metaphysics of the differential and 
To render these happy develope- 
ents n easily understood, the professor associated 
self with his pupils. It was then that he composed 
his Analytical Functions, and his Lectures on that Cal- 
 culus, of which he published several editions, 
___ It-was then likewise that he published his treatise on 
the numerical solution of equations, with notes on seve- 
ral points of the theory of algebraic equations. 
- It is said that Archimedes, whose + de reputation, 
at least with the historians, is founded upon the ma- 
chines of all kinds, by means of which he retarded the 
taking of S , despised these mechanical inven- 
tions, on which he wrote nothing, and placed import- 
ance only in his works of pure . We may some- 
times conceive, that the great mathematicians of our 
entertained the same sentiments with Archimedes. 
consider a problem as solved when it presents no 
analytical difficulty, when nothing remains to be done 
but differentiations, substitutions, and reductions, ope- 
rations which require merely patience, and a certain 
dexterity derived from practice. Satisfied with having 
removed all the real difficulties, they:concern themselves 
lng too little with the embarrassments which they 
ve to the calculator, and with the long labour neces- 
sary in order to make use of their formula, even after it 
has been suitably reduced. M. la Grange had more 
than once attempted to abridge the usual calcula- 
tions, 
The resolution of algebraic equations is sub- 
- ject to difficulties which are considered ‘as insurmount- 
able ; but in practice every determinate problem. brings 
us to an equation, all the co-efficients of which are given 
in numbers. It would be sufficient, therefore, to have 
a sure method of finding all the roots of such an equa- 
tion, which is.called numerical. This was the object 
which M. la viange roposed to himself. He analyses 
all the known methods, and shews their uncertainty and 
insufficiency. He reduces the problem to the determi- 
nation of a quantity smaller than the smallest difference 
between the roots. This is something. We cannot 
ton much admire the analytical skill displayed through- 
out the whole work, But notwithstanding all the re- 
sources of the genius of M. la Grange, we cannot con- 
ceal that the labour. of his method is exceedingly great, 
and calculators will doubtless continue to prefer me- 
thods less direct indeed, but more expeditious. The 
author resumed this subject.no less than four times, It 
is to be feared, that a,commodious and general solution 
will never be discovered, or at least it must be sought 
for by other means. The author seems to have ac- 
knowledged this himself, as he recommends the me- 
thod of M. Budan as the most convenient and elegant 
for resolving equations whose roots are all real. 
__ The desire of multiplying useful applications induced 
him to undertake a new edition of the Mecanique Ana- 
ique. His project was to develope the most useful 
parts of it. He laboured at it with all the ardour and in- 
telleetual power which he could have applied at any 
iod of his life. But this application occasioned a 
d neaeer of fatigue, which threw him into a fainting fit. 
He was found in that state-by Madame La Grange. 
His head in falling had struck against the corner of a 
table, and this shock had not restored him to his senses. 
This was a warning to take more care of himself. He 
thought so at first ; but he was too anxious to finish his 
work. The first volume had ap some time be- 
fore his death. It had been followed by a new edition 
of his Fonctions Analytiques, So much labour exhaust- 
GRANGE: 
458 
ed him. Towards the end of March a fever came on, La Grange 
he lost his appetite, his sleep was uneasy, and his. wa. 
ied by alarming swoonings. He —@% 
| ger ; but, preserving his undisturbs 
able serenity, he studied what passed within him, and, 
as if he were assisting at a and uncommon expe« 
riment, he; bestowed all his attention on it. His ree 
marks have not been lost. Friendship conducted to 
his house on the 8th of April, in the morning, MM. Las 
cepede, Monge, and Chaptal, who took care to write 
down the principal points of a conversation which was 
his last. (We have scrupulously followed these notes, 
and the woe under inv commas are faithfully 
copied from the manuscript of M. Chaptal. ) 
“He received them with tenderness and cordiality. 
I was very ill, my friends, (said he,) the day before 
yesterday ; I perceived myself dying, my body became 
weaker, my moral and physical powers were gradually 
declining ; I observed with pleasure the gradual dimi- 
nution of my strength, and I arrived at the point with- 
ovt pain, without regret, and by a very decli- 
vity. Death is not to be ‘cared, and when it comes 
without violence, it isa last function, which is neither 
inful nor di bie.” Then he explained to them 
is ideas respecting life, the seat of which he consider- 
ed as spread over the whole body, in every organ and 
all parts of the machine, which in his case became 
equally feebler in, ev: by the same degrees, 
“A little longer, and there would have been no func« 
tions, death would have overspread the whole body, 
for death is.merely the absolute repose of the body ; I 
wished to die,” added:he with greater force, ‘I found:a 
ecg init; but my wife did not wish it. I should 
ve preferred at that time a wife less kind, less eager 
to restore my oe and who would have allowed 
me gently to have finished my career. I have perform- 
ed my task, I have acquired some celebrity in the ma- 
thematics, I have hated nobody, I have:done no ill ; it 
is now proper to finish.” 
As he was very animated, ially at these last 
words, his friends, notwithstanding the interest with 
which they listened to him, proposed to retire. He 
retained them, began to relate to them the history of 
his life, of his labours, of his success, of his residence 
at Berlin, where ‘he had often told us what he had seen 
neara king ; of his arrival at Paris, the tranquillity he 
had enj at first, the anxiety occasioned to him by 
the Revolution, and how he had been finally rewarded by 
a,powerful monarch, capable of appreciating his worth. 
He had neither been ambitious of riches nor honour; 
but he had received both with respectful gratitude, 
and rejoiced at the acquisition for the advantage of the 
sciences, 
La Grange had not lost all hope of cure’; he believed 
only thathis.convalescence woaltbe tedious. He offered, 
when he recovered his srs 9 and dine at M. La- 
cepede’s country house wi MM. Monge and Chaptal, 
and proposed to give them details er cree his li 
which could be found nowhere else. These details are 
irretrievably lost. We do not even know to what he al- 
luded, nor what he could have added to the second vo- 
volume: of the Mecanique Analytique, which was then 
in the press. We have just learned that the Countess 
la Grange has put into the hands of M. Prony the com- 
plete manuscript of the second volume, in which will 
be found important additions, and sections entirely 
written anew. By the care of an editor so skilful, and 
so devoted to the memory of the author, the philoso- 
phical world is sure of obtaining with the greatest ac- 
