Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
[Joseph-Louis Lagrange, comte de l’Empire](_blank)
''Encyclopædia Britannica'' or Giuseppe Ludovico De la Grange Tournier;
25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange
or Lagrangia, was an
Italian
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance language
*** Regional Ita ...
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 ...
and
astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...
, later naturalized
French. He made significant contributions to the fields of
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, and both
classical and
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
.
In 1766, on the recommendation of Swiss
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and French
d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
, Lagrange succeeded Euler as the director of mathematics at the
Prussian Academy of Sciences
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
in Berlin,
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an em ...
, where he stayed for over twenty years, producing volumes of work and winning several prizes of the
French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
. Lagrange's treatise on
analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
(''
Mécanique analytique'', 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
and formed a basis for the development of
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
in the nineteenth century.
In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in the
decimalisation
Decimalisation or decimalization (see spelling differences) is the conversion of a system of currency or of weights and measures to units related by powers of 10.
Most countries have decimalised their currencies, converting them from non-decimal ...
in
Revolutionary France
The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are considere ...
, became the first professor of analysis at the
École Polytechnique
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, Savoi ...
upon its opening in 1794, was a founding member of the
Bureau des Longitudes, and became
Senator
A senate is a deliberative assembly, often the upper house or chamber of a bicameral legislature. The name comes from the ancient Roman Senate (Latin: ''Senatus''), so-called as an assembly of the senior (Latin: ''senex'' meaning "the el ...
in 1799.
Scientific contribution
Lagrange was one of the creators of the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, deriving the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s for extrema of
functionals. He extended the method to include possible constraints, arriving at the method of
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
.
Lagrange invented the method of solving
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s known as
variation of parameters
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible t ...
, applied
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
to the
theory of probabilities and worked on solutions for
algebraic equations
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
. He proved that
every natural number is a sum of four squares. His treatise ''Theorie des fonctions analytiques'' laid some of the foundations of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, anticipating
Galois. In
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 ...
, Lagrange developed a novel approach to
interpolation and
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
. He studied the
three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as
Lagrangian point
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...
s. Lagrange is best known for transforming
Newtonian mechanics
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
into a branch of analysis,
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
. He presented the mechanical "principles" as simple results of the variational calculus.
Biography
Early years
Firstborn of eleven children as ''Giuseppe Lodovico Lagrangia'', Lagrange was of Italian and French descent.
[ His paternal great-grandfather was a French captain of cavalry, whose family originated from the French region of ]Tours
Tours ( , ) is one of the largest cities in the region of Centre-Val de Loire, France. It is the Prefectures in France, prefecture of the Departments of France, department of Indre-et-Loire. The Communes of France, commune of Tours had 136,463 ...
.[ After serving under ]Louis XIV
, house = Bourbon
, father = Louis XIII
, mother = Anne of Austria
, birth_date =
, birth_place = Château de Saint-Germain-en-Laye, Saint-Germain-en-Laye, France
, death_date =
, death_place = Palace of Vers ...
, he had entered the service of Charles Emmanuel II
Charles Emmanuel II ( it, Carlo Emanuele II di Savoia); 20 June 1634 – 12 June 1675) was Duke of Savoy from 1638 to 1675 and under regency of his mother Christine of France until 1648. He was also Marquis of Saluzzo, Count of Aosta, Geneva, M ...
, Duke of Savoy
The titles of count, then of duke of Savoy are titles of nobility attached to the historical territory of Savoy. Since its creation, in the 11th century, the county was held by the House of Savoy. The County of Savoy was elevated to a Duchy of Sav ...
, and married a Conti
Conti is an Italian surname.
