12 



MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



(42.) 



His birth 

 and educa- 

 tion ; 



his content 

 poraries. 



(43.) 



Euler and 

 Laplace. 



intimately connected with the most salient points of 

 the history of physical astronomy, down even to the 

 present time, and are so interwoven with the disco- 

 veries of Laplace, and represent altogether so much 

 of the substantive character of the progress of the 

 age, that I have thought it necessary to devote a 

 small space to the recital of a few of the most pro- 

 minent of them, having regard to the intellectual 

 portraiture of the man as one of the most pre-emi- 

 nent and successful reasoners of his class who have 

 ever done honour to their race. I shall repeat as 

 little as possible what has been said elsewhere, and 

 confine myself to only two or three topics. 



Joseph Louis Lagrange was born at Turin in 1736 ; 

 he died at Paris 10th April 1813. His first paper 

 was written at the age of 17 or 18, and his end was 

 accelerated by the unremitting ardour of his labours 

 at the age of 77. He was consequently an original 

 author during sixty years ; and for the greater part of 

 this period he, togetherwith Laplace, monopolized the 

 greatest discoveries connected with analysis and phy- 

 sical astronomy, and exercised an almost undisputed 

 authority in the more recondite sciences. Euler, 

 Lagrange, and Laplace, by a singular coincidence, 

 lived to the respective ages of 90, 77, and 79, and 

 all retained their activity nearly to the last. They 

 produced, by the continuity and friendly rivalry of 

 their labours, carried to an extent in each case which 

 only astonishing physical vigour united to astonish- 

 ing mental aptitude could have produced, during 

 almost a century, an impression on the progress of 

 science altogether remarkable. This coincidence 

 was ;;the more happy, because physical astronomy 

 was exactly in that predicament when nothing less 

 than such a combination of intelligence and intensity 

 of application systematically urged, could have car- 

 ried Newton's theory through the difficulties wliich 

 at that time beset it difficulties which left the Prin- 

 cipia for so many years alone, and far in advance of 

 the general intelligence of the age. 



The pregnant suggestions of Euler were developed 

 and applied by Lagrange, and the triumphs of La- 

 grange nay, even his occasional failures were the 

 immediate precursors of some of Laplace's happiest 

 efforts. 



Amongst the former we reckon the method of the (44.) 

 variation of parameters, expounded to a certain point Variation 

 by Euler, though, as in many other cases, his results of P arame> 

 were vitiated by the haste and inaccuracy of his cal- 

 culations. That Lagrange borrowed the idea from 

 Euler cannot admit of a doubt, any more than that 

 he was indebted to him for the principles of the Cal- 

 culus of Variations. Lagrange, with customary truth- 

 fulness, even to his latest days, always spoke of Euler 

 as his best instructor and model, and as the chief 

 of modern mathematicians, Newton only excepted. 

 We know that he so regarded him in the case of the 

 calculus of variations which he studied in Euler's 

 " Methodus inveniendi lineas curvas, &c.," during 

 the first two years of his application to the higher 

 mathematics j 1 whilst Euler, with equal candour, 

 acknowledged the transcendent genius of the rising 

 geometer, forcing its way where he himself had 

 failed. 



The method of the Variation of the arbitrary con- (45.) 

 stants or Parameters, though it may be regarded in its signifi- 

 one point of view as a merely analytical artifice for ca 

 effecting integrations, is in reality a conception purely 

 geometrical, first introduced by Newton 2 under the 

 name of " revolving orbits," and applied by him to 

 the explication of the conception (to use a recently 

 introduced phrase) of the lunar inequalities. Neither 

 the moon nor any planet really describes a mathema- 

 tical ellipse (in consequence of the mutual perturba- 

 tions of the heavenly bodies). They describe curves of 

 double curvature in space, of which we could form 

 no intelligible idea, except by referring them to the 

 very approximate type of the ellipse, of which the 

 eccentricity, line of apsides, inclination, &c., are con- 

 tinually varying, not only from one revolution to 

 another, but throughout every part of a revolution. 

 This representation is not only convenient, but 

 strictly accurate. At each instant the moon or 

 planet is describing a portion of an ellipse, which 

 may be called the instantaneous ellipse, and which instantam 

 may be defined as the particular ellipse which the ous 

 body would go on to describe if it were at that 

 instant freed from all perturbation, and allowed to 

 complete a revolution under the single influence of 

 its acquired motion and the central force. To take 



Lagrange's * The following is a list of the books he then read, taken from a paper probably little known, which appeared soon after the 

 early death of Lagrange in the Moniteur newspaper, and which was translated in Thomson's Annals of Philosophy, vol. iv. He 



studies. first read Euclid's Elements, Clairaut's Algebra ; then, in less than two years, and in the following order, Agnesi's Analy- 

 tical Institutions, Euler's Analysis of Infinites, John- Bernouilli's Lectures, Euler's Mechanics, the two first books of Newton's 

 Principia, D'Alembert's Dynamics, and Bougainville's Integral Calculus, Euler's Differential Calculus and Methodus Inveniendi 

 a pretty course of mathematical reading for a youth between 17 and 19. 



Prom the same paper we abridge a few practical directions given by Lagrange for the study of mathematics, which, if 

 tolerably obvious, are interesting from the extraordinary genius of the man, and from his singular reticence on subjects of a 

 personal nature. " I never," he said, " studied more than one book at a time ; if good, I read it to the end." " I did not 

 oerplex myself with difficulties, but returned to them twenty times if necessary. This failing ,1 examined another author." " / 

 considered reading large treatises of pure analysis quite useless. We ought to devote our time and labour chiefly to the applica- 

 tirmo Thus he read Euler's Mechanics when he had acquired a very slight knowledge of the differential and integral calculus. 



tions. 



" I always read with my pen in my hand, developing the calculations, and exercising myself on the questions." 



" From the very beginning of my career, I endeavoured to make myself master of certain subjects, that I might have an 



opportunity of inventing improvements ; and / always, as far as possible, made theories to myself of the essential points, in order 



to fix them more completely in my mind, to render them my own, and to accustom myself to composition." " Finally, I every 



day assigned myself a task for the next. I learned this custom from the King of Prussia." 

 2 This Lagrange himself points out in his Mecanique Analytique. 



