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founded on new theorems, whose object we could not in this place make known. The method which is derived from them leaves nothing vague and indeterminate in the solutions, it leads them up to the final numerical applications, a necessary condition of every investigation, without which we should only arrive at useless transformations.

The same theorems which have made known to us the equations of the movement of heat, apply directly to certain problems of general analysis and dynamics whose solution has for a long time been desired.

Profound study of nature is the most fertile source of mathematical discoveries. Not only has this study, in offering a determinate object to investigation, the advantage of excluding vague questions and calculations without issue; it is besides a sure method of forming analysis itself, and of discovering the elements which it concerns us to know, and which natural science ought always to preserve: these are the fundamental elements which are reproduced in all natural effects.

We see, for example, that the same expression whose abstract properties geometers had considered, and which in this respect belongs to general analysis, represents as well the motion of light in the atmosphere, as it determines the laws of diffusion of heat in solid matter, and enters into all the chief problems of the theory of probability.

The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things.

Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.

Its chief attribute is clearness; it has no marks to express con

fused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them. If matter escapes us, as that of air and light, by its extreme tenuity, if bodies are placed far from us in the immensity of space, if man wishes to know the aspect of the heavens at successive epochs separated by a great number of centuries, if the actions of gravity and of heat are exerted in the interior of the earth at depths which will be always inaccessible, mathematical analysis can yet lay hold of the laws of these phenomena. It makes them present and measurable, and seems to be a faculty of the human mind destined to supplement the shortness of life and the imperfection of the senses; and what is still more remarkable, it follows the same course in the study of all phenomena; it interprets them by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes.

The problems of the theory of heat present so many examples of the simple and constant dispositions which spring from the general laws of nature; and if the order which is established in these phenomena could be grasped by our senses, it would produce in us an impression comparable to the sensation of musical sound.

The forms of bodies are infinitely varied; the distribution of the heat which penetrates them seems to be arbitrary and confused; but all the inequalities are rapidly cancelled and disappear as time passes on. The progress of the phenomenon becomes more regular and simpler, remains finally subject to a definite law which is the same in all cases, and which bears no sensible impress of the initial arrangement.

All observation confirms these consequences. The analysis from which they are derived separates and expresses clearly, 1o the general conditions, that is to say those which spring from the natural properties of heat, 2o the effect, accidental but continued, of the form or state of the surfaces; 3° the effect, not permanent, of the primitive distribution.

In this work we have demonstrated all the principles of the theory of heat, and solved all the fundamental problems. They could have been explained more concisely by omitting the simpler problems, and presenting in the first instance the most general results; but we wished to shew the actual origin of the theory and

its gradual progress. When this knowledge has been acquired and the principles thoroughly fixed, it is preferable to employ at once the most extended analytical methods, as we have done in the later investigations. This is also the course which we shall hereafter follow in the memoirs which will be added to this work, and which will form in some manner its complement'; and by this means we shall have reconciled, so far as it can depend on ourselves, the necessary development of principles with the precision which becomes the applications of analysis.

The subjects of these memoirs will be, the theory of radiant heat, the problem of the terrestrial temperatures, that of the temperature of dwellings, the comparison of theoretic results with those which we have observed in different experiments, lastly the demonstrations of the differential equations of the movement of heat in fluids.

The work which we now publish has been written a long time since; different circumstances have delayed and often interrupted the printing of it. In this interval, science has been enriched by important observations; the principles of our analysis, which had not at first been grasped, have become better known; the results which we had deduced from them have been discussed and confirmed. We ourselves have applied these principles to new problems, and have changed the form of some of the proofs. The delays of publication will have contributed to make the work clearer and more complete.

The subject of our first analytical investigations on the transfer of heat was its distribution amongst separated masses; these have been preserved in Chapter III., Section II. The problems relative to continuous bodies, which form the theory rightly so called, were solved many years afterwards; this theory was explained for the first time in a manuscript work forwarded to the Institute of France at the end of the year 1807, an extract from which was published in the Bulletin des Sciences (Société Philomatique, year 1808, page 112). We added to this memoir, and successively forwarded very extensive notes, concerning the convergence of series, the diffusion of heat in an infinite prism, its emission in spaces

1 These memoirs were never collectively published as a sequel or complement to the Théorie Analytique de la Chaleur. But, as will be seen presently, the author had written most of them before the publication of that work in 1822. [A. F.]

void of air, the constructions suitable for exhibiting the chief theorems, and the analysis of the periodic movement at the surface of the earth. Our second memoir, on the propagation of heat, was deposited in the archives of the Institute, on the 28th of September, 1811. It was formed out of the preceding memoir and the notes already sent in; the geometrical constructions and those details of analysis which had no necessary relation to the physical problem were omitted, and to it was added the general equation which expresses the state of the surface. This second work was sent to press in the course of 1821, to be inserted in the collection of the Academy of Sciences. It is printed without any change or addition; the text agrees literally with the deposited manuscript, which forms part of the archives of the Institute'.

