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Joseph-Louis Lagrange

Joseph-Louis (Giuseppe Lodovico),
comte de Lagrange
Born 25 January 1736(1736-01-25)
Turin, Piedmont
Died 10 April 1813 (aged 77)
Paris, France
Residence Piedmont

France

Prussia
Nationality Italian
French
Fields Mathematics
Mathematical physics
Institutions École Polytechnique
Doctoral advisor Leonhard Euler
Doctoral students Joseph Fourier
Giovanni Plana
Simeon Poisson
Known for Analytical mechanics
Celestial mechanics
Mathematical analysis
Number theory
Notes
Note he did not have a doctoral advisor but academic genealogy authorities link his intellectual heritage to Leonhard Euler, who played the equivalent role.

Joseph-Louis Lagrange (25 January 1736 – 10 April 1813), born Giuseppe Lodovico (Luigi) Lagrangia, was an Italian-born mathematician and astronomer, who lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. On the recommendation of Euler and D'Alembert, in 1766 Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing a large body of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique Analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1888-89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.

Born Giuseppe Lodovico Lagrangia in Turin of Italian parents, Lagrange had French ancestors on his father's side. In 1787, at age 51, he moved from Berlin to France and became a member of the French Academy, and he remained in France until the end of his life. Therefore, Lagrange is alternatively considered a French and an Italian scientist. Lagrange survived the French Revolution and became the first professor of analysis at the École Polytechnique upon its opening in 1794. Napoleon named Lagrange to the Legion of Honour and made him a Count of the Empire in 1808. He is buried in the Panthéon.

Contents

Scientific contribution

Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of 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, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem 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 are now known as Lagrangian points. But above all he impressed on mechanics, having transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and exhibited the so-called mechanical "principles" as simple results of the variational calculus.

Biography

Early years

Lagrange was born, of French and Italian descent (a paternal great grandfather was a French army officer who then moved to Turin)[1], as Giuseppe Lodovico Lagrangia in Turin. His father, who had charge of the Kingdom of Sardinia's military chest, was of good social position and wealthy, but before his son grew up he had lost most of his property in speculations, and young Lagrange had to rely on his own abilities for his position. He was educated at the college of Turin, but 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 Edmund Halley 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, and was made a lecturer in the artillery school.

Variational calculus

Lagrange is one of the founders of calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximizing and minimizing functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler's earlier analysis.[2] Lagrange also applied his ideas to problems of classical mechanics, generalizing the results of Euler and Maupertuis.

Euler was very impressed with Lagrange's results. It has sometimes 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 come to be disputed.[3] 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, 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, obtains the general differential equation 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, 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 (mx) \sin (nt)\,. The article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in this volume are on recurring series, probabilities, and the calculus of variations.

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, 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; a solution of Fermat's problem mentioned above: given an integer n which is not a perfect square, to find a number x such that x2n + 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 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. His solution is especially interesting as containing the germ of the idea of generalized equations of motion, equations which he first formally proved in 1780.

Berlin Academy

Already in 1756 Euler, with support from Maupertuis, made an attempt to bring Lagrange to the Berlin Academy. Later, D'Alambert interfered on Lagrange's behalf with Frederick of Prussia and wrote to Lagrange asking him to leave Turin for a considerably more prestigious position in Berlin. Lagrange turned down both offers, responding in 1765 that

It seems to me that Berlin would not be at all suitable for me while M.Euler is there.

In 1766 Euler left Berlin for Saint Petersburg, and Frederick 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 and he spent the next twenty years in Prussia, where he produced not only the long series of papers published in the Berlin and Turin transactions, but his monumental work, the Mécanique analytique. His residence at Berlin commenced with an unfortunate mistake. Finding most of his colleagues married, and assured by their wives that it was the only way to be happy, he married; his wife soon died, but the union was not a happy one.

Lagrange was a favourite of the king, who used frequently to discourse to him on the advantages of perfect regularity of life. The lesson went home, and thenceforth Lagrange studied his mind and body as though they were machines, and found by experiment the exact amount of work which he was able to do without breaking down. 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 always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.

