JOSEPH Louis, a mathematician and astronomer of the first rank, born at Turin on the 25th of January 1736, was the son of Joseph Louis Lagrange, treasurer at war, and Maria Theresa Gros, only daughter of a rich physician at Cambiano.

He was the eldest of eleven children, but nine of them died young. His family was of French extraction on both sides; and his French biographers have dwelt with pleasure on the minute particulars of their emigration, in order the more fully to authenticate their own claim to the honour of calling him their natural as well as adopted countryman. It was his great-grandfather that first settled at Turin, in the service of Emmanuel II., who married him to a Roman lady of the family of Conti. They had at one time acquired considerable affluence, but his father had ruined himself by his expenses and speculations; and Lagrange used frequently to observe, that he owed his own success in life to his father's misfortunes, since, if he had been rich, he should never have applied to the mathematics as a profession. The classics were at first his favourite study at the college of Turin; he began his scientific education with reading the works of the ancient geometers, and at first preferred their methods of investigation to the more modern analysis; but being convinced, as it is said, by a paper of Halley in the Philosophical Transactions, of the superiority of the algebraical mode of representation, he applied with redoubled ardour, at the age of seventeen, to the study of the later improvements in the methods of investigation; and in his subsequent works he abandoned, wherever it was practicable, all geometrical considerations, and seems to have valued himself on having produced a complete system of mechanics, free from the incumbrance of any diagram whatever. When he was only nineteen, he was made professor of geometry in the Royal School of Artillery, but not before he had exhibited, in his first publication, a specimen of the improvements which he was throughout his life to contribute to the mathematical sciences.

The friendships which he formed with his pupils, most of whom were his seniors, led to the establishment of a society which afterwards received the sanction of the royal authority, and to the publication of their memoirs, in which Lagrange not only took the most active part as a contributor of original papers, but also by materially assisting in the demonstrations of Foncenex, and promoting the researches of Cigna and Saluces. Foncenex was soon rewarded by being placed at the head of the maritime establishment which the king was then forming; and Lagrange received in a short time a still more flattering remuneration, in the panegyrics which were liberally bestowed on him by his great rivals, Euler and D'Alembert; the former procured him, in 1759, the compliment of being made a foreign member of the Academy of Berlin, having become well acquainted with his merits by an epistolary intercourse, which began as early as the year 1754, when Lagrange communicated to him his first ideas of the solution of isoperimetrical problems, which Euler had the delicacy to allow him time to complete, before the publication of his own further researches on the subject.

In 1764 he obtained a prize from the Academy of Sciences at Paris, for a memoir on the difficult subject of the libration of the moon, having treated it by an original method, derived from the principle of virtual velocities, which he afterwards applied so successfully to other branches of Lagrange mechanics. Soon after this time he found an agreeable relief from the monotony and retirement of his life at Turin, in accompanying his friend the Marquis Caraccioli, who was appointed ambassador at the court of London, as far as Paris, where he had the delight of becoming personally acquainted with a number of the most distinguished mathematicians of the age, who were capable of appreciating his merits, especially with Clairaut, D'Alembert, Condorcet, Fontaine, Nollet, and the Abbé Marie; but indisposition prevented his going on to England, as he had intended, and he returned to Turin after a short stay in France. A second prize, on the subject of the satellites of Jupiter, was awarded him in 1766; and the same tribute was again paid to his merit on three subsequent occasions. It was in this year that he was invited to Berlin, as a successor to Euler in the place of mathematical director of the academy, Euler having been induced to remove to Petersburg, by a better prospect of providing for his numerous family. The appointment of president of the academy, held by Mannertius, had been given but in part to Euler; the whole was offered to D'Alembert, who declined it; but both he and Euler united in recommending Lagrange as the fittest person for the situation. It was, however, with some difficulty that he obtained his sovereign's leave to quit Turin; and the favour was at last granted to him partly in pique, on account of the terms of the invitation, which expressed the desire of the "greatest king in Europe" to have the greatest mathematician at his court.

At Berlin he pursued his career of study in tranquillity, and without interruption, upon a competent income of about L300 a-year, with the advantage of such demonstrations of the royal protection as were still more important than income to his rank in society. The king seems to have preferred him to Euler, more tolerant in his opinions, though by no means joining in all the innovations of the day, and rather avoiding every discussion relating to them, as well as any great familiarity with his patron. He was made, in 1772, one of the eight foreign associates of the Parisian academy. He is said to have married more for the sake of complying with the universal custom of his friends and colleagues at Berlin, than for any desire of female society; and he invited a relation of his own from Turin, who became his first wife; but she was soon after carried off by a lingering disease. He was about this time very closely employed on his greatest and best work, the Mécanique Analytique; but it was with some difficulty that the Abbé Marie found a bookseller at Paris, who agreed to undertake its publication, and only upon condition of engaging himself to divide the loss, in case of failure in the sale. He also procured the valuable assistance of Mr Legendre as a corrector of the press.

Upon the death of Frederic in 1786, Lagrange no longer felt the same interest in remaining at Berlin, though he was not treated by the new court with any thing like disrespect. While the ministers of Naples, Sardinia, and Tuscany, were making him offers on behalf of their respective sovereigns, Mirabeau persuaded the French ambassador at Berlin to recommend M. de Vergennes to invite him to Paris; but it was in reality through M. de Breteuil's interest, and at the suggestion of the Abbé Marie, that he was ultimately induced to settle there in 1787, having received a grant of an income equal to that which he had enjoyed at Berlin, under the name of a veteran pensioner of the academy, with a vote in its deliberations. He was kindly received by Marie Antoinette, on account of his connection with Germany; and, until the Revolution, he had the use of apartments in the Louvre.

