Guillaume de l'Hôpital
Encyclopedia
Guillaume François Antoine, Marquis
de l'Hôpitalmaʁki də lopiˈtal (?, 1661, Paris
– February 2, 1704, Paris) was a French
mathematician
. His name is firmly associated with l'Hôpital's rule
for calculating limit
s involving indeterminate form
s 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. 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
.
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. Among his accomplishments were the determination of the arc length
of the logarithm
ic 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'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 section
s") was published posthumously in Paris in 1707.
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 March 17, 1694, l'Hôpital made the following proposal to Johann Bernoulli
: 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 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
, 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'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 (one should also note that the book was first published anonymously), Analyse was remarkably successful in popularizing the ideas of differential calculus stemming from Leibniz.
Marquis
Marquis is a French and Scottish title of nobility. The English equivalent is Marquess, while in German, it is Markgraf.It may also refer to:Persons:...
de l'Hôpitalmaʁki də lopiˈtal (?, 1661, Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
– February 2, 1704, Paris) was a French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. His name is firmly associated with l'Hôpital's rule
L'Hôpital's rule
In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
for calculating limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
s involving indeterminate form
Indeterminate form
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...
s 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
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
, entitled Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. This book was a first systematic exposition of differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
. Several editions and translations to other languages were published and it became a model for subsequent treatments of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
.
Biography
L'Hôpital was born into a noble family. His father was Anne-Alexandre de l'Hôpital, a Lieutenant-General of the King's army, ComteComte
Comte is a title of Catalan, Occitan and French nobility. In the English language, the title is equivalent to count, a rank in several European nobilities. The corresponding rank in England is earl...
de Sainte-Mesme and the first squire of Gaston, Duke of Orléans
Gaston, Duke of Orléans
Gaston of France, , also known as Gaston d'Orléans, was the third son of King Henry IV of France and his wife Marie de Medici. As a son of the king, he was born a Fils de France. He later acquired the title Duke of Orléans, by which he was generally known during his adulthood...
. 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, which was apparent since his childhood. For a while, he was a member of Nicolas Malebranche
Nicolas Malebranche
Nicolas Malebranche ; was a French Oratorian and rationalist philosopher. In his works, he sought to synthesize the thought of St. Augustine and Descartes, in order to demonstrate the active role of God in every aspect of the world...
's circle in Paris and it was there that in 1691 he met young Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
, who was visiting France and agreed to supplement his Paris talks on infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
with private lectures to l'Hôpital at his estate at Oucques
Oucques
Oucques is a commune in the Loir-et-Cher department of central France....
. In 1693, l'Hôpital was elected to the French academy of sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
and even served twice as its vice-president. Among his accomplishments were the determination of the arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
of the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
ic graph, one of the solutions to the brachistochrone problem, and the discovery of a turning point singularity
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
on the involute
Involute
In the differential geometry of curves, an involute is a curve obtained from another given curve by attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve; or in reverse, unwound. It is a roulette wherein the rolling curve is a straight...
of a plane curve near an inflection point
Inflection point
In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...
.
L'Hôpital exchanged ideas with Pierre Varignon
Pierre Varignon
Pierre Varignon was a French mathematician. He was educated at the Jesuit College and the University in Caen, where he received his M.A. in 1682. He took Holy Orders the following year....
and corresponded with Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, Christiaan Huygens, and Jacob and Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
. 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
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s") was published posthumously in Paris in 1707.
Calculus textbook
In 1696 l'Hôpital published his book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes ("Infinitesimal calculus with applications to curved lines"). This was the first textbook on infinitesimal calculusInfinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
and it presented the ideas of differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
and their applications to differential geometry of curves
Differential geometry of curves
Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....
in a lucid form and with numerous figures; however, it did not consider integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
. The history leading to the book's publication became a subject of a protracted controversy. In a letter from March 17, 1694, l'Hôpital made the following proposal to Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
: 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 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
Hieronymus Georg Zeuthen
Hieronymus Georg Zeuthen was a Danish mathematician.He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics.-Biography:...
and Moritz Cantor
Moritz Cantor
Moritz Benedikt Cantor was a German historian of mathematics.He was born at Mannheim, Germany. He came from a family that had emigrated to the Netherlands from Portugal, another branch of which had established itself in Russia, where Georg Cantor was born...
, 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'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 (one should also note that the book was first published anonymously), Analyse was remarkably successful in popularizing the ideas of differential calculus stemming from Leibniz.