Part 5 (1/2)

Let us next consider a point which belongs to pure mechanics, the history of the principle of virtual motions or virtual velocities. This principle was not first enunciated, as is usually stated, and as Lagrange also a.s.serts, by Galileo, but earlier, by Stevinus. In his Trochleostatica of the above-cited work, page 72, he says: ”Observe that this axiom of statics holds good here: ”As the s.p.a.ce of the body acting is to the s.p.a.ce of the body acted upon, so is the power of the body acted upon to the power of the body acting.”[49]

Galileo, as we know, recognised the truth of the principle in the consideration of the simple machines, and also deduced the laws of the equilibrium of liquids from it.

Torricelli carries the principle back to the properties of the centre of gravity. The condition controlling equilibrium in a simple machine, in which power and load are represented by weights, is that the common centre of gravity of the weights shall not sink. Conversely, if the centre of gravity cannot sink equilibrium obtains, because heavy bodies of themselves do not move upwards. In this form the principle of virtual velocities is identical with Huygens's principle of the impossibility of a perpetual motion.

John Bernoulli, in 1717, first perceived the universal import of the principle of virtual movements for all systems; a discovery stated in a letter to Varignon. Finally, Lagrange gives a general demonstration of the principle and founds upon it his whole a.n.a.lytical Mechanics. But this general demonstration is based after all upon Huygens and Torricelli's remarks. Lagrange, as is known, conceives simple pulleys arranged in the directions of the forces of the system, pa.s.ses a cord through these pulleys, and appends to its free extremity a weight which is a common measure of all the forces of the system. With no difficulty, now, the number of elements of each pulley may be so chosen that the forces in question shall be replaced by them. It is then clear that if the weight at the extremity cannot sink, equilibrium subsists, because heavy bodies cannot of themselves move upwards. If we do not go so far, but wish to abide by Torricelli's idea, we may conceive every individual force of the system replaced by a special weight suspended from a cord pa.s.sing over a pulley in the direction of the force and attached at its point of application. Equilibrium subsists then when the common centre of gravity of all the weights together cannot sink. The fundamental supposition of this demonstration is plainly the impossibility of a perpetual motion.

Lagrange tried in every way to supply a proof free from extraneous elements and fully satisfactory, but without complete success. Nor were his successors more fortunate.

The whole of mechanics, thus, is based upon an idea, which, though unequivocal, is yet unwonted and not coequal with the other principles and axioms of mechanics. Every student of mechanics, at some stage of his progress, feels the uncomfortableness of this state of affairs; every one wishes it removed; but seldom is the difficulty stated in words. Accordingly, the zealous pupil of the science is highly rejoiced when he reads in a master like Poinsot (ThAorie gAnArale de l'Aquilibre et du mouvement des systAmes) the following pa.s.sage, in which that author is giving his opinion of the a.n.a.lytical Mechanics: ”In the meantime, because our attention in that work was first wholly engrossed with the consideration of its beautiful development of mechanics, which seemed to spring complete from a single formula, we naturally believed that the science was completed or that it only remained to seek the demonstration of the principle of virtual velocities. But that quest brought back all the difficulties that we had overcome by the principle itself. That law so general, wherein are mingled the vague and unfamiliar ideas of infinitely small movements and of perturbations of equilibrium, only grew obscure upon examination; and the work of Lagrange supplying nothing clearer than the march of a.n.a.lysis, we saw plainly that the clouds had only appeared lifted from the course of mechanics because they had, so to speak, been gathered at the very origin of that science.

”At bottom, a general demonstration of the principle of virtual velocities would be equivalent to the establishment of the whole of mechanics upon a different basis: for the demonstration of a law which embraces a whole science is neither more nor less than the reduction of that science to another law just as general, but evident, or at least more simple than the first, and which, consequently, would render that useless.”[50]

According to Poinsot, therefore, a proof of the principle of virtual movements is tantamount to a total rehabilitation of mechanics.

Another circ.u.mstance of discomfort to the mathematician is, that in the historical form in which mechanics at present exists, dynamics is founded on statics, whereas it is desirable that in a science which pretends to deductive completeness the more special statical theorems should be deducible from the more general dynamical principles.