It was at this period of his life, when his success had been the most gratifying, and his fame had become per- Lagrange, secretly established, that he appeared to suffer under a degree of melancholy or apathy which was absolutely morbid. He confessed that all his enthusiasm was extinguished, and that he no longer felt the least relish for mathematical researches. He had not even the curiosity for two years to open the printed volume of his Mechanics, which he had never seen except in manuscript. It is a consolation to think that this annihilation of his energies was only partial and temporary. He amused himself in the meantime with metaphysics, "with the history of religions" and of languages, and with medical and botanical, and especially chemical studies; and the alarms and agitations of the Revolution, which soon followed, instead of overwhelming his broken spirit, seem to have roused his dormant powers, and to have revived his satisfied ambition, exciting him to new labours and new triumphs.

In 1791 his name appeared on the list of the foreign members of the Royal Society of London. Mr Maurice has asserted, that all the scientific bodies of Europe, except the Royal Society, received him with open arms; if the remark was intended as a censure of that society, it is right that its injustice should not pass unnoticed.

Notwithstanding the public embarrassments which attended the Revolution, Lagrange's pension was confirmed by the National Assembly, upon the proposition of M. Duséjour, in the most flattering manner; and when the depreciation of the currency materially reduced its amount, he received a partial indemnification, by being appointed a member of a committee for examining useful inventions, and afterwards a director of the mint, in conjunction with Berthollet and Monge; but this employment he found too laborious, and resigned it six months afterwards. He was greatly interested at this period in the establishment of the new system of weights and measures; he was so violently bent on decimation, that he scarcely forgave Borda for having made a measure of a quarter of a metre; and he thought so little of the advantage of integral subdivisions, that he sometimes declared he should have preferred the number eleven to twelve, for the very reason that it admitted no subdivision at all, and caused all lesser quantities to be expressed in units comparable to each other only. This opinion seems, however, to have been advanced rather as an exaggerated objection to the introduction of twelve, which was suggested by some more ardent innovators, than as seriously attributing a real advantage to the employment of a prime number.

When the academies were suppressed, the Jacobins purified the commission of weights and measures by striking out the names of many of its most distinguished members, while they retained that of Lagrange, probably because he was of no political party whatever, and had always been particularly cautious in expressing his sentiments of the events of the day. In October 1793, however, a decree was passed, which ordered all persons not born in France to leave the country. Guyton, who was a member of the Committee of Public Safety, advised him to claim an exemption from its operation, by a requisition of that committee, on the pretext of his being employed in preparing a report on Dr Hutton's Treatise on Gunnery; and he actually received an injunction from the committee, requiring his stay, "in order to complete the calculations which he had undertaken respecting the theory of projectiles." He was attempting to reunite the experiments of Dr Hutton with a more correct theory than had before been applied to them; but he published nothing of importance on the subject. After the murder of Bailly and Lavoisier, he had agreed to return to Berlin, and to resume his former situation there; and he was on the point of obtaining a passport, and even a public mission from Hérault de Séchelles. But the establishment, first of the Normal School, in which he was a professor, and then of the École Polytechnique, induced him to remain at Paris, and again directed his activity into its ancient channels. In the Normal Schools the masters were mixed with their pupils, in order that the facilities of conversation might produce a development of the subjects discussed in the most elementary manner that was possible; but the conversation was by no means supported in the form of incessant questions and answers; Lagrange's explanations were often interrupted by moments of silence, in which his inventive faculties were deeply engaged in reflection, and the whole of his powers were concentrated on a new train of ideas. It was amidst these discussions that the Theory of Analytical Functions originated, a work certainly not destitute of the marks of great mathematical talent, but which, when considered as a substitute for the method of fluxions and its kindred doctrines, resembles very much the suggested introduction of an undecimal in preference to a duodecimal scale of notation, with which the author had before amused himself.

Upon the re-establishment of the Institute, Lagrange was made one of the original members; and he was the first on the list of the Board of Longitude, which was then first instituted at Paris. He received about this time a compliment highly grateful both to his love of fame and to his filial affection, in the person of his father, then past ninety, and continuing to reside at Turin. By the direction of Talleyrand, who was minister for foreign affairs, the commissary of the directory of Piedmont, attended by the generals of the French army, and several other persons of distinction, went in procession to congratulate this venerable person on the merits of his son, whom he had not seen for more than thirty years, whom they declared "to have done honour to mankind by the brilliancy of his genius, and whom Piedmont was proud to have produced, and France to possess as a citizen." The old man lived to the age of ninety-five, and was sincerely regretted by his son.

Under the consular and imperial government, Lagrange was made a senator, a grand officer of the legion of honour, a count of the empire, and a knight grand cross of the order of reunion, in addition to the personal marks of friendship and intimacy which Bonaparte habitually conferred upon him at the meetings of the Institute, and on other occasions.

He applied with so much zeal to the republication of the first part of his Mechanics in 1811, and of his Analytical Functions in 1813, that his health is supposed to have suffered from the fatigue; which, in conjunction with a predisposition, not uncommon in advanced life, may very possibly have been the immediate cause of a fit that attacked him in the beginning of the latter year. In the month of March he was subject to frequent returns of fainting, accompanied by some fever. On the 8th of April he had a last conversation with Lapéde, Monge, and Chaptal, all the parties being aware that it was to be the last. He felt the approach of death, but he declared that it was in that form neither painful nor even disagreeable. He spoke with proper gratitude of the favours he had received from Bonaparte, who afterwards provided very liberally for his widow and his brother. The interview lasted more than two hours; and though his memory often failed him with respect to names and dates, yet his language was correct and energetic. He survived this effort only two days, and died on the morning of the 10th April 1813. He was buried at the Panthéon, or the church of St Geneviève, and his friends Lapéde and Laplace paid the last honours to his memory in a funeral oration.

