Allen Shields*

Lagrange and the M~canique Analytique

This is the 200th anniversary of the publication of Md- chanique Analitique (Paris, 1788), by Joseph-Louis La- grange. (Note the spelling of the title of the original edition.) Much of the following is taken from G. Sarton [1944], where extensive references to the orig- inal letters and books are given. Another source is the article by M. Delambre, which was written after La- grange's death. Delambre was a younger colleague of Lagrange. We recommend both articles highly.

Lagrange was born in Turin, Piedmont, on 25 Jan- uary 1736. On the certificate of baptism he is named Giuseppe Lodovico Lagrangia. His great-grandfather, a cavalry captain, was French, but had gone into the service of Sardinia, and the family had lived in Pied- mont since before 1675. He spent the first 30 years of his life in Turin, the next 21 in Berlin, and the last 26 in Paris (he died in 1813). He is somet imes called a French mathematician. This seems incorrect, although all but one or two of his publications were in French. During the Revolution he was in danger of banish- ment as an enemy alien, but was specially exempted from the law.

Delambre states that a chance reading of a memoir by E. Halley (Sarton gives the reference as Halley [1693]) awakened Lagrange to mathematics. Sarton states that the best introduction to mathematical anal- ysis then available was Maria Gaetana Agnesi [1748], which Lagrange must have seen. Sarton (footnote 5) writes:

There is no reason to believe that Lagrange knew that re- markable woman personally, but he refers to her great book in a letter to Fagnano dated 30 October 1754 (Fag- nano, 1912: 203). She was only 30 when her famous Insti-

* Column editor's address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003 USA

tuzione appeared, but a few years later (ca. 1752) she with- drew from the world and devoted herself entirely to reli- gion and charity. In 1762 the academy of Turin tried to bring her back to mathematics but did not succeed.

After s tudying L. Euler 's book on isoperimetric problems (Euler [1744]), on 12 August 1755 Lagrange wrote to Euler and presented new ideas on how to formulate and solve such problems in greater gener- ality by purely analytical methods. (Euler's book intro- duced what is today known as Euler's equation, or the Euler-Lagrange equation, a necessary condition in the form of a differential equation that an extremal func- tion must satisfy. Euler's derivation was not rigorous, however . He replaced the curve that is assumed to give the minimum by a polygonal line, and then con- sidered variations of the ordinates of the vertices, one at a time; see Sagan [1969], p. 52.) Lagrange's letter is in the Oeuvres, Vol. 14, 138-144. Euler contributed the name for this new field: calculus of variations.

In 1755 Lagrange was appointed as a teacher at the Royal School of Mathematics and Artillery in Turin. In 1759 he helped to found the Academy of Turin. La- grange's first published paper (as opposed to results contained in letters) appeared in 1759 in the first volume of the proceedings of the new academy (see Oeuvres, vol. 1, 3-20). It is entitled Recherches sur la m~thode de maximis et minimis , and in it he presents conditions for maxima and minima for functions of several variables: the first partial derivatives vanish, and the matrix of second partials is negative (or posi- tive) definite. Linear algebra was not available, so he wrote out the equivalent inequalities on the partial de- rivatives for functions of two and three variables, and states that one who has understood this will be able to handle four or more variables by the same method. Apparently he was the first to give these conditions (on p. 9 he writes: "'Comme je crois cette thdorie enti~re- ment nouvelle . . . ").

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He goes on to say that his methods apply equally to more general maximum and minimum problems in- volving "'formules int~grales ind~finies,'" by which he seems to mean integrals, perhaps with fixed limits of integration, but where the integrand involves an un- known function, which is to be chosen to maximize (or minimize) the value of the integral. Of course this is the basic problem studied in the calculus of variations, but Lagrange was the first to give this formulation. He states (w that using these methods he can deduce, by the principle of least action, all the mechanics of solid bodies and of fluids. Lagrange was 23 when he wrote this article.

