IX
The Conservation of Matter and Energy
The notion that in all the multifarious changes which we perceive to take place in the material world there must be permanent elements which persist unaltered through all these changes has been throughout the history of Science one of the guiding ideas which have ultimately given rise to such formulations as those contained in the expressions Conservation of Matter Conservation of Weight Conservation of Mass and Conservation of Energy. The principle of permanence expressed in such a formula as that “nothing is created and nothing destroyed” has usually been regarded as an a priori principle closely related to the principle of causation. The very generality of this a priori principle has prevented it from functioning as an efficient guide to the determination of the precise elements in the perceptual world to which the characteristic of persistence through all transformations appertains. Thus the actual successes of scientific investigation in this order of ideas have consisted in the ascertainment by experiment and observation of empirical laws the law of conservation of mass and the law of conservation of energy. Such empirical laws are ascertained to express definite facts relating to a considerable but limited range of observed phenomena the conservation which they express being of such a character that it is expressed quantitatively in a numerical form. The laws are then adopted as hypothetical principles in conceptual theories which relate to ranges of phenomena wider than those to which the empirical verification has been in the first instance applied.
The value of the laws in their general form must depend upon their success in performing their functions of description and prediction in relation to new classes of phenomena to which they are tentatively applied. The a priori principle in its general form to which I have referred as a metaphysical principle expressing a supposed necessity of thought need not be accepted as a part of the foundations of Natural Science whatever its actual influence may have been upon the minds of scientific investigators in the past. The main difficulty as regards the principle of the conservation of matter the principle that matter is neither created nor destroyed is that of forming a clear conception of what it exactly is that is conserved. If we regard matter as a construct including a complex of physical properties extension colour hardness conductibility of heat and electricity etc. we have ample and obvious evidence that these properties do not persist unchanged but are subject to the largest changes in what we regard as one and the same material system. What then is to be understood by the statement that matter can be neither created nor destroyed; that is by the principle of the conservation of matter? If we assert that what persists unchanged is a sub-stratum substance itself not identified with any or all of these physical properties but regarded as their bearer not only do we reduce the principle to one dependent upon a metaphysical theory but we remove from it all possibility of verification. It then becomes a bare philosophical assertion which has no direct relation with the world of percepts and is thus outside the domain of Natural Science. A real scientific law of conservation must contain an indication of some measurable quality or property of matter which can be ascertained to remain unaltered in magnitude during the actual chemical and motional transformations occurring in the physical world. Not even what is regarded as a primary quality of matter that of extension is conserved as a measurable quantity unaltered through all transformations.
There is however one other property which we have come to associate with all matter that of weight; this is estimated by the balance the systematic employment of which by Lavoisier brought about what has been described as a revolution in chemical Science. If however we understand conservation of matter to mean conservation of weight we are at once met with the difficulty that the weight of what we regard as one and the same piece of matter when estimated by the spring balance varies with the latitude of the place at which it is measured. Moreover in accordance with the theory of gravitation the weight would be very greatly altered if the matter were transported to another planet. However in the chemical transformations which take place in any one locality a verification of the principle of conservation of matter consists in a verification of the principle of conservation of weight. For different localities the differences of weight of one and the same object are eliminated in accordance with the Newtonian and Galilean Dynamics by dividing the weight by the acceleration due to gravity; this division yielding the measure of the mass of the body. Thus the conservation of matter now comes to be taken to mean the conservation of mass.
With the concept of mass I have already dealt more fully in connection with Dynamics. The actual mass of a body can be regarded as a quality which can be measured as derivative from the two measurements of weight and of acceleration. That the mass of a body is the amount of matter in it is a tautological statement which can only be taken to denote that the meaning assigned to the term quantity of matter is that it is the mass regarded as a measurable quality of the body. The principle of the conservation of matter regarded as mass· has however a much wider meaning than that it is unalterable for one and the same body in whatever position it be placed or however it be moving. It includes the assertion that mass as a measurable quantity is unchanged in amount throughout all the chemical and thermal changes that may take place in an isolated material system.