Geographical distribution
As of 2014, 63.5% of all known bearers of the surname ''Conti'' were residents of Italy (frequency 1:756), 11.8% of the United States (1:24,071), 9.2% of Brazil (1:17,439), 6.3% of Argentina ...
from the noble Roman family.[ Lagrange's father, Giuseppe Francesco Lodovico, was doctor in Law at the ]University of Torino
The University of Turin (Italian language, Italian: ''Università degli Studi di Torino'', UNITO) is a public university, public research university in the city of Turin, in the Piedmont (Italy), Piedmont region of Italy. It is one of the List ...
, while his mother was the only child of a rich doctor of Cambiano
Cambiano ( pms, Cambiagn) is a ''comune'' (municipality) in the Metropolitan City of Turin in the Italian region Piedmont, located about southeast of Turin.
Cambiano borders the following municipalities: Pino Torinese, Chieri, Pecetto Torinese, M ...
, in the countryside of Turin
Turin ( , Piedmontese language, Piedmontese: ; it, Torino ) is a city and an important business and cultural centre in Northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital ...
.[Lagrange](_blank)
St. Andrew University He was raised as a Roman Catholic (but later on became an agnostic
Agnosticism is the view or belief that the existence of God, of the divine or the supernatural is unknown or unknowable. (page 56 in 1967 edition) Another definition provided is the view that "human reason is incapable of providing sufficient ...
).
His father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father,[ and certainly Lagrange seems to have accepted this willingly. He studied at the ]University of Turin
The University of Turin (Italian: ''Università degli Studi di Torino'', UNITO) is a public research university in the city of Turin, in the Piedmont region of Italy. It is one of the oldest universities in Europe and continues to play an impo ...
and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.
It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley
Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720.
From an observatory he constructed on Saint Helena in 1676–77, H ...
from 1693
which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician. Charles Emmanuel III
Charles Emmanuel III (27 April 1701 – 20 February 1773) was Duke of Savoy and King of Sardinia from 1730 until his death.
Biography
He was born in Turin to Victor Amadeus II of Savoy and his first wife the French Anne Marie d'Orléans. Hi ...
appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" (mathematics assistant professor) at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins
Benjamin Robins (170729 July 1751) was a pioneering British scientist, Newtonian mathematician, and military engineer.
He wrote an influential treatise on gunnery, for the first time introducing Newtonian science to military men, was an early en ...
and Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange unfortunately proved to be a problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications. In this academy one of his students was François Daviet.
Variational calculus
Lagrange is one of the founders of the calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. Starting in 1754, he worked on the problem of the tautochrone
A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independe ...
, discovering a method of maximizing and minimizing functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s of variational calculus and considerably simplifying Euler's earlier analysis. Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
.
Euler was very impressed with Lagrange's results. It has been stated that "with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus"; however, this chivalric view has been disputed. Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773.
''Miscellanea Taurinensia''
In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the Turin Academy of Sciences, and most of his early writings are to be found in the five volumes of its transactions, usually known as the ''Miscellanea Taurinensia''. Many of these are elaborate papers. The first volume contains a paper on the theory of the propagation of sound; in this he indicates a mistake made by Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
, obtains the general differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a string vibrating transversely; in this paper he points out a lack of generality in the solutions previously given by Brook Taylor, D'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
, and Euler, and arrives at the conclusion that the form of the curve at any time ''t'' is given by the equation . The article concludes with a masterly discussion of echo
In audio signal processing and acoustics, an echo is a reflection of sound that arrives at the listener with a delay after the direct sound. The delay is directly proportional to the distance of the reflecting surface from the source and the lis ...
es, beat
Beat, beats or beating may refer to:
Common uses
* Patrol, or beat, a group of personnel assigned to monitor a specific area
** Beat (police), the territory that a police officer patrols
** Gay beat, an area frequented by gay men
* Battery (c ...
s, and compound sounds. Other articles in this volume are on recurring
Recurring means occurring repeatedly and can refer to several different things:
Mathematics and finance
*Recurring expense, an ongoing (continual) expenditure
*Repeating decimal, or recurring decimal, a real number in the decimal numeral syste ...
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
, probabilities, and the calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
.
