In the formal sciences we began the specialization of the object from the most general concept of thing conceivable, possessing no other characteristic attribute than its capability of being distinguished from other things; and we carried the specialization so far that we could follow in its movements an object definite as to time and space. This object, to be sure, was defined only in that it occupied a definite space, and accordingly had a definite form. As a matter of fact, the spacial thing of geometry and phoronomy reveals no further attributes.

It is here that the physical sciences enter into their dominion one after the other, and fill the empty space of the geometric thing with definite attributes. These are the secondary qualities of Locke, of which he assumed that they do not belong so much to the bodies themselves as that they merely appear to us so on account of the nature of our human sense organs. Now that our knowledge concerning the nature of those properties as well as the structure of our sense organs is much more thorough, we have more definite ideas also of the subjective part of the corresponding experiences, and in a large measure are able to separate it from the objective part.

All properties which physical bodies in contradistinction to geometric bodies possess can be traced back to a fundamental concept, which, in conjunction with the concepts explained in the former chapter, serves to characterize and distinguish the physical structure. For example, the fact that we can distinguish cubes of equal size but of different material, different temperature, and different luminosity, can be traced back always and entirely to the different kinds of energy acting in the geometric space in question. The concept of energy, therefore, plays approximately the same rÔle in the physical sciences as the concept of thing in the formal sciences, and the essentials of this new field of science are the comprehensive knowledge and development of this concept. Because of its great importance it has long been known and applied in individual forms. But the systematization of the physical sciences relative to energy is a matter of only recent date.

43. Mechanics.

Recently many scientists have taken exception to the traditional division of mechanics into statics, or the science of equilibrium, and dynamics, or the science of motion, because it does not correspond to the essence of the thing, equilibrium being only the limit-case of motion. However, the classic presentations of this science are based on that division, so that it must express an essential difference. This difference we can clearly recognize through the application of the concept of energy to mechanics. We then learn that statics is the science of work, or the energy of position, and that dynamics is the science of living force, or of the energy of motion.

By work in the mechanical sense we mean the expenditure of force required for the locomotion of physical bodies. While a cube of lead is geometrically equal to a cube of glass, we experience a great difference between them when we lift them from the floor to a table. We call the cube of lead heavier than the glass cube, and we find it requires more work to raise the former than the latter. For psychologic reasons this judgment becomes especially clear when the work required to lift the lead cube marks the limit of our physical capacity.

Work depends not only upon the difference described above, but also upon the distance through which it is exerted. It increases in proportion as the distance increases. In mechanics work is proportional both to the distance and to that peculiar property which in the given example we call weight. But a more general concept has been formed for that property in the mechanical sense, called force, of which weight constitutes but a special instance. Whenever there is a resistance combined with a change of place we speak of a force, and the product of the force and the distance we call work.

The cause of this kind of concept formation is the following: There are a great number of different machines, all of them possessing the peculiarity that work can be put into them at a definite place and taken out at another place. Now, centuries of experience have shown that it is impossible to obtain more work from such mechanical machines than has been put into them. As a matter of fact, the work obtained is always less than the work put in, and the two approach equality as the machine approaches perfection. It is to such ideal machines, therefore, that the law of the conservation of work applies. This law states that, though a given quantity of work may be changed in the most manifold ways as to direction, force, etc., it is impossible to change its quantity.

The reason we can judge of this fact with such certainty is because for many centuries a number of the ablest mechanicians have sought for a solution of the problem of perpetual motion, that is, for the construction of a machine from which more work can be gotten than is put into it. All such attempts have failed. But the positive result secured from these apparently futile efforts is the law of the conservation of work. The greatness and importance of this result will become apparent in the further course of our study.

Here for the first time we meet with a law expressing the quantitative conservation of a thing, which may none the less undergo the most varied qualitative changes. With the knowledge of this fact we involuntarily combine the notion that it is the "same" thing that passes through all these transformations, and that it only changes its outward form without being changed in its essence. Such ideas, it is true, are widespread, but they have a very doubtful side to them, since they correspond to no distinct concept. If we want to call the quantitative magnitude of the product of the force and distance the "essence" of work, and the determination of the force and the distance according to magnitude and direction, which come under consideration for each special value, as its "form," then, of course, there is no objection to be made to mere nomenclature. But we must bear in mind that the difference obtaining here lies exclusively in the fact that the amount of work measured quantitatively remains unchanged, while its factors undergo simultaneous and opposite changes.