Thus the principle implies that matter however it be sub-divided actually or conceptually may be regarded as having a quality the mass or quantity of matter which is measurable and remains unchanged in total amount during all motions and all chemical thermal or other transformations. The only means we have in an abstract conceptual scheme of representing this assumed quality is by the employment of numbers in relation to conceptual bodies in geometrical space. In its abstract form the principle asserts an invariant property of the sum of such numbers for the conceptual elements of a limited system. Only to a limited extent has this general law been verified experimentally; for the difficulties of measurement and of securing complete isolation of the substances which undergo chemical transformation are very great being subject to errors difficult to take completely into account. Moreover the conservation of mass through all motions is capable only of indirect verification in connection with the verification of the adequacy of a particular dynamical scheme.
Accordingly the principle must be regarded as an hypothesis which has been verified approximately in a large number of cases and is assumed by the Chemist and the Physicist to hold good as descriptive of relations in a large range of actual phenomena but as subject to possible refutation in cases in which more refined methods of measurement are employed in connection with electromagnetic or other phenomena.
In the new electron theory of matter of which I shall speak later mass occupies a wholly different position from that which is assigned to it in the mechanical theory which has been here discussed. In accordance with the electron theory the mechanical masses of bodies are no longer constant but have a sensible variation when the bodies are set in motion with velocities comparable with the velocity of light.
A sketch of the history of the doctrine of the conservation of matter is sufficient to show that the establishment of the principle in the modern form of that of the conservation of mass was the result of a gradual evolution. The ancient atomists not being in possession of the principle of inertia made no distinction between mass and weight. In this they are in accord with the popular view held even to-day. For example Lucretius following Democritus in this order of ideas appears to have regarded weight as an unalterable characteristic of all matter; in his poem the atoms are in motion on account of their weight; and this involves an identification of their masses with their weights. On the other hand the school of Aristotle radically opposed as it was in almost all respects to the views of the atomists did not believe that all matter has weight. With the Aristotelians the notions of matter and weight are kept quite distinct from one another; weight being regarded as an accidental quality of matter like colour or temperature. In their view weight is the resultant of two opposed qualities heaviness and lightness. Fire has no heaviness and earth no lightness. Water and air have both one being preponderant in water and the other in air. Plato had observed that the four elements are constantly transformed into one another; thus air and fire are concerned with the transformation of matter as when water boils or when wood burns.
In the middle ages the Aristotelian views about matter were prevalent although traces are to be found of the influence of the ancient atomists. Some of the Alchemists used reasoning founded on the consideration of weights; but they did not attach to weight the primary importance which it came later to possess and they did not believe in its unchangeableness during transformations. Indeed some of them expressly mention change of weight as occurring in the transmutation of matter. Thus for example Geber writes (? eighth century A.D.): “By our artifice we easily form silver out of lead; in the transformation the latter does not preserve its own weight but changes into a new weight.” This view of the Alchemists cannot be explained away as involving merely a reference to change of specific weight or to the change of weight produced by absorption of matter from air or fire; indeed specific weight and absolute weight were constantly confused with one another even as late as the seventeenth century. As long as the weight of a body was regarded as a merely accidental quality like its colour it was quite natural to suppose that it might be changed without the addition or subtraction of matter. Even Bacon held opinions not very different from those of the Alchemists. In some statements he affirmed the existence of absolutely light bodies and that change of weight may accompany a change of state; but in other statements probably following the ideas of the atomists he affirmed the constancy of weight. It must always be remembered that with the Alchemists and generally with those under the influence of the Aristotelian conception of substantial forms the question whether the weight remained quantitatively constant or not during a transformation seemed a matter of subordinate importance; the smallest change of quality was in their eyes of much greater interest. A knowledge of the relations of quantity being of little importance to the adherents of the philosophical doctrine of substantial forms it is hardly possible to regard the notion of the permanence of mass considered quantitatively