The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
, and by solutions of various problems in dynamics.
The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the 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 ...
; a solution of a Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
's problem: given an integer which is not a perfect square, to find a number such that is a perfect square; and the general differential equations of motion for three bodies moving under their mutual attractions.
The next work he produced was in 1764 on the libration
In lunar astronomy, libration is the wagging or wavering of the Moon perceived by Earth-bound observers and caused by changes in their perspective. It permits an observer to see slightly different hemispheres of the surface at different tim ...
of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for ...
. His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in 1780.
Berlin
Already by 1756, Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
, seeing Lagrange's mathematical talent, tried to persuade Lagrange to come to Berlin, but he shyly refused the offer. In 1765, d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
interceded on Lagrange's behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. He again turned down the offer, responding that
: ''It seems to me that Berlin would not be at all suitable for me while M.Euler is there''.
In 1766, after Euler left Berlin for Saint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
, Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange was finally persuaded. He spent the next twenty years in Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an em ...
, where he produced a long series of papers published in the Berlin and Turin transactions, and composed his monumental work, the ''Mécanique analytique''. In 1767, he married his cousin Vittoria Conti.
Lagrange was a favourite of the king, who frequently lectured him on the advantages of perfect regularity of life. The lesson was accepted, and Lagrange studied his mind and body as though they were machines, and experimented to find the exact amount of work which he could do before exhaustion. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or in the subject-matter were capable of improvement. He carefully planned his papers before writing them, usually without a single erasure or correction.
Nonetheless, during his years in Berlin, Lagrange's health was rather poor, and that of his wife Vittoria was even worse. She died in 1783 after years of illness and Lagrange was very depressed. In 1786, Frederick II died, and the climate of Berlin became difficult for Lagrange.
Paris
In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and Naples
Naples (; it, Napoli ; nap, Napule ), from grc, Νεάπολις, Neápolis, lit=new city. is the regional capital of Campania and the third-largest city of Italy, after Rome and Milan, with a population of 909,048 within the city's adminis ...
, and he accepted the offer of Louis XVI
Louis XVI (''Louis-Auguste''; ; 23 August 175421 January 1793) was the last King of France before the fall of the monarchy during the French Revolution. He was referred to as ''Citizen Louis Capet'' during the four months just before he was ...
to move to Paris. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
, which later became part of the Institut de France
The (; ) is a French learned society, grouping five , including the Académie Française. It was established in 1795 at the direction of the National Convention. Located on the Quai de Conti in the 6th arrondissement of Paris, the institute m ...
(1795). At the beginning of his residence in Paris he was seized with an attack of melancholy, and even the printed copy of his ''Mécanique'' on which he had worked for a quarter of a century lay for more than two years unopened on his desk. Curiosity as to the results of the French revolution
The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are considere ...
first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed.
It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend, the astronomer Pierre Charles Le Monnier
Pierre Charles Le Monnier (; 23 November 1715 – 3 April 1799) was a French astronomer. His name is sometimes given as Lemonnier.
Biography
Le Monnier was born in Paris, where his father Pierre (1675–1757), also an astronomer, was professor ...
. She insisted on marrying him, and proved a devoted wife to whom he became warmly attached.
In September 1793, the Reign of Terror
The Reign of Terror (french: link=no, la Terreur) was a period of the French Revolution when, following the creation of the First Republic, a series of massacres and numerous public executions took place in response to revolutionary fervour, ...
began. Under intervention of Antoine Lavoisier, who himself was by then already thrown out of the academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October 1793 that ordered all foreigners to leave France. On 4 May 1794, Lavoisier and 27 other tax farmers
Farming or tax-farming is a technique of financial management in which the management of a variable revenue stream is assigned by legal contract to a third party and the holder of the revenue stream receives fixed periodic rents from the contract ...
were arrested and sentenced to death and guillotined on the afternoon after the trial. Lagrange said on the death of Lavoisier:
: ''It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.''
Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time, Napoleon
Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader who ...
