448 
LaGrange, construction given by Euler, a construction true in it- 
Joseph 
Louis. 
—_——— 
self, although its first author had arrived at it by calen= 
lations which were not quite rigorous. He answers 
the objections of D’Alembert.. He demonstrates that 
whatever figure is given to the cord, the duration of 
the oscillations is always the same: a truth derived 
from experiment, which D’Alembert considered as very 
difficult, if not impossible, to demonstrate, He passes 
to the propagation of sound, treats of simple and com- 
pound echos, of the mixture of sounds, of the possibility 
of their spreading in the same space without interfering 
with each other. He demonstrates rigorously the ge- 
neration of harmonious sounds. Finally, he announces 
that his intention is to destroy the prejudices of those 
who still doubt whether the mathematics can ever 
throw a real light upon physics. 1 
We have given this long account of that memoir, be- 
cause it is the first by which M. la Grange became 
known. If the analytical reasoning in it be of. the 
most transcendent kind, the object at least, has some- 
thing sensible. He recals names and facts which are 
well known to most people. What is surprising is, that 
such a first essay should be the production of a young 
man, who took. possession of a subject treated by New- 
ton, Taylor, Bernoulli, D’Alembert, and Euler. He 
appears all at once in the midst of these great mathe- 
maticians as their equal, as a judge, who, in order to 
put an end to a difficult dispute, points out how far 
each of them is in the right, and how far they have 
deceived themselves; determines the dispute between 
them, corrects their errors, and gives them the true so- 
lution, which they had perceived without knowing it 
to be so. 
Euler saw the merit of the new method, and took it 
for the object of his profoundest meditations. D’Alem- 
bert did not yield the point in dispute. In his private 
letters, as well as in his printed memoirs, he proposed. 
numerous objections, to which La Grange afterwards 
answered. But these objections may give rise to this 
question: How comes it that, in a science in which 
every one admits the merit of exactness, geniuses of the 
first order take different sides, and continue to dispute 
for along time? The reason is, that in problems of this 
kind, the solutions of which cannot be subjected to the 
proof of experiment, besides the part of the calculation 
which is subjected to rigorous laws, and respecting 
which it is not possible to entertain two opinions, 
there is always a metaphysical part which leaves doubt 
and obscurity. It is because in. the calculations them- 
selves, mathematicians are often content with pointing 
out the way in which the demonstration may be made ; 
they suppress the developements, which are not always 
so superfluous as they think. The care of filling up 
these blanks would require a labour which the author 
alone would have the courage to accomplish. Even 
he himself, drawn on by his subject and by the habits 
which he has acquired, allows himself to leap over the 
intermediate ideas. He defines his definitive equation, 
instead of arriving at it step by step with an attention 
that would prevent every mistake. Hence it happens 
that more timid calculators sometimes point out mis- 
takes in the calculations of an Euler, a D’ Alembert, a 
La Grange. Hence it happens that men of very great 
genius do not at first agree, from not having studied 
each other with sufficient attention to understand each 
other’s meaning. 
The first answer of Euler was to make La Grange 
an associate of the Berlin academy. When he an- 
nounced to him this nomination on the 20th of Octoa 
3 
GRANGED j 
ber, 1759, he said, ‘* Your solution of the problem of iso- 
perimeters leaves gothine NeeerSeneeaEr am happy 
that this subject, with which I was almost alone occu 
pied since the first attempts, has been carried by you 
to the highest degree of perfection, . The import 
of the matter has induced me to draw up, with your 
assistance, an’analytical solution of it. But I shall not 
publish it till you yourself have published the sequel of, 
your researches, that I may not deprive you of any part, 
of the glory which is your due” 
If these delicate proceedings, and the testimonies of 
the highest esteem, were very flattering toa young man, 
of 24 years of age, they do no less honour to the great: 
man, who at that time swayed the sceptre of mat 
‘mathemas 
tics, and who thus accurately estimated the merit of a 
work that announced to him a successor, 
. But these praises are to be found in a letter. It. 
vey be supposed that the great and good Euler has in~ 
dulged in some of those exaggerations which the epis~ 
tolary style permits. Let,us see then how he has exe 
pressed himself in the dissertation which his letter an-: 
nounced. It begins as follows: © 5 
“ After having fatigued myself for a long time and. 
to no purpose, In endeavouring to. find this integral, 
what was my astonishment when I learnt that in the. 
Turin Memoirs the problem was resolved with as much 
facility as felicity ! This fine discovery produced in me; 
so much the more admiration, as it is very different: 
from the methods which I had given, and far surpasses 
them all in simplicity.” . . 
It is thus that Euler be 
explains with his usual clearness the fo of the: 
method of his young rival, and the theory of the new. 
calculus, which he cailed the calculus of variations, 
To make the motives of this admiration which Euler: 
bestowed with so much frankness better understood, it 
will not be useless to go back to the origin of the re-: 
searches of La Grange, such as he stated them himself 
two days before his death. _ ; : 1 
The first attempts to determine the maximum and: 
minimum in all indefinite integral formule, were 
made upon the occasion of the curve of swiftest de- 
scent, and the isoperimeters of Bernoulli, Euler had, 
brought them to a general method, in an original,work,, 
in which the profoundest. knowledge of’ the calculus is. 
conspicuous. But however ingenious his method was, 
it had not all the simplicity which one would wish to, 
see in a work of pure analysis.. The author admitted. 
this himself. He allowed the necessity of a demonstra-: 
tion independent of geometry. . He appeared to doubt 
the resources of analysis, and terminated his work by 
saying, “ If my principle be not sufficiently demon- 
Ftc as it is ni ae pin. ae 
doubt that, by means of a rigid metaphysical Anas. 
tion, it may be put in the clearest light, and I leave that. 
task to the metaphysicians.” Fe.8 ya 
F mon 
This appeal, to which the metaphysicians ss 
attention, was listened to. by La Grange, and. 
his emulation. In a short time the. young 
the solution of which Euler had despaired.. He found. 
it by analysis. And in giving an_account.of the way. 
in which he had been led to that disbovery, he said ex- 
pressly, and as it were in answer to Euler’s doubt, that, 
e regarded it not asa mney ea principle, but as a- 
necessary result of the laws of m ics, as a simp 
corollary from a more general law, which he afterwards 
made the foundation of his Mechanique Analylique. 
(See that work, page 189 of the first edition.) 
This noble emulation, which excited him to triumph 
gins the memoir, in which he- ; 
undation 
: 
i 
