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ABSTRACT
In this paper, we discuss results in spherical geometry that were obtained by a remarkable mathematician of the XVIIIth century, Anders Johan Lexell. We also present a short note on the place of these results in the history of this field as well as a short biography of Lexell.
Keywords and phrases: Spherical geometry, History of mathematics, Lexell's theorem.
La Suede a toujours eté féconde en grands hommes et Mr Lexell soutient, on ne peut pas mieux, la gloire de sa Patrie.
D. Bernoulli, letter to J. A. Euler on 6 July 1776
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1.Introduction
This paper is concerned with the contribution in spherical geometry of Anders Johann Lexell, a prominent XVIIIth-century European geometer who worked at the Saint Petersburg Academy of Sciences at the time when Euler was the leading figure there. Lexell was a distinguished astronomer and a collaborator of Euler. The two men became also friends, and Lexell helped Euler writing his scientific memoirs during the years where the latter was completely blind. Today, Lexell's work on spherical geometry is poorly known, except for the result that carries the name "Lexell's theorem" which several mathematicians, including famous ones like Jacob Steiner, Eugene Catalan and Henri Lebesgue, rediscovered and published proofs of it several decades after Lexell's memoir appeared in print ([3, 5, 16, 35, 34]). Even today, Lexell's theorem still gives rise to new developments ([39, 27]). Besides this theorem, Lexell has several other works on spherical geometry which we present in this paper.
All of Lexell's works on this subject are published in latin, in Acta academiae scientiarum imperialis petropolitanae in the following memoirs:
* Solutio problematis analytici, presented and published in 1772 [23];
* Solutio problematis geometrici ex doctrina sphaericorum, presented in 1781, published in 1784 [24];
* Demonstratio nonnullorum theorematum ex doctrina sphaerica, presented in 1782, published in 1786 [26];
* De proprietatibus circulorum in superficie sphaerica descriptorum, presented in 1784, published in 1786 [25].
These memoirs contain an amount of theorems of spherical geometry [25, 24, 26] and an application of spherical geometry to the solution of an analytical problem [23]. In his proofs Lexell used only the intrinsic geometry of the sphere and for the significant results of the memoirs (in...