) gave him honours and distinctions. This luckiness or safety may to some extent be due to his life attitude he expressed many years before: "''I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable''". A striking testimony to the respect in which he was held was shown in 1796 when the French commissary in Italy was ordered to attend in full state on Lagrange's father, and tender the congratulations of the republic on the achievements of his son, who "had done honor to all mankind by his genius, and whom it was the special glory of Piedmont
it, Piemontese
, population_note =
, population_blank1_title =
, population_blank1 =
, demographics_type1 =
, demographics1_footnotes =
, demographics1_title1 =
, demographics1_info1 =
, demographics1_title2 ...
to have produced." It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them. Appointed senator
A senate is a deliberative assembly, often the upper house or chamber of a bicameral legislature. The name comes from the ancient Roman Senate (Latin: ''Senatus''), so-called as an assembly of the senior (Latin: ''senex'' meaning "the el ...
in 1799, he was the first signer of the Sénatus-consulte
A ( French translation of la, senatus consultum, lit=decree of the senate) was a feature of French law during the French Consulate (1799–1804), First French Empire (1804–1814, 1815) and Second French Empire (1852–1870).
Consulate and Fir ...
which in 1802 annexed his fatherland Piedmont to France.[ He acquired French citizenship in consequence.][ The French claimed he was a French mathematician, but the Italians continued to claim him as Italian.'']
Units of measurement
Lagrange was involved in the development of the metric system
The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures ('' la Commission des Poids et Mesures'') when he was preparing to escape. After Lavoisier's death in 1794, it was largely Lagrange who influenced the choice of the metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
and kilogram
The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially ...
units with decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
subdivision, by the commission of 1799. Lagrange was also one of the founding members of the Bureau des Longitudes in 1795.
École Normale
In 1795, Lagrange was appointed to a mathematical chair at the newly established École Normale
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, Savoi ...
, which enjoyed only a short existence of four months. His lectures there were elementary; they contain nothing of any mathematical importance, though they do provide a brief historical insight into his reason for proposing undecimal
The undecimal numeral system (also known as the base-11 numeral system) is a positional numeral system that uses eleven as its base. While no known society counts by elevens, two are purported to have done so: the Māori, one of the two Polynes ...
or Base 11 as the base number for the reformed system of weights and measures. The lectures were published because the professors had to "pledge themselves to the representatives of the people and to each other neither to read nor to repeat from memory" .html" ;"title="Les professeurs aux Écoles Normales ont pris, avec les Représentans du Peuple, et entr'eux l'engagement de ne point lire ou débiter de mémoire des discours écrits"">Les professeurs aux Écoles Normales ont pris, avec les Représentans du Peuple, et entr'eux l'engagement de ne point lire ou débiter de mémoire des discours écrits" The discourses were ordered taken down in shorthand to enable the deputies to see how the professors acquitted themselves. It was also thought the published lectures would interest a significant portion of the citizenry .html" ;"title="Quoique des feuilles sténographiques soient essentiellement destinées aux élèves de l'École Normale, on doit prévoir quיelles seront lues par une grande partie de la Nation"">Quoique des feuilles sténographiques soient essentiellement destinées aux élèves de l'École Normale, on doit prévoir quיelles seront lues par une grande partie de la Nation"
École Polytechnique
In 1794, Lagrange was appointed professor of the École Polytechnique
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, Savoi ...
; and his lectures there, described by mathematicians who had the good fortune to be able to attend them, were almost perfect both in form and matter. Beginning with the merest elements, he led his hearers on until, almost unknown to themselves, they were themselves extending the bounds of the subject: above all he impressed on his pupils the advantage of always using general methods expressed in a symmetrical notation.
But Lagrange does not seem to have been a successful teacher. Fourier, who attended his lectures in 1795, wrote:
:his voice is very feeble, at least in that he does not become heated; he has a very marked Italian accent and pronounces the ''s'' like ''z'' ..The students, of whom the majority are incapable of appreciating him, give him little welcome, but the ''professeurs'' make amends for it.