France

In 1786, Frederick died, and Lagrange, who had found the climate of Berlin trying, gladly accepted the offer of Louis XVI to move to Paris. He received similar invitations from Spain and Naples. 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, which later became part of the National Institute. At the beginning of his residence in Paris he was seized with an attack of the 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 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 a young girl who insisted on marrying him, and proved a devoted wife to whom he became warmly attached. Although the decree of October 1793 that ordered all foreigners to leave France specifically exempted him by name, he was preparing to escape when he was offered the presidency of the commission for the reform of weights and measures. The choice of the units finally selected was largely due to him, and it was mainly owing to his influence that the decimal subdivision was accepted by the commission of 1799. In 1795, Lagrange was one of the founding members of the Bureau des Longitudes.

Though Lagrange had determined to escape from France while there was yet time, he was never in any danger; and the different revolutionary governments (and at a later time, Napoleon) loaded him with honours and distinctions. 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 honour to all mankind by his genius, and whom it was the special glory of Piedmont 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.

École normale

In 1795, Lagrange was appointed to a mathematical chair at the newly-established École normale, which enjoyed only a brief existence of four months. His lectures here were quite elementary, and contain nothing of any special importance, but they 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," and the discourses were ordered to be taken down in shorthand in order to enable the deputies to see how the professors acquitted themselves.

École Polytechnique

Lagrange was appointed professor of the École Polytechnique in 1794; and his lectures there are described by mathematicians who had the good fortune to be able to attend them, as almost perfect both in form and matter.[citation needed] 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.

On the other hand, 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 pronounced 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 professors make amends for it.

Late years

Lagrange's tomb in the crypt of the Panthéon.

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 in 1813. He was buried that same year in the Panthéon in Paris. The French inscription on his tomb there reads:

JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of Réunion. 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 splendid 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 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 one ought to particularly mention his paper on the 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

Classical mechanics
History of ...
Scientists
Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut
Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson

Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics.

Algebra

The greater number of his papers during this time were, however, contributed to the Prussian Academy of Sciences. 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 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 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 of any degree is also treated in these papers.
  • In 1773, Lagrange considered a functional determinant of order 3, a special case of a 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 to prove that Pell's equation x2ny2 = 1 has a nontrivial solution in the integers for any non-square natural number n.[4]
  • 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 that if n is a prime, then (n − 1)! + 1 is always a multiple of n, 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 to handle the general problem of when an integer is representable by the form ax2 + by2 + cxy.

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 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 were collected in the second edition of Euler's integral calculus which was published in 1794.

He made contributions to the theory of continued fractions.

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 resulting in the discovery of Lagrangian points, 1772
  • On the attraction of ellipsoids, 1773: this is founded on 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 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.

Mécanique analytique

Over and above these various papers he composed his great treatise, the Mécanique analytique. In this 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 generalized 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 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{d}{dt} \frac{\partial T}{\partial \dot{\theta}} - \frac{\partial T}{\partial \theta} + \frac{\partial V}{\partial \theta} = 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—though this is not the first time that method was published—as a means to solve this equation.[5] Amongst other minor theorems here given it may 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. All the analysis is so elegant that Sir William Rowan Hamilton said the work could only be described as a scientific poem. It may be interesting to note that 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 Legendre at last persuaded a Paris firm to undertake it, and it was issued under his supervision in 1788.

Work in France

Differential calculus and calculus of variations

Lagrange's lectures on the differential calculus 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. 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 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, Jacobi, and Weierstrass.

Infinitesimals

At a later period Lagrange reverted to 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, 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.

Continued fractions

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

ap−1 − 1 ≡ 0 (mod 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.

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 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 Comte of the Empire. He was awarded the Grand Croix of the Ordre Impérial de la Réunion in 1813, a week before his death in Paris.

Lagrange was awarded the 1764 prize of the French Academy of Sciences for his memoir on the libration 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 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 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 also bears his name.

Apocrypha

  • He was of medium height and slightly formed, with pale blue eyes and a colorless complexion. He was nervous and timid, he detested controversy, and, to avoid it, willingly allowed others to take credit for what he had done himself.[6]
  • Due to thorough preparation, he was usually able to write out his papers complete without a single crossing-out or correction.[6]

See also

Notes

The initial version of this article was taken from the public domain resource A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball.