Lagrange was habitually of delicate health, and extremely temperate in his diet and mode of life, limiting his food almost entirely to vegetables, and taking his exercise very punctually in the open air. At the age of fifty-six he mar- ried the young and handsome Madlle. Lemonnier, who appears to have felt the splendour of his celebrity and the goodness of his heart, as affording much more than a compensation for the great inequality of their ages. He was deeply sensible of her affectionate attachment, which he considered as the greatest happiness of his life, and on account of which alone he regretted its termination. He had no children, and he was perfectly contented to be without them.

In the midst of the most brilliant societies he was generally absorbed in his own reflections; and especially when there was music, in which he delighted, not so much for any exquisite pleasure that he received from it, as because, after the first three or four bars, it regularly lulled him into a train of abstract thought, and he heard no more of the performance, except as a sort of accompaniment assisting the march of his most difficult investigations, which he thus pursued with comfort and convenience. He was less fond of the theatre, from which he often returned without knowing what piece had been represented. His manner in conversation was gentle and timid; he was more in the habit of interrogating than of giving his opinion, and his favourite expression was, "I don't know." He was not, however, easily induced to change his sentiments when they were once fixed, having generally adopted them upon mature consideration. As a writer, whenever any controversy occurred, he was always calm in defending himself, and respectful in speaking of his antagonists. Notwithstanding that his person was striking and characteristic, as well as pleasing, he would never consent to have his portrait painted, thinking it unworthy of a man of intellectual excellence to wish to be remembered for the external form of his features. But a sketch of him was once obtained by stealth at a sitting of the Institute, and a mask of his face was taken after his death. His works bear witness, that for fifty-four years he occupied either the first or very nearly the first place among all the mathematicians of his age and of all ages. "Of all the inventors," says Laplace, "who have the most contributed to the advancement of human knowledge, Newton and Lagrange appear to me to have possessed in the highest degree that happy tact, which enabled them to distinguish general principles among a multitude of objects enveloping them, and which is the true characteristic of scientific genius. This tact, in Lagrange, was united with a singular elegance in the method of explaining the foundations of the most abstract truths of analysis." Lagrange was a great admirer of Euler, who perhaps excelled him in the adroitness with which he employed the most refined artifices of calculation, though his views and methods were less original and less powerful. D'Alembert was highly esteemed by Lagrange, as a man of abundant ingenuity and talent, though less accurate in his conclusions, and in his modes of reasoning, than either Euler or Newton. Newton he envied almost as much as he admired, for having found a system of the world in existence, and the principles of its modification not yet understood; but when it is remembered that the places of the heavenly bodies are now ascertained to seconds more nearly than they were to minutes during the life of Newton, it cannot be thought that Newton left too little for his successors to accomplish.

1. His first publication, at the age of eighteen, was a Letter to C. J. Fagnano, 23d June 1754. It contains series for fluxions and fluents of different orders, somewhat resembling the binomial theorem of Newton.

2. The series of his papers in the Miscellanea of Turin is continued from 1759 to 1785. The first is on Maxima and Minima, Misc. Turin., I., 1759, p. 16. It is founded on the principles laid down by Marquis de L'Hospital, and illustrates the successive approximation of an impulse through a series of elastic bodies, comprehending the combination of a number of variable quantities.