Late years
In 1810, Lagrange started a thorough revision of the ''Mécanique analytique'', but he was able to complete only about two-thirds of it before his death at Paris in 1813, in 128 rue du Faubourg Saint-Honoré
The Rue du Faubourg Saint-Honoré () is a street located in the 8th arrondissement of Paris, France. Relatively narrow and nondescript, especially in comparison to the nearby Avenue des Champs-Élysées, it is cited as being one of the most luxu ...
. Napoleon honoured him with the Grand Croix of the Ordre Impérial de la Réunion just two days before he died. He was buried that same year in the Panthéon
The Panthéon (, from the Classical Greek word , , ' empleto all the gods') is a monument in the 5th arrondissement of Paris, France. It stands in the Latin Quarter, atop the , in the centre of the , which was named after it. The edifice was b ...
in Paris. The inscription on his tomb reads in translation:JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of the Reunion
The Order of the Reunion (french: link=no, Ordre de la Réunion) was an order of merit of the First French Empire, set up to be awarded to Frenchmen and foreigners to reward services in the civil service, magistracy and army, particularly those fro ...
. Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January 1736. Died in Paris on 10 April 1813.
Work in Berlin
Lagrange was extremely active scientifically during twenty years he spent in Berlin. Not only did he produce his ''Mécanique analytique'', but he contributed between one and two hundred papers to the Academy of Turin, the Berlin Academy, and the French Academy. Some of these are really treatises, and all without exception are of a high order of excellence. Except for a short time when he was ill he produced on average about one paper a month. Of these, note the following as amongst the most important.
First, his contributions to the fourth and fifth volumes, 1766–1773, of the ''Miscellanea Taurinensia''; of which the most important was the one in 1771, in which he discussed how numerous astronomical
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
observations should be combined so as to give the most probable result. And later, his contributions to the first two volumes, 1784–1785, of the transactions of the Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, and the kind of problems for which it is suitable.
Most of the papers sent to Paris were on astronomical questions, and among these including his paper on the Jovian
Jovian is the adjectival form of Jupiter and may refer to:
* Jovian (emperor) (Flavius Iovianus Augustus), Roman emperor (363–364 AD)
* Jovians and Herculians, Roman imperial guard corps
* Jovian (lemur), a Coquerel's sifaka known for ''Zoboomafo ...
system in 1766, his essay on the problem of three bodies in 1772, his work on the secular equation of the Moon in 1773, and his treatise on cometary perturbations in 1778. These were all written on subjects proposed by the Académie française
An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or tertiary education, tertiary higher education, higher learning (and generally also research or honorary membershi ...
, and in each case the prize was awarded to him.
Lagrangian mechanics
Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
.
Algebra
The greater number of his papers during this time were, however, contributed to the Prussian Academy of Sciences
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
. Several of them deal with questions in algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
.
*His discussion of representations of integers by quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s (1769) and by more general algebraic forms (1770).
*His tract on the Theory of Elimination, 1770.
* Lagrange's theorem that the order of a subgroup H of a group G must divide the order of G.
*His papers of 1770 and 1771 on the general process for solving an algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
of any degree via the ''Lagrange resolvents
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rati ...
''. This method fails to give a general formula for solutions of an equation of degree five and higher, because the auxiliary equation involved has higher degree than the original one. The significance of this method is that it exhibits the already known formulas for solving equations of second, third, and fourth degrees as manifestations of a single principle, and was foundational in Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
. The complete solution of a binomial equation (namely an equation of the form ± ) is also treated in these papers.
*In 1773, Lagrange considered a functional determinant In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the i ...
of order 3, a special case of a Jacobian. He also proved the expression for the volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
with one of the vertices at the origin as the one sixth of the absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
formed by the coordinates of the other three vertices.
Number theory
Several of his early papers also deal with questions of number theory.
