action of the sun and moon upon the terrestrial spheroid._--The tropical
year will not continue indefinitely to diminish in length. When it has
once attained a certain minimum value, it will then increase until
finally having attained an extreme value in the opposite direction, it
will again begin to diminish, and thus it will perpetually oscillate
between certain fixed limits. Laplace found that the extent to which the
tropical year is liable to vary from this cause, amounts to thirty-eight
seconds. If it were not for the effect produced upon the inclination of
the equator to the ecliptic by the mean disturbing action of the sun and
moon upon the terrestrial spheroid, the extent of variation would amount
to 162 seconds.
_Motion of the perihelion of the terrestrial orbit._--The major axis of
the orbit of each planet is in a state of continual movement from the
disturbing action of the other planets. In some cases, it makes the
complete tour of the heavens; in others, it merely oscillates around a
mean position. In the case of the earth's orbit, the perihelion is
slowly advancing in the same direction as that in which all the planets
are revolving around the sun. The alteration of its position with
respect to the stars amounts to about 11" in a year, but since the
equinox is regressing in the opposite direction at the rate of 50" in a
year, the whole annual variation of the longitude of the terrestrial
perihelion amounts to 61". Laplace has considered two remarkable epochs
in connection with this fact; viz: the epoch at which the major axis of
the earth's orbit coincided with the line of the equinoxes, and the
epoch at which it stood perpendicular to that line. By calculation, he
found the former of these epochs to be referable to the year 4107,
B.C., and the latter to the year 1245, A.D. He accordingly suggested
that the latter should be used as a universal epoch for the regulation
of chronological occurrences.
(B.)
The _Mécanique Céleste_.--This stupendous monument of intellectual
research consists, as stated by the author, of five quarto volumes. The
subject-matter is divided into sixteen books, and each book again is
subdivided into several chapters. Vol. I. contains the first and second
books of the work; Vol. II. contains the third, fourth, and fifth books;
Vol. III. contains the sixth and seventh books; Vol. IV. contains the
eighth, ninth, and tenth books; and, finally, Vol. V. contains the
remaining six books. In the first book the author treats of the general
laws of equilibrium and motion. In the second book he treats of the law
of gravitation, and the movements of the centres of gravity of the
celestial bodies. In the third book he investigates the subject of the
figures of the celestial bodies. In the fourth book he considers the
oscillations of the ocean and the atmosphere, arising from the
disturbing action of the celestial bodies. The fifth book is devoted to
the investigation of the movements of the celestial bodies around their
centres of gravity. In this book the author gives a solution of the
great problems of the precession of the equinoxes and the libration of
the moon, and determines the conditions upon which the stability of
Saturn's ring depends. The sixth book is devoted to the theory of the
planetary movements; the seventh, to the lunar theory; the eighth, to
the theory of the satellites of Jupiter, Saturn, and Uranus; and the
ninth, to the theory of comets. In the tenth book the author
investigates various subjects relating to the system of the universe.
Among these may be mentioned the theory of astronomical refractions;
the determination of heights by the barometer; the investigation of the
effects produced on the movements of the planets and comets by a
resisting medium; and the determination of the values of the masses of
the planets and satellites. In the six books forming the fifth volume of
the work, the author, besides presenting his readers with an historical
exposition of the labours of Newton and his successors on the theory of
gravitation, gives an account of various researches relative to the
system of the universe, which had occupied his attention subsequently to
the publication of the previous volumes. In the eleventh book he
considers the subjects of the figure and rotation of the earth. In the
twelfth book he investigates the attraction and repulsion of spheres,
and the laws of equilibrium and motion of elastic fluids. The thirteenth
book is devoted to researches on the oscillations of the fluids which
cover the surfaces of the planets; the fourteenth, to the subject of the
movements of the celestial bodies around their centres of gravity; the
fifteenth, to the movements of the planets and comets; and the
sixteenth, to the movements of the satellites. The author published a
supplement to the third volume, containing the results of certain
researches on the planetary theory, and a supplement to the tenth book,
in which he investigates very fully the theory of capillary attraction.
There was also published a posthumous supplement to the fifth volume,
the manuscript of which was found among his papers after his death.
JOSEPH FOURIER.
BIOGRAPHY READ AT A PUBLIC ASSEMBLY OF THE ACADEMY OF SCIENCES, ON THE
18TH OF NOVEMBER, 1833.
Gentlemen,--In former times one academician differed from another only
in the number, the nature, and the brilliancy of his discoveries. Their
lives, thrown in some respects into the same mould, consisted of events
little worthy of remark. A boyhood more or less studious; progress
sometimes slow, sometimes rapid; inclinations thwarted by capricious or
shortsighted parents; inadequacy of means, the privations which it
introduces in its train; thirty years of a laborious professorship and
difficult studies,--such were the elements from which the admirable
talents of the early secretaries of the Academy were enabled to execute
those portraits, so piquant, so lively, and so varied, which form one of
the principal ornaments of your learned collections.
