It is easy to judge:

• how much these researches concern the physical sciences and civil economy
• what is their influence on the progress of the arts which require heat

They have also a necessary connection with the system of the world.

Their relations become known when we consider the grand phenomena which take place.

In fact, the radiation of the sun incessantly penetrates the air, the earth, and the waters.

Its elements are divided and change in direction every way.

Heat penetrates the Earth and would raise its mean temperature more and more had not they not escaped from the surface.*

Different climates result from the unequal exposure to the action of solar heat.

Climate is modified by several accessory causes, such as:

• elevation
• the form of the ground
• the neighbourhood and extent of continents and seas
• the state of the surface
• the direction of the winds.

The succession of day and night, the alternations of the seasons occasion in the solid earth periodic variations, which are repeated every day or every year.

But these changes become less and less sensible as the point at which they are measured recedes from the surface.

No diurnal variation can be detected at the depth of about 3 metres.

The annual variations cease to be appreciable at a depth much less than 60 metres.

The temperature at great depths is then sensibly fixed at a given place: but it is not the same at all points of the same meridian; in general it rises as the equator is approached.

The heat which the sun has communicated to the terrestrial globe, and which has produced the diversity of climates, is now subject to a movement which has become uniform.

It advances within the interior of the mass which it penetrates throughout, and at the same time recedes from the plane of the equator, and proceeds to lose itself across the polar regions.

In the higher regions of the atmosphere the air is very rare and transparent, and retains but a minute part of the heat of the solar rays: this is the cause of the excessive cold of elevated places.

The lower layers, denser and more heated by the land and water, expand and rise up: they are cooled by the very fact of expansion. The great movements of the air, such as the trade winds which blow between the tropics, are not determined by the attractive forces of the moon and sun.

The action of these celestial bodies produces scarcely perceptible oscillations in a fluid so rare and at so great a distance. It is the changes of temperature which periodically displace every part of the atmosphere.

The waters of the ocean are differently exposed at their surface to the rays of the sun.

The bottom of the basin which contains them is heated very unequally from the poles to the equator.

These 2 causes, ever present, and combined with gravity and the centrifugal force, keep up vast movements in the interior of the seas.

They displace and mingle all the parts, and produce those general and regular currents. which navigators have noticed.

Radiant heat which escapes from the surface of all bodies, and traverses elastic media, or spaces void of air, has special laws, and occurs with widely varied phenomena.

My mathematical theory gives an exact measure of them.

It:

• uses new catoptrics which has its own theorems
• determines by analysis all the effects of heat direct or reflected.

The enumeration of the chief objects of the theory sufficiently shews the nature of the questions which I have proposed to myself.

What are the elementary properties which it is requisite to observe in each substance, and what are the experiments most suitable to determine them exactly?

If the distribution of heat in solid matter is regulated by constant laws, what is the mathematical expression of those laws, and by what analysis may we derive from this expression the complete solution of equations, the laws of such a compound effect.

What is the resulting change in the general equations of hydrodynamics?

Such are the chief problems which I have solved, and which have never yet been submitted to calculation.

If we consider further the manifold relations of this mathematical theory to civil uses and the technical arts, we shall recognize completely the extent of its applications.

It includes an entire series of distinct phenomena, and that the study of it cannot be omitted without losing a notable part of the science of nature.

The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts, the causes of which are not considered by geometers, but which they admit as the results of common observations confirmed by all experiment.

The differential equations of the propagation of heat express the most general conditions, and reduce the physical questions to problems of pure analysis, and this is the proper object of theory.

They are not less rigorously established than the general equations of equilibrium and motion. In order to make this comparison more perceptible, we have always preferred demonstrations analogous to those of the theorems which serve as the foundation of statics and dynamics.

These equations still exist, but receive a different form, when they express the distribution of luminous heat in transparent bodies, or the movements which the changes of temperature and density occasion in the interior of fluids. The coefficients which they contain are subject to variations whose exact measure is not yet known; but in all the natural problems which it most concerns us to consider, the limits of temperature differ so little that we may omit the variations of these coefficients.

The equations of the movement of heat, like those which express the vibrations of sonorous bodies, or the ultimate oscillations of liquids, belong to one of the most recently discovered branches of analysis, which it is very important to perfect.

After having established these differential equations their integrals must be obtained; this process consists in passing from a common expression to a particular solution subject to all the given conditions.

This difficult investigation requires a special analysis 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 pro- blems 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.

This study:

• offers a determinate object to investigate.

• is a sure method of forming analysis itself
• discovers 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.

The motion of light in the atmosphere:

• determines the laws of diffusion of heat in solid matter.
• enters into all the chief problems of the theory of probability, when its abstract properties are used.

The analytical equations:

• were unknown to the ancient geometers
• were introduced by Descartes into the study of curves and surfaces
• are not restricted to the properties of shapes and objects 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 confused 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 imperfec- tion 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, 2° the effect, accidental but continued, of the form or state of the surfaces; 3° the effect, not permanent, of the primitive distribution.

Here I:

• demonstrate all the principles of the theory of heat
• solve all the fundamental problems.

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 beat 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 con- firmed. 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 3, Section 2.

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 for- warded very extensive notes, concerning the convergence of series, the diffusion of heat in an infinite prism, its emission in spaces void of air, the constructions suitable for exhibiting the chief theorems, and the analysis of the periodic movement at the sur- face of the earth.

My second memoir was:

• on the propagation of heat
• deposited in the archives of the Institute, on September 28, 1811.
• formed out of the preceding memoir

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 does not contain; they will be treated in the subsequent memoirs ’ at greater length, and, if it be in our power, with greater clearness.

The 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 Iso- thermal 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).

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’Aca- démie 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 to the analytical subject which we have considered.

My new theories are united forever to the mathematical sciences.

Instruments will be perfected and experiments multiplied.

The analysis which we have formed will be deduced from more general, that is to say, more simple and more fertile methods common to many classes of phenomena.

For all substances, solid or liquid, for vapours and permanent gases, determinations will be made of all the specific qualities relating to heat, and of the variations of the coefficients which express them'.

At different stations on the earth observations will be made, of the temperatures of the ground at different depths, of the intensity of the solar heat and its effects, constant or variable, in the atmosphere, in the ocean and in lakes; and the constant temperature of the heavens proper to the planetary regions will become known.

The theory itself will direct all these measures, and assign their precision. No considerable progress can hereafter be made which is not founded on experiments such as these; for mathematical analysis can deduce from general and simple phenomena the expression of the laws of nature; but the special application of these laws to very complex effects demands a long series of exact observations.