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The battle of Wattignies, considered as to its results,
will always occupy one of the foremost places in the
annals of the French Revolution. I should probably be
less positive on the difficulties of that day, compared
with so many others, if I could not support myself by
the opinion of the Prince of Cobourg himself When he
saw the French battalions begin to break, that general
could not find terms too strong to express, in presence of
bis staff, the confidence that he felt in the number and
ardour of his troops, and in the obstacles of all sorts, both
natural and artificial, that the uneven ground occupied by
the Austrians presented to the assailants. He exclaimed :
" The Republicans are excellent soldiers ; but if they dis-
lodge me from this position, I will consent to become a
Republican myself." Certainly nothing more decided or
more energetic could issue from the mouth of Cobourg.
For ray part, I could not conceive a more glorious bulletin
of the battle of Wattignies !

The German author from whom I have borrowed this
anecdote does not say whether, after having dislodged him,
the French summoned the Austrian general to keep his
word. I have some reason to suppose that, notwithstand-
ing their spirit of propaganda, they disdained a recruit



54 CARXOT.

who would have submitted, but whose vocation seemed
very uncertain.

STATISTICS OP THE OPERATIONS OF THE ARMIES.

Carnot felt the propriety, the want, of showing towards
the national armies a deference from which absolute gov-
ernments formerly felt themselves free, whilst their sol-
diers were enrolled at a money price : each year he had
to unroll to the eyes of the nation a detailed table of the
battles given by our legions, and of the effects that had
thence resulted. Here follows the conclusion of the reci-
tal of the campaign of seventeen months, during which
the troops of the Republic never laid down their arms for
a single day.

Twenty-seven victories, eight of which were pitched
battles ; 120 combats of minor importance ; 80,000 ene-
mies killed; 91,000 pi-isoners ; 116 fortresses or strong
towns taken, 36 of which had required to be besieged or
blockaded ; 230 forts or redoubts taken ; 3,800 guns of
various sizes ; 70,000 muskets ; 1,900 milHers (tons) of
gunpowder ; 90 flags. Let people, if they dare, after
reading this table, say that statistics are not eloquent !

CARNOT, NAMED BY FOURTEEN DEPARTMENTS, ENTERS
THE COUNCIL OF THE ELDERS, AND THEN THE EXEC-
UTIVE DIRECTORY. HOCHE SENT TO LA VENDEE,

MOREAU AND JOURDAN TO THE RHINE, AND BONA-
PARTE TO ITALY.

Carnot quitted the Council of Public Safety shortly
before the insurrection of the Parisian sections against
the Convention. Carry back your recollection towards
the military events that followed the forced, though legal
reti'eat of our colleague, and you will see almost every-



COUNCIL OF THE ELDERS. 55

where that victory abanJoned the standards of the Repub-
lic, and reverses succeeded each other, as triumphs did
before ; all the springs were unbent, mistrust and dis-
couragement took possession of every mind ; and you
will then understand, better than by an uninterrupted
series 'of brilHant successes, of wliat importance the
genius of one man alone may be to the destiny of na-
tions.

Carnot was called to the legislature which succeeded
to the National Convention by fourteen departments. If
I were allowed to express a personal sentiment, I would
say how pleased I have been to find the name of the de-
partment of the Eastern Pyrenees, in the list of those
which tried to reward our great citizen for the outrages
that a handful of members, excited by the butcher Le-
gendre, cast upon him on several occasions. A short
time after he entered the Council of the Elders, Carnot,
on the refusal of Sieyes, became one of the five members
of the Executive Directory.

At the moment when he for the second time was thus
called to direct our armies, the Republic had reached the
verge of an abyss. The public treasury was empty. The
Directory had great trouble even in procuring clerks and
servants in their office, so much was it thought to be in-
solvent. The despatching of a courier was often delayed
on account of the impossibility of providing for the ex-
penses of the journey ; the generals themselves no longer
received the eujld francs (I am not mistaken), the
eight francs ])er month "en numeraire," (in cash,) that
had been granted to them, as a supplement to their pay
in assignats ; the agricultural producers no longer sup-
plied the markets ; the manufacturers refused to sell their
products, because there was a right to pay for them in



