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Guillaume de l'Hôpital
Guillaume François Antoine, Marquis de l'Hôpital〔In the 17th and 18th centuries, the name was commonly spelled "l'Hospital", and he himself spelled his name that way. However, French spellings have been altered: the silent 's' has been removed and replaced with the circumflex over the preceding vowel. The former spelling is still used in English where there is no circumflex.〕 (:ɡijom fʁɑ̃swa ɑ̃twan maʁki də lopiˈtal) (?, 1661, Paris – 2 February 1704, Paris) was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞. Although the rule did not originate with l'Hôpital, it appeared in print for the first time in his treatise on the infinitesimal calculus, entitled ''Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes''.〔Answering l'Hôpital's question, in a letter of 22 July 1694 Johann Bernoulli described the rule of computing the limit of a fraction whose numerator and denominator tend to 0 by differentiating the numerator and denominator. A commonly made claim that l'Hôpital attempted to get credit for discovering the l'Hôpital's rule is inaccurate, since in the preface to his textbook, l'Hôpital generally acknowledged Leibniz, Jakob Bernoulli and Johann Bernoulli as the sources of the results in it.〕 This book was a first systematic exposition of differential calculus. Several editions and translations to other languages were published and it became a model for subsequent treatments of calculus.
== Biography ==
L'Hôpital was born into a military family. His father was Anne-Alexandre de l'Hôpital, a Lieutenant-General of the King's army, Comte de Sainte-Mesme and the first squire of Gaston, Duke of Orléans. His mother was Elisabeth Gobelin, a daughter of Claude Gobelin, Intendant in the King's Army and Councilor of the State.
L'Hôpital abandoned a military career due to poor eyesight and pursued his interest in mathematics, which was apparent since his childhood. For a while, he was a member of Nicolas Malebranche's circle in Paris and it was there that in 1691 he met young Johann Bernoulli, who was visiting France and agreed to supplement his Paris talks on infinitesimal calculus with private lectures to l'Hôpital at his estate at Oucques. In 1693, l'Hôpital was elected to the French academy of sciences and even served twice as its vice-president.〔Yushkevich, p. 270.〕 Among his accomplishments were the determination of the arc length of the logarithmic graph,〔Unbeknownst to him, a solution had already been obtained by James Gregory in letters to Collins (1670–1671), ''ibid''.〕 one of the solutions to the brachistochrone problem, and the discovery of a turning point singularity on the involute of a plane curve near an inflection point.〔This singularity is a cusp of the second kind, in which both branches are concave from the same side. It was described in l'Hôpital's letter to Johann Bernoulli from May 1694, Yushkevich, p. 275.〕
L'Hôpital exchanged ideas with Pierre Varignon and corresponded with Gottfried Leibniz, Christiaan Huygens, and Jacob and Johann Bernoulli. His ''Traité analytique des sections coniques et de leur usage pour la résolution des équations dans les problêmes tant déterminés qu'indéterminés'' ("Analytic treatise on conic sections") was published posthumously in Paris in 1707.
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