This discovery, that there is a magnitude which can be quantitatively determined, and which, as experience shows, remains unchanged, however much its factors may change, invariably results not only in a very simple and clear formulation of the corresponding natural law, but also corresponds to the general tendency of the human mind to work out conceptually "the permanent in change." If, in accordance with the word-sense, we denote everything which persists under changing conditions by the name of substance, we encounter in work the first substance of which we have attained knowledge in our scientific journeys. In the history of the evolution of human thought this substance has been preceded by others, especially by the weight and mass of ponderable bodies (which are also subject to a law of conservation), so that at present we are inclined to connect with the word substance a tacit secondary sense of ponderability. But this is a remnant of the still very widely spread mechanistic theory of the universe, which, though it has almost finished its rÔle in physics, will presumably continue to persist for a long time to come in the popularly scientific consciousness in accordance with the laws of collective thought.

44. Kinetic Energy.

The law of the conservation of work is by no means true of all cases in which work is expended or converted, but, as has been said, only of ideal machines, that is, of such cases which do not exist in reality. But while in imperfect machines there is at least an approximation to this law, there are besides countless normal cases in which we cannot even speak of an approximation. When, for example, a stone falls to the ground from a certain height, a certain quantity of work is expended, which is equal to that by means of which the stone can be raised again to its original height. This quantity of work apparently disappears entirely when the stone remains lying on the ground. We shall discuss this case later. Or the falling of the stone can be so guided that it can lift itself again. This happens, for instance, when, by fastening the stone to a thread, it is forced to move in a curved path, or to perform pendular oscillations. In that case, it is true, the stone will fall to the lowest point which the thread permits, and so will there have lost its work without having done any other work in the meantime. But it has entered a condition by virtue of which it raises itself again, so that (as before, only in the ideal limit-case) it reaches its former height, and so has lost no work. For this moment, too, then, the law of the conservation of work obtains. But in the meantime new relations have arisen.

What distinguishes the stone moving like a pendulum from the stone which simply falls is, that at its lowest point it has not remained lying still, but possesses a certain velocity. By means of this it lifts itself again, and after it has reached its former height, it has lost its velocity. Therefore, there is a reciprocal relation between the work which it loses and the velocity which it gains, and the question may therefore be put, How can this relation be represented mathematically? Experience teaches that in every such case a function of the velocity and of another property of the body, called mass, can be established in such a way that this function, called the kinetic energy of the body, increases precisely as much as the amount of work the body has expended, and vice versa. The sum of the kinetic energy of the body and of the work is therefore constant, and the clearest mode of conceiving of this relation is by assuming that work can be transformed into kinetic energy and vice versa in such a way that given amounts of the two magnitudes are equal or equivalent to one another. Naturally, this is only an abbreviated way of expressing the actual relations, for it might just as well be assumed that the work really disappears and the kinetic energy really originates anew, and that the disappearance of the one substance only happens regularly to coincide with the origin of the other. But it is this regular conjunction of phenomena that constitutes the sole ground of every causal relation, and in such a sense we are justified in regarding the disappearing work as the cause of the kinetic energy that arises, and to designate this relation summarily as a transformation.

By the inclusion of cases in which work is converted into kinetic energy the law of the conservation of work therefore becomes the law of the conservation of the sum of work and kinetic energy. We are thereby compelled to extend the concept of substance, which at first contains only work, to the sum of both magnitudes, and to introduce a new name for this enlarged concept.

It will soon appear that all cases of imperfect machines, in which work disappears without giving rise to an equivalent amount of kinetic energy, can, with a corresponding enlargement of the concept, be likewise included in the law of conservation. For experience has shown that in such cases something else arises, heat, light, or electric force, etc. This generalized concept, which embraces all natural processes and permits the sum of all corresponding values to be expressed by a law of conservation, we call energy. The law in question, therefore, is:

In all processes the sum of the existing energies remains unchanged.

The principle of the conservation of work in perfect machines proves to be an ideal special instance of this general law. A perfect machine is one in which work changes into nothing but work of another kind, and not into a different kind of energy. Then each side of the equation which expresses the general law of energy, namely,

Energy that has disappeared = energy that has arisen,

contains only the magnitude of the work, and expresses the law of the conservation of work. If, on the other hand, as in the case of the pendulum, the work increasingly changes part by part into kinetic energy, and vice versa, the equation during the first period is:

Work that has disappeared = kinetic energy that has arisen,

and during the second period in which the pendulum rises again,

Kinetic energy that has disappeared = work that has arisen.

