810 
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 1818. 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 sivty years ; and for the greater part of 
his contem- this period he, together with Laplace, monopolized the 
Poraries. 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- 
tied Newton's theory through the difficulties which 
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. 
(43.) The pregnant suggestions of Huler were developed 
Euler and and applied by Lagrange, and the triumphs of La- 
Laplace. grange—nay, even his occasional failures—were the 
immediate precursors of some of Laplace’s happiest 
efforts. 
(42.) 
His birth 
and educa- 
tion ; 
MATHEMATICAL AND PHYSICAL SCIENCE. 
[Diss. VI. 
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 Parame- 
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 Buler’s 
‘**Methodus inveniendi lineas curvas, &c.,” during 
the first two years of his application to the higher 
mathematics ;1 whilst Huler, 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°*"™ 
effecting integrations, is in reality a conception purely 
geometrical, first introduced by Newton? 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, &e., 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 instantancous ellipse, and which Instantane- 
may be defined as the particular ellipse which thes ellipse. 
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 
early 
studies. 
1 The following is a list of the books he then read, taken from a paper probably little known, which appeared soon after the 
death of Lagrange in the Moniteur newspaper, and which was translated in Thomson’s Annals of Philosophy, vol. iv. He 
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, Buler’s Mechanics, the two first books of Newton's 
Principia, D’Alembert’s Dynamics, and Bougainville’s Integral Calculus, Buler’s Diferential Calculus and Methodus Inveniendi 
—a pretty course of mathematical reading for a youth between 17 and 19. 
From 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. “TI never,” he said, “studied more than one book at a time; if good, I read it to the end.” “TI did not 
verplex myself with difficulties, but returned to them twenty times if necessary. This failing ,I examined another author.” “J 
considered reading large treatises of pure analysis quite useless. We ought to devote our time and labour chiefly to the applica- 
tions.” 
Thus he read Euler’s Mechanics when he had acquired a very slight knowledge of the differential and integral calculus. 
“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 I 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,” 
* This Lagrange himself points out in his Mécanique Analytique. 
