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Guillaume Francois Antoine, Marquis de l'Hospital (1661 – 1704) was a French mathematician. His name is firmly associated with l'Hospital's rule for calculating limits involving indeterminate forms 0/0 and . Although the rule did not originate with l'Hospital, 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". 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.
L'Hospital was born into a noble family. His father was a Lieutenant-General of the King's army, Comte de Sainte-Mesme and the first squire of Gaston, Duke of Orleans. His mother was a daughter of Claude Gobelin, Intendant in the King's Army and Councilor of the State.
L'Hospital 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’Hospital at his estate at Oucques. In 1693, L'Hospital was elected to the French academy of sciences and even served twice as its vice-president. Among his accomplishments were the determination of the arc length of the logarithmic graph, 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.
L'Hospital exchanged ideas with Pierre Varignon and corresponded with Gottfried Leibniz, Christiaan Huygens, and Jacob and Johann Bernoulli. His "Analytic treatise on conic sections" was published posthumously in Paris in 1707.
In 1696 L'Hospital published his book "Infinitesimal calculus with applications to curved lines". This was the first textbook on infinitesimal calculus and it presented the ideas of differential calculus and their applications to differential geometry of curves in a lucid form and with numerous figures; however, it did not consider integration. The history leading to the book's publication became a subject of a protracted controversy. In a letter from 17 March 1694, L'Hospital made the following proposal to Johann Bernoulli: in exchange for an annual payment of 300 Francs, Bernoulli would inform L'Hospital of his latest mathematical discoveries, with holding them from correspondence with others, including Varignon. Bernoulli's immediate response has not been preserved, but he must have agreed soon, as the subsequent letters show. L'Hospital may have felt fully justified in describing these results in his book, after acknowledging his debt to Leibniz and the Bernoulli brothers, "especially the younger one" (Johann). Johann Bernoulli grew increasingly unhappy with the accolades bestowed on L'Hospital's work and complained in private correspondence about being sidelined. After L'Hospital's death, he publicly revealed their agreement and claimed credit for the statements and portions of the text of Analysis, which were supplied to L'Hospital in letters. Over a period of many years, Bernoulli made progressively stronger allegations about his role in the writing of Analysis, culminating in the publication of his old work on integral calculus in 1742: he remarked that this is a continuation of his old lectures on differential calculus, which he discarded since L'Hospital had already included them in his famous book. For a long time, these claims were not regarded as credible by many historians of mathematics, because L'Hospital's mathematical talent was not in doubt, while Bernoulli was involved in several other priority disputes. For example, both H.G.Zeuthen and Moritz Cantor, writing at the cusp of the 20th century, dismissed Bernoulli's claims on these grounds. However, in 1921 Paul Schafheitlin discovered a manuscript of Bernoulli's lectures on differential calculus from 1691–1692 in the Basel University library. The text showed remarkable similarities to L'Hospital's writing, substantiating Bernoulli's account of the book's origin.
L'Hospital's pedagogical brilliance in arranging and presenting the material remains universally recognized. Regardless of the exact authorship (one should also note that the book was first published anonymously), Analysis was remarkably successful in popularizing the ideas of differential calculus stemming from Leibniz.

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