Thus, while work can be called a substance only in a limited sense, since its conservation is limited only to perfect machines, we may call energy a substance unqualifiedly, since in every instance of which we know the principle has been maintained that a quantity of any energy never disappears unless an equivalent quantity of another energy arises. Accordingly, this law of the conservation of energy must be taken as a fundamental law of the physical sciences. But not only do all the phenomena of physics, including chemistry, occur within the limits of the law of conservation, but until the contrary is proved the law of conservation must also be regarded as operative in all the later sciences, that is, in all the activities of organisms, so that all the phenomena of life must also take place within the limits of the law of conservation. This corresponds to the general fact, which I have emphasized a number of times, that all the laws of a former science find application in all the following sciences, since the latter can only contain concepts which by specialization, that is, by the addition of further characteristics, have sprung from the concepts of the former or more general sciences.

45. Mass and Matter.

It has been noted above that kinetic energy depends upon another magnitude beside velocity. A conception of its nature can be obtained when we try to put different bodies in motion. In doing so the muscles of the arm perform certain quantities of work, and we feel whether the quantities are greater or smaller. In this way we obtain a clear consciousness of the fact that different bodies require quite different quantities of work for the same velocity. The property which comes into play here is called mass, and mass is proportional to the work which the various bodies require to attain the same velocity. Since the work and the velocity can be measured very accurately by appropriate means, mass also lends itself to a correspondingly accurate measurement.

All known ponderable bodies have mass. That means there is a regular connection between the property which makes a body tend to the earth with a certain definite force (called weight) and the property by virtue of which a body assumes certain velocities under the influences of motive causes. We can readily conceive that it is possible for us to learn only of such bodies as are heavy, that is, bodies which are held by the earth, since the others, if they exist at all, would naturally have left the earth long ago. That all these bodies also have mass is to be explained in a similar way. For a body of mass zero would at each impulse assume infinitely great velocity, and could therefore never be the object of our observation. Consequently, by reason of the physical conditions obtaining on the earth's surface, the bodies known to us must combine both properties, mass and weight.

The name given to this concept of the combined presence of mass and weight in space is matter. Experience shows that there is a law of conservation for these magnitudes also, according to which whatever changes we may produce in bodies possessing weight and mass, no change will occur in the sum of their weight and mass. According to the nomenclature previously introduced we must therefore call weight and mass substances, since they remain the same as to quantity, no matter what changes they may undergo. However, it is usual to apply the name substance to the concept of matter composed of mass and weight. In fact, scientists often go so far as to limit the name to this single instance of the various laws of conservation, and to take substance to mean exclusively the combination of mass and weight. This is connected with the conception which we are about to discuss, that all natural phenomena can ultimately be conceived as the motion of matter. Through the greater part of the nineteenth century this conception, called scientific materialism, was accepted almost without opposition. At present it is being more and more recognized that it was only an unproved assumption, which the development of science daily proves to be more untenable.

46. Energetic Mechanics.

In the light of our previous observations the branch of science traditionally known as mechanics appears as the science of work and of kinetic energy. Furthermore, statics is shown to be the science of work, while dynamics, besides treating of kinetic energy in itself, also treats of the phenomena of the change of work into kinetic energy, and vice versa. We shall find the same relation again later, only in more manifold forms. Every branch of physics proves to be the science of a special kind of energy, and to the knowledge of each kind of energy must be added the knowledge of the relations by which it changes to the other forms of energy and vice versa. It is true that in the traditional division of physics this system has not been strictly carried out, since an additional and very influential motive for classification has been the regard paid to the various human sense organs.

Nevertheless this ground does not lie in the field of physics, but in that of physiology, and must, therefore, be abandoned in the interest of strict systematization.

Of the physical sciences mechanics was the first to develop in the course of historical evolution. A number of factors contributed to this end—the wide distribution of mechanical phenomena, their significance to human life, and the comparative simplicity of the principles of mechanics, which made it possible to discover them at an early date. Most to be noted is, that of all departments of physics mechanics is the first which lent itself to comprehensive mathematical treatment. It is true that the mathematical treatment of mechanics was possible only after idealizing assumptions had been made—perfect machines and the like—so that the results of this mathematical treatment not infrequently had very little to do with reality. The mistake of losing sight of the physical problem and of making mechanics a chapter of mathematics has not always been avoided, and it is only in most recent times that the consciousness has again arisen that the classical mechanics, in arbitrarily limiting itself to extreme idealized cases, sometimes runs the risk of losing sight of the aim of science.