as a part of the stock of ideas of those who were under the influence of Aristotelian conceptions. For the notion of matter as underlying substance differentiated from the accidental quality of weight did not admit of quantitative measurement although the substance was regarded as in some sense persisting through all changes. As the gradual emancipation from the Aristotelian conceptions took place the conception of mass became clarified. It was formulated with tolerable clearness by Kepler and also by Descartes although with the latter as with the Aristotelians weight remained an accidental property not possessed by all matter. That air has weight was generally recognized in the time of Descartes but fire was still regarded as devoid of weight. With Descartes matter being a plenum the quantity of matter was indicated by its volume; thus he makes the assertion that “when a jar is full of gold or of lead it does not contain more matter than when we think it empty.” But he regarded the terrestrial matter as the only kind to be taken into account in mechanical action and thus in reality he made a distinction between mass and volume the mass appertaining to terrestrial matter only. He did not however recognize that mass and weight are in a fixed ratio; in fact the title of one of the chapters of his Principles is “That their weight has not always the same proportion to their matter” and here matter is to be understood as terrestrial matter. For a long period after the time of Descartes the notion that weight is an accidental quality of matter prevented the general acceptance of the conservation of weight although opinion on the point was by no means unanimous even in Descartes' own time. Jean Rey in his essays which were published before the actual appearance of Descartes' Principles attempted to give an a priori demonstration that weight is conserved in every transformation. Moreover he gave an experimental proof that air is heavy and that in the formation of lime the increase of weight is due to material taken from the air. The persistence of disbelief in the invariability of weight is exhibited in the utterances of many writers even until the end of the eighteenth century. Thus Hobbes declares that” all accidents other than greatness or extension can be engendered or destroyed “thereby leaving no room for the conservation of weight or of mass. Leibniz who had a clear conception of mechanical mass states that” water contains in equal volume as much matter as mercury only to the matter belonging to the water there is added a foreign non-heavy matter which is between its pores “for it is a strange fiction to make all matter heavy.”
Newton who did not admit the existence of imponderable matter showed experimentally that the weight is proportional to the mass of a body; and Huygens stated definitely that quantity of matter is measured by its weight. The separate lines of work of Physicists and Chemists make it difficult to ascertain the views of the Physicists of the seventeenth and eighteenth centuries on the nature of chemical phenomena. Almost the only exception to this separation was the work of Robert Boyle both Physicist and Chemist who appears to admit the principle of conservation of weight without however explicitly formulating it. In the seventeenth century although it was generally admitted that air has weight it was not generally believed that this is the case for fire. In the eighteenth century however we find that Berkeley regarded the increase of weight of some metals when heated for example in the case of antimony as due to the fire in the sun's rays; he remarked that we do not know the weight of a solar ray. Diderot stated that “the fire of our furnaces considerably augments the weight of some matter such as calcinated lead.”
The special form which the notion of imponderable substance took in the minds of the Chemists of this period was the theory of phlogiston a substance in vented to account for thermal phenomena. Phlogiston was endowed with negative weight and since it intervened in all chemical reactions there was no difficulty in conceiving that its admixture with matter prevented the conservation of weight in a chemical transformation. How little attention was paid by Chemists to all questions relating to quantities is illustrated by the fact that one of the chief French Chemists of the day Macquer on hearing that Lavoisier was preparing an attack upon the theory of phlogiston stated that he had been disquieted for a moment but was reassured when he learned that Lavoisier's objections were based solely on quantitative considerations. The definite establishment of the principle of conservation of matter by the systematic use of the balance is mainly due to Lavoisier and may be dated from his memoir on The change of water into earth published by the French Academy in 1773. This was followed in 1774 by a work in which by the employment of the balance he decided between the rival theories of Black and Meyer as to what happens in chemical transformations. First applying the principle without explicitly “any matter can furnish nothing in an experiment beyond the totality of its weight” and further “the determination of the weight of materials and their products before and after experiments is the basis of everything useful and exact in Chemistry.” “In every operation there is an equal amount of matter before and after the operation.”
