emissive power operate? Leslie first sought the solution of this
important question. His observations seem to show that the intensities
of the radiating rays are proportional (it is necessary, Gentlemen, that
I employ the scientific expression) to the sines of the angles which
these rays form with the heated surface. But the quantities upon which
the experimenter had to operate were too feeble; the uncertainties of
the thermometric estimations compared with the total effect were, on the
contrary, too great not to inspire a strong degree of distrust: well,
Gentlemen, a problem before which all the processes, all the instruments
of modern physics have remained powerless, Fourier has completely solved
without the necessity of having recourse to any new experiment. He has
traced the law of the emission of caloric sought for, with a perspicuity
which one cannot sufficiently admire, in the most ordinary phenomena of
temperature, in the phenomena which at first sight appeared to be
entirely independent of it.
Such is the privilege of genius; it perceives, it seizes relations where
vulgar eyes see only isolated facts.
Nobody doubts, and besides experiment has confirmed the fact, that in
all the points of a space terminated by any envelop maintained at a
constant temperature, we ought also to experience a constant
temperature, and precisely that of the envelop. Now Fourier has
established, that if the calorific rays emitted were equally intense in
all directions, if the intensity did not vary proportionally to the sine
of the angle of emission, the temperature of a body situated in the
enclosure would depend on the place which it would occupy there: _that
the temperature of boiling water or of melting iron, for example, would
exist in certain points of a hollow envelop of glass!_ In all the vast
domain of the physical sciences, we should be unable to find a more
striking application of the celebrated method of the _reductio ad
absurdum_ of which the ancient mathematicians made use, in order to
demonstrate the abstract truths of geometry.
I shall not quit this first part of the labours of Fourier without
adding, that he has not contented himself with demonstrating with so
much felicity the remarkable law which connects the comparative
intensities of the calorific rays, emanating under all angles from
heated bodies; he has sought, moreover, the physical cause of this law,
and he has found it in a circumstance which his predecessors had
entirely neglected. Let us suppose, says he, that bodies emit heat not
only from the molecules of their surfaces, but also from the particles
in the interior. Let us suppose, moreover, that the heat of these latter
particles cannot arrive at the surface by traversing a certain thickness
of matter without undergoing some degree of absorption. Fourier has
reduced these two hypotheses to calculation, and he has hence deduced
mathematically the experimental law of the sines. After having resisted
so radical a test, the two hypotheses were found to be completely
verified, they have become laws of nature; they point out latent
properties of caloric which could only be discerned by the eye of the
intellect.
In the second question treated by Fourier, heat presents itself under a
new form. There is more difficulty in following its movements; but the
conclusions deducible from the theory are also more general and more
important.
Heat excited, concentrated into a certain point of a solid body,
communicates itself by way of conduction, first to the particles nearest
the heated point, then gradually to all the regions of the body. Whence
the problem of which the following is the enunciation.
By what routes, and with what velocities, is the propagation of heat
effected in bodies of different forms and different natures subjected to
certain initial conditions?
Fundamentally, the Academy of Sciences had already proposed this problem
as the subject of a prize as early as the year 1736. Then the terms heat
and caloric were not in use; it demanded _the study of nature, and the
propagation_ OF FIRE! The word _fire_, thrown thus into the
programme without any other explanation, gave rise to a mistake of the
most singular kind. The majority of philosophers imagined that the
question was to explain in what way _burning_ communicates itself, and
increases in a mass of combustible matter. Fifteen competitors presented
themselves; _three_ were crowned.
This competition was productive of very meagre results. However, a
singular combination of circumstances and of proper names will render
the recollection of it lasting.