3. On an Equation of Finite Differences, and on the Theory of recurring Series, p. 33. The equation is resolved by an exponential Integral, and the sum of the series is obtained by the principles of fluxions; the same mode of calculation is also applied to the laws of chance. 4. Researches on the Nature and Propagation of Sound; end of the volume. The researches of Taylor and of Newton were true and correct as particular solutions only of the problems of chords and of undulations; though mistaken for general solutions, and are consequently fully combated by Cramer's researches, though certainly too far extended, is here approved by Lagrange. Daniel Bernoulli very successfully defended them both, not only as particular solutions, but as capable of being rendered universal by proper modifications and combinations. Euler had proposed a more general construction for the case of chords; D'Alembert insisted that this method required a limitation to figures exempt from angles and from abrupt changes of curvature, and Lagrange is inclined to admit his exceptions. But after all, the question is merely a metaphysical refinement, since no abrupt changes can ever occur in actual forms of a chord; and a chord affording a harmonic of well-selected oscillations will approach without limit to a mathematical ellipse. The author begins this essay, with considering the motions of a finite number of bodies, and then proceeds to the affections of a fluid, which he reduces to the same equations as are applicable to the motions of chords, and these he integrates in D'Alembert's manner. He lastly examines the phenomena of the grave harmonics observed by Tartini, and explains them very satisfactorily from the analogy of the beats of discordant sounds. 5. New Researches on Sound, Misc. Turin., II., 1760-1, p. 11. The same subject is here continued, and extended to the divergence of sound, which had before been examined by Euler. The author concludes that there was no necessity in the demonstration of Newton and Euler, which deduce the same quantity from different laws of the supposititious motion, since the velocity is really uniform in all cases. The oscillations of a heavy chain are computed, and some remarks are made in conclusion respecting the sounds of flutes. 6. On the Maxima and Minima of Indefinite Integrals, p. 173. This essay contains the foundation of the method of independent variations, which has excited so much attention, for the universality of its application and the utility of its results. It was received with distinguished applause by Euler, as fulfilling his own wishes for the extension of a similar method; and it was Euler who more fully explained its principles, and gave it the name of the method of variation, which has since been generally applied to it. In fact, the possibility of the method had long before been laid by Leibnitz, under the name of differentiation from curve to curve; and he had proved that the process of integration, with respect to one kind of variation, might be applied to the differentials or fluxions taken in another manner, without the necessity of first obtaining the fluent; and Euler had employed this consideration in treating of the geometrical properties of curves affording maxima or minima; but his method is less simple and less general than that of Lagrange, who first pointed out the universality of the principle, that the variation of the fluxion is equal to the fluxion of the variation, and showed its utility in many cases of such integrations, as leave the expression containing still a function of another kind; and the principal application of the method, he made the fluxion of the ordinate of a vibrating chord represent its inclination to the axis at any given time, whilst its variation indicated its velocity or its change of place in successive intervals of time, and the fluxion of a revolving solid to relate to the magnitude of its different parts, whilst its variation depended on its rotatory velocity. The steps of the method are generally simple and easily understood, at least they may and ought to be rendered so; but the merit of the invention is not the less because it admits of a very ready application, and because it might have occurred to a less distinguished mathematician; as indeed something nearly resembling it seems to have been employed by Fourier in 1784, under the name of his famous differential calculus in the investigation of a tautochronous curve. 7. It was particularly in demonstrating the law which is called the law of the least action, that Lagrange completed the theory of variations, where Euler had felt its deficiency; and the Application of the Method to several Mechanical Problems constitutes the second part of the Memoir, p. 108. The author takes occasion also to correct an error of D'Alembert, who had imagined that there was no necessity that the different strata of a given density, in a body like the earth, supposed to be in a state of fluidity, should all be level, which, however, is here shown to be unnecessary consequence of D'Alembert's own equations. 8. Adding to the Memoir on Sound, p. 173; admitting the difficulty placed by D'Alembert respecting the continuity of the figure of a chord, and acknowledging that the initial figure must not be supposed angular; which in fact, as M. Fourier has lately demonstrated, and as had been remarked many years ago in this country, an infinite series of harmonic curves may approach infinitely near to two right lines meeting in an angle. 9. Problems relating to the Integral Calculus, Misc. Turin., III., 1762-5, p. 179; a miscellaneous paper, containing remarks on the resolution of equations, containing fluxions. Lagrange, of different orders; on some cases of the motions of fluids; on the vibrations of chords; on the properties of small oscillations in general; on loaded threads; on central forces; and on the theory of Jupiter and Saturn. 10. An Arithmetical Problem, *Mém. Taur.,* iv., 1766-9, p. 44. This paper, dated at Berlin, contains a complete resolution of equations of the second degree, having whole numbers for their roots; a problem the most of those of Fermat, of more curiosity than utility, but very calculated to exercise the powers of minds like those of Euler, Lagrange, and others. The question which is the particular subject of this paper was proposed as a challenge by Fermat, to his contemporaries in England, and correctly answered by Wallis, though without a very satisfactory demonstration. 11. Integration of an Equation, p. 368: a case in which the whole equation is integrable, though its parts, even when properly separated, are incapable of perfect integration. 12. On the Method of Variations, p. 163: in answer to Fontaine, and to de Saur and Jourquier, who had attacked him in their Integration Calculus. 13. On the motion of a Body attracted by two fixed Centres, p. 183-216, including the effects of different supposed laws of attraction. 14. On the Figure of Columns, p. 415. This memoir contains an attempt to demonstrate that the cone is a more advantageous figure for the strength of a column than any conoid, and the cylinder than any cone. But the calculations are founded on the erroneous supposition that the column must bend before it breaks; and, even upon this hypothesis, it appears possible to assign a stronger form than a cylinder, since the summit and the base of the cylinder must certainly contain some useless matter. 15. On the Mean of a number of Observations, p. 167: showing the advantage of taking the mean from the theory of probabilities. 16. On the Motion of Fluids, *Mém. Taur.,* 1765, p. 1, p. 75. The author observes, that this investigation will be assisted by a formula of twice the height due to the velocity when the whole impulse of the jet is received by an obstacle, but of the simple height when a limited surface is exposed to the force of a larger stream. 17. On the Integration of some Irrational Fluxions, ii., p. 218: involving the square root of an expression ascending to the fourth power of the variable quantity. 18. Some of the later of these papers are subsequent in date to those which are found in the Memoirs of the Academy of Berlin; but the order of composition is of little consequence. The first communication of Lagrange to the Academy, of which he was made director, is on Tautochromon Curves, *Mém. Taur.,* 1765, p. 364. The paper is dated 1767; and it contains a demonstration of Fermat's investigation of the subject. 19. On the expected Transit of Venus, 1766, p. 265. The author has here analytically investigated the curves of immersion and emersion for the different parts of the earth. But, as Mr Delambre observes, in order to arrive at the very easy and tolerably accurate solution previously given by Delille and Lalande, he is obliged to employ in succession several elaborate expedients, founded on some very subtle principles, accompanied by various transformations of his ordinates, while by a trigonometrical calculation of a few lines we may obtain a more complete formula, containing all the terms which Fermat has neglected, and which, although very small, are not absolutely negligible. At the same time, he has certainly applied his formula to the calculation of the parallax of the sun in a very convenient manner, which had accidentally escaped both Delille and Lalande, though it follows readily from the trigonometrical calculation. 20. On Indeterminate Problems of the Second Degree, 1767, p. 165. This is the first of a numerous series of papers relating to this difficult branch of analysis, which, notwithstanding its perfect insufficiency, has afforded sufficient scope for the exercise of talent, to give celebrity to the names of Diophantus and Fermat among the most ingenious of mathematicians. 21. On Numerical Equations, p. 311. This subject was also much cultivated by the author at a subsequent period; he here finds an equation for the difference between roots, and exhibits the result in the form of a continued fraction. 22. Continuation of the Memoir on Numerical Equations, 1768, p. 111. The method of continued fractions is still further improved. 23. On the Resolution of Indeterminate Problems in whole Numbers, p. 181. 24. On the Resolution of Literal Equations by Series, p. 251. The contents of these memoirs have been principally merged in the author's later productions. 25. On the Force of Springs, 1769, p. 167. It is demonstrated in this interesting paper, that the source of hair-spring approaches to the law of a circular pendulum, the more nearly as its length is greater. 26. On Kepler's Problem, 1769, p. 204: an application of the methods explained in the last volume, especially of a very elegant formula for the reversion of series. 27. On Elimination, p. 303: a refined and general method of exterminating a quantity from an equation, which, however, is somewhat intricate, even in the simplest cases. 28. Remarks on Isochronous Curves, 1770, p. 97: chiefly in answer to Fontaine, who had attacked him, and who had claimed the invention of the test of integrability of an expression containing several variable quantities. Lagrange observes, that he might very possibly have rediscovered it, but that it was published by Nicolas Bernoulli in 1720; by Fontaine not till 1738. 29. On Arithmetical Theorems, p. 123: relating to the decomposition of a number into squares. 30. On the Resolution of Equations, p. 134. 31. On a Theorem respecting Prime Numbers, 1771, p. 125. A demonstration of the property of prime numbers discovered by Mr Wilson, and published by Waring; and of some other theorems. 32. On Equations, p. 138: in continuation. 33. On a New Mode of Differentiation and Integration, 1772, p. 185. The novelty consists in considering the differential symbol of a function as a quantity multiplying the letter to which it is attached, and inferring by induction that the result of the combinations obtained will in general remain unaltered by the supposition. The grounds of this method have been of late more fully explained by Arbogast and others. The results are here applied to interpolations, and to differences of various orders. 