In the present day, biographies are less confined in their object. The
convulsions which France has experienced in emancipating herself from
the swaddling-clothes of routine, of superstition and of privilege, have
cast into the storms of political life citizens of all ages, of all
conditions, and of all characters. Thus has the Academy of Sciences
figured during forty years in the devouring arena, wherein might and
right have alternately seized the supreme power by a glorious sacrifice
of combatants and victims!
Recall to mind, for example, the immortal National Assembly. You will
find at its head a modest academician, a patern of all the private
virtues, the unfortunate Bailly, who, in the different phases of his
political life, knew how to reconcile a passionate affection for his
country with a moderation which his most cruel enemies themselves have
been compelled to admire.
When, at a later period, coalesced Europe launched against France a
million of soldiers; when it became necessary to organize for the crisis
fourteen armies, it was the ingenious author of the _Essai sur les
Machines_ and of the _Géométrie des Positions_ who directed this
gigantic operation. It was, again, Carnot, our honourable colleague, who
presided over the incomparable campaign of seventeen months, during
which French troops, novices in the profession of arms, gained eight
pitched battles, were victorious in one hundred and forty combats,
occupied one hundred and sixteen fortified places and two hundred and
thirty forts or redoubts, enriched our arsenals with four thousand
cannon and seventy thousand muskets, took a hundred thousand prisoners,
and adorned the dome of the Invalides with ninety flags. During the same
time the Chaptals, the Fourcroys, the Monges, the Berthollets rushed
also to the defence of French independence, some of them extracting from
our soil, by prodigies of industry, the very last atoms of saltpetre
which it contained; others transforming, by the aid of new and rapid
methods, the bells of the towns, villages, and smallest hamlets into a
formidable artillery, which our enemies supposed, as indeed they had a
right to suppose, we were deprived of. At the voice of his country in
danger, another academician, the young and learned Meunier, readily
renounced the seductive pursuits of the laboratory; he went to
distinguish himself upon the ramparts of Königstein, to contribute as a
hero to the long defence of Mayence, and met his death, at the age of
forty years only, after having attained the highest position in a
garrison wherein shone the Aubert-Dubayets, the Beaupuys, the Haxos, the
Klebers.
How could I forget here the last secretary of the original Academy?
Follow him into a celebrated Assembly, into that Convention, the
sanguinary delirium of which we might almost be inclined to pardon, when
we call to mind how gloriously terrible it was to the enemies of our
independence, and you will always see the illustrious Condorcet occupied
exclusively with the great interests of reason and humanity. You will
hear him denounce the shameful brigandage which for two centuries laid
waste the African continent by a system of corruption; demand in a tone
of profound conviction that the Code be purified of the frightful stain
of capital punishment, which renders the error of the judge for ever
irreparable. He is the official organ of the Assembly on every occasion
when it is necessary to address soldiers, citizens, political parties,
or foreign nations in language worthy of France; he is not the tactician
of any party, he incessantly entreats all of them to occupy their
attention less with their own interests and a little more with public
matters; he replies, finally, to unjust reproaches of weakness by acts
which leave him the only alternative of the poison cup or the scaffold.
The French Revolution thus threw the learned geometer, whose discoveries
I am about to celebrate, far away from the route which destiny appeared
to have traced out for him. In ordinary times it would be about Dom[40]
Joseph Fourier that the secretary of the Academy would have deemed it
his duty to have occupied your attention. It would be the tranquil, the
retired life of a Benedictine which he would have unfolded to you. The
life of our colleague, on the contrary, will be agitated and full of
perils; it will pass into the fierce contentions of the forum and amid
the hazards of war; it will be a prey to all the anxieties which
accompany a difficult administration. We shall find this life intimately
associated with the great events of our age. Let us hasten to add, that
it will be always worthy and honourable, and that the personal qualities
of the man of science will enhance the brilliancy of his discoveries.
FOOTNOTE:
[40] An abbreviation of Dominus, equivalent to the English prefix
Reverend.--_Translator_.
BIRTH OF FOURIER.--HIS YOUTH.
Fourier was born at Auxerre on the 21st of March, 1768. His father, like
that of the illustrious geometer Lambert, was a tailor. This
circumstance would formerly have occupied a large place in the _éloge_
of our learned colleague; thanks to the progress of enlightened ideas, I
may mention the circumstance as a fact of no importance: nobody, in
effect, thinks in the present day, nobody even pretends to think, that
genius is the privilege of rank or fortune.
