The Analytic Theory of Heat

The Analytic Theory of Heat (1878) is a translation by Alexander Freeman, M.A., with notes, of Joseph Fourier's Théorie Analytique de la Chaleur (1822). Fourier based his reasoning on Newton's law of cooling: the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. In this work Fourier claims that any function of a variable, can be expanded in a series of sines of multiples of the variable. Though not correct without additional conditions, Fourier's observation that some discontinuous functions are the sum of infinite series was a breakthrough.

When a Fourier series converges has been fundamental question for centuries. Joseph-Louis Lagrange gave particular cases and implied that the method was general, but did not pursue the subject. Peter Gustav Lejeune Dirichlet was the first to give a satisfactory demonstration, with some restrictive conditions. This work provides the foundation for what is today known as the Fourier transform.

Fourier made important contributions to dimensional analysis. The book utilizes the important physical concept of dimensional homogeneity in equations; i.e. an equation can be formally correct only if the dimensions match on either side of the equality. Fourier's partial differential equation for conductive diffusion of heat is now taught to every student of mathematical physics. Théorie Analytique de la Chaleur was edited and republished, with corrections, by Jean Gaston Darboux in 1888.

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Translators Preface

In preparing this version in English of Fourier's celebrated treatise on Heat, the translator has followed faithfully the French original. He has, however, appended brief foot-notes, in which will be found references to other writings of Fourier and modern authors on the subject, distinguished by the initials [Alexander Freeman] A. F.

The notes marked R.L.E. are... from... memoranda on the margin of a copy of... Robert Leslie Ellis.

Preliminary Discourse

Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.

Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.

Archimedes... explained the mathematical principles of the equilibrium of solids and fluids. ... Galileo, the originator of dynamical theories, discovered the laws of motion of heavy bodies. Within this new science Newton comprised the whole system of the universe.

The successors of these philosophers have extended these theories, and given them an admirable perfection: they have taught us that the most diverse phenomena are subject to a small number of fundamental laws...

[T]he same principles regulate all the movements of the stars, their form, the inequalities of their courses, the equilibrium and the oscillations of the seas, the harmonic vibrations of air and sonorous bodies, the transmission of light, capillary actions, the undulations of fluids, in fine the most complex effects of all the natural forces, and thus has the thought of Newton been confirmed: quod tam paucis tam multa praestet geometria gloriatur [from so little to so much stands the glory of Geometry.]

[M]echanical theories... do not apply to the effects of heat... a special order of phenomena, which cannot be explained by the principles of motion and equilibrium.

We have... instruments adapted to measure many of these effects... but... not the mathematical demonstration of the laws...

I have deduced these laws... in the course of several years with the most exact instruments...

To found the theory, it was... necessary to distinguish and define... the elementary properties which determine the action of heat... a very small number of general and simple facts; whereby every... problem... is brought back to... mathematical analysis.

[T]o determine... movements of heat, it is sufficient to submit each substance to three fundamental observations. ...[B]odies ...do not possess in the same degree the power to contain heat, to receive or transmit it across their surfaces, nor to conduct it through the interior of their masses. These are the three... qualities... our theory... distinguishes and shews how to measure.

No diurnal variation can be detected at the depth, of about three metres [ten feet]; and the annual variations cease to be appreciable at a depth much less than sixty metres.

Radiant heat which escapes from the surface of all bodies, and traverses elastic media, or spaces void of air, has special laws... The mathematical theory... I... formed gives an exact measure of them. It consists... in a new catoptrics which... serves to determine... effects... direct or reflected.

The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts...

The differential equations of... heat [propagation] express the most general conditions, and reduce... physical questions to... pure analysis... not less rigorously established than... equations of equilibrium and motion. ...[W]e have always preferred demonstrations analogous to... the theorems... of statics and dynamics. These equations... receive a different form, when they express the distribution of luminous heat in transparent bodies, or the movements in the interior of fluids occasioned by changes of temperature and density. ...[I]n... natural problems which... most concerns us... the limits of temperature differ so little that we may omit... variations of... coefficients.