  1. ^ Lagrange biography
  2. ^ Although some authors speak of general method of solving "isoperimetric problems", the eitghteenth century meaning of this expression amounts to "problems in variational calculus", reserving the adjective "relative" for problems with isoperimetric-type constraints. The celebrated method of Lagrange multipliers, which applies to optimization of functions of several variables subject to constraints, did not appear until much later, see Craig Fraser, Isoperimetric Problems in the Variational Calculus of Euler and Lagrange, Historia Mathematica, 19(1992), pp.4–23.
  3. ^ Galletto, D., The genesis of Mècanique analytique, La Mècanique analytique de Lagrange et son héritage, II (Turin, 1989). Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 126 (1992), suppl. 2, 277--370, MR1264671.
  4. ^ Oeveres, t.1, 671–732
  5. ^ Marco Panza, "The Origins of Analytic Mechanics in the 18th Century", in Hans Niels Jahnke (editor), A History of Analysis, 2003, p. 149
  6. ^ a b W. W. Rouse Ball, 1908, "Joseph Louis Lagrange (1736 - 1813)," A Short Account of the History of Mathematics, 4th ed. pp. 401 - 412

References

External links


Quotes

Up to date as of January 14, 2010

From Wikiquote

Joseph-Louis Lagrange, comte de l'Empire (January 25, 1736April 10, 1813) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics.

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  • It took them only an instant to cut off that head, but France may not produce another like it in a century.

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1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

JOSEPH LOUIS LAGRANGE (1736-1813), French mathematician, was born at Turin, on the 25th of January 1736. He was of French extraction, his great grandfather, a cavalry captain, having passed from the service of France to that of Sardinia, and settled in Turin under Emmanuel II. His father, Joseph Louis Lagrange, married Maria Theresa Gros, only daughter of a rich physician at Cambiano, and had by her eleven children, of whom only the eldest (the subject of this notice) and the youngest survived infancy. His emoluments as treasurer at war, together with his wife's fortune, provided him with ample means, which he lost by rash speculations, a circumstance regarded by his son as the prelude to his own good fortune; for had he been rich, he used to say, he might never have known mathematics.

The genius of Lagrange did not at once take its true bent.

His earliest tastes were literary rather than scientific, and he learned the rudiments of geometry during his first year at the college of Turin, without difficulty, but without distinction. The perusal of a tract by Halley (Phil. Trans. xviii. 960) roused his enthusiasm for the analytical method, of which he was destined to develop the utmost capabilities. He now entered, unaided save by his own unerring tact and vivid apprehension, upon a course of study which, in two years, placed him on a level with the greatest of his contemporaries. At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the Calculus of Variations. It was eagerly welcomed by the Berlin mathematician, who had the generosity to withhold from publication his own further researches on the subject, until his youthful correspondent should have had time to complete and opportunity to claim the invention. This prosperous opening gave the key-note to Lagrange's career. Appointed, in 1754, professor of geometry in the royal school of artillery, he formed with some of his pupils - for the most part his seniors - friendships based on community of scientific ardour. With the aid of the marquis de Saluces and the anatomist G. F. Cigna, he founded in 1758 a society which became the Turin Academy of Sciences. The first volume of its memoirs,' published in the following year, contained a paper by Lagrange entitled Recherches sur la nature et la propagation du son, in which the power of his analysis and his address in its application were equally conspicuous. He made his first appearance in public as the critic of Newton, and the arbiter between d'Alembert and Euler. By considering only the particles of air found in a right line, he reduced the problem of the propagation of sound to the solution of the same partial differential equations that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject. He further treated in a masterly manner of echoes and the mixture of sounds, and explained the phenomenon of grave harmonics as due to the occurrence of beats so rapid as to generate a musical note. This was followed, in the second volume of the Miscellanea Taurinensia (1762) by his "Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies," together with the application of this important development of analysis to the solution of several dynamical problems, as well as to the demonstration of the mechanical principle of "least action." The essential point in his advance on Euler's mode of investigating curves of maximum or minimum consisted in his purely analytical conception of the subject. He not only freed it from all trammels of geometrical construction, but by the introduction of the symbol b gave it the efficacy of a new calculus. He is thus justly regarded as the inventor of the "method of variations" - a name supplied by Euler in 1766.