56 CARNOT.

paper-money, and paper-money then was of no value.
From one end of France to the other, famine had thrown
people into an extreme state of irritation, which daily
manifested itself in sanguinary disorders. The army
offered a no less deplorable aspect : it was deficient in
means of transport, in clothing, in shoes, in munitions.
Misery had engendered a want of discipline. Pichegru
was weaving criminal relations with the Prince of Conde,
allowed himself to be beaten at Heidelberg, compromised
the array of Jourdan, evacuated Mannheim, raised the
siege of Mayence, and ceded the frontier of the Rhine
to the Austrians. War recommenced in La Vendee ;
the English threatened us with a descent in the Pays-
Bas, and on our own coasts. In a word, on our Alpine
frontier, Scherer and Kellermann painfully sustained a
defensive war against the united forces of the Emperor of
Austria, the King of Sardinia, and the confederated
Italian princes.

Gentlemen, the great strength of mind, united to the
most ardent patriotism, was requisite, under such cruel
circumstances, to induce men to accept the burden of
public affairs. Let us add that Carnot was so little blind
to the faults of the Constitution of the year IIL, and,
above all, to the inconveniences of a multiple executive
power, that he had publicly pointed them out in the
midst of the Convention, at the time when this constitu-
tion was discussed. He then exclaimed : " The destinies
of the state will henceforward depend only on the personal
character of five men. The more these characters differ,
the more dissimilar will be the views of these five direc-
tors, and the more will the state have to suffer from their
alternate influence." The majority disdained tliese just
apprehensions ; faithful to a line of conduct from which



BONAPARTE IN ITALY. 57

he was never seen to swerve, Carnot submitted without a
murmur ; and, as soon as the new government had re-
ceived the legal sanction, he served it with the same
energy, zeal, and devotion that he had before displayed
as a member of the Committee of Public Safety.

La V endee was on fire ; Iloche receives orders from
Carnot to pacify it, together with a new system of opera-
tions. This republican general complies, triumphs over
Charette, takes possession of Stofflet, and clears the Moi*-
bihan of the numerous bands of chouans who ravaged it.
In less than eight months, the civil war, that impious
war, in which, however, great courage was displayed on
both sides, ceased to desolate our territory.

On the Rhine, our armies are placed under the com-
mand of Jourdan and Moreau. A scientific and pro-
found plan of the campaign connected the movements of
those two generals, and soon carried their victorious
troops into the heart of Germany.

In La Vendee, in Germany, on the Rhine, Carnot, as
we have shown, had infused confidence into officers
already celebrated by memorable triumphs. The com-
mand of the army of Italy he gave, on the contrary, to a
general only twenty-five years of age, whose known
claims were then restricted to some secondary services
that he achieved during the siege of Toulon, and to the
easy defeat of the Parisian Sectionaries, on the 13th
Vendumiaire, year III., on the humble fields of battle of
the Pont Royal and the Rue St.-IIonore, and the steps
of St.-Roch. I here claim for Carnot the honour of
having personally pointed out and selected the young
Genei-al Bonaparte for the command of our third army,
because it legitimately belongs to him ; because this
choice was long unjustly considered as the result of a
3 *



58 CARNOT.

boudoir intrigue ; and because every one, I think, must
be glad to see the liistory of the incomparable campaign
of Italy purified from such a stain. I have thought, in
short, that I ought not to neglect to show you your col-
league discerning with infinite perspicacity the hero of
Eivoli, of Arcole, of Castiglione, through the bark of
timidity, of reserve, — let us out with the true word, of
awkwardness, — that everybody then remarked in the
protege of Barras.

I foresee all the incredulity I should meet with, if I
were to venture on still farther extending the limits of
the influence that our colleague exercised over the Italian
campaign ; and yet, should I not have found, even in the
small number of official documents already known to the
public, under date of the 10th Floreal, year IV., for
example, a despatch from the head-quarters of Cherasco,
in which Bonaparte writes to Carnot : — " The armistice
concluded between the King of Sardinia and ourselves,
enables me to communicate through Turin, that is to say,
to spare half the journey ; / could therefore quickly re-
ceive YOUR ORDERS AND LEARN YOUR INTENTIONS, AS
TO THE DIRECTION TO BE GIVEN TO THE ARMY." A

letter to the Minister of Finance, of the 2d Prairial, year
IV., dated from head-quarters at Milan, would afibrd the
following sentence : — " The Executive Dii-ectory, who
named me to the command of this army, has arranged
A PLAN OF offensive WARFARE which requires prompt
measures and extraordinary I'esources."