The same theorems which have made known... the equations of... [heat] movement.., apply... to... problems of general analysis and dynamics whose solution has... long... been desired.

[T]he same expression whose abstract properties geometers had considered, and which... belongs to general analysis, represents... the motion of light in the atmosphere... determines the laws of diffusion of heat in solid matter, and enters into... the theory of probability.

The analytical equations... which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to... figures, and... rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and... obscurities, [i.e.,] more worthy to express the invariable relations of natural things.

[M]athematical analysis is as extensive as nature... it defines all perceptible relations, measures times, spaces, forces, temperatures; this... science is formed slowly, but it preserves every principle... acquired; it grows and strengthens... incessantly in the midst of... variations and errors of... 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... in the immensity of space, if man wishes to know the... heavens at successive epochs... if the actions of gravity and of heat are exerted in the interior of the earth at depths... inaccessible, mathematical analysis can yet lay hold of the laws of these phenomena. It makes them present and measurable, and seems... a faculty of the... mind destined to supplement the shortness of life and... imperfection of... senses... more remarkable, it follows the same course in the study of all phenomena; it interprets... by the same language, as if to attest the unity and simplicity of the... the universe, and to make... evident that... order which presides over all natural causes.

The problems of the theory of heat present... simple and constant dispositions which spring from the general laws of nature; and if the order... in these phenomena could be grasped... it would produce... impression comparable to... musical sound.

In this work we have demonstrated all the principles of the theory of heat, and solved all the fundamental problems... [W]e wished to shew the actual origin of the theory and its gradual progress.

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... experiments, lastly the demonstrations of the differential equations of the movement of heat in fluids.

The new theories explained in our work are united for ever to the mathematical sciences, and rest like them on invariable foundations; all the elements... they... possess they will preserve, and... acquire greater extent. Instruments will be perfected and experiments multiplied. The analysis which we have formed will be deduced from more general, ...[i.e,] more simple and more fertile methods... For all substances... determinations will be made of all... 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... depths, of the intensity of the solar heat and its effects... 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... will direct... these measures, and assign their precision. No considerable progress can... be made... not founded on experiments... for mathematical analysis can deduce from general and simple phenomena the expression of the laws of nature; but... application of these laws... demands... exact observations.

Ch. I Introduction

The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory... is to demonstrate these laws; it reduces... researches on the propagation of heat, to problems of the integral calculus whose elements are given by experiment.

[T]he action of heat is always present, it penetrates all bodies and spaces, it influences the processes of the arts, and occurs in all the phenomena of the universe.

When heat is unequally distributed among the different parts of a solid mass, it tends to attain equilibrium, and passes slowly from the parts which are more heated to those which are less; and... it is dissipated at the surface, and lost in the medium or in the void.

The tendency [of heat] to uniform distribution and the spontaneous emission which acts at the surface of bodies, change continually the temperature at their different points.

The problem of the propagation of heat consists in determining what is the temperature at each point of a body at a given instant, supposing that the initial temperatures are known.

If we expose to... continued... uniform... source of heat, the same part of a metallic ring, whose diameter is large, the molecules nearest... the source will be first heated, and, after a... time, every point of the solid will have... nearly the highest temperature... it can attain... not the same at different points... [and] less and less... [the] more distant from that [source] point...

When the temperatures have become permanent, the source... supplies, at each instant, a quantity of heat which... compensates for that... dissipated at all the points of the external surface of the ring.

If now the source be suppressed, heat will continue to be propagated in the [ring's] interior... but that... lost in the... void, will no longer be compensated... by the... source, so that all... temperatures will... diminish... until... equal to the temperatures of the surrounding medium.

Whilst the temperatures are permanent and the source remains [continued and uniform], if at every point of the mean circumference of the ring an ordinate be raised perpendicular to the plane of the ring, whose length is... the fixed temperature at that point, the curved line which passes through the ends of these ordinates will represent the... state of the temperatures...