In 1774 Lavoisier described the commencement of his fundamental discoveries relating to combustion. He verified that various metals when heated in a closed vessel receive an increase of weight and that the amount of air in the vessel is diminished; he showed that the loss of weight of the air is nearly equivalent to the increase of weight of the metal. A slight increase in the weight of the whole vessel he properly attributes to an exterior deposit due to the fire. In this way he provides a refutation of the idea of the intervention of the element fire and shows that the increase of weight can only come from the air. Even after the composition of water became known and the phenomenon we call oxidization the new conceptions of Lavoisier only triumphed slowly. They do not appear to have been completely accepted by Priestley or Cavendish. Scheele regarded heat as a compound of phlogiston and oxygen; both of them he thought of as heavy but supposed that together they give rise to an imponderable substance. Heat united with very little phlogiston is transformed into light but united with a great quantity it becomes inflammable air that is hydrogen. Even Lavoisier shows traces of analogous conceptions. He regarded oxygen as resulting from a combustion of ponderable matter and an imponderable fluid caloric. Heat he regarded as a material element contained in a gas and the conception which he had of gases was related by means of intermediate hypotheses with that of imponderable fluids.
After the vicissitudes which I have sketched the principle of the conservation of matter regarded as measured by dynamical mass has come to be accepted as an empirical law which is applicable within a large range of phenomena although some Chemists have maintained that it is possible to detect deviations from the law which cannot be assigned to the effect of instrumental errors or to disturbing factors not easily taken fully into account. Moreover the flood of light which has lately been thrown upon the properties of radioactive substances has suggested views in which dynamical mass no longer holds its former position as fundamental and irreducible. It has indeed been suggested that in accordance with the electrical theory of matter there would be nothing surprising in a change of weight owing to chemical reactions.
The origin of the principle of the Conservation of Energy is much more modern than that of the principle of the Conservation of Matter. In its general form the principle of the Conservation of Energy dates back only to the middle of the nineteenth century but in its restricted form as a principle of Mechanics in the narrower sense of the term it is in the ideas of Descartes Leibniz and especially of Huygens that we find its origin. The notion of matter is one formed by common sense but the conception of energy has been created by Science for its own special purposes. It seems therefore quite natural that the doctrine of the conservation of energy should have arisen at a much later stage in the history of Science than that of the conservation of matter at least in a crude form. The notion of work as measured by the product of a force into the displacement in the direction of the force of the body on which it acts is due to Galileo who showed that in simple mechanical machines the work of the resistance in a displacement is equal to that of the power. He concluded that by the aid of such machines it is impossible to create work but he did not show that work cannot be destroyed. For the case of a falling body he gave the formula which expresses the principle of energy.
The next step in the direction of setting up a general principle relating to the movement of bodies was taken by Descartes who attempted to set up a principle of the conservation of motion through all changes in the physical world. In this attempt he made the mistake of taking the sum of the products of the masses into their velocities instead of the squares of the velocities as representing the quantity which is conserved. This error was pointed out by Leibniz in a treatise bearing the title “A short demonstration of a Remarkable Error of Descartes and others concerning the Natural Law by which they think that the Creator always preserves the same Quantity of Motion; by which however the Science of Mechanics is totally perverted.” Leibniz distinguished between simple pressure (vis mortua) and the force of a moving body (vis viva) but he confused the question of the right measure of force with that of the constancy of momentum and of the kinetic energy of a system. Neither the Cartesian nor the Leibnizian measure of the effectiveness of a body in motion is as Leibniz observed to be identified with the Newtonian measure of force. Leibniz like Descartes regarded the principle formulated by him as embracing all the phenomena of the Universe. He justified the principle of the conservation of vis viva by an appeal to the principle of causation in the form that the effect is equal to the cause.