47. The Mechanistic Theories.

Because the evolution of mechanics antedates that of the other branches of physics, mechanics has largely served as a model for the formal organization of the other physical sciences, just as geometry, which has been handed down to us from antiquity in the very elaborate form of Euclid, has largely been used as a model for scientific work in general. Such methods of analogy prove to be extremely useful at first because they serve as a guide to indicate when and where new sciences, in which all possibilities are open, can be got hold of. But later on such analogies are apt to be harmful. For each new science soon requires new methods, by reason of the peculiar manifoldness which it has to deal with, and the finding and the introduction of these new methods are easily delayed, and, as a matter of fact, often have been delayed, because scientists could not free themselves soon enough from the old analogy.

By its being based upon memory the human mind is so constructed that it cannot assimilate something entirely new. The new must in some way be connected with the known in order that it may be organically embodied in the aggregate of concepts. Therefore, it is the first involuntary impulse of our mind, in the presence of new experiences or thoughts, to look about for such points at which a linking of the unknown to the known seems possible. In the case of mechanics this necessity for finding connecting links has acted in such a way that the attempt has been made, and is still being made, to conceive and represent all physical phenomena as mechanical.

The impulse to this was first given by the extraordinary successes which mechanics has attained in the generalization and prediction of the motions of the heavenly bodies. The names of Copernicus, Kepler, and Newton mark the individual steps in the mechanization of astronomy. The cause of this lies in the fact that the heavenly bodies actually approximate very closely the ideal of the purely mechanical form with which classical mechanics deals. These successes encourage the attempt to apply these mental instruments that were productive of such rich results to all other natural phenomena. An old theory, according to which all physical things are composed of the most minute solid particles of matter called atoms, supported these tendencies and invited the attempt to regard the little world of atoms as subject to the same laws as had been found to apply so successfully to the great world of the stars.

Thus we see how this mechanistic hypothesis, the assumption that all natural phenomena can be reduced to mechanical phenomena, comes as if it were a self-understood matter, and with its claim to be a profound interpretation of nature it scarcely permits the question as to its justification to be raised at all. And the effects here have been the same as I described above in cases in which inferences from analogy are accepted too extensively or too credulously. While it is true, no doubt, that the mechanical hypothesis at first was fruitful of results in special research, because it facilitated the putting of the question—for example, we need think only of the atomic hypothesis in chemistry—later, the efforts to find further hypothetic help for the inadequacies of the hypothesis that gradually came to light, have not infrequently led scientific research to pseudo-problems, that is, to questions which are questions only in hypothesis, but to which no actual reality can be shown to correspond. Such problems, therefore, are by their very nature insoluble, and constitute an inexhaustible source of differences of scientific opinion.

The most flagrant of the injurious consequences of the mechanistic hypothesis appear in the scientific treatment of the mental phenomena. Ready as scientists were to represent all other life phenomena, such as digestion, assimilation, and even generation and propagation, as the consequence of an extremely complicated play of certain atoms, their courage never went so far as to apply this principle to mental life and to consider that by mechanics the last word had been said on the subject.

It is because of this hesitancy to bring mental phenomena under the same mechanistic principle as all the other phenomena that the philosophical systems had to search for some other means to connect the mental world with the mechanical, and the efforts of the philosophers to bring about this end have been most varied. Of the various doctrines that have come down to us, that of the pre-established harmony proposed by Leibnitz is in the ascendant in our day, and is now called the theory of the psycho-physical parallelism. According to this theory it is assumed that the mental world exists alongside, and quite independent of, the mechanical, but that the things have been so prearranged that mental processes take place simultaneously with certain mechanical processes (according to some, with all mechanical processes) in such a way that, although the two series do not influence each other in the least, they always correspond to each other precisely. How such a relation has come about and how it is maintained remains unsaid, or is left to future explanation.

We need only think of the content of this hypothesis with an unbiased mind to lose all relish for it at once. In fact, it has no other raison d'Être than the presumption that the mental and the mechanical world are opposed to each other. As soon as we abandon the thesis that the non-mental world is exclusively mechanical, we acquire the possibility again of finding for the theory of mental phenomena a constant and regular connection with the theories of all other phenomena, especially with the phenomena of life. Therefore it will be found most expedient in every respect, instead of rendering scientific research one-sided and almost blind to nonconforming facts by preconceived hypotheses, such as the mechanistic hypothesis, to seek, as hitherto, from step to step, the new elements of manifoldness which must be taken account of in the progressive upbuilding of science and to limit ourselves faithfully to them in the formation of general ideas.

48. Complementary Branches of Mechanics.

The field of pure or classical mechanics is limited to the above two kinds of energy, work and kinetic energy, though these do not exhaust the manifoldness of the mechanical energies. Accordingly, other branches of mechanics dealing with the corresponding phenomena are added to the classical mechanics described above.