In this memoir, and in the writings which preceded it, will be found a first explanation of applications which our actual work

It appears as a memoir and supplement in volumes IV. and V. of the Mémoires de l'Académie des Sciences. For convenience of comparison with the table of contents of the Analytical Theory of Heat, we subjoin the titles and heads of the chapters of the printed memoir:

THÉORIE DU MOUVEMENT DE LA CHALEUR DANS LES CORPS SOLIDES, PAR M. FOURIER. [Mémoires de l'Académie Royale des Sciences de l'Institut de France. Tome IV. (for year 1819). Paris 1824.]

I. Exposition.

II. Notions générales et définitions préliminaires.

III. Equations du mouvement de la chaleur.

IV. Du mouvement linéaire et varié de la chaleur dans une armille.

V. De la propagation de la chaleur dans une lame rectangulaire dont les températures

sont constantes.

VI. De la communication de la chaleur entre des masses disjointes.

VII. Du mouvement varié de la chaleur dans une sphère solide.

VIII. Du mouvement varié de la chaleur dans un cylindre solide.

IX. De la propagation de la chaleur dans un prisme dont l'extrémité est assujettie à une température constante.

X. Du mouvement varié de la chaleur dans un solide de forme cubique.

XI. Du mouvement linéaire et varié de la chaleur dans les corps dont une dimension est infinie.

SUITE DU MÉMOIRE INTITULÉ: THEORIE DU MOUVEMENT DE LA CHALEUR DANS LES CORPS SOLIDES; PAR M. FOURIER. [Mémoires de l'Académie Royale des Sciences de l'Institut de France. Tome V. (for year 1820). Paris, 1826.]

XII. Des températures terrestres, et du mouvement de la chaleur dans l'intérieur d'une sphère solide, dont la surface est assujettie à des changemens périodiques de température.

XIII. Des lois mathématiques de l'équilibre de la chaleur rayonnante.

XIV.

Comparaison des résultats de la théorie avec ceux de diverses expériences. [A. F.]

ness.

does not contain; they will be treated in the subsequent memoirs' at greater length, and, if it be in our power, with greater clearThe results of our labours concerning the same problems. are also indicated in several articles already published. The extract inserted in the Annales de Chimie et de Physique shews the aggregate of our researches (Vol. III. page 350, year 1816). We published in the Annales two separate notes, concerning radiant heat (Vol. IV. page 128, year 1817, and Vol. VI. page 259, year 1817).

Several other articles of the same collection present the most constant results of theory and observation; the utility and the extent of thermological knowledge could not be better appreciated than by the celebrated editors of the Annales".

In the Bulletin des Sciences (Société philomatique year 1818, page 1, and year 1820, page 60) will be found an extract from a memoir on the constant or variable temperature of dwellings, and an explanation of the chief consequences of our analysis of the terrestrial temperatures.

M. Alexandre de Humboldt, whose researches embrace all the great problems of natural philosophy, has considered the observations of the temperatures proper to the different climates from a novel and very important point of view (Memoir on Isothermal lines, Société d'Arcueil, Vol. III. page 462); (Memoir on the inferior limit of perpetual snow, Annales de Chimie et de Physique, Vol. v. page 102, year 1817).

3

As to the differential equations of the movement of heat in fluids mention has been made of them in the annual history of the Academy of Sciences. The extract from our memoir shews clearly its object and principle. (Analyse des travaux de l'Académie des Sciences, by M. De Lambre, year 1820.)

The examination of the repulsive forces produced by heat, which determine the statical properties of gases, does not belong

1 See note, page 9, and the notes, pages 11-13.

2 Gay-Lussac and Arago. See note, p. 13.

3 Mémoires de l'Académie des Sciences, Tome XII., Paris, 1833, contain on pp. 507–514, Mémoire d'analyse sur le mouvement de la chaleur dans les fluides, par M. Fourier. Lu à l'Académie Royale des Sciences, 4 Sep. 1820. It is followed on pp. 515 530 by Extrait des notes manuscrites conservées par l'auteur. The memoir is signed Jh. Fourier, Paris, 1 Sep. 1820, but was published after the death of the author. [A. F.]

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