TheInfoList

Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJoseph-Louis Lagrange, comte de l’Empire
''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 may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, a Romance language *** Regional Italian, regional variants of the ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

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 is the process of breaking a complexity, complex topic or Substance theory, 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 Ari ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, and both
classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...
and
celestial mechanics Celestial mechanics is the branch of astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and cel ...
. 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 made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

and French
d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician A mechanician is an engineer or a scientist working in the field of mechanics, or in a related or sub-field: engineering or com ...
, 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 An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), ...
in Berlin,
Prussia Prussia, , Old Prussian Distribution of the Baltic tribes, circa 1200 CE (boundaries are approximate). Old Prussian was a Western Baltic language belonging to the Balto-Slavic branch of the Indo-European languages The Indo-Europ ...

, 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 A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discipli ...
. Lagrange's treatise on
analytical mechanics Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry * ...
('''', 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), ''Newton'' (film), a 2017 Indian fil ...

and formed a basis for the development of mathematical physics 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 (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Engli ...
in
Revolutionary France The French Revolution ( ) was a period of radical political and societal change in France France (), officially the French Republic (french: link=no, République française), is a spanning and in the and the , and s. Its ...
, became the first professor of analysis at the
École Polytechnique File:Statue X DSC08329.JPG, upA statue in the courtyard of the school commemorates the cadets of ''Polytechnique'' rushing to the Battle of Paris (1814), defence of Paris in 1814. A copy was installed in West Point. The École Polytechnique (Fr ...
upon its opening in 1794, was a founding member of the
Bureau des Longitudes The ''Bureau des Longitudes'' () is a France, French scientific institution, founded by decree of 25 June 1795 and charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation. During th ...
, and became
Senator The Curia Julia in the Roman Forum A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicameral legislature. The name comes from the Ancient Rome, ancient Roman Senate (Latin: ''Senatus''), so-call ...
in 1799.

# Scientific contribution

Lagrange was one of the creators of the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
, deriving the
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
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 equation In mathematics, an ...
. Lagrange invented the method of solving
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

s known as
variation of parameters Variation or Variations may refer to: Science and mathematics * Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon * Genetic variation, the difference in DNA among individuals ...
, applied
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
to the theory of probabilities and worked on solutions for algebraic equations. 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, anticipating Galois. In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

, Lagrange developed a novel approach to
interpolation In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...
and Taylor theorem. He studied the
three-body problem In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through 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, L-points, or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbit, orbiting bodies. Mathematically, this involves th ...
s. Lagrange is best known for transforming
Newtonian mechanics Newton's laws of motion are three Scientific law, 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: ''Law 1''. A body continues ...
into a branch of analysis,
Lagrangian mechanics Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
, and 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 Centre-Val de Loire Centre-Val de Loire (, , ,In isolation, ''Centre'' is pronounced . ; Occitan Occitan (; oc, occitan, link=no ,), also known as ''lenga d'òc'' (; french: langue d'oc) ...

. After serving under
Louis XIV Louis XIV (Louis Dieudonné; 5 September 16381 September 1715), also known as Louis the Great () or the Sun King (), was King of France from 14 May 1643 until his death in 1715. His reign of 72 years and 110 days is the List of longest-reigning mo ...

, 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 the Duke of Savoy from 1638 to 1675 and under regency of his mother Christine of France Christine of France (10 February 1606 – 27 December 1663) was t ...
,
Duke of SavoyThe following is a list of rulers of Savoy Savoy (; frp, Savouè ; french: Savoie ; it, Savoia ; pms, Savòja ; ) is a cultural-historical region in the Western Alps. Situated on the cultural boundary between Franco-Provençal, Occitan ...