[T]he thickness of the ring is supposed... sufficiently small for the temperature to be... equal at all points of the same section perpendicular to the mean circumference.

When the [heat] source is removed, the line which bounds the ordinates... at the different points will change its form continually.

The problem consists in expressing, by one equation, the variable form of this curve, and in thus including in a single formula all the successive [temperature] states of the solid.

Let $z$ be the constant temperature at point $m$ [on] the mean circumference [of the ring], $x$ the distance of this point from the [heat] source [point $o$ ], that is to say the length of the arc of the mean circumference, included between the point $m$ and the point $o$ ... $z$ is the highest temperature which the point $m$ can attain by virtue of the constant action of the source, and this permanent temperature $z$ is function $f(x)$ of the distance $x$ . The first part of the problem consists in determining the function $f(x)$ which represents the permanent [temperature] state of the solid.

Consider next the variable state... as soon as the [heat] source has been removed; denote by $t$ the time... passed since the... source [removal], and by $v$ the... temperature at... $m$ after the time $t$ . $v$ will be a... function $F(x,t)$ of the distance x and the time $t$ ; the object... is to discover this function $F(x,t)$ , of which we only know as yet that the initial value... $f(x)=F(x,o)$ .

If we place a solid homogeneous... sphere or cube, in a medium... [of] constant temperature... for a... long time, it will acquire at all its points... [the] temperature... of the fluid. Suppose the mass to be withdrawn... to transfer... to a cooler medium, heat will begin to be dissipated at its surface; the temperatures at different points of the mass will not be... the same, and if we suppose it divided into an infinity of layers by surfaces parallel to its external surface, each of those layers will transmit, at each instant, a certain quantity of heat to the layer which surrounds it. If... each molecule carries a separate thermometer... the state of the solid will from time to time be represented by the variable system of... these thermometric heights. It is required to express the successive states by analytical formulae, so that we may know at any... instant the temperatures... and compare the quantities of heat which flow during the same instant, between two adjacent layers, or into the surrounding medium.

If the mass is spherical, and we denote by $x$ the distance... from the centre... $t$ the time... cooling, and by $v$ the variable temperature of the point $m$ ... all points... at the same distance $x$ ... have the same temperature $v$ . This quantity $v$ is a certain function $F(x,t)$ of the radius $x$ and... time $t$ ... such that it becomes constant whatever... value of $x$ , when... [ $t=0$ ]; for... the temperature at all points is the same at... emersion. The problem consists in determining... [ $F(x,t)$ ].

[D]uring... cooling... heat escapes, at each instant, through the external surface, and passes into the medium... [and] this quantity is not constant; it is greatest at the beginning of... cooling. If... we consider the variable state of the internal spherical surface... [at] radius... $x$ ... there must be at each instant a... quantity of heat which traverses that surface, and passes through that part... more distant from the centre. This continuous flow of heat is variable like that through the external surface, and both are quantities comparable with each other; their ratios are numbers whose varying values are functions of the distance $x$ , and of the time $t$ ... elapsed. It is required to determine these functions.

[T]he effects of the propagation of heat depend in... every solid substance, on three elementary qualities... its capacity for heat, its own conducibility, and the exterior conducibility.

[I]f two bodies of the same volume and of different nature have equal temperatures, and if the same quantity of heat be added to them, the increments of temperature are not the same; the ratio of these increments is the, ratio of their capacities for heat.

The proper or interior conducibility of a body expresses the facility with which heat is propagated in passing from one internal molecule to another.

The external or relative conducibility of a solid body depends on the facility with which heat penetrates the surface, and passes from this body into a given medium, or... from the medium into the solid. The last property is modified by the... polished state of the surface... also according to the medium in which... immersed; but the interior conducibility can change only with the nature of the solid.

These three elementary qualities are represented... by constant[s], and the theory... indicates experiments suitable for measuring their values. As soon as... determined... problems relating to the propagation of heat depend only on numerical analysis.