34. On the Forms of Imaginary Roots, p. 222: in general reducible to $A + \sqrt{-1}$. 35. On Astronomical Refractions, p. 259: without any practical applications. 36. On Equations of Partial Differences, p. 333: especially in finding multipliers to make them integrable. 37. On undisturbed Tides, 1773, p. 65: a more direct method of investigation than that of Edmond D'Alembert, but without any new results. 38. On the Attraction of Elliptic Spheroids, p. 162. The author observes, that MacLaurin's prize essay in a masterpiece of geometry, comparable to the best works of Archimedes, though D'Alembert had once doubted of the accuracy of some of his propositions. Thomas Simpson's was the first analytical solution of the problem, but it was indirect, and depending on series only. In this paper the method of demonstration only is varied. Legendre and Laplace subsequently continued the inquiry. "But Mr Ivory," says Delambre, "has lately shown us that a very simple consideration may in some cases supersede a multitude of calculations, and even afford us theorems to which the most prolix computations could scarcely have led." 39. On the Lengths of Parallels, p. 149: an analytical determination of the contents, and of the figures that may be inscribed in the pyramids, when their six sides are given. 40. Arithmetical Researches, p. 265: on the integral roots, in equations of the second degree. 41. On Particular Integrals, 1774, p. 197. Laplace had already pointed out the occasional occurrence of integrals not included in the general and direct expression obtained by the usual modes of integration. Such values are here deduced from the variation of the quantities originally considered as constant, which often affords us an equation of a different form, and leads to values not comprehended in the regular expression of the result. 42. On the Motions of the Planetary Orbits, p. 276. Euler, Lalande, and Delisle had found some expressions for the temporary change of position of the moon; the equations are here integrated, and the total change determined. 43. On Recurring Series varying in two ways, or on Partial Finite Differences, 1775, p. 183: with an application to the theory of chances, upon Laplace's principles. 44. On Spheroids, p. 273: a demonstration of MacLaurin's theorem (Fluxions, art. 633) concerning the attraction of a compressed spheroid or an amygdaloid; derived from the formula contained in the former paper. 45. Arithmetical Researches continued, p. 323: demonstrating some theorems of Fermat with which Euler had not succeeded; yet leaving others still unsolved. 46. On the Mean Motions of the Planets, 1776, p. 209: showing that all such changes are periodical. Laplace had detected an error in the author's reasoning when he attributed secular equations to the motions of Jupiter and Saturn, the expressions containing the terms in question being compensated by others which he had neglected. 47. Cases of Spherical Trigonometry solved by Series, p. 214: without any apparent advantage. 48. On Integration by continued Fractions, p. 236: giving an example of the binomial theorem converted into a continued fraction, which, however, exhibits no particular elegance nor utility. 49. On the Number of Imaginary Roots of Equations, 1777, p. 111. Harriot was the father of the doctrine of equations. Newton made great improvements in it; but his rule remains imperfect with regard to the higher equations, even with the additions of MacLaurin and Campbell. In the present paper the theorem of Waring is demonstrated, without any material attempt to extend it. 50. On the Diophantine Analysis, p. 140. It is remarked that Fermat left all his propositions underenounced, except this theorem, that the sum of two biquadratic numbers can never be a square. 51. On Escapements, p. 173: an investigation of the best forms of pallets for the dead beat and the receding escapement. 52. On determining the orbits of Comets by three observations, 1778, p. 111-124. The first part of this memoir is historical and critical, and the author allows due credit to the ingenuity of Newton's method: his own does not appear to have been of any practical utility. 53. On the Theory of Telescopes, p. 163: comparing the general theorems of Cotes and Euler, and applying Lagrange, the method of recurring series to their demonstration; with a rule for determining the magnitude of the field. 54. On the Expression of the Time in a cubic Section, p. 181; after Lambert, who determines it from the shape of the described, or the form of the revolving fluid, and the great axis; the theorem is here analytically demonstrated. 55. On Particular Integrals, 1779, p. 121: examples from some mechanical curves. 56. On Geographical Projections for Maps, p. 161-185. The methods here proposed for the construction of maps have been found too intricate for adoption. 57. On the Theory of the Libration of the Moon, 1780, p. 203. In the prize essay on the moon's libration, the author had made the first application of the method of variations; the investigation is here continued; and it is observed that the moon cannot be of homogeneous matter, nor its form such as would afford equilibrium to a fluid diverging, or that the effects of the attractions concerned would be much less perceptible than they are. 58. Report on a Quadrature of the Circle, Hist. Ac. Bel. 1781, p. 17. This paper only requires to be noticed as a specimen of the author's condensation. 59. Theory of the Motion of Fluids, Mem., p. 151: an application of D'Alembert's principles to the phenomena of running fluids, and to the motion of waves; but founded on an arbitrary assumption with respect to the depth affected by the waves. 60. On the Secular Variations of the Elements of the Planets, p. 199. The theory of perturbations is here examined by two methods, either comprehending the general form of the orbit, or regarding the local effects only. 61. Report on a Mode of finding the Form of a Body, 1783, p. 39: a memoir which has been already mentioned. 62. On the Secular Variations of the Planets, Mem., p. 169: a continuation of the former memoir, with all the details of the application, and a determination of the change of the place of the ecliptic, together with a determination of the permanency of the general arrangement of the system, depending on the exemption of the mean distances from all variations not periodical, while the other elements are liable to greater alterations. 63. On the Periodical Variations of the Planetary Motions, 1783, p. 161: a sequel to the memoirs on the secular variations. 64. Additions respecting the Secular Variations, p. 191: containing the examination, and extending it to the case of Jupiter and Saturn, which had before been investigated by Laplace. 65. On the Corrections of the Former Astronomical Approximations, p. 224. The errors here considered arise from the employment of the powers of the arcs described in the equations concerned, these arcs increasing without limit; they may be avoided by means of approximations founded on the supposition of the variation of the elements. Laplace had before employed a method still more refined. 66. On a Particular Mode of Approximation, 279: resembling that which Briggs employed for making logarithms. 67. On a New Property of the Centre of Gravity, p. 290: relating to the mutual distances of the bodies. 68. A direct and precise Determination of the Solution of a certain p. 316: in his third memoir the problem reduces to equations of the eighth or seventh degree. 69. Theory of the Periodical Variations of the Planetary Motions, 1784, p. 187: continuation of the memoir of the preceding year, containing the independent variations of the eccentricities and inclinations for the six principal planets; with a numerical application of the formulae demonstrated in the first part. 70. On the Integration of Equations of Linear Partial Differences, 1785, p. 174: entering into further details of the method laid down in a former paper, which is here applied to the problem of trajectories, a problem once proposed by Leibnitz as a trial of strength to Newton, who was not fully aware of the nature of the difficulty intended to be computed: it was, however, solved in England by Taylor, in France by Nieuwenhuis, Delisle, and Hermann gave a more complete solution, and Euler added still more to the generality of the investigation. The author observes that the problem is a mere curiosity; there is, however, one case in which a trajectory of the kind here considered is actually applicable to a natural phenomenon of common occurrence, which is that of a wave diverging from a point in a gradually shelving shore; for the figure or direction of the collateral parts of such a wave may be shown to be the orthogonal trajectory cutting an infinite number of cycloids beginning at the given point. 71. On the Motion of the Apheles of the Planets, 1786, p. 1: a geometrical investigation, in the manner that Newton himself included in the appendix to the Principia. 72. On the Theory of Sound and Waves. This paper is also intended to complete the demonstrations contained in the same work. The volume in which both these interesting memoirs appear seems to have been published out of the regular order, from some circumstances connected with the death of Frederick; and it is wanting in many of the British libraries. 73. Note accompanying a Memoir of Duval le Rol, 1786-7, p. 253. On the Secular Equations of the Georgian Planet. 74. On a Question relating to Annuities, 1792-3, p. 235: the case of an annuity supposed to commence after a death, and to cease at a given age. 75. Additions to former Memoirs, p. 247: on recurring series (m. 43; on elliptic spheroids (m. 33): on interpolations, in Mouton's manner, Lagrange, comprehending the inequality of the distances of the observations; on the secular equation of the moon (m. 64). After Laplace's great discovery in the case of the solar system, Laplace found that it might have been easily deduced from his own calculations almost in the same form, if he had not accidentally neglected the application, from having assured himself, in 1763, that the results of a similar computation were nearly inseparable in the case of Jupiter and Saturn. It was in 1787 that the discovery of Laplace was announced. The acceleration here computed is $10^{-5}$ for the first century after 1800. Mayer found $14.9^{\circ}$, from a comparison of observations. 76. On a general Law of Optics, 1803, Mem., p. 3: a demonstration of the foundation of the method long used by English opticians for determining the magnifying power of telescopes of all kinds, which form an image of the object beyond the eyepiece by reflecting that of the object of that image. The author hazards, in this paper, the very singular assertion, that the illumination of the object may be the same in all telescopes whatever, notwithstanding the common opinion, that it depends on the magnitude of the object-glass; and his reasoning would be correct if the pupil of the eye were always less than the image of the object-glass in question; since, as he observes, the density of the light in this image is always inversely as the magnifying power; but he forgets to consider that the illumination on the retina, when the whole pencil is taken in, is in the joint ratio of the density and the extent; a consideration which justifies the common opinion on this subject. He shows that the above-mentioned maxim is consequently mistaken in his conclusions, if he proceeds to calculate upon erroneous grounds. It deserves, however, to be remembered, that the brightness of any given angular portion of a magnified image must always be somewhat less than that of an equal portion of the object seen by the naked eye; because it can be no greater if the pencil fills the pupil, and will be less in proportion as the pencil is smaller than the pupil, besides the unavoidable loss of light at the refracting surfaces.

