Abstract
Starting in the late 1700s, Legendre spent several decades developing a general theory of the integrals which he called elliptic functions, and which satisfy an addition formula analogous to that established by Euler in Theorem 7.1. Here is what he wrote about his motivations in the foreword of his 1825 book [128]
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That is, non-algebraic functions.
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Here are the exact words of Euler (Novi Com. Petrop., tom. X, pag. 4): “Imprimis autem hic idoneus signandi modus desiderari videtur, cujus ope arcus elliptici œque commode in calculo exprimi queant ac jam logarithmi et arcus circulares, ad insigne analyseos incrementum, in calculum sunt introducti. Talia signa novam quamdam calculi speciem supeditabunt.”
References
A.-M. Legendre, Traité des fonctions elliptiques, Tome Premier (Huzard-Courcier, Paris, 1825)
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Popescu-Pampu, P. (2016). Legendre and Elliptic Functions. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_8
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DOI: https://doi.org/10.1007/978-3-319-42312-8_8
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