To attempt to demonstrate this law (he writes) would obscure it. Indeed everyone regards it as an incontestable axiom that every efficient cause cannot perish either totally or in part without producing an effect equal to the loss. The idea that we have of the vis viva as it exists in a body in motion is something absolute independent and positive; that it remains in the body even if the rest of the Universe were annihilated. It is then clear that if the vis viva of a body diminishes or increases on impact with another body the vis viva of this other body must change increase or diminish by the same quantity.
His view of the scope of the principle appears clearly in the following passage:
I had maintained that active forces are conserved in the world. It has been objected that two soft or inelastic bodies when they collide lose part of their force. I answer that this is not so. It is true that the “wholes” lose it in respect of their total motion but the parts receive it being agitated internally by the force of the collision. Thus the loss ensues only in appearance. The forces are not destroyed but dissipated amongst the minute parts. That is not as if they were lost but it is like the changing of large coins into small ones.
The truth of the principle had previously been demonstrated by Huygens who had however formulated it without indicating the great generality of its scope. He distinguished between the conservation of vis viva and that of momentum in his statement:
The quantity of motion possessed by two bodies may be augmented or diminished by their encounter; but there remains always the same quantity on the same side if we subtract the quantity of opposite motion. The sum of the products of every hard body multiplied by the square of its velocity is always the same before and after the encounter.
It must be observed that both for Descartes and for Leibniz the world consists only of matter in motion and there exists no action at a distance. Consequently they did not admit the existence of what we call potential energy so that for them the principle of energy consisted in the constancy of the total kinetic energy. It should also be observed that Leibniz in the passage I have quoted in speaking of the dissipation of the molar energy amongst the smallest particles of a body does not seem to have considered this transformation as equivalent to the production of heat.
In the eighteenth century the conception of heat as a substance gradually gained upon the Cartesian idea of heat-motion. The amount of this substance was supposed to be conserved in its passage from one body to another. When it ceased to manifest itself by means of the thermometer heat was regarded by Black as still present but as latent heat capable of manifesting itself in certain conditions and thus his conception of latent heat was analogous to our conception of potential energy. Even the invention of the steam engine produced no immediate change in this conception of the substantiality of heat. Watt and his successors failed to attain to the view that thermal changes indicate any relation between heat and mechanical motion. However towards the end of the century Lavoisier and Laplace tentatively related the production of heat by friction with the conception of heat-motion and defined the amount of heat as the sum of the products of the masses of the molecules into the squares of their velocities.
Early in the nineteenth century direct experimental demonstrations were obtained by Rumford and by Humphry Davy of the transformation of motion into heat. The concept of latent vis viva called by Poncelet work was formulated (1803) by Lazare Carnot; this concept is that now known as potential energy. In 1839 the engineer Séguin in a work on the construction of railways remarked that:
As the theory at present adopted would lead however to this result (perpetual motion) it appears to me more natural to suppose that a certain number of calories disappear in the act of producing mechanical force or power and conversely; and that the two phenomena are bound together by conditions which assign to them invariable relations.
It appears however that before Séguin the principle of the equivalence of heat and mechanical energy had been conceived in its generality by Sadi Carnot who obtained by calculation an estimate of the mechanical equivalent of heat. In his earlier work Carnot had employed the material theory of heat and his later formulation of the modern theory was preserved only in manuscript notes which remained unpublished until 1871. But it was the experimental researches of Joule published in 1843 that brought prominently before the scientific world the theory of the equivalence of heat and mechanical energy. The earlier estimates obtained by Joule of the number of foot-bounds of work equivalent to the heat required to raise the temperature of a pound of water one degree Fahrenheit were widely discordant varying between 742 and 1040; but as the result of a later series of experiments he obtained 770 foot-bounds as the equivalent; and this is not very different from the value now accepted. Joule did not however doubt that the value of the equivalent exists as a definite number notwithstanding the considerable variation in his experimental determinations of it. His certitude on the matter was derived from his conviction of its a priori necessity.