Fourier became an orphan at the age of eight years. A lady who had
remarked the amiability of his manners and his precocious natural
abilities, recommended him to the Bishop of Auxerre. Through the
influence of this prelate, Fourier was admitted into the military school
which was conducted at that time by the Benedictines of the Convent of
St. Mark. There he prosecuted his literary studies with surprising
rapidity and success. Many sermons very much applauded at Paris in the
mouth of high dignitaries of the Church were emanations from the pen of
the schoolboy of twelve years of age. It would be impossible in the
present day to trace those first compositions of the youth Fourier,
since, while divulging the plagiarism, he had the discretion never to
name those who profited by it.
At thirteen years Fourier had the petulance, the noisy vivacity of most
young people of the same age; but his character changed all at once, and
as if by enchantment, as soon as he was initiated in the first
principles of mathematics, that is to say, as soon as he became sensible
of his real vocation. The hours prescribed for study no longer sufficed
to gratify his insatiable curiosity. Ends of candles carefully collected
in the kitchen, the corridors and the refectory of the college, and
placed on a hearth concealed by a screen, served during the night to
illuminate the solitary studies by which Fourier prepared himself for
those labours which were destined, a few years afterwards, to adorn his
name and his country.
In a military school directed by monks, the minds of the pupils
necessarily waver only between two careers in life--the church and the
sword. Like Descartes, Fourier wished to be a soldier; like that
philosopher, he would doubtless have found the life of a garrison very
wearisome. But he was not permitted to make the experiment. His demand
to undergo the examination for the artillery, although strongly
supported by our illustrious colleague Legendre, was rejected with a
severity of expression of which you may judge yourselves: "Fourier,"
replied the minister, "not being noble, could not enter the artillery,
although he were a second Newton."
Gentlemen, there is in the strict enforcement of regulations, even when
they are most absurd, something respectable which I have a pleasure in
recognizing; in the present instance nothing could soften the odious
character of the minister's words. It is not true in reality that no one
could formerly enter into the artillery who did not possess a title of
nobility; a certain fortune frequently supplied the want of parchments.
Thus it was not a something undefinable, which, by the way, our
ancestors the Franks had not yet invented, that was wanting to young
Fourier, but rather an income of a few hundred livres, which the men who
were then placed at the head of the country would have refused to
acknowledge the genius of Newton as a just equivalent for! Treasure up
these facts, Gentlemen; they form an admirable illustration of the
immense advances which France has made during the last forty years.
Posterity, moreover, will see in this, not the excuse, but the
explanation of some of those sanguinary dissensions which stained our
first revolution.
Fourier not having been enabled to gird on the sword, assumed the habit
of a Benedictine, and repaired to the Abbey of St. Benoît-sur-Loire,
where he intended to pass the period of his noviciate. He had not yet
taken any vows when, in 1789, every mind was captivated with beautifully
seductive ideas relative to the social regeneration of France. Fourier
now renounced the profession of the Church; but this circumstance did
not prevent his former masters from appointing him to the principal
chair of mathematics in the Military School of Auxerre, and bestowing
upon him numerous tokens of a lively and sincere affection. I venture to
assert that no event in the life of our colleague affords a more
striking proof of the goodness of his natural disposition and the
amiability of his manners. It would be necessary not to know the human
heart to suppose that the monks of St. Benoît did not feel some chagrin
upon finding themselves so abruptly abandoned, to imagine especially
that they should give up without lively regret the glory which the order
might have expected from the ingenious colleague who had just escaped
from them.
Fourier responded worthily to the confidence of which he had just become
the object. When his colleagues were indisposed, the titular professor
of mathematics occupied in turns the chairs of rhetoric, of history, and
of philosophy; and whatever might be the subject of his lectures, he
diffused among an audience which listened to him with delight, the
treasures of a varied and profound erudition, adorned with all the
brilliancy which the most elegant diction could impart to them.
MEMOIR ON THE RESOLUTION OF NUMERICAL EQUATIONS.
About the close of the year 1789 Fourier repaired to Paris and read
before the Academy of Sciences a memoir on the resolution of numerical
equations of all degrees. This work of his early youth our colleague, so
to speak, never lost sight of. He explained it at Paris to the pupils of
the Polytechnic School; he developed it upon the banks of the Nile in
presence of the Institute of Egypt; at Grenoble, from the year 1802, it
was his favourite subject of conversation with the Professors of the
Central School and of the Faculty of Sciences; this finally, contained
the elements of the work which Fourier was engaged in seeing through the
press when death put an end to his career.