*Lagrange (1766–1769) was the first European to prove that Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
has a nontrivial solution in the integers for any non-square natural number .
*He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squares, 1770.
*He proved Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
that (for any integer ): is a prime if and only if is a multiple of , 1771.
*His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved.
*His Recherches d'Arithmétique of 1775 developed a general theory of binary quadratic forms
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
to handle the general problem of when an integer is representable by the form .
*He made contributions to the theory of continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s.
Other mathematical work
There are also numerous articles on various points of analytical geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineer ...
. In two of them, written rather later, in 1792 and 1793, he reduced the equations of the quadrics (or conicoids) to their canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
s.
During the years from 1772 to 1785, he contributed a long series of papers which created the science of partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. A large part of these results was collected in the second edition of Euler's integral calculus which was published in 1794.
Astronomy
Lastly, there are numerous papers on problems in astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. Of these the most important are the following:
*Attempting to solve the general three-body problem, with the consequent discovery of the two constant-pattern solutions, collinear and equilateral, 1772. Those solutions were later seen to explain what are now known as the Lagrangian point
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...
s.
*On the attraction of ellipsoids, 1773: this is founded on Maclaurin
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ( ...
's work.
*On the secular equation of the Moon, 1773; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point. Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in 1777.
*On the motion of the nodes of a planet's orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
, 1774.
*On the stability of the planetary orbits, 1776.
*Two papers in which the method of determining the orbit of a comet
A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ar ...
from three observations is completely worked out, 1778 and 1783: this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject.
*His determination of the secular and periodic variations of the elements of the planets, 1781–1784: the upper limits assigned for these agree closely with those obtained later by Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted.
*Three papers on the method of interpolation, 1783, 1792 and 1793: the part of finite differences dealing therewith is now in the same stage as that in which Lagrange left it.
Fundamental treatise
Over and above these various papers he composed his fundamental treatise, the ''Mécanique analytique''.
In this book, he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, both of solids and fluids.
The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalised co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables ''x'', called generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
, whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation. For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form
:
where ''T'' represents the kinetic energy and ''V'' represents the potential energy of the system.
He then presented what we now know as the method of Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
—though this is not the first time that method was published—as a means to solve this equation.[Marco Panza, "The Origins of Analytic Mechanics in the 18th Century", in Hans Niels Jahnke (editor), ''A History of Analysis'', 2003, p. 149]
Amongst other minor theorems here given it may suffice to mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
. All the analysis is so elegant that Sir William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
said the work could be described only as a scientific poem. Lagrange remarked that mechanics was really a branch of pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram. At first no printer could be found who would publish the book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under the supervision of Laplace, Cousin, Legendre (editor) and Condorcet in 1788.
Work in France
Differential calculus and calculus of variations
Lagrange's lectures on the differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
at École Polytechnique form the basis of his treatise ''Théorie des fonctions analytiques'', which was published in 1797. This work is the extension of an idea contained in a paper he had sent to the Berlin papers in 1772, and its object is to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series, relying in particular on the principle of the generality of algebra In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange,. particularl ...
.
A somewhat similar method had been previously used by John Landen
John Landen (23 January 1719 – 15 January 1790) was an English mathematician.
Life
He was born at Peakirk, near Peterborough in Northamptonshire, on 28 January 1719. He was brought up to the business of a surveyor, and acted as land agent to ...
in the ''Residual Analysis'', published in London in 1758. Lagrange believed that he could thus get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
, the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics.
Another treatise on the same lines was his ''Leçons sur le calcul des fonctions'', issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
, in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as the starting point for the researches of Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
, Jacobi Jacobi may refer to:
* People with the surname Jacobi (surname), Jacobi
Mathematics:
* Jacobi sum, a type of character sum
* Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
* Jacobi eigenva ...
, and Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
.