His correspondence with Euler continued. In 1756 Euler (who was then at the Berlin Academy) had La- grange appointed as a foreign member of the acad- emy. On 2 October 1759 Euler wrote a famous letter to Lagrange:

Your solution of the isoperimetric problems leaves nothing to be desired and I rejoice that this subject, with which I have been so completely occupied since my first efforts, has been carded by you to such a high degree of perfection. The importance of the subject has stimulated me to develop, aided by your lights, an analytical solution that I will keep secret as long as your own meditations are not published, lest I take away from you a part of the glory you deserve.

Euler was almost 30 years older than Lagrange. Lagrange's work appeared in 1760-61 (Oeuvres, vol.

1, 335-362). On the second page we find: "Problem I. Given an indefinite integral formula represented by

fZ, where Z is a function of the variables x, y, z and the i r d i f f e r ences dx, dy, dz, d2x, d2y, d2z . . . . . find the relation that these variables must have among themselves in order that the formula fZ become a maximum or minimum." He introduces the symbol and shows how to calculate with it just as one calcu- lates with d in elementary calculus. The method is then applied to several examples. In Euler's subse- quent memoir (promised in the letter above) he begins by writing:

After having exhausted myself in a long and unsuccessful search . . . what was my astonishment when I learned that in the Turin Memoirs the problem had been solved easily and successfully. I admired this beautiful discovery even more because it is quite different from the methods I had used, and considerably surpasses them in simplicity.

Meanwhile Frederick the Great was apparently be- coming a little tired of Euler. In writing to Voltaire he referred to his G~om~tre borgne (one-eyed), whose ears were not made to feel the delicacies of poetry. And Euler longed to return to St. Petersburg, where he had been at the academy before coming to Berlin. At this, d'Alembert in Paris became fearful that Fred- erick would call him to replace Euler. So both Euler and d'Alembert praised Lagrange to Frederick and eventual ly Lagrange was appointed to the Berlin Academy in 1766, at an annual salary equivalent to 6,000 French francs, to replace Euler. (See Delambre, pp. xxii-xxiii.) His inaugural speech to the Academy was only about 200 words! (Oeuvres, vol. 14, 316.) La- grange seems to have been kind, and was careful to offend no one; he just wanted to be left alone to do his mathematics.

Sarton (p. 466) writes that the leading academies were highly competitive institutions, each trying to have the top scholars of the age. The Prussian (Berlin) Academy was founded by Leibniz in 1700, but it was Frederick the Great, who ruled from 1740 to 1786, who built it into a leading research institution. Sarton quotes (p. 467) from an account of the Prussian court by Dieudonn6 Thi6bault (Mes souvenirs de vingt ans de sdjour d Berlin, vol. 5, 39-42).

J o s e p h - L o u i s L a g r a n g e

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At Berlin he [Lagrange] married a lady who was his rela- tion, and who possessed a n . . . extraordinary sweetness of temper. His domestic life was tranquil and retired, as they both preferred the . . . society of a small but select circle of friends . . . . He was on no occasion accessible to any kind of intrigue or party spirit . . . . Nor should this trait be ascribed to timidity. I perfectly remember that M. de Schxxx the minis ter . . , having prevailed on the king to adopt the project of a fund for widows, and M. de la Grange, having read a memoir to the academy in which he demonstrated that this fund would necessarily and speedily end in a bankruptcy, the minister sent a message to the former importing that instead of publishing his memoir he should have communicated it to the govern- ment; to which the academician replied that first, he had not published his memoir . . , but had merely warned his

co l l eagues . . , and second, that not having engaged him- self in the academy for the purpose of receiving the com- mands of m i n i s t e r s . . , he had nothing to do with waiting in their antechambers to offer them advice they had not required him to give; that it was their own business to look out for persons capable of furnishing principles for the calculations they stood in need of.

The s ame au tho r wri tes as follows abou t Lagrange ' s daily rout ine and w o r k habits in Berlin.

M. de la Grange observed a regular and uniform plan for the occupation of every day. His mornings were conse- crated to reading and writing letters; immediately after dinner he devoted a few hours to different visits, or to his accustomed walk which he took alone that he might pursue the pace he judged salutary for his health. At six in the evening he returned to his cabinet where he shut him- self in, that he might be sure of remaining undisturbed till midnight, when he took several dishes of tea before he went to bed.