By these performances Lagrange found himself, at the age of twenty-six, on the summit of European fame. Such a height bad not been reached without cost. Intense application during early youth had weakened a constitution never robust, and led to accesses of feverish exaltation culminating, in the spring of 1761, in an attack of bilious hypochondria, which permanently lowered the tone of his nervous system. Rest and exercise, however, temporarily restored his health, and he gave proof of the undiminished vigour of his powers by carrying off, in 1764, the prize offered by the Paris Academy of Sciences for the best essay on the libration of the moon. His treatise was remarkable, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his radical formula of mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities. His success encouraged the Academy to propose, in 1766, as a theme for competition, the hitherto unattempted theory of the Jovian system. The prize was again awarded to Lagrange; and he earned the same distinction with essays on the problem of three bodies in 1772, on the secular equation of the moon in 1774, and in 1778 on the theory of cometary perturbations.

He had in the meantime gratified a long felt desire by a visit to Paris, where he enjoyed the stimulating delight of conversing with such mathematicians as A. C. Clairault, d'Alembert, Condorcet and the Abbe Marie. Illness prevented him from visiting London. The post of director of the mathematical department of the Berlin Academy (of which he had been a member since 1759) becoming vacant by the removal of Euler to St Petersburg, the latter and d'Alembert united to recommend Lagrange as his successor. Euler's eulogium was enhanced by his desire to quit Berlin, d'Alembert's by his dread of a royal command to repair thither; and the result was that an invitation, conveying the wish of the "greatest king in Europe" to have the "greatest mathematician" at his court, was sent to Turin. On the 6th of November 1766, Lagrange was installed in his new position, with a salary of 6000 francs, ample leisure for scientific research, and royal favour sufficient to secure him respect without exciting envy. The national jealousy of foreigners, was at first a source of annoyance to him; but such prejudices were gradually disarmed by the inoffensiveness of his demeanour. We are told that the universal example of his colleagues, rather than any desire for female society, impelled him to matrimony; his choice being a lady of the Conti family, who, by his request, joined him at Berlin. Soon after marriage his wife was attacked by a lingering illness, to which she succumbed, Lagrange devoting all his time, and a considerable store of medical knowledge, to her care.

The long series of memoirs - some of them complete treatises of great moment in the history of science - communicated by Lagrange to the Berlin Academy between the years 1767 and 1787 were not the only fruits of his exile. His Mecanique analytique, in which his genius most fully displayed itself, was produced during the same period. This great work was the perfect realization of a design conceived by the author almost in boyhood, and clearly sketched in his first published essay.' Its scope may be briefly described as the reduction of the theory of mechanics to certain general formulae, from the simple development of which should be derived the equations necessary for the solution of each separate problem. 2 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid and harmonious as to constitute, in Sir William Hamilton's words, "a kind of scientific poem." This unification of method was one of matter also. By his mode of regarding a liquid as a material system characterized by the unshackled mobility of its minutest parts, the separation between the mechanics of matter in different forms of aggregation finally disappeared, and the fundamental equation of forces was for the first time extended to hydrostatics and hydrodynamics.' Thus a universal science of matter and motion was derived, by an unbroken sequence of deduction, from one radical principle; and analytical mechanics assumed the clear and complete form of logical perfection which it now wears.

A publisher having with some difficulty been found, the book appeared at Paris in 1788 under the supervision of A. M. Legendre. But before that time Lagrange himself was on the spot. After the death of Frederick the Great, his presence was competed for by the courts of France, Spain and Naples, and a residence in Berlin having ceased to possess any attraction for him, he removed to Paris in 1787. Marie Antoinette warmly patronized him. He was lodged in `the Louvre, received the grant of an income equal to that he had hitherto enjoyed, and, with the title of "veteran pensioner" in lieu of that of "foreign associate" (conferred in 1772), the right of voting at the deliberations of the Academy. In the midst of these distinctions, a profound melancholy. seized upon him. His mathematical enthusiasm was for the time completely quenched, and during two years the printed volume of his Mecanique, which he had seen only in manuscript, lay unopened beside him. He relieved his dejection ' Ouvres, i. 15.2 Mec. An., Advertisement to 1st ed. ' E. Diihring, Kritische Gesch. der Mechanik, 220, 367; Lagrange, Mec. An. i. 166-172, 3rd ed.