Has not the public a right to be surprised upon reading this Academic
declaration: "the question affords no handle to geometry!" In matter of
inventions, to attempt to dive into the future, is to prepare for one's
self striking mistakes. One of the competitors, the great Euler, took
these words in their literal sense; the reveries with which his memoir
abounds, are not compensated in this instance by any of those brilliant
discoveries in analysis, I had almost said of those sublime
inspirations, which were so familiar to him. Fortunately Euler appended
to his memoir a supplement truly worthy of his genius. Father Lozeran de
Fiesc and the Count of Créqui were rewarded with the high honour of
seeing their names inscribed beside that of the illustrious geometer,
although it would be impossible in the present day to discern in their
memoirs any kind of merit, not even that of politeness, for the courtier
said rudely to the Academy: "the question, which you have raised,
interests only the curiosity of mankind."
Among the competitors less favourably treated, we perceive one of the
greatest writers whom France has produced; the author of the _Henriade_.
The memoir of Voltaire was, no doubt, far from solving the problem
proposed; but it was at least distinguished by elegance, clearness, and
precision of language; I shall add, by a severe style of argument; for
if the author occasionally arrives at questionable results, it is only
when he borrows false data from the chemistry and physics of the
epoch,--sciences which had just sprung into existence. Moreover, the
anti-Cartesian colour of some of the parts of the memoir of Voltaire was
calculated to find little favour in a society, where Cartesianism, with
its incomprehensible vortices, was everywhere held in high estimation.
We should have more difficulty in discovering the causes of the failure
of a fourth competitor, Madame the Marchioness du Châtelet, for she also
entered into the contest instituted by the Academy. The work of Emilia
was not only an elegant portrait of all the properties of heat, known
then to physical inquirers, there were remarked moreover in it,
different projects of experiments, among the rest one which Herschel has
since developed, and from which he has derived one of the principal
flowers of his brilliant scientific crown.
While such great names were occupied in discussing this question,
physical inquirers of a less ambitious stamp laid experimentally the
solid basis of a future mathematical theory of heat. Some established,
that the same quantity of caloric does not elevate by the same number of
degrees equal weights of different substances, and thereby introduced
into the science the important notion of _capacity_. Others, by the aid
of observations no less certain, proved that heat, applied at the
extremity of a bar, is transmitted to the extreme parts with greater or
less velocity or intensity, according to the nature of the substance of
which the bar is composed; thus they suggested the original idea of
_conductibility_. The same epoch, if I were not precluded from entering
into too minute details, would present to us interesting experiments. We
should find that it is not true that, at all degrees of the thermometer,
the loss of heat of a body is proportional to the excess of its
temperature above that of the medium in which it is plunged; but I have
been desirous of showing you geometry penetrating, timidly at first,
into questions of the propagation of heat, and depositing there the
first germs of its fertile methods.
It is to Lambert of Mulhouse, that we owe this first step. This
ingenious geometer had proposed a very simple problem which any person
may comprehend. A slender metallic bar is exposed at one of its
extremities to the constant action of a certain focus of heat. The parts
nearest the focus are heated first. Gradually the heat communicates
itself to the more distant parts, and, after a short time, each point
acquires the maximum temperature which it can ever attain. Although the
experiment were to last a hundred years, the thermometric state of the
bar would not undergo any modification.
As might be reasonably expected, this maximum of heat is so much less
considerable as we recede from the focus. Is there any relation between
the final temperatures and the distances of the different particles of
the bar from the extremity directly heated? Such a relation exists. It
is very simple. Lambert investigated it by calculation, and experience
confirmed the results of theory.
In addition to the somewhat elementary question of the _longitudinal_
propagation of heat, there offered itself the more general but much more
difficult problem of the propagation of heat in a body of three
dimensions terminated by any surface whatever. This problem demanded the
aid of the higher analysis. It was Fourier who first assigned the
equations. It is to Fourier, also, that we owe certain theorems, by
means of which we may ascend from the differential equations to the
integrals, and push the solutions in the majority of cases to the final
numerical applications.