To this he adds ,

A philosopher remarkable for the equanimity of his con- duct, ever wise, ever tolerant, blending with his genius for mathematics a knowledge no less extensive than various in the different branches of literature, and with these the most genuine simplicity and gentleness of manners.

W h e n the d e a t h of Freder ick s e e m e d i m m i n e n t , Char les de Calonne , the Minister of Finances unde r Louis XVI, sent an agen t to Berlin to invest igate. The m a n he sent was Coun t Mirabeau (1749-1791). Fred- erick d ied in A u g u s t 1786 and the new king was not a m a n of the E n l i g h t e n m e n t . Mi rabeau s a w tha t La- g range was not h a p p y in this a t m o s p h e r e and wrote to France on 28 N o v e m b e r 1786 (Sarton, p. (468):

There is h e r e . . , an acquisition to be made, worthy of the King of France . . . . The illustrious La Grange, the greatest mathematician that has appeared since Newton, and who, by his understanding and genius, is the man in all Europe who has most astonished me . . . . He is much disgusted, silently but irremediably disgusted, because his disgust originates in contempt. The passions, brutalities, and lu- natic boasting of Hertzberg [the Prussian minister], the addition of so many academicians with whom Lagrange cannot, without blushing, a s soc i a t e . . , all induce him to retire from a country where the crime of being a foreigner is not to be forgiven . . . . It cannot be doubted but that he would willingly exchange the sun and coin of Prussia for the sun and coin of France . . . . The ambassador from Naples to Copenhagen has made him the handsomest of offers . . . he has received pressing invitations from the Grand Duke and the King of Sardinia. But all these pro- posals would easily be forgotten if put in competition with ours. La Grange here receives a pension of six thousand livres. And cannot the King of France dedicate that sum to the first mathematician of the age? . . . De Boynes has given eighteen thousand livres a year . . . to one Bosco- v i c h . . , despised by all the learned of Europe as a literary quack of poor abilities . . . and why will not M. de Ca- lonne grant a pension of two thousand crowns to the first man in Europe of his dass, and probably the last great genius the mathematical sciences shall possess; the pas- sion for which diminishes because of the excessive diffi-

L e o n h a r d E u l e r

culties that are to be surmounted, and the infinitely few means of acquiring fame by d i scovery? . . . I entreat I may have an immediate answer, for I own I have induced M. de la Grange to suspend his declarations on the proposi- tions that have been made him, till he has heard what ours may be.

(Not e v e r y o n e shares this op in ion of Bo~covi~: the nu- clear inst i tute in Zagreb is n a m e d for him.)

Thus it was that Lagrange, to escape the t roubles in Prussia and to find a peaceful place to work , accepted the French offer and ar r ived in Paris in 1787. He was r e c e i v e d b y Mar ie A n t o i n e t t e a n d l o d g e d in t he Louvre . W h a t was p e r h a p s his grea tes t work , the Md- canique Analytique, was p u b l i s h e d in Paris in 1788. Most of the wri t ing had b e e n done in Berlin. He states in the p re face tha t one will f ind no f igures in this work . Ins tead he deve lops a s imple app roach to dy- namical p rob lems , a lmos t an algori thm. He bases the t r ea tmen t on kinetic and potent ia l energy, ra ther t han on forces, as in Newton . Of course the t rea tments are u l t imate ly the same, bu t for m a n y kinds of p rob lems , for example , mot ion on a cu rved surface, Lagrange ' s a p p r o a c h is easier to apply.

We p a s s ove r L a g r a n g e ' s exper iences du r ing the Revo lu t i on except for one i tem. Lavois ier w a s con-

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The title page from M~canique Analytique.

demned to death, and executed on 8 May 1794. The next day Lagrange said to Delambre (p. xl), "It has taken them only a moment to cause that head to fall, and a hundred years may not suffice to produce a sim- ilar one."