with miscellaneous studies, especially with that of chemistry, which, in the new form given to it by Lavoisier, he found "aisee comme l'algebre." The Revolution roused him once more to activity and cheerfulness. Curiosity impelled him to remain and watch the progress of such a novel phenomenon; but curiosity was changed into dismay as the terrific character of the phenomenon unfolded itself. He now bitterly regretted his temerity in braving the danger. "Tu l'as voulu" he would repeat self-reproachfully. Even from revolutionary tribunals, however, the name of Lagrange uniformly commanded respect. His pension was continued by the National Assembly, and he was partially indemnified for the depreciation of the currency by remunerative appointments. Nominated president of the Academical commission for the reform of weights and measures, his services were retained when its "purification" by the Jacobins removed his most distinguished colleagues. He again sat on the commission of 1799 for the construction of the metric system, and by his zealous advocacy of the decimal principle largely contributed to its adoption.

Meanwhile, on the 31st of May 1792 he married Mademoiselle Lemonnier, daughter of the astronomer of that name, a young and beautiful girl, whose devotion ignored disparity of years, and formed the one tie with life which Lagrange found it hard to break. He had no children by either marriage. Although specially exempted from the operation of the decree of October 1 793, imposing banishment on foreign residents, he took alarm at the fate of J. S. Bailly and A. L. Lavoisier, and prepared to resume his former situation in Berlin. His design was frustrated by the establishment of and his official connexion with the Ecole Normale, and the Ecole Polytechnique. The former institution had an ephemeral existence; but amongst the benefits derived from the foundation of the Ecole Polytechnique one of the greatest, it has been observed, 4 was the restoration of Lagrange to mathematics. The remembrance of his teachings was long treasured by such of his auditors - amongst whom were J. B. J. Delambre and S. F. Lacroix - as were capable of appreciating them. In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions. He seemed, not a professor amongst students, but a learner amongst learners; pauses for thought alternated with luminous exposition; invention accompanied demonstration; and thus originated his Theorie des fonctions analytiques (Paris, 1797). The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions. By means of this "calculus of derived functions" Lagrange hoped to give to the solution of all analytical problems the utmost "rigour of the demonstrations of the ancients"; 6 but it cannot be said that the attempt was successful. The validity of his fundamental position was impaired by the absence of a well-constituted theory of series; the notation employed was inconvenient, and was abandoned by its inventor in the second edition of his Mecanique; while his scruples as to the admission into analytical investigations of the idea of limits or vanishing ratios have long since been laid aside as idle. Nowhere, however, were the keenness and clearness of his intellect more conspicuous than in this brilliant effort, which, if it failed in its immediate object, was highly effective in secondary results. His purely abstract mode of regarding functions, apart from any mechanical or geometrical considerations, led the way to a new and sharply characterized development of the higher analysis in the hands of A. Cauchy, C. G. Jacobi, and others.' The Theorie des fonctions is divided into three parts, of which the first explains the general doctrine of functions, the second deals with its 4 Notice by J. Delambre, Ouvres de Lagrange, i. p. xlii. Ouvres, iii. 441. e Theorie des fonctions, p. 6.7 H. Suter, Geschichte der math. Wiss. ii. 222-223.

application to geometry, and the third with its bearings on mechanics.

On the establishment of the Institute, Lagrange was placed at the head of the section of geometry; he was one of the first members of the Bureau des Longitudes; and his name appeared in 1791 on the list of foreign members of the Royal Society. On the annexation of Piedmont to France in 1796, a touching compliment was paid to him in the person of his aged father. By direction of Talleyrand, then minister for foreign affairs, the French commissary repaired in state to the old man's residence in Turin, to congratulate him on the merits of his son, whom they declared "to have done honour to mankind by his genius, and whom Piedmont was proud to have produced, and France to possess." Bonaparte, who styled him "la haute pyramide des sciences mathematiques," loaded him with personal favours and official distinctions. He became a senator, a count of the empire, a grand officer of the legion of honour, and just before his death received the grand cross of the order of reunion.