The 2d of Prairial, year IV. (May 21, 1796,) Carnot
wrote thus to the young general: — "Attack Beaulieu
before reinforcements can reach liira ; do not neglect
any thing to prevent this junction ; you must not weaken
yourself before him, and above all, do not, by disastrously



BONAPARTE IN ITALY. 59

dividing your force, give him the means of beating us in
detail, and retaking the ground he has lost. After the
defeat of Beaulieu, you will make the expedition to

Leghorn The intention of the Directory is, that

the army shall not pass beyond the Tyrol, until after the
expedition to the south of Italy."

Doubtless, these general orders are not the campaign
of Italy. No human intelligence could foresee either the
route that General Beaulieu would follow after his sep-
aration from the Piedmontese army, nor the manoeuvres
of "Wurmsur, nor the long resistance made at Mantua by
that old general, nor the marches of Alvinzi, nor many
glorious incidents which I abstain from recalling ; with-
out doubt it required all the hardihood and genius of
Bonaparte, and the cooperation of such officers as Mas-
sena, Augereau, Lannes, Murat, Rampon, to annihilate
in a few months three large Austrian armies. Finally,
all that I have wished to say is, that it would be unjust
to entirely omit the name of Carnot in reciting those
immortal campaigns.

I should have a right to say even more were we study-
ing another phase of those wars, — their moral and civil-
izing phase. Who does not remember those treaties of
peace, in which masterpieces of painting and of sculpture
wei-e inducements to pardon perfidy and treachery in our
enemies, and the official visits of our victorious generals
to diffident learned men, rendered illustrious by important
discoveries ? Well, Gentlemen, all this, whatever peo-
ple may say of it, was prescribed by Carnot. Will any
doubts still be entertained if I transcribe the following
letter from our colleague, dated

" 24tli of Prairial, year IV.

" General, in recommending you, by our letter of the



60 CAKNOT.

26th Floreal, to visit and receive the celebrated artists
of the countries in which you happen to be, we have
especially designated the great astronomer Oriani, of
Milan, as deserving of being protected and honoured by
the republican troops. The Directory will learn with
satisfaction that you have fulfilled its intentions respect-
ing this learned and distinguished man, and it requests
you consequently to relate what you have done to prove
to citizen Oi'iani the interest and the esteem that the
French have always felt for him, and to testify that they
know how to unite the love of glory and liberty with a
love for the arts and for talent."

PUBLICATION OF THE WORK ENTITLED " REFLECTIONS
ON THE METAPHYSICS OF THE INFINITESIMAL CAL-
CULUS."

The word science, which the series of events has just
brought to my pen, reminds me that this epoch is that of
the publication of one of Carnot's mathematical works.
I am aware how fatiguing it will be to you to listen to
the analysis of it ; but it is quite necessary that the
savant also should be occasionally represented in this
assembly. The early and very remarkable work on
machines of which we gave an idea, has sufficiently in-
dicated how much we may expect from the firm, lucid,
and penetrating mind of Carnot. It was then a brilliant
and glorious future which the young officer brought as
an offei'ing to his country, when, obeying the voice of his
fellow-citizens, he exchanged the smooth, tranquil life of
the mathematician, for the adventurous and rock-bestrewn
career of the tribune. This sacrifice, moreover, he did
not make without regret ; for geometry was always his
favourite relaxation. Debarred by imperious daily du-



THE INFINITESIMAL CALCULUS. 61

ties from the pleasure of " measuring himself with " the
grand problems whose solution requires years of con-
tinuous and persevering effort, Carnot chose those diffi-
cult but circumscribed questions which may be taken up,
abandoned, and taken up again, by fits and starts ; which
an elevated mind capable of coping with difficult sub-
jects, develops and fathoms without paper or pencil,
either during a walk, in the midst of the excitements of
a crowd, the gayeties of a l:)anquet, or the vigils of labo-
rious nights ; in a word, he directed his meditations to-
wards the " metaphysics of the calculus." In the present
day such researches would be. I fear, but little relished ;
nevertheless, if we recur to the times when mathematical
studies gradually led to the consideration of quantities of
such diffiirent natures, we shall be amply aware of the
apprehension with which they inspired exact philosophers,
and must acknowledge that, on many points, it is rather
habit than true science which has rendered us more con-
fident.