If by mechanical energies we understand all energies in which changes of space are connected with changes of energy, there are as many different forms as there are spacial concepts that seem applicable. Form, Volume, and Surface of bodies in space are especially recognizable as the field of action for energy, which shows different properties or manifoldnesses according to each of these relations.

The energy of form is manifested in bodies (solid or rigid bodies) that maintain a definite shape because every change of shape is connected with work or with the expenditure of some other energy. If the changes are small, the bodies are of such a nature that they return to their former condition of their own accord after the force exerted upon them has ceased to act. This property is called elasticity. However, the theory of elasticity, which has been extensively and rationally developed, is regarded as belonging rather to mathematical physics in general than to mechanics in particular. In greater changes of shape the energy of form, or elastic energy, passes into other forms, and the body does not return to its former shape after the force has been removed.

Other bodies have no energy of form (or only in an infinitesimally slight degree), so that they allow of changes of form without the expenditure of work, but their volume can be changed only by work. These are divided into two classes. First, the liquids, which have a definite volume (corresponding to the definite shape of solids), the changes of which in every sense, both compression and expansion, require work. Secondly, the gases with volume energy in only one sense of the word, in which only the compression of volume requires work, while in expansion a certain amount of work is thrown off. Such bodies can exist only so long as the expenditure of their volume energy by spontaneous expansion is prevented by the presence of a counter energy, as, for example, the elasticity of the walls of a vessel. This tendency is called pressure.

Finally, there are energy qualities at the surfaces between various kinds of bodies which come into play at the change of these surfaces. They always lie in such a direction that the enlargement of the surfaces requires work, and hence, by reason of the law of conservation of energy, cannot proceed by itself. In cases where there has been an inverse kind of energy present, that is, one which diminishes with increasing surface, it also has been active as a rule, thus bringing about the disappearance of the existing boundaries.

Since the seat of this kind of energy is in the surfaces (or superficies), it is called surface-energy. The phenomena depending upon it manifest themselves most clearly at the surface boundaries between liquids and gases. They are called capillary phenomena. This strange name, derived from the word capilla, hair, has its origin in the fact that because of surface-energy liquids rise in tubes which they wet, and the narrower the tube the higher they rise. If the lumen of the tube is as fine as a hair, a considerable rise can be observed. This is the entire connection between the name and the thing.

The mechanics of liquids is called hydromechanics, that of gases, aeromechanics, after the most familiar liquid, water, and the most familiar gas, air. The study of surface-energy under the name of the capillary theory forms part of theoretical physics. While formerly this branch, too, was regarded as a working part, or, rather, as a playing part, of mathematical problems, in more recent times extensive experimental research has made its entry in this province also, and has demonstrated the necessity of passing from the former abstractions or idealizations, which were carried altogether too far, to a better and profounder regard for the actually existing complexities.

49. The Theory of Heat.

The various forms of energies the aggregate of which is comprehended in physics, have very different special characters. A systematic investigation has not yet been made of the characters of manifoldness by which, for example, work is distinguished from heat, electrical energy from kinetic energy, etc., nor of what are the essential properties peculiar to each individual energy. We feel certain that differences do exist, for otherwise the energies could not be distinguished, and we feel certain that these differences are very important, for doubt seldom arises as to the kind of energy to which a certain phenomenon is to be assigned. But just as we have no systematic table of the elementary concepts, so we are still without a systematic natural history of the forms of energy in which the peculiarities of every species are characterized, and in which the entire material is so arranged according to these characteristics that we can take a general survey of it.

As regards heat energy, its foremost and most striking characteristic is its physiological effect. In our skin there are organs for the perception of heat as well as of cold, that is, for temperatures above and below the temperature of the skin. However, the temperature that these organs can bear without injury to themselves is of a very small range, beyond which physical apparatuses of all kinds must be used, such as "thermometers."

Heat is the simplest kind of energy from the point of view of manifoldness. Every heat quantity is marked by a temperature, just as a kinetic energy is marked by velocity. But while a velocity is determined in space so that velocities of equal magnitude have in addition a threefold infinite manifoldness in reference to direction, a temperature is characterized completely and unambiguously by a simple number, the degree of temperature. Two temperatures of equal degree can in no wise be distinguished, since temperature possesses no other possible manifoldness than degree.

The same property is found in heat energy itself. In heat energy we measure the quantity of energy itself and call it the heat quantity, while in some of the other kinds of energy, only the factors into which they can be divided are measured, and no habitual conception of the energy itself is developed. A heat quantity is likewise fully indicated by its measure number.