In fact, a great master, Gauss, gave expression to this desire in his presentment of the principle of least constraint (Crelle's Journal fAr reine und angewandte Mathematik, Vol. IV, p. 233) in the following words: ”Proper as it is that in the gradual development of a science, and in the instruction of individuals, the easy should precede the difficult, the simple the complex, the special the general, yet the mind, when once it has reached a higher point of view, demands the contrary course, in which all statics shall appear simply as a special case of mechanics.” Gauss's own principle, now, possesses all the requisites of universality, but its difficulty is that it is not immediately intelligible and that Gauss deduced it with the help of D'Alembert's principle, a procedure which left matters where they were before.

Whence, now, is derived this strange part which the principle of virtual motion plays in mechanics? For the present I shall only make this reply. It would be difficult for me to tell the difference of impression which Lagrange's proof of the principle made on me when I first took it up as a student and when I subsequently resumed it after having made historical researches. It first appeared to me insipid, chiefly on account of the pulleys and the cords which did not fit in with the mathematical view, and whose action I would much rather have discovered from the principle itself than have taken for granted. But now that I have studied the history of the science I cannot imagine a more beautiful demonstration.

In fact, through all mechanics it is this self-same principle of excluded perpetual motion which accomplishes almost all, which displeased Lagrange, but which he still had to employ, at least tacitly, in his own demonstration. If we give this principle its proper place and setting, the paradox is explained.

The principle of excluded perpetual motion is thus no new discovery; it has been the guiding idea, for three hundred years, of all the great inquirers. But the principle cannot properly be based upon mechanical perceptions. For long before the development of mechanics the conviction of its truth existed and even contributed to that development. Its power of conviction, therefore, must have more universal and deeper roots. We shall revert to this point.

II. MECHANICAL PHYSICS.

It cannot be denied that an unmistakable tendency has prevailed, from Democritus to the present day, to explain all physical events mechanically. Not to mention earlier obscure expressions of that tendency we read in Huygens the following:[51]

”There can be no doubt that light consists of the motion of a certain substance. For if we examine its production, we find that here on earth it is princ.i.p.ally fire and flame which engender it, both of which contain beyond doubt bodies which are in rapid movement, since they dissolve and destroy many other bodies more solid than they: while if we regard its effects, we see that when light is acc.u.mulated, say by concave mirrors, it has the property of combustion just as fire has, that is to say, it disunites the parts of bodies, which is a.s.suredly a proof of motion, at least in the true philosophy, in which the causes of all natural effects are conceived as mechanical causes. Which in my judgment must be accomplished or all hope of ever understanding physics renounced.”[52]

S. Carnot,[53] in introducing the principle of excluded perpetual motion into the theory of heat, makes the following apology: ”It will be objected here, perhaps, that a perpetual motion proved impossible for purely mechanical actions, is perhaps not so when the influence of heat or of electricity is employed. But can phenomena of heat or electricity be thought of as due to anything else than to certain motions of bodies, and as such must they not be subject to the general laws of mechanics?”[54]

These examples, which might be multiplied by quotations from recent literature indefinitely, show that a tendency to explain all things mechanically actually exists. This tendency is also intelligible. Mechanical events as simple motions in s.p.a.ce and time best admit of observation and pursuit by the help of our highly organised senses. We reproduce mechanical processes almost without effort in our imagination. Pressure as a circ.u.mstance that produces motion is very familiar to us from daily experience. All changes which the individual personally produces in his environment, or humanity brings about by means of the arts in the world, are effected through the instrumentality of motions. Almost of necessity, therefore, motion appears to us as the most important physical factor. Moreover, mechanical properties may be discovered in all physical events. The sounding bell trembles, the heated body expands, the electrified body attracts other bodies. Why, therefore, should we not attempt to grasp all events under their mechanical aspect, since that is so easily apprehended and most accessible to observation and measurement? In fact, no objection is to be made to the attempt to elucidate the properties of physical events by mechanical a.n.a.logies.

But modern physics has proceeded very far in this direction. The point of view which Wundt represents in his excellent treatise On the Physical Axioms is probably shared by the majority of physicists. The axioms of physics which Wundt sets up are as follows: 1. All natural causes are motional causes.