, and married a
Conti Conti is an surname. Geographical distribution As of 2014, 63.5% of all known bearers of the surname ''Conti'' were residents of (frequency 1:756), 11.8% of the (1:24,071), 9.2% of (1:17,439), 6.3% of (1:5,300), 2.5% of (1:21,201) and 1.3% ...
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: ''Università degli Studi di Torino'', or often abbreviated to UNITO) is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tert ...
, while his mother was the only child of a rich doctor of
Cambiano Cambiano is a ''comune The (; plural: ) is a basic Administrative division, constituent entity of Italy, roughly equivalent to a township or municipality. Importance and function The provides many of the basic civil functions: Civil regi ...
, in the countryside of
Turin Turin ( , Piedmontese Piedmontese (autonym: or , in it, piemontese) is a language spoken by some 700,000 people mostly in Piedmont it, Piemontese , population_note = , population_blank1_title = , population_blank1 = ...

.Lagrange
St. Andrew University
He was raised as a Roman Catholic (but later on became an
agnostic Agnosticism is the view that the existence of God, of the divinity, divine or the supernatural is unknown or Uncertainty, unknowable. (page 56 in 1967 edition) Another definition provided is the view that "human reason is incapable of providing ...

). 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 language, Italian: ''Università degli Studi di Torino'', or often abbreviated to UNITO) is a university in the city of Turin in the Piedmont (Italy), Piedmont region of north-western Italy. It is one of the oldes ...
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 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 su ...

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 the Duchy of Savoy, Duke of Savoy and List of monarchs of Sardinia, King of Sardinia from 1730 until his death. Biography He was born in Turin to Victor Amadeus II of Savoy and 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, mathematics, Newtonian mathematician, and military engineer. He wrote an influential treatise on gunnery, for the first time introducing Newtonian science to military men, was ...

and
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

. 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 is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
. Starting in 1754, he worked on the problem of the
tautochroneImage:Tautochrone curve.gif, 300px, Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position diagram ...
, 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 made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
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 An intellectual is a person who engages in critical thinking, research, and Human self-reflection, reflection to adv ...
. 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), ''Newton'' (film), a 2017 Indian fil ...

, obtains the general
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

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 Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Life and work Brook Taylor ...
,
D'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician A mechanician is an engineer or a scientist working in the field of mechanics, or in a related or sub-field: engineering or com ...
, and Euler, and arrives at the conclusion that the form of the curve at any time ''t'' is given by the equation $y = a \sin \left(mx\right) \sin \left(nt\right)\,$. The article concludes with a masterly discussion of
echo In audio signal processing Audio signal processing is a subfield of signal processing that is concerned with the electronic manipulation of audio signals. Audio signals are electronic representations of sound waves—longitudinal waves which ...
es,
beat Beat, beats or beating may refer to: Common meanings Assigned activity or area * Patrol, an area (usually geographic) that one is responsible to monitor, including: ** Beat (police), the territory and time that a police officer patrols ** Beat ...
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 system ...
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 i ...
,
probabilities Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

, and the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
. 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 :''This article discusses the history of the principle of least action. For the application, please refer to action (physics) In physics, action is an attribute of the dynamics (physics), dynamics of a physical system from which the equations ...

, and by solutions of various problems in
dynamics Dynamics (from Greek language, Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power (disambiguation), power") or dynamic may refer to: Physics and engineering * Dynamics (mechanics) ** Aerodynamics, the study o ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
; a solution of
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the '' Parlement'' of Toulouse Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

's problem mentioned above: given an integer ''n'' which is not a perfect square, to find a number ''x'' such that ''x''2''n'' + 1 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 File:MoonVisibleLibration.jpg, Theoretical extent of visible lunar surface (in green) due to libration, compared to the extent of the visible lunar surface without libration (in yellow). The projection is the Winkel tripel projection, Winkel ...