That heat is an energy, that is, that it is developed in equal quantities from other kinds of energy, and can change back again into them, is a discovery which, despite its fundamental and general character, was not made before the forties of the nineteenth century. As often happens in cases of important scientific advances, the same idea came simultaneously to a number of investigators. The first to grasp and fully comprehend this idea was Julius Robert Mayer of Heilbronn, who published his results in 1842. Mayer not only showed that the imperfect machines (p. 134), which limit the validity of the law of the conservation of work, owe this peculiarity to the fact that they transform a part of the work into heat, and that when we take account of this part, the law of conservation holds perfectly good, but he also calculated, with extraordinary acumen, the mechanical equivalent of heat from the then existing data of physics. That is to say, he determined how many units of heat (in the measure then in use) correspond to a unit of work (in its specific measure) in the change from one to the other, and back. And this fundamental knowledge of the existence of a quantitatively unchangeable substance, arising from work, and capable of being transformed into it, Mayer did not limit in its application merely to heat. He was the first to construct a table, which he made as complete as possible, of all the forms of energy then known, and to assert and prove the possibility of their reciprocal change into each other.

In view of this relation of the quantitative equivalent of the various forms of energy when transformed into one another, an attempt is being made at present to measure them all with the same unit. That is, some easily obtained quantity of energy is arbitrarily chosen as a unit and it is determined that in every other form of energy the unit shall be equal to the quantity obtained from that unit on its transformation into the energy in question. For formal reasons the kinetic energy of a mass of two grams which moves with the velocity of one centimeter in a second has been chosen as the unit. It is called erg, an abbreviation of energy. The amount is very small, and for technical reasons 10¹⁰ times greater unit is used. To raise the temperature of a gram of water one degree a quantity of energy equal to 41,830,000 ergs is required.

50. The Second Fundamental Principle.

Another fundamental discovery has been made in connection with the heat form of energy, which, like the law of conservation, relates to all forms of energy, but has found its first and most important application in heat. While the law of conservation answers the question, how much of the new form of energy is developed if a given quantity of energy changes, but gives no clue as to when such a change occurs, this second law asserts the condition under which such changes arise, and is therefore called the second fundamental principle.

The discovery of this law antedates Mayer's discovery of the law of conservation by about twenty years, and was made by a French military engineer, Sadi Carnot, who died soon afterward without having lived to see the recognition his great work obtained. Carnot asked himself the question, Upon what does the action of the steam engine, which had just then come into use, depend? This led him first to the more general question of the action of heat engines in general. He found that no heat engine could work unless the heat dropped from a higher to a lower temperature, just as no water wheel can work unless the water flows from a higher to a lower level, and he determined the conditions which an ideal heat engine must fulfil, that is, a machine in which the greatest possible value in work is obtained from heat. However, an ideal machine of this nature can be constructed in very different ways, and Carnot's discovery consists in the recognition of the fact that the quantity of work obtained from the heat unit does not at all depend upon the peculiar construction of the ideal machine, but is determined solely by the temperature between which the heat transition takes place. This follows from the following considerations:

In the first place an ideal engine must be reversible, that is, it must be capable of working both ways, changing heat into work and work back into heat. Now, if we have two ideal engines between the same temperatures, and if we assume that engine A produces more work from the same quantity of heat than engine B, then let A move one way and let B move the other way with the work obtained from A. Since B produces less work from a given amount of heat, hence more heat from an equal amount of work, there will in the end be more heat at the higher temperature than was originally there. But experience teaches that there is no means in nature by which heat in the absence of concomitant change could be caused to rise to a higher temperature. Therefore an engine so constructed as to produce this result is impossible, And B cannot be of such a nature as to produce less work from the same quantity of heat than A.

The reverse is also impossible. For then we need merely couple the engines in the reverse way in order to obtain the same effect. Therefore, since B can do neither less nor more work than A, the two must do the same amount of work—which was to be proved.

It is obvious that this process of proof is similar to that by which the law of conservation was established. Because the arbitrary creation of energy from nothing is impossible there must be definite and immutable relations of change between the forms of energy. Because energy at rest does not spontaneously pass into conditions in which it can do work, the efficiencies of the machines must have definite and unchangeable values. If, for example, we could cause heat of its own accord to rise to a higher temperature, we could also construct a perpetual motion machine which would always yield work at no expense. But this perpetual motion would not be one that creates work out of nothing, but one that extracts it from energy at rest. A perpetual motion machine of this nature, too, is, according to our experience, impossible, and this impossibility forms the content of the second fundamental principle.