The first memoir of Fourier on the theory of heat dates from the year
1807. The Academy, to which it was communicated, being desirous of
inducing the author to extend and improve his researches, made the
question of the propagation of heat the subject of the great
mathematical prize which was to be awarded in the beginning of the year
1812. Fourier did, in effect, compete, and his memoir was crowned. But,
alas! as Fontenelle said: "In the country even of demonstrations, there
are to be found causes of dissension." Some restrictions mingled with
the favourable judgment. The illustrious commissioners of the prize,
Laplace, Lagrange, and Legendre, while acknowledging the novelty and
importance of the subject, while declaring that the real differential
equations of the propagation of heat were finally found, asserted that
they perceived difficulties in the way in which the author arrived at
them. They added, that his processes of integration left something to be
desired, even on the score of rigour. They did not, however, support
their opinion by any arguments.
Fourier never admitted the validity of this decision. Even at the close
of his life he gave unmistakable evidence that he thought it unjust, by
causing his memoir to be printed in our volumes without changing a
single word. Still, the doubts expressed by the Commissioners of the
Academy reverted incessantly to his recollection. From the very
beginning they had poisoned the pleasure of his triumph. These first
impressions, added to a high susceptibility, explain how Fourier ended
by regarding with a certain degree of displeasure the efforts of those
geometers who endeavoured to improve his theory. This, Gentlemen, was a
very strange aberration of a mind of so elevated an order! Our colleague
had almost forgotten that it is not allotted to any person to conduct a
scientific question to a definitive termination, and that the important
labours of D'Alembert, Clairaut, Euler, Lagrange, and Laplace, while
immortalizing their authors, have continually added new lustre to the
imperishable glory of Newton. Let us act so that this example may not be
lost. While the civil law imposes upon the tribunes the obligation to
assign the motives of _their judgments_, the academies, which are the
tribunes of science, cannot have even a pretext to escape from this
obligation. Corporate bodies, as well as individuals, act wisely when
they reckon in every instance only upon the authority of reason.
CENTRAL HEAT OF THE TERRESTRIAL GLOBE.
At any time the _Théorie Mathématique de la Chaleur_ would have excited
a lively interest among men of reflection, since, upon the supposition
of its being complete, it threw light upon the most minute processes of
the arts. In our time the numerous points of affinity existing between
it and the curious discoveries of the geologists, have made it, if I may
use the expression, a work for the occasion. To point out the ultimate
relation which exists between these two kinds of researches would be to
present the most important part of the discoveries of Fourier, and to
show how happily our colleague, by one of those inspirations reserved
for genius, had chosen the subject of his researches.
The parts of the earth's crust, which the geologists call the
sedimentary formations, were not formed all at once. The waters of the
ocean, on several former occasions, covered regions which are situated
in the present day in the centre of the continent. There they deposited,
in thin horizontal strata, a series of rocks of different kinds. These
rocks, although superposed like the layers of stones of a wall, must not
be confounded together; their dissimilarities are palpable to the least
practised eye. It is necessary also to note this capital fact, that
each stratum has a well-defined limit; that no process of transition
connects it with the stratum which it supports. The ocean, the original
source of all these deposits, underwent then formerly enormous changes
in its chemical composition to which it is no longer subject.
With some rare exceptions, resulting from local convulsions the effects
of which are otherwise manifest, the order of antiquity of the
successive strata of rocks which form the exterior crust of the globe
ought to be that of their superposition. The deepest have been formed at
the most remote epochs. The attentive study of these different envelops
may aid us in ascending the stream of time, even beyond the most remote
epochs, and enlightening us with respect to those stupendous revolutions
which periodically overwhelmed continents beneath the waters of the
ocean, or again restored them to their former condition. Crystalline
rocks of granite upon which the sea has effected its original deposits
have never exhibited any remains of life. Traces of such are to be found
only in the sedimentary strata.
Life appears to have first exhibited itself on the earth in the form of
vegetables. The remains of vegetables are all that we meet with in the
most ancient strata deposited by the waters; still, they belong to
plants of the simplest structure,--to ferns, to species of rushes, to
lycopodes.
