rectify or theories im 
elt have always directed M. la Grange in 
; ice of his subjects, 
_. D’Alembert had considered it as i ible to sub- 
‘to calculation the motions of a fluid inclosed in a 
, unless this vessel had a certain figure. La 
Grange demonstrates the contrary ; except in the case 
“when the fluid divides itself into different masses. But 
-even then we may determine the places where the 
fluid divides itself into different portions, and ascer- 
‘tain the motion of each as if it were alone. , 
_ D’Alembert had thought that in a fluid mass, such 
‘as the earth may have been at its origin, it was not ne- 
~eessary for the different beds to be on a level. La 
“Grange shows that the equations of D’Alembert are 
“themselves equations of on a level. 
_ In combating D’Alembert with all the delicacy due 
‘to'a mathematician of his rank, he often employs very 
‘beautiful theorems, for which he was indebted to his 
adversary. D’Alembert on his side added to the re- 
‘searches of La G . © Your problem appeared to 
‘me so ’beautiful,” says he in a letter to La Grange, 
that I have sought for another solution of it. I 
_ shave found a simpler method of arriving at your ele- 
“gant formula.” ese examples, which it would be 
‘easy to multiply, prove with what politeness these ce- 
lebrated rivals mded, who, opposing each other 
‘without intermission, whether conquerors or conquered, 
constantly found in their discussion reasons for esteem- 
‘ing each other more, and furnished to their antagonist 
“occasions which might lead them to new triumphs. 
“~The academy of sciences of Paris had proposed, as 
‘the subject of a prize, the theory of the libration of the 
‘moon. That is to'say, they demanded the cause why 
‘the moon, in revolving round the earth, always turns 
- ‘the same face to it, some variations excepted, observed 
astronomers, and of which Cassini had first ex- 
; ined the phenomena. The point was to calculate 
all the phenomena, and to deduce them from the prin- 
‘eiple of universal gravitation. Such a subject was an 
I to the genius of La Grange, an opportunity fur- 
to apply his analytical principles and discoveries. 
_ "The attempt of D’Alembert was not disappointed. The 
memoir of La Grange is one of his finest pieces. We 
_ see in it the first developement of his ideas, and the 
germ of his Mecanique Analytique. D'Alembert wrote 
to him: “I have read with as much pleasure as ad- 
‘vantage your excellent paper on the Libration, so wor- 
‘thy of the prize which it obtained.” 
is success enco the academy to propose, as 
oa the theory of the satellites of Jupiter. Euler, 
Clairaut, and D’Alembert had employed themselves 
about the problem of three bodies, as connected with 
‘the lunar motions. Bailly then applied the theory 
-of Clairaut to the problem of the satellites, and it had 
led him to very interesting results. But this theo- 
‘ty was insufficient. The earth has only one moon 
while by ae has four, which ought continually to act 
upon other, and alter their positions in their revo- 
lutions. The em was that of six bodies. La 
Grange attacked the difficulty and overcame it, demon- 
‘strated the cause of the i ities observed by astro- 
‘nomers, and pointed out some others too feeble to be 
‘ascertained by observations, The shortness of the time 
allowed, and the immensity of the calculations, both 
‘analytical and numerical, did not permit him to ex- 
‘haust the subject entirely in a first memoir. He was 
‘sensible of this himself, and promised further results, 
VOL. X. PART I. 
GRANGE. 
449 
ell le otbinn Dhnmnelnany: pesvewtet shins Sem Le Gang, 
giving. Twenty-four years M. La Place took up 
that difficult theory, and made im discoveries 
in it, which completed it, and put it in the power of as- 
tronomers to banish empiricism from their tables, 
About the same time a problem of \quite.a different 
kind attracted the attention of M. la Grange. Fermat, 
one of the greatest mathematicians of his time, had left 
very remarkable theorems respecting the properties of 
numbers, which he probably discovered. by induction. 
-Hehad promised the demonstrations of them ; but at his 
death no trace of them could be found. Whether he 
had suppressed them as insufficient, or from some other 
cause, cannot now be ascertained. ‘These theorems per-~ 
haps may appear more curious than useful. But it is 
well known that difficulty constitutes a strong attraction 
for all men, especially for mathematicians, Without 
such a motive would they have attached so much im- 
portance to the problems of the brachystrochonon, of the 
isoperimeters, and of the orthogonal trajectories? Cer- 
tainly not.. They wished to create the science of cal- 
culation, and to perfect methods which could not fail 
‘some day of finding useful applications. With this 
view, they attached themselves to the first question 
which required new resources. The system of the 
world discovered by Newton was a most fortunate 
event for them. Never could the transcendant calculus 
find a subject more worthy or more rich. Whatever 
progress 1s made in it, the first discoverer will always 
retain his rank. Accordingly, M. la Grange, w 
‘cites him often,as the greatest genius that ever existed, 
adds also, “ and the most fortunate. We do not find 
every day a system of the world to establish.” It has 
required 100 years of labours and discoveries to raise 
the edifice-of which Newton laid the foundation. But 
every thing is ascribed to him, and we suppose him to 
have traversed the whole country upon which he mere- 
ly entered. 
Many mathematicians doubtless employed themselves 
on the theorems of Fermat } but none had been success 
ful. Euler alone had penetrated into that difficult road 
in which M. dre and M. Gauss afterwards sig- 
nalized themselves. M. la Grange, in demonstrating 
or rectifying some opinions of Euler, resolved a pro- 
blem which ap to be the key of all the others ; 
and from which he deduced a useful result ; namely, 
the complete resolution of equations of the second de- 
gree, with two indeterminates, which must be whole 
numbers. 
This memoir, printed like the preceding, amon 
those of the Turin Academy, is notwithstanding da’ 
Berlin, the 20th September 1768. This date points out 
to us one of the few events which render the life of La 
Grange, not entirely a detail of his writings. 
His stay at Turin was not agreeable to him. He saw 
no'person there who cultivated the mathematics with 
success. He was impatient to see'the philosophers of 
Paris, with whom he corresponded. M. de Caraccioli, 
with whom he lived in the greatest intimacy, was ap- 
inted ambassador to London, and was to pass through 
Vetta on his way, where he intended to spend some 
time. He proposed this journey to M. la Grange, who 
consented to it with joy, and who was received as he 
had a right to ex by D’ Alembert, Clairaut, Condor- 
cet, Fontaine, Nollet, Marie, and the other philosophers. 
Falling dangerously ill after a dinner in the Italian 
style given him pA owes he was not able to accompa- 
ny his friend to London, who had received sudden or- 
ders to repair to his post, and who was obliged to leave 
aL 
to 
—~ 