On the face of it this apparently "self-evident" proposition does not reveal how fruitful of results it is when applied to the discovery of simple but not obvious relations. It can only be said here that the deductions from this principle form the chief content of the extensive science of thermodynamics, which deals with the changes of heat into other forms of energy. We must only emphasize the fact that the application of this law, as was already observed in stating it, is not confined to the changes of heat alone. It is a law rather which finds application in all the forms of energy. For in every form of energy there is a property which corresponds to temperature in heat, and upon the equality or the inequality of which depends whether the energy in question is at rest or ready for transformations. This property is called the intensity of the energy. In work, for instance, it is force, in volume-energy it is pressure. If once the intensity in a body is equal, its energy is at rest, and it never again moves of its own accord.

Another form in which to present these relations is to make a distinction between free energy and energy at rest. If we have a heat quantity the temperature of which is higher than that of the surrounding objects, it can be used to do work only until its temperature has dropped to that of the surrounding objects. Although energy in abundance is still present, there is no longer any energy capable of change, or free energy. Since differences of temperature, like other differences of intensity, have a constant tendency to diminish, the amount of free energy on earth is constantly decreasing, and yet it is only this free energy that has value. For since all phenomena depend upon change of energy, and change of energy is possible only through free energy, free energy is the condition of all phenomena.

51. Electricity and Magnetism.

While the knowledge of heat energy goes back to the most ancient periods of civilization, electrical and magnetic energies are relatively young acquisitions. The highly developed technical application of both with the rich harvests they have yielded belongs exclusively to most recent times.

Both these forms of energy, like those discussed above, are connected in the main with ponderable "matter," but in a much slighter and less regular measure. While it is not possible as yet to render any given body free of heat (although lately the absolute zero point has been considerably approximated), freedom from electrical and magnetic energy is the normal condition of most bodies. This is connected with the peculiarity that electrical and magnetic properties are decidedly bi-symmetrical or polar. This property is not found in any other form of energy, and can serve as the special scientific characteristic of electricity and magnetism. This peculiarity shows itself in the concepts of positive and negative magnetism, and positive and negative electricity, and is due to the fact that two equal opposite quantities of electricity or magnetism, when added together, do not produce double their value, but nullify each other.[G]

The fact that electrical and magnetic energies generally exist only in a transitory state (with the notable exception of the magnetic condition of the earth) is probably the cause of our not having developed a sense organ for them, especially since their phenomena as they occur in nature have only occasionally and in very rare instances (thunderstorms) an influence upon us. On the other hand, the modern development of electrotechnics is based upon that property of electrical energy by virtue of which large quantities of it can be conducted along a thin wire over great distances without any considerable loss, and at the point desired can be easily changed into any other forms of energy. But since the collection and conservation of large quantities of electrical energy is hardly possible technically, the electrical apparatus must be so constructed that the quantities each time required should be produced at the moment they are used. The chief source of electricity is the chemical energy of coal, which is first transformed into heat, then into mechanical energy, and finally into electrical energy. This extremely roundabout process is necessary because a method technically practicable of transforming the chemical energy of coal directly into electrical energy has not yet been invented. On the other hand, mechanical energy can be easily and completely changed into electrical energy. Upon this is based the exploitation of much "water power," the energy of which could not be utilized but for the great capacity for change of the electrical form.

52. Light.

The case of light in our day seems to be similar to that of sound, which, although it has its special sense organ in man, is yet no particular form of energy, but has been found to be a combination of mechanical energies in an oscillatory or mutually changing state. It seems highly probable that light, too, is not a special form of energy, but a peculiar oscillatory combination of electrical and magnetic energies. It is true that the circle of proof is not yet quite closed, but the gaps have become so small that the above conclusion may at any rate be accepted as highly probable.

However that may be, light is an energy which, according to the known laws, travels through space with tremendous rapidity. We will call it radiant energy, since the part optically visible, to which alone the name light in its original sense belongs, represents an extremely small portion of a vast field, the properties of which change quite continuously from one end to the other.

Radiant energy is characterized as an oscillatory or wave-like process. So long as this fact was unknown (up to the beginning of the nineteenth century) it was thought that light consisted of minute spherical particles, which shot through space in a straight line with the tremendous velocity mentioned above. Later, in order to "explain" its wave nature, which in the meantime has come to be recognized, it was assumed to be due to the elastic vibrations of an all-pervading thing called ether, of which we know nothing else. This elastic undulatory theory has been abandoned in our time in favor of an electromagnetic theory supported by quite considerable experiential grounds. Whether it will be spared the fate that has overtaken the older theories (or rather hypotheses) of light cannot as yet be predicted with any degree of certainty.