We might reason a priori (he writes)1 that such an absolute destruction of living force cannot possibly take place because it is manifestly absurd to suppose that the powers with which God has endowed matter can be destroyed any more than that they can be created by man's agency; but we are not left with this argument alone decisive as it must be to every unprejudiced mind.
The work of J. R. Mayer on the conservation of energy which appeared in the year before the first publication of Joule's experiments was the earliest publication on the subject in its modern form. In 1843 there also appeared a work on the same subject by a Danish savant A. Colding. The work of Mayer was mainly guided by his philosophical ideas and did not include any experimental verifications of the principle such as those of Joule. He appeals to the old idea that forces are causes and that the cause is equal to the effect. His determination of the equivalent of heat by a calculation presupposes the existence of the constant relation of equivalence as was the case in the similar calculation made by Sadi Carnot. Colding makes the assumption that energy persists as a kind of indestructible non-material substance. Thus he writes2:
Since forces are spiritual and immaterial beings since they are entities which are known to us only by their empire over nature these entities must doubtless be very superior to every existing material thing; and as it is evident that it is by forces alone that the wisdom that we perceive is expressed and that we admire in nature these powers must be in relation with the spiritual immaterial and intellectual power itself which guides the course of nature; but if this be so it is absolutely impossible to conceive that these forces should be anything mortal or perishable. Without doubt consequently they must be regarded as absolutely imperishable.
So far the scope of the principle of the Conservation of Energy has been confined to the mechanical domain and to the equivalence of mechanical energy and heat but in the well-known treatise published by Helmholtz in 1847 there is consistently developed the doctrine that the conservation of energy is applicable to all departments of Physics. This work gives ample evidence that the author like Joule Mayer and Colding originally regarded the principle as one which follows from the principle of causation.
In a very interesting passage in the introduction to his treatise Helmholtz writes:
It is the object of these sciences (the physical sciences) to seek for the laws by which the different processes in nature are reduced to general rules and from these rules can be re-determined. These rules for example the laws of the refraction or reflection of light and that of Mariotte and Gay-Lussac for the volume of gases are clearly nothing but general notions by which all the phenomena concerned are embraced. The search for them is the business of the experimental part of our sciences. Their theoretical part on the other hand attempts to find the unknown causes of the processes from the visible effects; it attempts to bring them under the law of causality. We are compelled and authorized to do this by the principle that every change in nature must have a sufficient cause. The immediate causes to which we attribute phenomena may be invariable or variable; in the latter case the same principle compels us to seek for other causes of this variability and so on until we have arrived at ultimate causes which work according to an invariable law which consequently at every time with the same external conditions produce the same effect. The final goal of the theoretical sciences is thus to search for the final invariable causes of the processes in nature. Whether all processes can be reduced to such (causes) that is whether nature is completely comprehensible or whether there are changes in her which do not obey the law of necessary causation and which therefore fall into the domain of spontaneity or freedom cannot here be decided. It is certainly clear that science whose aim it is to comprehend nature must start from the hypothesis that she is comprehensible and must examine and form conclusions in accordance with this hypothesis until it is perhaps compelled by irrefutable facts to acknowledge the limits of the hypothesis.
It will be observed that Helmholtz's opinions as to the search for efficient causation in nature being the function of the theoretical parts of Natural Science are in divergence with the view that I have maintained in these lectures. It is however very interesting to notice that at a later time Helmholtz declared that he had modified the opinions expressed in the passage I have quoted. In fact in a note appended to a later edition of his work he says:
The philosophical discussions in the introduction were more strongly influenced by Kant's epistemological views than I at the present time would recognize as correct. I have later made clear to myself that the Principle of Causality is in fact nothing else than the hypothesis that all natural phenomena are subject to law.
It would appear from this statement that Helmholtz emancipated himself from the idea that efficient causation is to be found in nature by the aid of science and that he finally identified the term causality with the recognition of invariability in sequences of phenomena.