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 Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...
. 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

Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

and
Maupertuis Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters An intellectual is a person who engages in critical thinking, research, and Human self-reflection, reflection to adv ...
, 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 A mechanician is an engineer or a scientist working in the field of mechanics, or in a related or sub-field: engineering or com ...
interceded on Lagrange's behalf with 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

, 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 Distribution of the Baltic tribes, circa 1200 CE (boundaries are approximate). Old Prussian was a Western Baltic language belonging to the Balto-Slavic branch of the Indo-European languages The Indo-Europ ...

, 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 and the third-largest city of , after and , with a population of 967,069 within the city's administrative limits as of ...

, and he accepted the offer of
Louis XVI Louis XVI (Louis-Auguste; ; 23 August 175421 January 1793) was the last King of France The monarchs of the Kingdom of France The Kingdom of France ( fro, Reaume de France, frm, Royaulme de France, french: link=no, Royaume de France) wa ...

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 A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discipli ...
, which later became part of the
Institut de France The (; Institute of France) is a French learned society A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discipline (academia), academic discipli ...
(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 France (), officially the French Republic (french: link=no, République française), is a spanning and in the and the , and s. Its ...

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 (20 November 1715 – 3 April 1799) was a French 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 ast ...
. 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, commonly called The Terror (french: link=no, la Terreur), was a period of the French Revolution The French Revolution ( ) refers to the period that began with the Estates General of 1789 and ended in coup of 18 Br ...
began. Under intervention of Antoine
Lavoisier Antoine-Laurent de Lavoisier ( , ,; 26 August 17438 May 1794), When reduced without charcoal, it gave off an air which supported respiration and combustion in an enhanced way. He concluded that this was just a pure form of common air and th ...

, 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 contract, legal contract to a third party and the holder of the revenue stream receives fixed periodic rents from the ...
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 Napoléon Bonaparte (15 August 1769 – 5 May 1821) was a French military and political leader. He rose to prominence during the French Revolution The French Revolution ( ) refers to the period that began with the Estates General o ...
) loaded him with 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 Debating chamber, chamber of a bicameral legislature. The name comes from the Ancient Rome, ancient Roman Senate (Latin: ''Senatus''), so-called as an assembly of the senior (Lat ...
in 1799, he was the first signer of the Sénatus-consulte 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 A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purpose ...

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 ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...
and
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...
units with
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
subdivision, by the commission of 1799. Lagrange was also one of the founding members of the
Bureau des Longitudes The ''Bureau des Longitudes'' () is a France, French scientific institution, founded by decree of 25 June 1795 and charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation. During th ...
in 1795.

### École Normale

In 1795, Lagrange was appointed to a mathematical chair at the newly established École Normale, 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
undecimalThe undecimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is the ...
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 File:Statue X DSC08329.JPG, upA statue in the courtyard of the school commemorates the cadets of ''Polytechnique'' rushing to the Battle of Paris (1814), defence of Paris in 1814. A copy was installed in West Point. The École Polytechnique (Fr ...
; 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. , 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 commenced 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 Champs-Élysées, avenue des Champs Élysées, it is cited as being one o ...
. 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 Montagne Sainte-Geneviève, in the center of the Place du Panthéo ...

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. 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 astronomy, astronomical 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, 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 Jupiter, Jovian 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, 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 Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
.

## 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 An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), ...
. Several of them deal with questions in algebra. *His discussion of representations of integers by quadratic forms (1769) and by more general algebraic forms (1770). *His tract on the Elimination theory, Theory of Elimination, 1770. *Lagrange's theorem (group theory), 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 of any degree via the ''Lagrange resolvents''. 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. The complete solution of a binomial equation (namely an equation of the form $ax^n$ ± $b=0$) is also treated in these papers. *In 1773, Lagrange considered a functional determinant of order 3, a special case of a Jacobian matrix and determinant, Jacobian. He also proved the expression for the volume of a tetrahedron with one of the vertices at the origin as the one sixth of the absolute value of the determinant 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 has a nontrivial solution in the integers for any non-square natural number . *He proved the theorem, stated by Bachet without justification, that Lagrange's four-square theorem, every positive integer is the sum of four squares, 1770. *He proved Wilson's theorem 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 List of important publications in mathematics#Recherches d'Arithmétique, Recherches d'Arithmétique of 1775 developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form . *He made contributions to the theory of continued fractions.