2. Every motional cause lies outside the object moved.

3. All motional causes act in the direction of the straight line of junction, and so forth.

4. The effect of every cause persists.

5. Every effect involves an equal countereffect.

6. Every effect is equivalent to its cause.

These principles might be studied properly enough as fundamental principles of mechanics. But when they are set up as axioms of physics, their enunciation is simply tantamount to a negation of all events except motion.

According to Wundt, all changes of nature are mere changes of place. All causes are motional causes (page 26). Any discussion of the philosophical grounds on which Wundt supports his theory would lead us deep into the speculations of the Eleatics and the Herbartians. Change of place, Wundt holds, is the only change of a thing in which a thing remains identical with itself. If a thing changed qualitatively, we should be obliged to imagine that something was annihilated and something else created in its place, which is not to be reconciled with our idea of the ident.i.ty of the object observed and of the indestructibility of matter. But we have only to remember that the Eleatics encountered difficulties of exactly the same sort in motion. Can we not also imagine that a thing is destroyed in one place and in another an exactly similar thing created? After all, do we really know more why a body leaves one place and appears in another, than why a cold body grows warm? Granted that we had a perfect knowledge of the mechanical processes of nature, could we and should we, for that reason, put out of the world all other processes that we do not understand? On this principle it would really be the simplest course to deny the existence of the whole world. This is the point at which the Eleatics ultimately arrived, and the school of Herbart stopped little short of the same goal.

Physics treated in this sense supplies us simply with a diagram of the world, in which we do not know reality again. It happens, in fact, to men who give themselves up to this view for many years, that the world of sense from which they start as a province of the greatest familiarity, suddenly becomes, in their eyes, the supreme ”world-riddle.”

Intelligible as it is, therefore, that the efforts of thinkers have always been bent upon the ”reduction of all physical processes to the motions of atoms,” it must yet be affirmed that this is a chimerical ideal. This ideal has often played an effective part in popular lectures, but in the workshop of the serious inquirer it has discharged scarcely the least function. What has really been achieved in mechanical physics is either the elucidation of physical processes by more familiar mechanical a.n.a.logies, (for example, the theories of light and of electricity,) or the exact quant.i.tative ascertainment of the connexion of mechanical processes with other physical processes, for example, the results of thermodynamics.

III. THE PRINCIPLE OF ENERGY IN PHYSICS.

We can know only from experience that mechanical processes produce other physical transformations, or vice versa. The attention was first directed to the connexion of mechanical processes, especially the performance of work, with changes of thermal conditions by the invention of the steam-engine, and by its great technical importance. Technical interests and the need of scientific lucidity meeting in the mind of S. Carnot led to the remarkable development from which thermodynamics flowed. It is simply an accident of history that the development in question was not connected with the practical applications of electricity.

In the determination of the maximum quant.i.ty of work that, generally, a heat-machine, or, to take a special case, a steam-engine, can perform with the expenditure of a given amount of heat of combustion, Carnot is guided by mechanical a.n.a.logies. A body can do work on being heated, by expanding under pressure. But to do this the body must receive heat from a hotter body. Heat, therefore, to do work, must pa.s.s from a hotter body to a colder body, just as water must fall from a higher level to a lower level to put a mill-wheel in motion. Differences of temperature, accordingly, represent forces able to do work exactly as do differences of height in heavy bodies. Carnot pictures to himself an ideal process in which no heat flows away unused, that is, without doing work. With a given expenditure of heat, accordingly, this process furnishes the maximum of work. An a.n.a.logue of the process would be a mill-wheel which scooping its water out of a higher level would slowly carry it to a lower level without the loss of a drop. A peculiar property of the process is, that with the expenditure of the same work the water can be raised again exactly to its original level. This property of reversibility is also shared by the process of Carnot. His process also can be reversed by the expenditure of the same amount of work, and the heat again brought back to its original temperature level.

Suppose, now, we had two different reversible processes A, B, such that in A a quant.i.ty of heat, Q, flowing off from the temperature taCA to the lower temperature taCC should perform the work W, but in B under the same circ.u.mstances it should perform a greater quant.i.ty of work W + W'; then, we could join B in the sense a.s.signed and Ain the reverse sense into a single process. Here A would reverse the transformation of heat produced by B and would leave a surplus of work W', produced, so to speak, from nothing. The combination would present a perpetual motion.

