Abstract
In the Principia there are two purely mathematical pages containing an astonishingly modern topological proof of a remarkable theorem on the transcendence of Abelian integrals (42).
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Notes
This example was mentioned by Huygens in a letter to Leibniz in 1691. See also H. Brougham and E.J. Routh, Analytical view of Sir Isaac Newton’s Principia, London, 1855. Leibniz, in his reply to Huygens, formulated the problem of transcendence of the areas of the segments cut off from an algebraic curve, defined by an equation with rational coefficients, by straight lines with algebraic coefficients (for instance, of the transcendence of the number π and of logarithms of algebraic numbers). The problem of Leibniz is more general than Hilbert’s 7th problem, but unlike the last is still, it seems, unsolved.
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© 1990 Birkhäuser Verlag Basel
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Arnol’d, V.I. (1990). Kepler’s Second Law and the Topology of Abelian Integrals. In: Huygens and Barrow, Newton and Hooke. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9129-5_6
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DOI: https://doi.org/10.1007/978-3-0348-9129-5_6
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-2383-7
Online ISBN: 978-3-0348-9129-5
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