ipsi vel ejus portioni cuicunque exhibeatur rectangulum æquale, interventu extensionis in rectam curvæ, per Geometriam ordinariam construendæ., Miscellanea Berolinensia ad incrementum scientiarum ex scriptis Societati Regiae Scientiarum exhibitis edita III, 1727, pp. 285–287
Pierre Varignon, Schediasma de Dimensione Superficiei Coni ad basim circularem obliqui, ope longitudinis Curvæ, cujus constructio à sola Circuli quadratura pendet, Miscellanea Berolinensia ad incrementum scientiarum ex scriptis Societati Regiae Scientiarum exhibitis edita III, 1727, pp. 280–284
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Curtin, D.J. (2018). Euler’s Work on the Surface Area of Scalene Cones. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90983-7_4
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