He wrote at least one non-mathematical article during the Revolution. It is entitled Essai d'arithmetique politique sur les premiers besoins de l'intdrieur de la Rdpub- lique (Oeuvres, vol. 7, 571-578). It presents statistical data to show the need for more meat in the national diet, to raise the proportion of meat to wheat up to that enjoyed by soldiers.

Lagrange and Laplace and Monge were among the first professors of mathematics at the new Ecole Poly- technique. Two days before his death, Lagrange spoke with Lac6p~de, Monge, and Chaptal for more than two hours, and afterwards Chaptal wrote an extensive summary of this conversation; the reader will find ex- tensive quotes in the article by Delambre, starting on p. xliv.

Bibl iography

M. G. Agnesi [1748], Instituzione analitiche ad uso della gio- ventf~ italiana, 2 vols., Milano.

M. Delambre, Notice sur la vie et les ouvrages de M. le Comte J.-L. Lagrange, Oeuvres de Lagrange, vol. 1, ix-li.

L. Euler [1744], Methodus inveniendi lineas curvas maximi minimi proprietate gaudentes, sive solutio problematis isoperimetras latissimo sensu accepti, Lausanne,

Geneva. German translation in the book Variationsrech- nung, ed. P. G. St/ickel, Wissenschaftliche Buchgesell- schaft, Darmstadt, 1976.

E. Halley [1693], An instance of the excellence of the modern algebra in the resolution of the problem of finding the foci of optick glasses universally, Philos. Trans. 17 (205), 960-969.

J-L. Lagrange, Oeuvres, 14 vol., Paris, Gauthier-Villars, 1867-1897, ed. A. Serret, G. Darboux.

H. Sagan [1969], Calculus of variations, McGraw-Hill, N.Y. G. Sarton [1944], Lagrange's personality, Proc. Amer. Philos.

Soc. 88, 457-496. D.J. Struik, Lagrange, Joseph-Louis, Encyclopaedia Britan-

nica, 15th edition (1976), vol. 7, 101-102.

Notes on past columns. Volume 10, Number 1: Car- ath~odory and conformal mapping. Our paraphrase of Er- hard Schmidt's obituary contained the following:

In 1918 Carath6odory went to Berlin. However he left after two years at the request of the Greek government. He was given full power to organize the newly planned Greek university in Smyrna. After some initial success these efforts came to an abrupt end when Turkey marched in and occupied the area.

Selman Akbulut of Michigan State University pointed out that the phrase 'Turkey marched in and occupied the area' is not a fair summary of what happened. Also, Smyrna is the Greek name; the Turkish name is Izmir. The city is on the Medi te r ranean coast of Turkey. We apologize. The following description is taken from the current (15th) edition of the Encyclo- paedia Britannica (1976), vol. 20, p. 385-6, and vol. 28, p. 933-4.

Greece was neutral at the start of the World War, and in January 1915 the Allies tried to tempt Greece with promises of territory in Anatolia. Greece did fi- nally enter the war in June 1917; after the war the Al- lied Supreme Council in Paris authorized the landing of Greek troops at Smyrna (May 1919). In 1920 the Al- lied peace terms were presented to the Ottoman Em- pire. The government of Sultan Mehmed VI decided that they could not resist and so they signed the Treaty of S~vres (August 1920), which called for strip- ping Turkey of much of its territory. However Greece was the only country to ratify the treaty. In Turkey, resistance was organized by a Turkish general, Mus- tafa Kemal. The Ottoman government, under pressure from the Allies, sent troops against Kemal but they were defeated. Meanwhile the Greek army started to march inland and nearly reached Ankara, before being defeated by Kemal in August 1921; the Greek army evacuated Izmir on 9 September 1922. The Treaty of Lausanne (July 1923) finally fixed the frontiers.

According to Erhard Schmidt's obi tuary--see also vol. 2, p. 842 of the Britannica--Carath6odory, at some personal risk, removed the university library from Izmir when the Greeks left. The books went to the University of Athens, where he taught until going to Munich in 1924.

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