Amongst the quantities to which I have alluded, the
" eVra^tonaZ" presented themselves first. The ancients
scrupulously avoided using them ; the moderns would
also have wished to avoid the use of them ; " but they"
(the quantities) " gained the day hy their numbers" says
the ingenious author of the " Geometry of Infinites."

To the quantities which were not numerically assign-
able, succeeded the impossible quantities, the " imaginary
quantities" regular symbols of which it would be vain to
attempt to give, not only the exact values, but even mere
approximations. These imaginaries are nevertheless used
in combination by addition and subtraction ; they are
multiplied and divided, the one by the other in the same
manner as real quantities ; at the end of the calculation



62 CARNOT.

the imaginaries sometimes disappear amongst the trans-
formations which they undergo, and the result is then
held to be quite as certain as if it had been arrived at
without the help of these algebraic hieroglyphics. It
must be confessed that, though thousands on thousands
of applications of the calculus justify this confidence, few
geometers fail to take credit to themselves for the ab-
sence of imaginary quantities in the demonstrations
where they have been able to avoid them.

The " irifinite " first made its irruption into geometry
on the day when Archimedes determined the approxi-
mate proportion of the diameter to the circumference by
assimilating the circle to a polygon " ^vith an infinite
number of sides." Bonaventura Cavalieri afterwards
went much farther in the same field of research ; various
considerations led him to distinguish some " infinitely
great quantities " of several orders, from some infinite
quantities which were nevertheless infinitely smaller than
other quantities. Can we be astonished that, at sight of
such results, and notwithstanding his lively predilection
for combinations, which had led hira to veritable dis-
coveries, the ingenious Italian author should have ex-
claimed, in the style of that period, " Here are difip^cidties
of which even the arms of Achilles coidd make nothing!"

The '■'■infinitely small" quantities had, for their part,
slipped into geometry even before the " infinitely great"
and this not only to facilitate or abridge such and such
demonstrations, but as the immediate and necessary re-
sult of certain elementary properties of curves.

Let us examine, in effect, the properties of the most
simple of all — the circumference of a circle ; and by that
we will not understand the rugged clumsy curve which
we should succeed in drawing by the aid of our com-



THE DIFFERENTIAL CALCULUS. 63

passes or best geometrical drawing-pens ; but really the
circumference of a circle endowed with an ideal per-
fection, really a curve without thickness and without
roughness of any sort. Let us, in imagination, draw a
tangent to this curve. At the point where the tangent
and the curve touch one another, tliey will form an angle,
which has been called the " angle of contact." This
angle, since the first origin of mathematical science, has
been the object of the most serious reflections of geom-
eters. Since two thousand years ago it has been rigor-
ously demonstrated that no straight line, drawn from the
apex of the angle of contact, can be included between its
two sides, and that it cannot pass between the curve and
the tangent. Well, I ask, what else is that angle into
which an infinitely fine straight line cannot be introduced
or insinuated, but an infinitely small quantity.

The infinitely small angle of contact, into which no
straight line can be introduced, may nevertheless include
between its two sides milHons of circumferences of cir-
cles, all greater than the first. This truth is established
by reasoning of an incontestable and uncontested force.
Here, then, we have, in the very heart of elementary
geometry, an infinitely small quantity, and, what is still
more incomprehensible, susceptible of being divided as
much as we please ! The human intellect was humili-
ated and lost in face of such results ; but, at any rate,
these were results, and it submitted.

The infinitely small quantities which Leibnitz intro-
duced into his differential calculus excited moi*e scruples.
This great geometer distinguished several orders of them,
those of the second order might be neglected in relation
with the infinitely small of the first ; these infinitely
small of the first order in their turn disappeared before



64 CARNOT.

finite quantities. At eacli transformation of the formulae
it miglit be possible, according to this hierarchy, to dis-
embarrass one's self of fresh quantities ; and, nevertheless,
one was obliged to believe, to admit, that the definitive
results were i-igorously exact ; tliat the infinitesimal cal-
culus was not mei'ely a mere method of approximation.
Such was, considering the whole thing, the origin of the
strong and tenacious opposition which the new calculus
raised up at its birth ; such was also the difficulty which
a man equally celebrated as a geometer and a theologian,
Berkeley, bishop of Cloyne, had in view when he ex-
claimed, addressing himself to the incredulous in matters
of religion, " Look at the science of mathematics ; does
it not admit mysteries more incomprehensible than those
of religion ? "