The later works of Lagrange have principally been published at Paris, and most of them in the various collections of the academy. The following memoirs are his principal discoveries:—first, the Essay on the Libration of the Moon, which obtained the prize in 1764, Ac. Par. Prix, ix., 1772. It is in this memoir that the method of variations was first practically applied to a mechanical problem. 78. On the Inequalities of the Satellites of Jupiter, in 1776; including the consideration of their mutual perturbations, and consequently a case of the problem of six bodies. The author never resumed the subject, but its investigation was completed by Laplace. 79. A New Method of solving the Problem of Three Bodies, in 1772. 80. On the Secular Equation of the Moon, in 1774, M. Sav. Ecr. vii. for 1773: an unsuccessful attempt, with conjectures respecting the existence of a regular motion, and even doubts of the correctness of the formulae of Halley's discovery. 81. A prize memoir, On the Perturbation of Comets passing near Planets, M. Sav. Ecr. x., Par. 1785, p. 65: finding the path directly, without regard to the conic sections, and employing three different modes of computation for the different parts of the orbit. 82. On forming Tables by observations only, Mem. Ac. Par., 1772, l., p. 513. The method of recurring series is principally employed, and the author observes, that the problem is more useful than difficult, giving an experimental example in the equation of time, for which he obtains, from the results of the tables, an expression very near the truth. Delambre remarks that it is only a continuance of the system adopted by Poincare and the other French astronomers, who, doing what we ought have done in a circumspect manner by pure mathematics if Newton and the laws of gravity had not existed; and he thinks the paper only valuable as a specimen of Lagrange's talent for overcoming difficulties which he might more easily have avoided. 83. On the Nodes and Inclinations of the Planetary Orbits, 1774, p. 97: with details of the calculations for all the planets. 84. On the Variation of the Elements of the Planets, Mem. Math. Inst. 1808, p. 1. The object of this paper is to show, as Poisson had done before, that all the changes of the system are periodical. The method is more general, but less simple, than that of Poisson, who first discovered the principle by induction; and lunar accuracy is given as an example. M. Poisson has extended his calculations to quantities of the second order, which do not enter into Lagrange's investigations. 85. On the Variation of Independent Quantities in General Mechanical Problems, p. 257. The author observes, that many of the modern improvements of mathematics depend on doing away the distinction between constant and variable quantities, which was so valuable when it was first enforced by Descartes. 86. A second Memoir on the Variation of Independent Quantities, 1809, p. 343: simplifying the general application of the doctrine. 87. Lectures on Arithmetic and Algebra, Sciences des Écoles Lagrange, Normalis, Year III., 1794-95. The first lecture relates to the elements of arithmetic, the second to the lower orders of equations, and the third to the higher. All these lectures, under the name of "conversations," were taken down in shorthand by some of the students, and afterwards corrected by the professors.

