769 



LA GRANGE, JOSEPH-LOUIS DE. 



LAGRANGE, JOSEPH-LOUIS DE. 



r?o 



delight consisted in the perusal of the various Latin authors, and 

 more especially the works of Cicero and Virgil. These however in 

 his second year were superseded by the synthetical writings of the 

 ancient geometricians, and these in their turn gave place to the more 

 powerful analysis of modern times. The perusal of a memoir by 

 Dr. Halley ('Phil. Trans.,' 1693) 'On the superiority of modern 

 algebra in determining the foci of object-glasses,' is said by his bio- 

 graphers to have convinced him of the utter inadequacy of geometrical 

 methods as instruments of investigation, and it is not improbable 

 that tliis might have been the occasion of hia selecting the path which 

 he thenceforth pursued with so much honour to himself and BO great 

 advantage to science. 



Before he attained the age of nineteen he was appointed to the 

 professorship of mathematics at the military college of Turin, where 

 by far the greater part of his pupils were older than himself. The year 

 following (1755) he addressed a letter to Euler relative to the isoperi- 

 metrical problems, and that of the curve of quickest descent, which had 

 engrossed so much of the attention of the principal mathematicians of 

 the day, and of Euler in particular; but, owing to the want of general 

 methods, their labours had proved but partially successful. Each 

 problem had been resolved by methods peculiar to itself, and the 

 solutions rested upon artifices unsatisfactorily indirect. In this letter 

 Lagrange communicates the germs of his calculus of variations, to 

 which his recent analytical researches had led, and shows with what 

 advantage and facility it may be applied to the problems in question. 

 Euler, in his reply, expresses his entire concurrence in the correct- 

 ness of its principles, and hails the discovery as the harbinger of 

 others of yet greater importance; he acknowledges how much the 

 application of these principles had promoted the success of his own 

 recent investigations, which however he refrained from publishing 

 until the remainder of the researches of Lagrange were made known, 

 lest he should thereby deprive him of any portion of the glory which 

 was so justly his due, and concludes by announcing the nomination 

 of Lagrange as a member of the Academy of Berlin. 



In 1758 he took an active part in the foundation of the Royal 

 Academy of Turin, in which he was unanimously chosen the director 

 of the pbysico-mathematical sciences. The following year appeared 

 the 6rst volume of the Transactions of that Society, consisting prin- 

 cipally of the researches of Lagrange on the propagation of sound, 

 and on the integration of differential equations, and those of finite 

 differences. He here also proves, on the subject of vibrating chords, 

 that the time of oscillation is independent of the figure of the chord, 

 an empirical truth, the demonstration of which D'Alembert believed 

 to bo impossible (see the preface to D'Alembert's ' Opuscules Mathd- 

 matique*,' Paris, 4to, 1761, tome L) [D'ALEMUERT.] Lagrange and 

 D'Alembert were rivals, but not opponents. Their cause was a 

 common one, which each laboured to promote with indefatigable zeal. 

 The manner in which their controversies were conducted shows that 

 they were prepared to sacrifice every personal feeling to their love of 

 truth and the advantage of science. When either attempts the refu- 

 tation of his rival's theory, it is frequently by means of the beautiful 

 theorems to which the researches of the other has already led. On 

 the other hand, a discovery of importance, by whichever party it may 

 happen to be made, is immediately followed by the congratulations of 

 him from whom congratulation is due. Thus D'Alembert, in one 

 of his letters to Lagrange, says, " Your problem appeared to me so 

 beautiful, that I have investigated a solution upon different prin- 

 ciples ; " and upon another occasion, when the Academy had proposed 

 the ' Theory of the Libration of the Moon ' as the subject of one of its 

 prizes, and the medal had been awarded (1764) to the memoir of 

 Lagrange, we find D'Alembert writing to him solely to express the 

 pleasure and advantage which he had derived from its perusal, and 

 his acquiescence in the justice of the award. 



The calculus of variations, upon the discovery of which the fame of 

 Lagrange may be permitted to rest, is eminently important in many 

 branches of the mathematics, as in the determination of the maxima 

 and minima values of indefinite integral formula), &c. ; but its utility is 

 most conspicuous in the higher branches of physical astronomy. The 

 space allotted to this article admits of our giving but one illustration of 

 it* importance in this respect. Euler, in his ' Treatise of Isoperirueters,' 

 printed at Lausanne in 1744, had shown, that in the case of trajecto- 

 ries described about a central force, the product of the integral of 

 the velocity and the element of the curve was either a maximum or 

 minimum ; but when he attempted to extend this principle to a 

 system of bodies acting upon one another, he found that the highest 

 analysis of which he could avail himself was insufficient to overcome 

 the difficulties of the problem. This failure on the part of Euler 

 excited the emulation of Lagrange, whose chief objects appear gene- 

 rally to have been the extension and generalisation of existing 

 theories. By a beautiful application of his method of variations to 

 a principle of dynamics discovered by Huyghens, and known by the 

 name of the Conservation of vis viva, he was led to the following 

 general theorem : " In every system of bodies acted upon by forces 

 proportional to any function of the distance, the curves described by 

 the bodies are necessarily such that the sum of the products of the 

 mass, the integral of the velocity and the element of the curve, is 

 always either a maximum or minimum." This theorem, the proof 

 of which offered to much difficulty to Euler, has been denominated 



BIOO. DIV. VOL. Ill, 



the principle of ' least action,' and is frequently regarded as one of the 

 four great principles of dynamics, although Lagrange has shown that 

 it is merely a corollary to a still more general formula given by him. 

 in the second section of the second part of his ' Mecauique Aualy tique.' 