The preparation of a new edition of his Mecanique exhausted his already failing powers. Frequent fainting fits gave presage of a speedy end, and on the 8th of April 1813 he had a final interview with his friends B. Lacepede, G. Monge and J. A. Chaptal. He spoke with the utmost calm of his approaching death; "c'est une derniere fonction," he said, "qui n'est ni penible ni desagreable." He nevertheless looked forward to a future meeting, when he promised to complete the autobiographical details which weakness obliged him to interrupt. They remained untold, for he died two days later on the 10th of April, and was buried in the Pantheon, the funeral oration being pronounced by Laplace and Lacepede.

Amongst the brilliant group of mathematicians whose magnanimous rivalry contributed to accomplish the task of generalization and deduction reserved for the 18th century, Lagrange occupies an eminent place. It is indeed by no means easy to distinguish and apportion the respective merits of the competitors. This is especially the case between Lagrange and Euler on the one side, and between Lagrange and Laplace on the other. The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements. Finally, of the grand series of researches by which the stability of the solar system was ascertained, the glory must be almost equally divided between Lagrange and Laplace. In analytical invention, and mastery over the calculus, the Turin mathematician was admittedly unrivalled. Laplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way. But Laplace unquestionably surpassed his rival in practical sagacity and the intuition of physical truth. Lagrange saw in the problems of nature so many occasions for analytical triumphs; Laplace regarded analytical triumphs as the means of solving the problems of nature. One mind seemed the complement of the other; and both, united in honourable rivalry, formed an instrument of unexampled perfection for the investigation of the celestial machinery. What may be called Lagrange's first period of research into planetary perturbations extended from 1774 to 1784 (see Astronomy: History). The notable group of treatises communicated, 1781-1784, to the Berlin Academy was designed, but did not prove to be his final contribution to the theory of the planets. After an interval of twenty-four years the subject, re-opened by S. D. Poisson in a paper read on the 10th of June 1808, was once more attacked by Lagrange with all his pristine vigour and fertility of invention. Resuming the inquiry into the invariability of mean motions, Poisson carried the approximation, with Lagrange's formulae, as far as the squares of the disturbing forces, hitherto neglected, with the same result as to the stability of the system. He had not attempted to include in his calculations the orbital variations of the disturbing bodies; but Lagrange, by the happy artifice of transferring the origin of coordinates from the centre of the sun to the centre of gravity of the sun and planets, obtained a simplification of the formulae, by which the same analysis was rendered equally applicable to each of the planets severally. It deserves to be recorded as one of the numerous coincidences of discovery that Laplace, on being made acquainted by Lagrange with his new method, produced analogous expressions, to which his independent researches had led him. The final achievement of Lagrange in this direction was the extension of the method of the variation of arbitrary constants, successfully used by him in the investigation of periodical as well as of secular inequalities, to any system whatever of mutually interacting bodies.' "Not 1 CEuvres, vi. 77 1.

without astonishment," even to himself, regard being had to the great generality of the differential equations, he reached a result so wide as to include, as a particular case, the solution of the planetary problem recently obtained by him. He proposed to apply the same principles to the calculation of the disturbances produced in the rotation of the planets by external action on their equatorial protuberances, but was anticipated by Poisson, who gave formulae for the variation of the elements of rotation strictly corresponding with those found by Lagrange for the variation of the elements of revolution. The revision of the Mecanique analytique was undertaken mainly for the purpose of embodying in it these new methods and final results, but was interrupted, when two-thirds completed, by the death of its author.

In the advancement of almost every branch of pure mathematics Lagrange took a conspicuous part. The calculus of variations is indissolubly associated with his name. In the theory of numbers he furnished solutions of many of P. Fermat's theorems, and added some of his own. In algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving algebraical equations of every degree. The method indeed fails for equations of an order above the fourth, because it then involves the solution of an equation of higher dimensions than they proposed. Yet it possesses the great and characteristic merit of generalizing the solutions of his predecessors, exhibiting them all as modifications of one principle. To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler. But it was in the application to mechanical questions of the instrument which he thus helped to form that his singular merit lay. It was his just boast to have transformed mechanics (defined by him as a "geometry of four dimensions") into a branch of analysis, and to have exhibited the so-called mechanical "principles" as simple results of the calculus. The method of "generalized coordinates," as it is now called, by which he attained this result, is the most brilliant achievement of the analytical method. Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many equations as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the differential equations of motion thence deduced by simple differentiation. Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor theorems of great elegance, - among which may be mentioned his theorem that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum. To this entire branch of knowledge, in short, he successfully imparted that character of generality and completeness towards which his labours invariably tended.