Leaving aside the supposed demonstration of the principle of energy by means of the a priori principle of causation it is possible to deduce the principle from the classical system of Mechanics if the assumption be made that all the phenomena of motion are governed by central forces; in which case they form what is called a conservative system. In accordance with this assumption the forces acting between every pair of corpuscles of a system are along the line joining them are equal in magnitude and opposite in direction; and the magnitude of such a force depends solely on the distance between the particles. The whole energy of such a system consists then of two parts the kinetic energy or energy of the motion of the system and the potential energy or energy of position which represents the capacity of the forces of the system to do mechanical work. That the principle in the form that the sum of the kinetic energy and the potential energy of the system is constant for such a system during its motion follows as a mathematical consequence of the dynamical scheme was demonstrated by Helmholtz in his treatise. But as regards the application of the principle in general physics it should be remarked that it is exceedingly doubtful whether in the molecular or sub-molecular domain it is possible to restrict the forces to those of the central type. For example it does not seem possible to regard the phenomena of permanent deformation and of crystallization as involving such forces only.
We have already seen that the desire to form picturable images of mechanical phenomena resulted in a reluctance to accept the notion of forces acting at a distance as part of a mechanical scheme. The same tendency has led to attempts to explain what is apparently potential or latent energy as really reducible to the kinetic energy of small parts of bodies or of a medium and thus ultimately to abolish the distinction between kinetic and potential energy; the latter being regarded as less concrete or picturable than the former. But in accordance with the view of the character of conceptual schemes which has been adopted in these lectures the concept of potential energy as a measurable quantity is really on the same footing as that of kinetic energy; and consequently even if it still appears to be a desirable simplification to reduce the two concepts to one such reduction has no longer the same urgency as with those who feel bound to a more realistic interpretation of the concepts of dynamical science.
The principle of energy has frequently been regarded as a consequence of the principle of the impossibility of perpetual motion. Very numerous attempts many of them very ingenious have been made to construct machines which should actualize perpetual motion. The conviction of the impossibility of perpetual motion however became gradually very strong among men of science. It was affirmed by Leonardo da Vinci Galileo Stevinus and by Leibniz the last of whom employed it to establish the principle of vis viva. In 1775 the French Academy of Sciences directed that solutions submitted to the Academy of the problems of the duplication of the cube the trisection of an angle the quadrature of the circle and of the construction of machines involving perpetual motion should no longer be examined. In the case of the last problem the Academy based its decision upon a priori grounds stated as follows:
The construction of perpetual motion is absolutely impossible: even if friction and the resistance of the medium did not ultimately destroy the resistance of the moving force that force can only produce an effect equal to its cause; if then it is desired that the effect of a finite force shall continue indefinitely it is necessary that the force should be infinitely small in a finite time. Making abstraction of the friction and the resistance a body on which motion has been once impressed will conserve it always; but it would be by not acting on other bodies and the only perpetual motion possible on this hypothesis (which moreover cannot be realized in nature) would be absolutely useless for the object at which the constructors of perpetual motion aim.
Helmholtz showed that the principle of energy is deducible from that of the impossibility of perpetual motion. That impossibility he regarded however as a fact of experience established by the numerous vain attempts to construct a perpetuum mobile. It has however been pointed out by Poincaré that it is only in the case of reversible phenomena that the conservation of energy follows from the impossibility of perpetual motion. The general principle of the conservation of the energy of an isolated system in all its various forms through all the physical and chemical changes which the system may undergo can only be regarded as an hypothetical principle to be used tentatively as a guide in our attempts to describe conceptually the various processes in the transformations. The fact that we have no assurance that all the possible forms of energy which may occur in physical phenomena are known to us makes it impossible to conceive that the principle should admit of anything like complete empirical verification.
