Guillaume François Antoine, Marquis de l'Hôpital (; sometimes spelled L'Hospital; 1661 – 2 February 1704), also known as Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont, and Seigneur d'Ouques-la-Chaise, was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. His name is firmly associated with

l'Hôpital's rule In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...

for calculating

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

s involving

indeterminate form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...

s 0/0 and ∞/∞. Although the rule did not originate with l'Hôpital, it appeared in print for the first time in his 1696 treatise on the infinitesimal calculus, entitled ''

Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes ''Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes'' (literal translation: ''Analysis of the infinitely small to understand curves''), 1696, is the first textbook published on the infinitesimal calculus of Leibniz. It was writ ...

''. 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

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 ''Comte'' is the French, Catalan and Occitan form of the word 'count' (Latin: ''comes''); ''comté'' is the Gallo-Romance form of the word 'county' (Latin: ''comitatus''). Comte or Comté may refer to: * A count in French, from Latin ''comes'' * A ...

de Saint-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 Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...

, 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 Oucques () is a former commune in the Loir-et-Cher department of central France. On 1 January 2017, it was merged into the new commune Oucques la Nouvelle.French academy of sciences and even served twice as its vice-president. Among his accomplishments were the determination of the

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

of the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

ic graph, one of the solutions to the

brachistochrone problem In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...

, and the discovery of a turning point singularity on the

involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or ...

of a plane curve near an

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...

. L'Hôpital exchanged ideas with Pierre Varignon and corresponded with

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

, Christiaan Huygens, and

Jacob Jacob (; ; ar, يَعْقُوب, Yaʿqūb; gr, Ἰακώβ, Iakṓb), later given the name Israel, is regarded as a patriarch of the Israelites and is an important figure in Abrahamic religions, such as Judaism, Christianity, and Islam. J ...

and

. 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 section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

s") was published posthumously in Paris in 1707. Acta Eruditorum - II geometria, 1695 – BEIC 13358210

Calculus textbook

In 1696 l'Hôpital 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 Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the ...

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'Hôpital made the following proposal to

: in exchange for an annual payment of 300 Francs, Bernoulli would inform l'Hôpital of his latest mathematical discoveries, withholding 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'Hôpital 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'Hôpital's work and complained in private correspondence about being sidelined. After l'Hôpital's death, he publicly revealed their agreement and claimed credit for the statements and portions of the text of ''Analyse'', which were supplied to l'Hôpital in letters. Over a period of many years, Bernoulli made progressively stronger allegations about his role in the writing of ''Analyse'', culminating in the publication of his old work on

integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

in 1742: he remarked that this is a continuation of his old lectures on differential calculus, which he discarded since l'Hôpital 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'Hôpital'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 Moritz Benedikt Cantor (23 August 1829 – 10 April 1920) was a German historian of mathematics. Biography Cantor was born at Mannheim. He came from a Sephardi Jewish family that had emigrated to the Netherlands from Portugal, another branch o ...

, 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 to 1692 in the

Basel University The University of Basel ( Latin: ''Universitas Basiliensis'', German: ''Universität Basel'') is a university in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest surviving universi ...

library. The text showed remarkable similarities to l'Hôpital's writing, substantiating Bernoulli's account of the book's origin. L'Hôpital's pedagogical brilliance in arranging and presenting the material remains universally recognized. Regardless of the exact authorship (the book was first published anonymously), ''Analyse'' was remarkably successful in popularizing the ideas of differential calculus stemming from Leibniz.

Personal life

L'Hôpital married Marie-Charlotte de Romilley de La Chesnelaye, also a mathematician and a member of the nobility, and inheritor of large estates in

Brittany Brittany (; french: link=no, Bretagne ; br, Breizh, or ; Gallo: ''Bertaèyn'' ) is a peninsula, historical country and cultural area in the west of modern France, covering the western part of what was known as Armorica during the period ...

. Together, they had one son and three daughters.

Notes

References

* G. L'Hôpital, E. Stone, ''The Method of Fluxions, both direct and inverse; the former being a translation from de l'Hospital's "Analyse des infinements petits," and the latter, supplied by the translator,

Edmund Stone Edmund Stone (c. 1700 – c. 1768) was an autodidact mathematician from Scotland in the 18th century. Life and work Practically nothing is known about the life of Edmund Stone. He was the son of the gardener of John Campbell, 2nd Duke of Arg ...

'', London, 1730 * G. L'Hôpital, ''Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes'', Paris, 1696 * G. L'Hôpital, ''Analyse des infinement petits'', Paris 1715 * William Fox,
Guillaume-François-Antoine de L'Hôpital
',

Catholic Encyclopedia The ''Catholic Encyclopedia: An International Work of Reference on the Constitution, Doctrine, Discipline, and History of the Catholic Church'' (also referred to as the ''Old Catholic Encyclopedia'' and the ''Original Catholic Encyclopedia'') i ...

, vol 7, New York, Robert Appleton Company, 1910 * C. Truesdel
The New Bernoulli Edition
Isis, Vol. 49, No. 1. (Mar., 1958), pp. 54–62, discusses the strange agreement between Bernoulli and de l'Hôpital on pages 59–62. * A.P. Yushkevich (ed), ''History of mathematics from the most ancient times to the beginning of the 19th century'', vol 2, ''Mathematics of the 17th century'' (in Russian). Moscow, Nauka, 1970

External links

* {{DEFAULTSORT:Lhopital, Guillaume de 1661 births 1704 deaths 17th-century French mathematicians 18th-century French mathematicians History of calculus Mathematical analysts Officers of the French Academy of Sciences