As we ascend into the upper strata, vegetation becomes more and more
complex. Finally, near the surface, it resembles the vegetation actually
existing on the earth, with this characteristic circumstance, however,
which is well deserving attention, that certain vegetables which grow
only in southern climates, that the large palm-trees, for example, are
found in their fossil state in all latitudes, and even in the centre of
the frozen regions of Siberia.
In the primitive world, these northern regions enjoyed then, in winter,
a temperature at least equal to that which is experienced in the present
day under the parallels where the great palms commence to appear: at
Tobolsk, the inhabitants enjoyed the climate of Alicante or Algiers!
We shall deduce new proofs of this mysterious result from an attentive
examination of the size of plants.
There exist, in the present day, willow grass or marshy rushes, ferns,
and lycopodes, in Europe as well as in the tropical regions; but they
are not met with in large dimensions, except in warm countries. Thus, to
compare together the dimensions of the same plants is, in reality, to
compare, in respect to temperature, the regions where they are produced.
Well, place beside the fossil plants of our coal mines, I will not say
the analogous plants of Europe, but those which grow in the countries of
South America, and which are most celebrated for the richness of their
vegetation, and you will find the former to be of incomparably greater
dimensions than the latter.
The _fossil flora_ of France, England, Germany, and Scandinavia offer,
for example, ferns ninety feet high, the stalks being six feet in
diameter, or eighteen feet in circumference.
The _lycopodes_ which, in the present day, whether in cold or temperate
climates, are creeping-plants rising hardly to the height of a decimètre
above the soil; which even at the equator, under the most favourable
circumstances, do not attain a height of more than _one_ mètre, had in
Europe, in the primitive world, an altitude of twenty-five mètres.
One must be blind to all reason not to find, in these enormous
dimensions, a new proof of the high temperature enjoyed by our country
before the last irruptions of the ocean!
The study of _fossil animals_ is no less fertile in results. I should
digress from my subject if I were to examine here how the organization
of animals is developed upon the earth; what modifications, or more
strictly speaking, what complications it has undergone after each
cataclysm, or if I even stopped to describe one of those ancient epochs
during which the earth, the sea, and the atmosphere had for inhabitants
cold-blooded reptiles of enormous dimensions; tortoises with shells
three feet in diameter; lizards seventeen mètres long; pterodactyles,
veritable flying dragons of such strange forms, that they might be
classed on good grounds either among reptiles, among mammiferous
animals, or among birds. The object, which I have proposed, does not
require that I should enter into such details; a single remark will
suffice.
Among the bones contained in the strata nearest the present surface of
the earth, are those of the hippopotamus, the rhinoceros, and the
elephant. These remains of animals of warm countries are to be found in
all latitudes. Travellers have discovered specimens of them even at
Melville Island, where the temperature descends, in the present day, 50°
beneath zero. In Siberia they are found in such abundance as to have
become an article of commerce. Finally, upon the rocky shores of the
Arctic Ocean, there are to be found not merely fragments of skeletons,
but whole elephants still covered with their flesh and skin.
I should deceive myself very much, Gentlemen, if I were to suppose that
each of you had not deduced from these remarkable facts a conclusion no
less remarkable, to which indeed the fossil flora had already habituated
us; namely, that as they have grown older, the polar regions of the
earth have cooled down to a prodigious extent.
In the explanation of so curious a phenomenon, cosmologists have not
taken into account the existence of possible variations of the intensity
of the solar heat; and yet the stars, those distant suns, have not the
constant brightness which the common people attribute to them. Nay, some
of them have been observed to diminish in a sufficiently short space of
time to the hundredth part of their original brightness; and several
have even totally disappeared. They have preferred to attribute every
thing to an internal or primitive heat with which the earth was at some
former epoch impregnated, and which is gradually being dissipated in
space.