The history of science exhibits the discovery of various forms of energy previously unrecognized. In particular our knowledge of the phenomena of electricity except in merely trivial manifestations dates only from the investigations of Gilbert three centuries ago. The recent discoveries in the last few decades; those of Hertzian waves Roentgen rays and of radioactive substances have made us acquainted with forms of energy whose existence had been previously unsuspected. Are we even now certain that we are acquainted with all the forms of energy that may be discovered in solar radiation and which may be beyond the known limits of the luminous thermal and actinic rays? The discovery of the Roentgen rays is one illustration of the fact that a form of energy may long remain undiscovered when it is so to speak before our eyes; for Crookes'tubes had been employed for a quarter of a century before Roentgen's discovery of the rays to which they give rise. We have seen that in the case of a system consisting of particles between which central forces act dependent only on their relative distances the principle of energy take the simple form that the sum of the kinetic energy and the potential energy is constant; the former depending only on the velocities of the particles and the latter upon their positions but not on their velocities; so that the total energy can be resolved only in one manner into the sum of the two components. But if as in the case of Weber's law of mutual action of two electric molecules the mutual action depends not only on their distance but also on their velocities and on their accelerations the second part of the energy would depend on the velocities and it might contain terms depending on the squares of the velocities. In such a case we have no means of distinguishing between terms which belong to the two parts respectively of the total energy. Poincaré has pointed out1 that we then have no means of defining the energy of the system because if the total energy of the system is constant so also is any function of that total energy; and such a function might be substituted for the energy itself and made the basis of an amended definition of the energy of the system. There exists in such a case no means of fixing upon a precise definition of the energy as such that it may be divided into two parts each of a specified form. Moreover if the principle of energy is to be of any use it is necessary to take account of the distinctions between the mechanical energy of molar bodies and the other forms of energy such as heat chemical and electrical energy. This can only be done if it is possible to divide the whole energy of the system into parts which are absolutely distinct in form; a part involving only the squares of the velocities of the bodies another part independent of these velocities and of the thermal and electric states of the system and a third part independent of the velocities and the positions of the bodies and dependent only on their internal states. But the case of electric energy due to the mutual electric action of the bodies suffices to show the impossibility of this division into such separate parts. For the electrostatic energy depends not only on the electric charges of the bodies but also upon their positions. If the bodies are in motion their electrodynamic energy depends not only on their states and positions but also on their velocities. We have therefore no obvious means of selecting and separating out the different parts of the total energy in the desired manner. The conclusion drawn by Poincaré from these considerations is that when an attempt is made to extend the principle of the conservation of energy so as to embrace all the phenomena with which Physics has to deal we are faced with the difficulty of defining the energy of the system in a unique manner so that different parts of it may be identified as referring to the different phenomena which occur in the system. He remarks that when this extreme generality is aimed at there appears to be nothing left of the principle except an enunciation: is something constant “and that in this form the principle lies outside the bounds of experiment and is reduced to a kind of tautology.
This criticism is pertinent in relation to the attempt made by Ostwald and others to set up a science of Energetics based upon the Principle of Energy and that of Least Action (or some other similar principle) with the view of avoiding the difficulties connected with the hypothesis of the existence of atoms. The fundamental conception of Energetics is that every change in an isolated system is regulated by two laws. The first is that the sum of the kinetic and potential energies is constant through all the transformations of the system. The second is that if the system passes from one configuration at one time to another configuration at another time the passage always takes place in such a manner that the mean value of the difference of the two kinds of energy in the interval of time between the two specified times is a minimum.
The lessons to be drawn from the history of the varying conceptions that have arisen at various times in connection with the sustained efforts that have been made to attain clear conceptions of what it is that is conserved in matter and its various transformations are mainly those of the inadequate character of a priori conceptions such as the principle of causation and of the partial character of the empirical verification of the principles. It would appear that when the utmost has been attained as regards clearness of statement of these principles as conceptual laws there remains an element of doubt and uncertainty and of tentativeness as regards the range of applicability both in practice and in theory of these laws in their function of describing the actual changes and transformations in the perceptual world.