[T]here is no mathematical theory which has a closer relation... with public economy, since it serves to give clearness and perfection to the practice of the numerous arts... founded on... heat.

To complete our theory it was necessary to examine the laws which radiant heat follows, on leaving the surface of a body. ...[T]he intensities of the different rays, which escape in all directions from any point in the surface of a heated body, depend on the angles which their directions make with the surface at the same point. We have proved that the intensity of a ray diminishes as the ray makes a smaller angle with the element of surface, and that it is proportional to the sine of that angle.
- Ref: Mem. Acad. d. Sc. Tome V. Paris, 1826, pp. 179—213.

[A] very extensive class of phenomena exists, not produced by mechanical forces, but resulting simply from the presence and accumulation of heat. This part of natural philosophy cannot be connected with dynamical theories, it has principles peculiar to itself...

In whatever manner the heat was at first distributed, the system of temperatures altering more and more, tends to coincide... with a definite state which depends only on the form of the solid. In the ultimate state the temperatures of all the points are lowered in the same time, but preserve amongst each other the same ratios: in order to express this property the analytical formulae contain terms composed of exponentials and of quantities analogous to trigonometric functions.

Several problems of mechanics present analogous results, such as the isochronism of oscillations, the multiple resonance of sonorous bodies. ...As to those results which depend on changes of temperature... mathematical analysis has outrun observation, it has supplemented our senses, and has made us in a manner witnesses of regular and harmonic vibrations in the interior of bodies.

These considerations present a singular example of the relations which exist between the abstract science of numbers and natural causes.

[T]he functions obtained by successive differentiations, which are employed in the development of infinite series and in the solution of numerical equations, correspond also to physical properties. The first of these functions, or the fluxion properly so called, expresses in geometry the inclination of the tangent of a curved line, and in dynamics the velocity of a moving body when the motion varies; in the theory of heat it measures the quantity of heat which flows at each point of a body across a given surface. Mathematical analysis has therefore necessary relations with sensible phenomena; its object is not created by human intelligence; it is a pre-existent element of the universal order, and is not in any way contingent or fortuitous; it is imprinted throughout all nature.

The theory of heat will always attract the attention of mathematicians, by the rigorous exactness of its elements and the analytical difficulties... and above all by the extent and usefulness of its applications; for all its consequences concern... general physics, the operations of the arts, domestic uses and civil economy.

Of the nature of heat uncertain hypotheses only could be formed, but the knowledge of the mathematical laws to which its effects are subject is independent of all hypothesis; it requires only an attentive examination of the chief facts which common observations have indicated, and which have been confirmed by... experiments.

The action of heat tends to expand all bodies, solid, liquid or gaseous; this is the property which gives evidence of its presence.

When all the parts of a solid homogeneous body... are equally heated, and preserve without any change the same quantity of heat, they have also and retain the same density.

The temperature of a body equally heated in every part, and which keeps its heat, is that which the thermometer indicates when it is and remains in perfect contact with the body in question.
Perfect contact is when the thermometer is completely immersed in a fluid mass, and, in general, when there is no point of the external surface of the instrument which is not touched by one of the points of the solid or liquid mass whose temperature is to be measured.

[D]ifferent bodies placed in the same region, all of whose parts are and remain equally heated, acquire also a common and permanent temperature.

Of... the action of heat, that which seems simplest and most conformable to observation, consists in comparing this action to that of light. Molecules separated from one another reciprocally communicate, across empty space, their rays of heat, just as shining bodies transmit their light.

All bodies have the property of emitting heat through their surface; the hotter they are the more they emit; the intensity of the emitted rays changes very considerably with the state of the surface.

Every surface which receives rays of heat from surround ing bodies reflects part and admits the rest : the heat which is not reflected, but introduced through the surface, accumulates within the solid; and so long as it exceeds the quantity dissipated by irradiation, the temperature rises.

[M]olecules which compose... bodies are separated by spaces void of air, and have the property of receiving, accumulating and emitting heat. Each of them sends out rays on all sides, and at the same time receives other rays from the molecules which surround it.