When the Academy of Berlin was threatened with the departure of 

 Euler for St. Petersburg, Frederick renewed his importunities to 

 D'Alembert to succeed him. [D'ALEMBEKT.] D'Alembert however 

 from various motives, being unwilling to quit his native country, sug- 

 gested that the proffered honour might be conferred upon Lagrange. 

 Lagrange was accordingly appointed professor of physical and mathe- 

 matical sciences to the Academy, and continued for more than twenty 

 years to enrich the memoirs of that society with his researches con- 

 nected with physical astronomy and other subjects of importance. 

 The insignificant stipend (1500 crowns) which was allotted to him, 

 when contrasted with the munificent offers made to D'Alembert, cannot 

 fail to strike every reader with surprise. Lagrange quitted Berlin 

 after the death of Frederick, not being satisfied with the treatment he 

 then received. He had previously been invited by the ministers of 

 Louis XVI. to settle in Paris. 



In 1772 M. Lagrango was elected foreign associate of the Royal 

 Academy of Paris, and ia 1787, on his arrival at the French capital, he 

 received the honorary title of veteran pensioner. Apartments were 

 allotted to him in the Louvre, and here, surrounded by the principal 

 mathematicians of the day, he continued to live happily up to the time 

 of the revolution. After this he began to be subject to fits of melan- 

 choly, which so far increased upon him that he has been heard to say 

 that his enthusiasm for the sciences was extinguished, and that his 

 love of physical research had disappeared. He was successively 

 appointed professor of mathematics to the normal and polytechnic 

 schools, member of the Institute, of the board of longitude, grand 

 officer of the legion of honour, and count of the empire. He died at 

 Paris, the 10th of April 1813, in his seventy-eighth year. His remains 

 were deposited in the Pantheon, and his funeral oration was spoken 

 by his illustrious friends Laplace and Lace'pede. 



"Among those who have most effectually extended the limits of our 

 knowledge," said Laplace, in his funeral oration, " Newton and La- 

 grange appear to have possessed in the highest degree the happy art of 

 detecting general principles, which constitutes the true genius of 

 science. This art, joined to a rare elegance in the exposition of the 

 most abstract theories, characterised Lagrange." His work on 

 Mechanic*, resting upon the method of variations of which he was the 

 inventor, flows wholly from a single formula, and from a principle 

 known before his time, but of which no one but himself was able to 

 appreciate the importance. "Among the successors of Galileo and 

 Newton," says Professor Hamilton, speaking of the theoretical develop- 

 ment of the laws of motion, " Lagrauga has perhaps done more than 

 any other analyst to give extent and harmony to such deductive 

 researches, by showing that the most varied consequences respecting 

 the motions of systems of bodies may be derived from one radical 

 formula ; the beauty of the method so suiting the dignity of the results 

 as to make of his great work a kind of scientific poem." 



We conclude this imperfect sketch of the life and writings of 

 Lagrange with a Hat of his published works, which we believe to be 

 complete : 



Letter dated 23rd June, 1754, addressed to Jules Charles Fagnano, 

 containing a series for the differentials and integrals of any order 

 whatever, and corresponding to the ' Binomial Theorem ' of Newton, 

 Turin, 1754 ; 'Analytical Mechanics,' 1st edit. 1788, 2nd edit. 1811-15 

 (the second volume of the last edition is edited by Messrs. Do Prony, 

 Gamier, and Binet). 'Theory of Analytical Functions,' Ist-edit, 1797, 

 2nd edit. 1813; 'Resolution of Numerical Equations,' 1st edit. 1798, 

 2nd edit. 1808, 3rd edit, (edited by Poinsot) 1826; 'Lessons on the 

 Calculus of Functions,' 1st edit. 1801, 2nd edit. 1804, 3rd edit. 1806 

 (printed in the ' Journal of the Polytechnic School,' tome 5). 



Memoirs in the Transactions of the Academy of Turin 1759, tome 1, 



Method of Maxima and Minima; Integration of Differential Equa- 

 tions and Equations of Finite Differences; On the Propagation of 

 Sound. 1762, tome 2, Supplement to the Researches on the Propa- 

 gation of Sound, contained in vol. 1 ; A new method of determining 

 the Maxima and Minima of Indefinite Integral Formula; ; application 

 of that method to Dynamics ; New Researches on the Propagation of 

 Sound. 1765, tome 3, Application of the Integral Calculus to Dyna- 

 mics, Hydrodynamics, and Physical Astronomy ; tome 4, Integration 

 of Differential Equations ; Method of Variations ; On the Motion of a 

 Body acted upon by two Central Forces ; tome 5, On the Percussion of 

 Fluids ; New Theory of the Integral Calculus. 



Memoirs in the Transactions of the Academy of Berlin. 1765, tome 

 21, On Tautochronous Curves. 1766, tome 22, On the Transit of 

 Venus, June 3, 1769. 1767, tome 23, On the Solution of Indeterminate 

 Problems of the second degree, and on Numerical Equations. 1768, 

 tome 24, Additions to the Memoir on the Resolution of Numerical 

 Equations ; New Method of Resolving Indeterminate Equations ; New 

 Method of Resolving Algebraic Equations by means of Series. 1769, 

 tome 25, On the Force of Springs ; On the Problem of Kepler ; and On 

 Elimination. 



Memoirs in the Transactions of the Berlin A cademy (new series). 



1770, On Tuutochronous Curves ; Algebraic Equations, and Arithmetic. 



1771, On Prime Numbers and Algebraic Equations. 1772, On Differcu- 



3D 