Upon this hypothesis the inhabitants of the polar regions, although
deprived of the sight of the sun for whole months together, must have
evidently enjoyed, at very ancient epochs, a temperature equal to that
of the tropical regions, wherein exist elephants in the present day.
It is not, however, as an explanation of the existence of elephants in
Siberia, that the idea of the intrinsic heat of the globe has entered
for the first time into science. Some savans had adopted it before the
discovery of those fossil animals. Thus, Descartes was of opinion that
originally (I cite his own words,) _the earth did not differ from the
sun in any other respect than in being smaller_. Upon this hypothesis,
then, it ought to be considered as an extinct sun.
Leibnitz conferred upon this hypothesis the honour of appropriating it
to himself. He attempted to deduce from it the mode of formation of the
different solid envelopes of which the earth consists. Buffon, also,
imparted to it the weight of his eloquent authority. According to that
great naturalist, the planets of our system are merely portions of the
sun, which the shock of a comet had detached from it some tens of
thousands of years ago.
In support of this igneous origin of the earth, Mairan and Buffon cited
already the high temperature of deep mines, and, among others, those of
the mines of Giromagny. It appears evident that if the earth was
formerly incandescent, we should not fail to meet in the interior
strata, that is to say, in those which ought to have cooled last, traces
of their primitive temperature. The observer who, upon penetrating into
the interior of the earth, did not find an increasing heat, might then
consider himself amply authorized to reject the hypothetical conceptions
of Descartes, of Mairan, of Leibnitz, and of Buffon. But has the
converse proposition the same certainty? Would not the torrents of heat,
which the sun has continued incessantly to launch for so many ages, have
diffused themselves into the mass of the earth, so as to produce there a
temperature increasing with the depth? This a question of high
importance. Certain easily satisfied minds conscientiously supposed that
they had solved it, when they stated that the idea of a constant
temperature was by far the _most natural_; but woe to the sciences if
they thus included vague considerations which escape all criticism,
among the motives for admitting and rejecting facts and theories!
Fontenelle, Gentlemen, would have traced their horoscope in these words,
so well adapted for humbling our pride, and the truth of which the
history of discoveries reveals in a thousand places: "When a thing may
be in two different ways, it is almost always that which appears at
first the least natural."
Whatever importance these reflections may possess, I hasten to add that,
instead of the arguments of his predecessors, which have no real value,
Fourier has substituted proofs, demonstrations; and we know what meaning
such terms convey to the Academy of Sciences.
In all places of the earth, as soon as we descend to a certain depth,
the thermometer no longer experiences either diurnal or annual
variation. It marks the same degree, and the same fraction of a degree,
from day to day, and from year to year. Such is the fact: what says
theory?
Let us suppose, for a moment, that the earth has constantly received all
its heat from the sun. Descend into its mass to a sufficient depth, and
you will find, with Fourier, by the aid of calculation, a constant
temperature for each day of the year. You will recognize further, that
this solar temperature of the inferior strata varies from one climate to
another; that in each country, finally, it ought to be always the same,
so long as we do not descend to depths which are too great relatively to
the earth's radius.
Well, the phenomena of nature stand in manifest contradiction to this
result. The observations made in a multitude of mines, observations of
the temperature of hot springs coming from different depths, have all
given an increase of one degree of the centigrade for every twenty or
thirty metres of depth. Thus, there was some inaccuracy in the
hypothesis which we were discussing upon the footsteps of our colleague.
It is not true that the temperature of the terrestrial strata may be
attributed solely to the action of the solar rays.
This being established, the increase of heat which is observed in all
climates when we penetrate into the interior of the globe, is the
manifest indication of an intrinsic heat. The earth, as Descartes and
Leibnitz maintained it to be, but without being able to support their
assertions by any demonstrative reasoning,--thanks to a combination of
the observations of physical inquirers with the analytical calculations
of Fourier,--is _an encrusted sun_, the high temperature of which may be
boldly invoked every time that the explanation of ancient geological
phenomena will require it.