The effects of heat can by no means be compared with those of an elastic fluid whose molecules are at rest.
It would be useless to attempt to deduce from this hypothesis the laws of [heat] propagation... The free state of heat is the same as that of light; the active state... is then entirely different from that of gaseous substances. Heat acts in the same manner in a vacuum, in elastic fluids, and in liquid or solid masses, it is propagated only by way of radiation, but its sensible effects differ according to the nature of bodies.

Heat is the origin of all elasticity; it is the repulsive force which preserves the form of solid masses, and the volume of liquids. In solid masses, neighbouring molecules would yield to their mutual attraction, if its effect were not destroyed by the heat which separates them.
This elastic force is greater according as the temperature is higher; which is the reason... bodies dilate or contract when their temperature is raised or lowered.

The equilibrium... in the interior of a solid mass, between the repulsive force of heat and the molecular attraction, is stable; [i.e.,] it re-establishes itself when disturbed... If the molecules are arranged at [equilibrium] distances.., and if an external force begins to increase this distance without any change of temperature, the effect of attraction begins by surpassing that of heat, and brings back the molecules to their original position, after a multitude of oscillations... A similar effect is exerted in the opposite sense when a mechanical cause diminishes the primitive distance of the molecules; such is the origin of the vibrations of sonorous or flexible bodies, and of all the effects of their elasticity.

[T]he mode of action of heat always consists, like... light, in... reciprocal communication of rays... but it is not necessary to consider the phenomena under this aspect... to establish the theory of heat. ...T[]he laws of equilibrium and propagation of radiant heat, in solid or liquid masses, can be rigorously demonstrated, independently of any physical explanation, as the necessary consequences of common observations.

[T]he quantity of heat which one of the molecules receives from the other is proportional to the difference of temperature of the two molecules... it is null, if the temperatures are equal...

Denoting by $v$ and $v^{\prime }$ the temperatures of two equal molecules $m$ and $n$ ... $p$ their extremely small distance [apart], and... $dt$ , the infinitely small... instant, the quantity of heat which $m$ receives from $n$ during this instant will be... $(v^{\prime }-v)\theta (p)\cdot dt$ . We denote by $\theta (p)$ a certain function of the distance p which, in solid bodies and in liquids, becomes [zero] nothing when $p$ has a sensible magnitude. The function is the same for every point of the same given substance... [but] varies with the nature of the substance.

The quantity of heat which bodies lose through their surface is subject to the same principle. If we denote by $\sigma$ the area, finite or infinitely small, of the surface, all of whose points have the temperature $v$ , and if $a$ represents the temperature of the... air, the coefficient $h$ being the... external conducibility, we shall have $\sigma h(v-a)dt$ as the expression for the quantity of heat which this surface $\sigma$ transmits to the air during... instant $dt$ . ... $h$ may... be considered as having a constant value, proper to each state of the surface, but independent of the temperature.

[C]onsider... the uniform movement of heat in the simplest case, which is... an infinite... solid body formed of some homogeneous substance... enclosed between two parallel and infinite planes; the lower plane A is maintained... at a constant temperature $a$ ... the upper plane B is... maintained... at... fixed temperature $b$ , ...less than... $a$ ; the problem is to determine... the result... if... continued for an infinite time. ...In the final and fixed state... the permanent temperature... is... the same at all points of the same section parallel to the base... [D]enoting by $z$ the height of an intermediate section... from the plane A... $e$ the whole height or distance AB, and... $v$ the temperature of the section whose height is $z$ , we must have $v=a+{\frac {b-a}{e}}z$ . ...[I]f the temperatures were at first established in accordance with this law, and... the... surfaces A and B... always kept at... temperatures $a$ and $b$ , no change would happen.