## Other mathematical work

There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in 1792 and 1793, he reduced the quadric, equations of the quadrics (or conicoids) to their canonical forms. During the years from 1772 to 1785, he contributed a long series of papers which created the science of partial differential equations. 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. Of these the most important are the following: *Attempting to solve the three-body problem, 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, L-points, or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbit, orbiting bodies. Mathematically, this involves th ...
s. *On the attraction of ellipsoids, 1773: this is founded on Colin Maclaurin, 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, 1774. *On the stability of the planetary orbits, 1776. *Two papers in which the method of determining the orbit of a comet 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 orbital elements, elements of the planets, 1781–1784: the upper limits assigned for these agree closely with those obtained later by Urbain Le Verrier, 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, 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, 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 :$\frac \frac - \frac + \frac = 0,$ 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 equation In mathematics, an ...
—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 :''This article discusses the history of the principle of least action. For the application, please refer to action (physics) In physics, action is an attribute of the dynamics (physics), dynamics of a physical system from which the equations ...

. All the analysis is so elegant that Sir William Rowan Hamilton said the work could be described only as a scientific poem. Lagrange remarked that mechanics was really a branch of pure mathematics 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 Adrien-Marie Legendre, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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. A somewhat similar method had been previously used by John Landen 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, 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 equation In mathematics, an ...
, 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 Augustin Louis Cauchy, Cauchy, Carl Gustav Jakob Jacobi, Jacobi, and Karl Weierstrass, Weierstrass.

## Infinitesimals

At a later period Lagrange fully embraced the use of infinitesimals 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 fractions, and enunciates several other theorems. In a note at the end he shows how Fermat's little theorem, that is : $a^-1 \equiv 0\pmod p$ 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 motions had formed the subject of some of the most remarkable of Lagrange's Berlin papers. In 1806 the subject was reopened by Siméon Poisson, 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 in 1790, a Fellow of the Royal Society and a foreign member of the Royal Swedish Academy of Sciences in 1806. In 1808, Napoleon made Lagrange a Grand Officer of the Legion of Honour and a Count of the Empire. He was awarded the Grand Croix of the Order of the Reunion, 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 Montagne Sainte-Geneviève, in the center of the Place du Panthéo ...

, a mausoleum 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 A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discipli ...
for his memoir on the
libration File:MoonVisibleLibration.jpg, Theoretical extent of visible lunar surface (in green) due to libration, compared to the extent of the visible lunar surface without libration (in yellow). The projection is the Winkel tripel projection, Winkel ...

of the Moon. In 1766 the Academy proposed a problem of the motion of the satellites of Jupiter, 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 List of the 72 names on the Eiffel Tower, 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower 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 Lagrange (crater), Lagrange and the asteroid 1006 Lagrangea also bear his name.

* List of things named after Joseph-Louis Lagrange * Four-dimensional space * Gauss's law * History of the metre * Undecimal#Base-11_in_the_history_of_measurement, Lagrange's role in measurement reform * Seconds pendulum

# References

## Sources

The initial version of this article was taken from the public domain resource ''Rouse History of Mathematics, A Short Account of the History of Mathematics'' (4th edition, 1908) by W. W. Rouse Ball. * * ''Columbia Encyclopedia'', 6th ed., 2005,
Lagrange, Joseph Louis.
* W. W. Rouse Ball, 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 Ivor Grattan-Guinness, Grattan-Guinness, I., ed., ''Landmark Writings in Western Mathematics''. Elsevier: 258–76. * Lagrange, Joseph-Louis. (1811). ''Mécanique Analytique''. Courcier (reissued by Cambridge University Press, 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. *

* *
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 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 People from Turin