File:Lagrange-6.jpg, 1813 copy of "Theorie des fonctions analytiques"
File:Lagrange-7.jpg, Title page to "Theorie des fonctions analytiques"
File:Lagrange-9.jpg, Introduction to "Theorie des fonctions analytiques"
File:Lagrange-10.jpg, First page of "Theorie des fonctions analytiques"
Infinitesimals
At a later period Lagrange fully embraced the use of infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
s in preference to founding the differential calculus on the study of algebraic forms; and in the preface to the second edition of the ''Mécanique Analytique'', which was issued in 1811, he justifies the employment of infinitesimals, and concludes by saying that:
: ''When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs.''
Number theory
His ''Résolution des équations numériques'', published in 1798, was also the fruit of his lectures at École Polytechnique. There he gives the method of approximating to the real roots of an equation by means of continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s, and enunciates several other theorems. In a note at the end he shows how Fermat's little theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
, that is
:
where ''p'' is a prime and ''a'' is prime to ''p'', may be applied to give the complete algebraic solution of any binomial equation. He also here explains how the equation whose roots are the squares of the differences of the roots of the original equation may be used so as to give considerable information as to the position and nature of those roots.
Celestial mechanics
The theory of the planetary motion
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
s had formed the subject of some of the most remarkable of Lagrange's Berlin papers. In 1806 the subject was reopened by Poisson, who, in a paper read before the French Academy, showed that Lagrange's formulae led to certain limits for the stability of the orbits. Lagrange, who was present, now discussed the whole subject afresh, and in a letter communicated to the academy in 1808 explained how, by the variation of arbitrary constants, the periodical and secular inequalities of any system of mutually interacting bodies could be determined.
Prizes and distinctions
Euler proposed Lagrange for election to the Berlin Academy and he was elected on 2 September 1756. He was elected a Fellow of the Royal Society of Edinburgh
The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was established i ...
in 1790, a Fellow of the Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
and a foreign member of the Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...
in 1806. In 1808, Napoleon
Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader who ...
made Lagrange a Grand Officer of the Legion of Honour
The National Order of the Legion of Honour (french: Ordre national de la Légion d'honneur), formerly the Royal Order of the Legion of Honour ('), is the highest French order of merit, both military and civil. Established in 1802 by Napoleon, ...
and a Count of the Empire
As Emperor of the French, Napoleon I created titles of nobility to institute a stable elite in the First French Empire, after the instability resulting from the French Revolution.
Like many others, both before and since, Napoleon found that t ...
. He was awarded the Grand Croix of the Ordre Impérial de la Réunion in 1813, a week before his death in Paris, and was buried in the Panthéon
The Panthéon (, from the Classical Greek word , , ' empleto all the gods') is a monument in the 5th arrondissement of Paris, France. It stands in the Latin Quarter, atop the , in the centre of the , which was named after it. The edifice was b ...
, a mausoleum
A mausoleum is an external free-standing building constructed as a monument enclosing the interment space or burial chamber of a deceased person or people. A mausoleum without the person's remains is called a cenotaph. A mausoleum may be consid ...
dedicated to the most honoured French people.
Lagrange was awarded the 1764 prize of the French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
for his memoir on the libration
In lunar astronomy, libration is the wagging or wavering of the Moon perceived by Earth-bound observers and caused by changes in their perspective. It permits an observer to see slightly different hemispheres of the surface at different tim ...
of the Moon. In 1766 the academy proposed a problem of the motion of the satellites of Jupiter
There are 82 known moons of Jupiter, not counting a number of moonlets likely shed from the inner moons. All together, they form a satellite system which is called the Jovian system. The most massive of the moons are the four Galilean moons: ...
, and the prize again was awarded to Lagrange. He also shared or won the prizes of 1772, 1774, and 1778.
Lagrange is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower
The Eiffel Tower ( ; french: links=yes, tour Eiffel ) is a wrought-iron lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed and built the tower.