By what precedes we see... Heat penetrates the mass gradually across the lower plane: the temperatures of the intermediate sections are raised, but can never exceed nor even quite attain a certain limit... this limit or final temperature is different for different intermediate layers, and decreases in arithmetic progression from the fixed temperature of the lower plane to the fixed temperature of the upper plane. ... [D]uring each division of time, across a section parallel to the base, or a... portion of that section, a certain quantity of heat flows, which is constant... the same for all the intermediate sections; it is equal to that which proceeds from the source, and to that which is lost... at the upper surface...

[T]o compare... the intensities of the constant flows of heat... propagated uniformly in the two solids, that is... the quantities of heat which, during unit of time, "cross unit of surface of each of these bodies. The ratio of these intensities is that of the two quotients ${\frac {a-b}{e}}$ and ${\frac {a^{\prime }-b^{\prime }}{e^{\prime }}}$ . ...[D]enoting the first flow by $F$ and the second by $F^{\prime }$ we... have ${\frac {F}{F^{\prime }}}={\frac {a-b}{e}}\div {\frac {a^{\prime }-b^{\prime }}{e^{\prime }}}$ .

Suppose... in the second solid, the permanent temperature $a^{\prime }$ ...is that of boiling water, 1... $b^{\prime }$ is that of melting ice, 0... distance $e^{\prime }$ is the unit of measure... [Then ${\frac {a^{\prime }-b^{\prime }}{e^{\prime }}}={\frac {1-0}{1}}=1$ .] [D]enote by $K$ the constant flow of heat which, during unit of time... would cross unit of surface in this [second] solid, if it were formed of a given substance; K expressing a certain number of units of heat, that~is to say a certain... [multiple] of the heat necessary to convert a kilogramme of ice into water... [T]o determine the constant flow $F$ , in a solid... of the same substance, the ${\frac {F}{K}}={\frac {a-b}{e}}\div 1$ or $F=K{\frac {a-b}{e}}$ . ... $F$ denotes the quantity of heat which, during the unit of time, passes across a unit of area of the surface taken on a section parallel to the base.

Thus the thermometric state of a solid enclosed between two parallel infinite plane sides whose perpendicular distance is $e$ , and which are maintained at fixed temperatures $a$ and $b$ , is represented by the two equations: $v=a+{\frac {b-a}{e}}z$ , and $F=K{\frac {a-b}{e}}$ or $F=-K{\frac {dv}{dz}}$ The first... expresses the law according to which the temperatures decrease from the lower side to the opposite side, the second indicates the quantity of heat which, during a given time, crosses a definite part of a section parallel to the base.

We have taken... coefficient K... to be the measure of the specific conducibility of each substance; this... has... different values for different bodies.
It represents... the quantity of heat which, in a homogeneous solid formed of a given substance and enclosed between two infinite parallel planes, flows, during one minute, across a surface of one square metre taken on a section parallel to the extreme planes, supposing that these two planes are maintained, one at the temperature of boiling water, the other at the temperature of melting ice, and that all the intermediate planes have acquired and retain a permanent temperature.

Ch. IX.

The chief elements of the method we have followed are these:
1st. We consider... the general condition given by the partial differential equation, and all the special conditions which determine the problem... and we... form the analytical expression which satisfies all... these conditions.

2nd. We first perceive that this expression contains an infinite number of terms, into which unknown constants enter, or that it is equal to an integral which includes one or more arbitrary functions. In the first instance, [i.e.], when the general term is affected by the symbol $\textstyle \sum$ , we derive from the special conditions a definite transcendental equation, whose roots give the values of an infinite number of constants.
The second instance... when the general term becomes... infinitely small... the sum of the series is... changed into a definite integral.

3rd. We can prove by the fundamental theorems of algebra, or even by the physical nature of the problem, that the transcendental equation has all its roots real, in number infinite.

4th. In elementary problems, the general term takes the form of a sine or cosine; the roots of the definite equation are either whole numbers, or real or irrational quantities, each... included between two definite limits.
In more complex problems, the general term takes the form of a function given implicitly by means of a differential equation integrable or not. However it may be, the roots of the definite equation exist, they are real, infinite in number. This distinction of the parts of which the integral must be composed, is very important, since it shews... the form of the solution, and the necessary relation between the coefficients.