Radiant energy is of very marked importance in human relations. As light it serves, with the aid of the corresponding receiving organs, the eyes, as a more manifold means of intercommunication between our bodies and the outer world than any other form of energy. The energy quantities penetrating to us from the extreme limits of the world space mark the outermost limits of which we have knowledge in any way whatsoever, and finally the energy quantities radiating to us from the sun constitute the supply of free energy at the expense of which all organic life on earth is maintained. Even the chemical energy stored up in coal represents nothing else than accumulations of former sun radiation, which had been transformed by the plants into the permanent form of chemical energy.

Very recently other newly discovered forms of radiant energy have been added to light. They are produced in manifold circumstances, and some bodies emit them constantly. The scientific elaboration of these extremely manifold and unusual phenomena has not yet been carried so far that they can be reduced to a doubt-free system. But so much, it seems, is already apparent, that they are presumably not purely new forms of energy, but rather very composite phenomena which may yield one or more new energies as component parts. But despite the peculiarity of these new rays, nothing certain has as yet been proved against the law of conservation itself.

53. Chemical Energy.

Since chemical energy is only one of several forms of energy, there seems to be no justification for allotting it to a special science, since all the other forms of energy must be incorporated in physics.

But the actual existence of chemistry as a special science which has already many subdivisions is justified in the first place by the external fact that in practical life and in industry chemistry occupies a very wide field comparable, if not superior, to that of the whole of physics. In the next place, from the psychological point of view, it is found that the chemist's methods of reasoning and working are so different from those of the physicist that a division seems to be in order for that reason also. Finally, there is in the nature of chemical energy itself an important distinction which marks it off from the other forms.

While, for example, there is only one form of heat or of kinetic energy, and in electricity there are only the two forms of polar opposites, chemistry, even after the greatest theoretical reduction, possesses at least about eighty forms. That is, it possesses as many forms as there are chemical elements. The experiential law, that the elements cannot be changed into one another,[H] also limits the corresponding changes of the chemical energies into one another, and thus characterizes the independence of these various forms. From this results a disproportionately greater manifoldness of relations, which find their expression in the many thousands of the individualized chemical substances or combinations.

This great manifoldness and the slight regularity hitherto found in connection with the properties and reciprocal relations of the numerous chemical elements renders modern chemistry more a descriptive than a rational science. It was no more than twenty years ago that an earnest and successful attempt was begun to apply the stricter methods of physics to the investigation of chemical phenomena. These labors, so far as they have gone, have yielded a great many far-reaching and comprehensive principles.

The significance of chemistry in human life is twofold. In the first place the energy of the human body, just as that of all other living organisms, depends chiefly upon the action of chemical energies in the most manifold forms. Of all the physical sciences, therefore, chemistry is the most important for biology, particularly for physiology. In the second place, as I have emphasized a number of times, it possesses the peculiar property which enables it to be preserved for a long time without passing into other forms and being dissipated. Furthermore, energy in this form permits of the most powerful concentration. More of chemical energy can be stored in a given space than of any other form of energy. Both these properties, then, may be considered as the reason why organic beings are constituted chiefly by means of chemical energy. At any rate, it is due to these two peculiarities that chemical energy serves as the primary source for almost all the energy used in industry.

Further, the manifoldness of chemical energy is the cause of the peculiar manner in which it is transformed into other forms. In the other forms of energy the transformation can be effected by the body itself. Nothing else is required. If a stone is thrown and it hits against a wall, it loses its kinetic energy, the greater part of which changes into heat. But in order to liberate the chemical energy of, say, coal, the coal alone is not sufficient; another chemical substance is required, the oxygen of the air. The interaction of the two substances produces a new substance, and it is only during this process that a corresponding part of the chemical energy is liberated. There are a few chemical processes also (allotropic and isomeric changes) in which a single substance without the co-agency of another substance can give off energy. But the quantity of energy thus obtained is infinitely small as compared to that liberated by the interaction of two or more substances. Because of the necessity of two or more substances to co-operate in giving off chemical energy, the opportunity for the transformation of chemical energy is less than for the transformation of the other forms of energy, and this is the main reason why it can be conserved so long and so easily. All that is necessary is to prevent contact with another substance. This is a problem, it is true, which from the point of view of strict theoretical rigor it is almost impossible to solve. In practice, however, it can be easily solved for periods of time long enough at least to require special means to enable us to recognize that it is only a temporary and not a fundamental solution. Scientifically expressed, the cause of this is that the diffusion of the various substances in one another can theoretically never be completely eliminated, while on the other hand the velocity of the diffusion over distances measured only by decimeters is extremely low.