83. An Essay on Political Arithmetic, Roederer, Collection de documents sur Paris, Year IV., 1795-96.

84. In the Journal de l'École Polytechnique, we find an Essay on Numerical Analysis, p. 232; a detailed explanation of the grounds of the theory laid down in the separate publication on this subject.

85. Analysis of Spherical Triangles, p. 270: giving all their essential properties in a concise form.

86. Lectures on the Calculus of Variations, published separately in 8vo; a commentary on the theory of functions, and a supplement to it, combined in twenty lectures.

90. On the Object of the Theory of Analytical Functions, vol. II., 1860, p. 232: a detailed explanation of the grounds of the theory laid down in the separate publication on this subject.

91. Analysis of Spherical Triangles, p. 270: giving all their essential properties in a concise form.

92. Lectures on the Calculus of Variations, published separately in 8vo; a commentary on the theory of functions, and a supplement to it, combined in twenty lectures.

94. Treatise on Light, xiv., 1808; at the end, chiefly relating to the method of variations. On a Difficulty respecting the Attraction of Spheroids, vii., xv.; remarks which may serve as a commentary on a passage of the Mécanique Céleste.

95. On the Origin of Comets, Connaissance des Temps, 1814.

97. On the Calculation of Eclipses, as affected by Parallax, 1817; from the Berlin Almanac for 1782. This memoir, as Delambre observes, is singularly attractive to a person previously unacquainted with the methods which are employed; but though the formulae first introduced are direct and rigorously accurate, the whole investigation ends in an approximation which wants both these properties.

98. The most important of all the works of Lagrange are those which have appeared in separate volumes; and among these we may reckon his Additions à l'Algèbre d'Éuler, 8vo, Lyons, 1774; vol. II., German, 1793; English, 1797. They relate chiefly to continued fractions and to indeterminate problems, and constitute the most valuable part of the whole work, which, in its abridged form at least, is far inferior to Macaulay's Algebra.

99. Mécanique Analytique, 4to, Paris, 1788, 2nd edit., vol. I., 1811; vol. II., by Prony, Lacroix, and Binet, 1813. This work exhibits a uniform and elegant system of mechanical problems, derived from the simple principle of virtual velocities, which was well known to former authors, but never so extensively applied. It has been said, that many parts of it may be read with advantage, even by those who are not competent to enter into any of the computations, exhibiting such a history of the progress of the science as could only have been sketched by a master. The new edition, begun at the age of seventy-five, comprehends all the improvements contained in the author's later memoirs on various subjects.