5th. It remains only to determine the constants which depend on the initial state; which is done by elimination of the unknowns from an infinite number of equations of the first degree. We multiply the equation which relates to the initial state by a differential factor, and integrate it between defined limits, which are most commonly those of the solid in which the movement is effected.
There are problems in which we have determined the coefficients by successive integrations, as may be seen in... the temperature of dwellings. In this case we consider the exponential integrals, which belong to the initial state of the infinite solid... The most remarkable of the problems... and the most suitable for shewing... our analysis, is... the movement of heat in a cylindrical body. In other researches, the determination of the coefficients would require processes of investigation... we do not... know. But... without determining the values of the coefficients, we can always acquire an exact knowledge of the problem, and of the natural course of the phenomenon... the chief consideration is that of simple movements.

6th. When the expression sought contains a definite integral, the unknown functions... under the... integration are determined, either by the theorems... we have given... or by a more complex process... in the Second Part.
These theorems can be extended to any number of variables. They belong in some respects to an inverse method of definite integration; since they serve to determine under the symbols $\textstyle \int$ and $\textstyle \sum$ unknown functions... such that the result of integration is a given function.
The same principles are applicable to... problems of geometry... general physics, or... analysis, whether the equations contain finite or infinitely small differences, or... both.
The solutions... obtained by this method are complete, and consist of general integrals. ...[O]bjections... are devoid of... foundation...

7th. ...[E]ach of these solutions gives the equation proper to the phenomenon, since it represents it distinctly throughout the... extent of its course, and... determine[s] with facility all its results numerically.
The functions... obtained by these solutions are then composed of a multitude of terms... finite or infinitely small: but the form of these expressions is... [not] arbitrary; it is determined by the physical character of the phenomenon. For this reason, when the value of the function is expressed by a series into which exponentials relative to the time enter, it is of necessity... since the natural effect whose laws we seek, is... decomposed into distinct parts, corresponding to the... terms of the series. The parts express so many simple movements compatible with the special conditions; for each one of these movements, all the temperatures decrease, preserving their primitive ratios. In this composition we ought not to see a result of analysis due to the linear form of the differential equations, but an actual effect which becomes sensible in experiments. It appears also in dynamical problems in which we consider the causes which destroy motion; but it belongs necessarily to all problems of the theory of heat, and determines the nature of the method which we have followed for the solution...

8th. The mathematical theory of heat includes : first, the exact definition of all the elements of the analysis; next, the differential equations; lastly, the integrals appropriate to the fundamental problems. The equations can be... [obtained] in several ways; the same integrals can also be obtained, or other problems solved, by introducing certain changes in the course of the investigation. ...[T]hese researches do not constitute a method different from our own; but confirm and multiply its results.

9th. ...[The objection] that the transcendental equations which determine the exponents having imaginary roots... would... [of necessity] employ the terms which proceed from them, and... would indicate a periodic character in part of the phenomenon... has no foundation, since the equations in question have.... all their roots real, and no part of the phenomenon can be periodic.

10th. It has been alleged that... to solve... problems of this kind, it is necessary to resort in all cases to a... form of the integral... denoted as general... but this distinction has no foundation... the use of a single integral... in most cases... complicating... unnecessarily.

11th. It has been supposed that the method which consists in expressing the integral by a succession of exponential terms, and in determining their coefficients by means of the initial state, does not solve the problem of a prism which loses heat unequally at its two ends; or that, at least, it would be very difficult to verify in this manner the solution derivable from the integral ( $\gamma$ ) by long calculations. We shall perceive, by a new examination, that our method applies directly to this problem, and that a single integration even is sufficient.

12th. We have developed in series of sines of multiple arcs functions which appear to contain only even powers of the variable, $\cos x$ for example. We have expressed by convergent series or by definite integrals separate parts of different functions, or functions discontinuous between certain limits, for example that which measures the ordinate of a triangle. Our proofs leave no doubt of the exact truth of these equations.