A scientific subject does not occupy so much space in the life of a man
of science of the first rank without being important and difficult. The
subject of algebraic analysis above mentioned, which Fourier had studied
with a perseverance so remarkable, is not an exception to this rule. It
offers itself in a great number of applications of calculation to the
movements of the heavenly bodies, or to the physics of terrestrial
bodies, and in general in the problems which lead to equations of a high
degree. As soon as he wishes to quit the domain of abstract relations,
the calculator has occasion to employ the roots of these equations; thus
the art of discovering them by the aid of an uniform method, either
exactly or by approximation, did not fail at an early period to excite
the attention of geometers.
An observant eye perceives already some traces of their efforts in the
writings of the mathematicians of the Alexandrian School. These traces,
it must be _acknowledged_, are so slight and so imperfect, that we
should truly be justified in referring the origin of this branch of
analysis only to the excellent labours of our countryman Vieta.
Descartes, to whom we render very imperfect justice when we content
ourselves with saying that he taught us much when he taught us to doubt,
occupied his attention also for a short time with this problem, and left
upon it the indelible impress of his powerful mind. Hudde gave for a
particular but very important case rules to which nothing has since been
added; Rolle, of the Academy of Sciences, devoted to this one subject
his entire life. Among our neighbours on the other side of the channel,
Harriot, Newton, Maclaurin, Stirling, Waring, I may say all the
illustrious geometers which England produced in the last century, made
it also the subject of their researches. Some years afterwards the names
of Daniel Barnoulli, of Euler, and of Fontaine came to be added to so
many great names. Finally, Lagrange in his turn embarked in the same
career, and at the very commencement of his researches he succeeded in
substituting for the imperfect, although very ingenious, essays of his
predecessors, a complete method which was free from every objection.
From that instant the dignity of science was satisfied; but in such a
case it would not be permitted to say with the poet:
"Le temps ne fait rien à l'affaire."
Now although the processes invented by Lagrange, simple in principle and
applicable to every case, have theoretically the merit of leading to the
result with certainty, still, on the other hand, they demand
calculations of a most repulsive length. It remained then to perfect the
practical part of the question; it was necessary to devise the means of
shortening the route without depriving it in any degree of its
certainty. Such was the principal object of the researches of Fourier,
and this he has attained to a great extent.
Descartes had already found, in the order according to which the signs
of the different terms of any numerical equation whatever succeed each
other, the means of deciding, for example, how many real positive roots
this equation may have. Fourier advanced a step further; he discovered a
method for determining what number of the equally positive roots of
every equation may be found included between two given quantities. Here
certain calculations become necessary, but they are very simple, and
whatever be the precision desired, they lead without any trouble to the
solutions sought for.
I doubt whether it were possible to cite a single scientific discovery
of any importance which has not excited discussions of priority. The new
method of Fourier for solving numerical equations is in this respect
amply comprised within the common law. We ought, however, to acknowledge
that the theorem which serves as the basis of this method, was first
published by M. Budan; that according to a rule which the principal
Academies of Europe have solemnly sanctioned, and from which the
historian of the sciences dares not deviate without falling into
arbitrary assumptions and confusion, M. Budan ought to be considered as
the inventor. I will assert with equal assurance that it would be
impossible to refuse to Fourier the merit of having attained the same
object by his own efforts. I even regret that, in order to establish
rights which nobody has contested, he deemed it necessary to have
recourse to the certificates of early pupils of the Polytechnic School,
or Professors of the University. Since our colleague had the modesty to
suppose that his simple declaration would not be sufficient, why (and
the argument would have had much weight) did he not remark in what
respect his demonstration differed from that of his competitor?--an
admirable demonstration, in effect, and one so impregnated with the
elements of the question, that a young geometer, M. Sturm, has just
employed it to establish the truth of the beautiful theorem by the aid
of which he determines not the simple limits, but the exact number of
roots of any equation whatever which are comprised between two given
quantities.
PART PLAYED BY FOURIER IN OUR REVOLUTION.--HIS ENTRANCE INTO THE CORPS
OF PROFESSORS OF THE NORMAL SCHOOL AND THE POLYTECHNIC
SCHOOL.--EXPEDITION TO EGYPT.
We had just left Fourier at Paris, submitting to the Academy of Sciences
the analytical memoir of which I have just given a general view. Upon
his return to Auxerre, the young geometer found the town, the
surrounding country, and even the school to which he belonged, occupied
intensely with the great questions relative to the dignity of human
nature, philosophy, and politics, which were then discussed by the
orators of the different parties of the National Assembly. Fourier
abandoned himself also to this movement of the human mind. He embraced
with enthusiasm the principles of the Revolution, and he ardently
associated himself with every thing grand, just, and generous which the
popular impulse offered. His patriotism made him accept the most
difficult missions. We may assert, that never, even when his life was at
stake, did he truckle to the base, covetous, and sanguinary passions
which displayed themselves on all sides.