100. Théorie des Fonctions Analytiques, 4to, Paris, 1797, 1813. The abstract theory of analytical functions has been very fashionable among modern mathematicians, but the mode of reasoning which contains are chiefly of a metaphysical nature, if they can with propriety be called improvements. The notation is less than that which is in common use, and has been abandoned by the author in some republications of the works in which he had at first employed it. The calculations, too, are often more intricate than others which afford the same conclusions.

101. Équation des Équations Numériques, 4to, Paris, 1798, 1808. The refined and abstract speculations contained in this volume are more calculated to promote the advancement of the abstract science of quantity, than to be applied, as the title would seem to denote, to the purposes of numerical computation. These methods are investigated in general laborious and complicated, though instructive, and capable of great application; and for equations of which all the roots are real, the author himself recommends M. Budan's method, as preferable to his own. The second edition contains a number of very interesting notes, which are full of ingenuity and novelty.

The author of so immense a series of laborious investigations must certainly have been a most extraordinary man. He had acquired the character of an illustrious mathematician almost in his boyhood; and he continued to apply the force of his powerful mind, far more than half a century, to the almost uninterrupted pursuit of his favourite science. It is true, that his earliest were also his greatest successes; and all that followed was as could well be expected from a continued employment of the means which he possessed at the beginning; for, in fact, the whole taken together appears to bear a stronger character of great industry than of great sagacity or talent.

"It was formerly usual," says Delambre, "for mathematicians to inquire, in every investigation, for some general considerations, which might be capable of simplifying it, or of reducing it to a problem already resolved, and to endeavour by these means either to abridge the calculation, or sometimes to supersede it altogether. But since the discovery of the infinitesimal calculus, the facility and universality of this method, which often renders the possession of any talent in the calculator wholly unnecessary, has made it more usual for mathematicians to direct their chief attention to the perfection of this all-powerful instrument. But at the present day, when researches of this kind appear to be completely exhausted by the labours of Euler, Lagrange, and other industrious contemporaries, it might perhaps be more advisable to return to the ancient method, and to follow the example of [Newton, surely, and of] Daniel Bernoulli, who, as Condorcet observes, was entitled to the praise of moderation in the intensity of his calculations. Lagrange was in the habit of employing his sublime talents in a different manner. He liked to make everything dependent on his analysis, though, in some instances, he united both methods in the highest degree, as his invention of the calculus of variations bears witness. His reducing the theory of sound to that of the vibration of chords, is a specimen of a very ingenious simplification; as well as his mode of computing the planetary motion by the variation of the elements of their orbits, which is also applicable to all other cases where opposing forces are small disturbing forces. But it must be confessed, on the other hand, that he has sometimes created difficulties where none existed, by applying his profound and ingenious methods to the solution of elementary problems, which have been obtained from a construction of the simplest kind; and the powerful agents which he employs, on many trifling occasions, remind us only of the man in the fable, who came to borrow the club of Hercules, and the thunder of Jupiter, for the purpose of destroying a flea;" or of the modern mathematician who, without any fable, or any figure of rhetoric, proposes to adjust a standard measure by placing it at a distance and viewing it with a good telescope. The habit of relying too confidently on calculation, and too little on common sense, will perpetually account for the mistakes of Lagrange which have been already noticed, respecting the forms of columns, and the illumination of optical instruments. There are also very instances of the kind which may be produced from the modern history of the sciences. It seems, indeed, as if mathematical learning were the enthusiasm of physical talent; and, unless Great Britain can succeed in stemming the torrent, and in checking the useless accumulation of weighty materials, the fabric of science will sink in a few ages under its own insupportable bulk. A splendid example has already been displayed by the author of the article APOLLONIUS; and, to do justice to our neighbours, it must be allowed that they have never been wanting gratitude, and acknowledged it by merited applause. "All the analytical difficulties of the problem," say Lefebvre and Delambre (Mém. Toulouse, 1812), "vanish at once before this method; and a theory, which before required the most abstruse analysis, may now be explained, in its whole extent, by considerations perfectly elementary." It is indeed only when a subject is so simplified that the investigation can be considered as complete, since we are never so sure that we understand the process of nature as when we can trace at once in our minds all the steps by which that process is conducted. It is not without some reason that a similar disposition to revel in the luxuries of mathematical sport has been sometimes objected to in Laplace, a man of equal analytical powers; but with Lagrange, but possessed, apparently, of greater capacity, and certainly more successful in his application of mathematics to physical researches, although he also seems, on some occasions, to have suffered his habits of abstract reasoning to lead away his attention from the true conditions of the problem, particularly in his first supplement respecting capillary attraction, which concludes with an equation so erroneous that he has been obliged to abandon it in silence. (See the article COMBUSTION.) Another instance of ill-applied competence has been pointed out in the article CHROMATICS; when Laplace attempted to deduce the laws of extraordinary refraction from the principle of the minimum of action, he seems to have forgotten that the demonstration of that principle, in his own words, rests expressly on a condition which is here wanting, that the forces concerned must be functions of the distances, and of course independent of the directions. These imperfections, however, do not deserve to be noticed as materially affecting the general merits either of Lagrange or of Laplace, but they may be considered as accidents which ought to warn us against relying too implicitly on authority, however high, when it appears to militate against clear, simple reasoning, and against common sense.

(Lagrange, Mém. Inst. 1812, t.; Journal de l'Empire, 25th April 1813; Virez et Potel, Précis Historique, Paris, 1813, in 4to; Cassini, Éloge, Pauze, in 8vo, 1813; Maurice in Biographie Universelle, xxiii., Paris, in 8vo, 1819.)