13th. We find in the works of many geometers results and processes of calculation analogous to those... we... employed. These are particular cases of a general method, which... it became necessary to establish in order to ascertain... the mathematical laws of the distribution of heat. This theory required an analysis... one principal element of which is the... expression of separate functions [ $f(x)$ ], or of parts of functions... $f(x)$ which has values existing when... $x$ is included between given limits, and whose value is always nothing, if the variable is not included between those limits. This function measures the ordinate of a line which includes a finite arc of arbitrary form and coincides with the axis of abscissae in all the rest of its course.
This motion is not opposed to the general principles of analysis; we might even find... first traces... in the writings of Daniel Bernouilli...Cauchy...Lagrange and Euler. It had always been regarded as manifestly impossible to express in a series of sines of multiple arcs, or at least in a trigonometric convergent series, a function which has no existing values unless the values of the variable are included between certain limits, all the other values of the function being nul. But this point of analysis is fully cleared up, and it remains incontestable that separate functions, or parts of functions, are exactly expressed by trigonometric convergent series, or by definite integrals. We have insisted on this... since we are not concerned... with an abstract and isolated problem, but with a primary consideration intimately connected with the most useful and extensive considerations. Nothing has appeared to us more suitable than geometrical constructions to demonstrate the truth of these new results, and to render intelligible the forms which analysis employs far their expression.

14th. The principles which have served to establish for us the analytical theory of heat, apply directly to the investigation of the movement of waves in fluids, a part of which has been agitated. They aid also the investigation of the vibrations of elastic laminae, of stretched flexible surfaces, of plane elastic surfaces of very great dimensions, and apply in general to problems which depend upon the theory of elasticity. The property of the solutions which we derive from these principles is to render the numerical applications easy, and to offer distinct and intelligible results, which really determine the object of the problem, without making that knowledge depend upon integrations or eliminations which cannot be effected. We regard as superfluous every transformation of the results of analysis which does not satisfy this primary condition.

About The Analytic Theory of Heat

In this groundbreaking study, arguing that previous theories of mechanics... did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to study the conduction of heat in solids. Known... as the 'Fourier Series', this work paved the way for modern mathematical physics. ...This book will be especially helpful for mathematicians... interested in trigometric series and their applications.
- Cambridge University Press: Frontmatter (2009) in Fourier, JBJ, Tr. A. Freeman, The Analytical Theory of Heat Cambridge Library Collection - Mathematics. pp. i-ii.

Between 1807 and 1811... Fourier... developed a mathematical theory of heat conduction... independent of the caloric hypothesis, but... was not published until 1822... as Théorie analytique de la chaleur... Fourier set the study of the theory of heat in the tradition of rational mechanics, basing it on differential equations... The heat transmitted between... molecules was proportional to the difference in their temperature and a function of the distance between them... [and] varied with the nature of the... substance. ...Fourier did not rely upon... speculation about the nature of heat. ...[W]hat was important was not what heat was, but what it did, in a given experimental setting.
- Diane Greco, "Fourier and the theory of heat" (Nov 16, 2020)

[O]ne can hardly imagine someone with a broader background than Fourier, more uniquely situated to simultaneously tackle problems of pure thought as well as in the physical world around him, perhaps in the same stroke of the pen. In the introduction of The Analytical Theory of Heat, he made no secret about the fact that he intended to do just that, with mathematics as his language and tool.
- Kenneth M. Monks, "Fourier’s Heat Equation and the Birth of Fourier Series" (2022) Digital Commons @Ursinus College.

External links

Archive.org: Jean Baptiste Joseph Fourier
- The Analytical Theory of Heat (1878) Tr. Alexander Freeman
YouTube videos
- The Revolutionary Genius Of Joseph Fourier Derivation of the Heat equation from Newton's law of cooling and specific heat capacity research of Lavoisier and Laplace, plus more @Dr. Will Wood channel.