Locally nicknamed "'' ...
when it first opened. ''Rue Lagrange'' in the 5th Arrondissement in Paris is named after him. In Turin, the street where the house of his birth still stands is named ''via Lagrange''. The lunar crater
Lunar craters are impact craters on Earth's Moon. The Moon's surface has many craters, all of which were formed by impacts. The International Astronomical Union currently recognizes 9,137 craters, of which 1,675 have been dated.
History
The wor ...
Lagrange and the asteroid 1006 Lagrangea also bear his name.
See also
* List of things named after Joseph-Louis Lagrange
* Four-dimensional space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
* Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
* History of the metre
The history of the metre starts with the Scientific Revolution that is considered to have begun with Nicolaus Copernicus's publication of '' De revolutionibus orbium coelestium'' in 1543. Increasingly accurate measurements were required, and ...
* Lagrange's role in measurement reform
* Seconds pendulum
A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
Notes
References
Citations
Sources
The initial version of this article was taken from the public domain
The public domain (PD) consists of all the creative work
A creative work is a manifestation of creative effort including fine artwork (sculpture, paintings, drawing, sketching, performance art), dance, writing (literature), filmmaking, ...
resource '' A Short Account of the History of Mathematics'' (4th edition, 1908) by W. W. Rouse Ball
Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...
.
*
* ''Columbia Encyclopedia'', 6th ed., 2005,
Lagrange, Joseph Louis.
* W. W. Rouse Ball
Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...
, 1908,
Joseph Louis Lagrange (1736–1813)
''A Short Account of the History of Mathematics'', 4th ed
also on Gutenberg
* Chanson, Hubert, 2007,
Velocity Potential in Real Fluid Flows: Joseph-Louis Lagrange's Contribution
" ''La Houille Blanche'' 5: 127–31.
* Fraser, Craig G., 2005, "Théorie des fonctions analytiques" in Grattan-Guinness, I., ed., ''Landmark Writings in Western Mathematics''. Elsevier: 258–76.
* Lagrange, Joseph-Louis. (1811). ''Mécanique Analytique''. Courcier (reissued by Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, 2009; )
* Lagrange, J.L. (1781) "Mémoire sur la Théorie du Mouvement des Fluides"(Memoir on the Theory of Fluid Motion) in Serret, J.A., ed., 1867. ''Oeuvres de Lagrange, Vol. 4''. Paris" Gauthier-Villars: 695–748.
* Pulte, Helmut, 2005, "Méchanique Analytique" in Grattan-Guinness, I., ed., ''Landmark Writings in Western Mathematics''. Elsevier: 208–24.
*
External links
*
*
Lagrange, Joseph Louis de: The Encyclopedia of Astrobiology, Astronomy and Space Flight
*
Derivation of Lagrange's result (not Lagrange's method)
* Lagrange's works (in French
Oeuvres de Lagrange, edited by Joseph Alfred Serret, Paris 1867, digitized by Göttinger Digitalisierungszentrum
(Mécanique analytique is in volumes 11 and 12.)
Joseph Louis de Lagrange – Œuvres complètes
Gallica-Math
Inventaire chronologique de l'œuvre de Lagrange
Persee
*
*
''Mécanique analytique'' (Paris, 1811-15)
{{DEFAULTSORT:Lagrange, Joseph-Louis
Lagrangian mechanics
1736 births
1813 deaths
Scientists from Turin
18th-century Italian mathematicians
19th-century Italian mathematicians
Burials at the Panthéon, Paris
Counts of the First French Empire
Italian people of French descent
French agnostics
18th-century French astronomers
18th-century Italian astronomers
Mathematical analysts
Members of the French Academy of Sciences
Members of the Prussian Academy of Sciences
Members of the Royal Swedish Academy of Sciences
Honorary members of the Saint Petersburg Academy of Sciences
Number theorists
French geometers
People from the Kingdom of Sardinia
Grand Officiers of the Légion d'honneur
Fellows of the Royal Society
18th-century French mathematicians
19th-century French mathematicians