His share in the gigantic task of verifying the Newtonian theory would alone suffice to immortalize his name. His co-operation was indeed more indispensable than at first sight appears. Much as was done by him, what was done through him was still more important. Some of his brilliant rival's most conspicuous discoveries were implicitly contained in his writings, and wanted but one step for completion. But that one step, from the abstract to the concrete, was precisely that which the character of Lagrange's mind indisposed him to make. As notable instances may be mentioned Laplace's discoveries relating to the velocity of sound and the secular acceleration of the moon, both of which were led close up to by Lagrange's analytical demonstrations. In the Berlin Memoirs for 1778 and 1783 Lagrange gave the first direct and theoretically perfect method of determining cometary orbits. It has not indeed proved practically available; but his system of calculating cometary perturbations by means of "mechanical quadratures" has formed the startingpoint of all subsequent researches on the subject. His determination 2 of maximum and minimum values for the slowly varying planetary eccentricities was the earliest attempt to deal with the problem. Without a more accurate knowledge of the masses of the planets than was then possessed a satisfactory solution was impossible; but the upper limits assigned by him agreed closely with those obtained later by U. J. J. Leverrier. 3 As a mathematical writer Lagrange has perhaps never been surpassed. His treatises are not only storehouses of ingenious methods, but models of symmetrical form. The clearness, elegance and originality of his mode of presentation give lucidity to what is obscure, novelty to what is familiar, and simplicity to what is abstruse. His genius was one of generalization and abstraction; and the aspirations of the time towards unity and perfection received, by his serene labours, an embodiment denied to them in the troubled world of politics.

B1s1.10GRAPHY. - Lagrange's numerous scattered memoirs have been collected and published in seven 4to volumes, under the title Ouvres, v. 211 seq.

3 Grant, History of Physical Astronomy, p. 117.

Ouvres de Lagrange, publiees sous les soins de M. J. A. Serret (Paris, 1867-1877). The first, second and third sections of this publication comprise respectively the papers communicated by him to the Academies of Sciences of Turin, Berlin and Paris; the fourth includes his miscellaneous contributions to other scientific collections, together with his additions to Euler's Algebra, and his Lecons elementaires at the Ecole Normale in 1795. Delambre's notice of his life, extracted from the Mem. de l'Institut, 1812, is prefixed to the first volume. Besides the separate works already named are Resolution des equations numeriques (1798, 2nd ed., 1808, 3rd ed., 1826), and Lecons sur le calcul des fonctions (1805, 2nd ed., 1806), designed as a commentary and supplement to the first part of the Theorie des fonctions. The first volume of the enlarged edition of the Mecanique appeared in 1811, the second, of which the revision was completed by MM Prony and Binet, in 1815. A third edition, in 2 vols., 4to, was issued in 1853-1855, and a second of the Theorie des fonctions in 1813.

See also J. J. Virey and Potel, Précis historique (1813); Th. Thomson's Annals of Philosophy (1813-1820), vols. ii. and iv.; H. Suter, Geschichte der math. Wiss. (1873); E. Diihring, Kritische Gesch. der allgemeinen Principien der Mechanik (1877, 2nd ed.); A. Gautier, Essai historique sur le probleme des trois corps (1817); R. Grant, History of Physical Astronomy, &c.; Pietro Cossali, Eloge (Padua, 1813); L. Martini, Cenni biogrdfici (1840); Moniteur du 26 Fevrier (1814); W. Whewell, Hist. of the Inductive Sciences, ii. passim; J. Clerk Maxwell, Electricity and Magnetism, ii. 184; A. Berry, Short Hist. of Astr., p. 313; J. S. Bailly, Hist. de l'astr. moderne, iii. 156, 185, 232; J. C. Poggendorff, Biog. Lit. Handweirterbuch. (A. M. C.)


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