Tiiese mysteries at the present day, exist no longer
for those who desire to become initiated in the knowl-
edge of the methods which constitute the differential
calculus in Newton's theory of fluxions ; in a paper
wherein D'Alembert introduces the consideration of the
limits towards which the ratios of the finite differences
of functions converge ; or, indeed, in Lagrange's Theory
of Analytical Functions. Nevertheless, Leibnitz's course
has prevailed, because it is more simple, easier to recol-
lect, and more convenient in practice. It is, then, im-
portant to study it in itself, to penetrate into its essence,
and to assure one's self of the perfect exactness of the
rules Avhich it furnishes, without the necessity of cor-
roborating them by the results of the calculus of fluxions,
or of limits, or of functions. That task, — I mean the
search for the true spirit of differential analysis, — forms
the principal oI)ject of the book which Carnot published,
in 1799, under the modest title of Rejlections on the



EPOCH OF THE REVOLUTION. G5

3Ietaphysics of the Infinitesimal Calculus. I am bold
enough to assert that the authors, otherwise so excellent,
of the best treatises on the differential calculus, have not
sutlicieutly consulted the work of our colleague. The
advantages which ought to result from the immediate in-
troduction of infinitely small or elementary quantities
into formulas ; the considerations by help of which it may
be proved that the calculator, by afterwards throwing
aside these quantities, will arrive nevertheless at mathe-
matically exact results, by means of certain compensations
for errors ; in a word, the fundamental and characteristic
traits of Leibnitz's method, are analyzed by Cax'not, with
a clearness, a certainty of judgment, and an ingenuity,
which we should look for in vain elsewhere, though the
question has been the object of the reflections and re-
searches of the greatest geometers of Europe.

CARNOT BEING "FRUCTIDORISE " IS OBLIGED TO RE-
CUR TO FLIGHT. HE IS ERASED FROM THE LIST

OF THE INSTITUTE, AND SUCCEEDED BY GENERAL
BONAPARTE.

France has always shown itself an idolater of military
glory. Satisfy this passion largely in a national war, and
you need not be uneasy about the administration of the
interior, however imperfect it may be. The sympathies
of the people, and in case of need even their entire sub-
mission, may be gained by any government that takes
care to adorn itself monthly with a new victory over its
external enemies. I perceive but one exception to this
rule in our annals. It is also requisite, however, that, by
an assimilation, too often deceitful, the legal representa-
tives of the country should be considered as the faithful
interpreters of the wishes, the sentiments, the opinions



66 CARNOT.

of the majority. The exception to Avhich I am about
to allude, will be furuii>hed by our Directorial govern-
ment.

When the elections of the year V. brought a reinforce-
ment of royalists to the two minorities of the Council
of Five-Hundred, and of the Elders, who till then had
limited themselves to making a very moderate opposition
to the Directory ; when, strong in what they thought the
popular support, the minority, fancying that they had be-
come the majority, took off the mask so far as to name
for the presidency of the Council of Five-Hundred that
same Pichegru, who not long before had branded with
treason the laurels that he had gained in Holland in the
name of the Republic ; when the enemies of the Direc-
torial power openly unveiled their projects in the saloons
of the celebrated Clichy Club ; when the recriminations,
the reciprocal accusations, that had reached the utmost
violence, were already succeeded by deeds of violence
against patriots, and the gainers of national property, —
our troops were yet everj'where triumphant. The army
of the Rhine and Moselle under the orders of Moreau,
the army of the Sambre and Meuse, commanded by
Jourdan, had gloriously crossed the Rhine ; they were
marching into the heart of Germany ; the army of Italy
was only twenty leagues from Vienna ; at Leoben, Bona-
parte signed the preliminaries of the much wished-for
treaty of peace. Without compi'omising the negotiations,
he could show himself touchy about mere questions of
etiquette ; he could bluntly refuse to let the name of
the Emperor of Germany precede that of the French
Republic in the protocols ; he could also, when General
Meerwald, and the Marquis del Gallo talked to him
about gratitude, answer, without a boast, in the following



EPOCH OF THE REVOLUTION. 67

memorable words : " The French Republic does not re-



Online LibraryF. (François) AragoBiographies of distinguished scientific men (Volume 2) → online text (page 5 of 38)