With the feeling, now, that it makes little difference whether the mechanical laws are broken directly or indirectly (by processes of heat), and convinced of the existence of a universal law-ruled connexion of nature, Carnot here excludes for the first time from the province of general physics the possibility of a perpetual motion. But it follows, then, that the quant.i.ty of work W, produced by the pa.s.sage of a quant.i.ty of heat Q from a temperature taCA to a temperature taCC, is independent of the nature of the substances as also of the character of the process, so far as that is unaccompanied by loss, but is wholly dependent upon the temperature taCA, taCC.

This important principle has been fully confirmed by the special researches of Carnot himself (1824), of Clapeyron (1834), and of Sir William Thomson (1849), now Lord Kelvin. The principle was reached without any a.s.sumption whatever concerning the nature of heat, simply by the exclusion of a perpetual motion. Carnot, it is true, was an adherent of the theory of Black, according to which the sum-total of the quant.i.ty of heat in the world is constant, but so far as his investigations have been hitherto considered the decision on this point is of no consequence. Carnot's principle led to the most remarkable results. W. Thomson (1848) founded upon it the ingenious idea of an ”absolute” scale of temperature. James Thomson (1849) conceived a Carnot process to take place with water freezing under pressure and, therefore, performing work. He discovered, thus, that the freezing point is lowered 00075 Celsius by every additional atmosphere of pressure. This is mentioned merely as an example.

About twenty years after the publication of Carnot's book a further advance was made by J. R. Mayer and J. P. Joule. Mayer, while engaged as a physician in the service of the Dutch, observed, during a process of bleeding in Java, an unusual redness of the venous blood. In agreement with Liebig's theory of animal heat he connected this fact with the diminished loss of heat in warmer climates, and with the diminished expenditure of organic combustibles. The total expenditure of heat of a man at rest must be equal to the total heat of combustion. But since all organic actions, even the mechanical actions, must be set down to the credit of the heat of combustion, some connexion must exist between mechanical work and expenditure of heat.

Joule started from quite similar convictions concerning the galvanic battery. A heat of a.s.sociation equivalent to the consumption of the zinc can be made to appear in the galvanic cell. If a current is set up, a part of this heat appears in the conductor of the current. The interposition of an apparatus for the decomposition of water causes a part of this heat to disappear, which on the burning of the explosive gas formed, is reproduced. If the current runs an electromotor, a portion of the heat again disappears, which, on the consumption of the work by friction, again makes its appearance. Accordingly, both the heat produced and the work produced, appeared to Joule also as connected with the consumption of material. The thought was therefore present, both to Mayer and to Joule, of regarding heat and work as equivalent quant.i.ties, so connected with each other that what is lost in one form universally appears in another. The result of this was a substantial conception of heat and of work, and ultimately a substantial conception of energy. Here every physical change of condition is regarded as energy, the destruction of which generates work or equivalent heat. An electric charge, for example, is energy.

In 1842 Mayer had calculated from the physical constants then universally accepted that by the disappearance of one kilogramme-calorie 365 kilogramme-metres of work could be performed, and vice versa. Joule, on the other hand, by a long series of delicate and varied experiments beginning in 1843 ultimately determined the mechanical equivalent of the kilogramme-calorie, more exactly, as 425 kilogramme-metres.

If we estimate every change of physical condition by the mechanical work which can be performed upon the disappearance of that condition, and call this measure energy, then we can measure all physical changes of condition, no matter how different they may be, with the same common measure, and say: the sum-total of all energy remains constant. This is the form that the principle of excluded perpetual motion received at the hands of Mayer, Joule, Helmholtz, and W. Thomson in its extension to the whole domain of physics.

After it had been proved that heat must disappear if mechanical work was to be done at its expense, Carnot's principle could no longer be regarded as a complete expression of the facts. Its improved form was first given, in 1850, by Clausius, whom Thomson followed in 1851. It runs thus: ”If a quant.i.ty of heat Q' is transformed into work in a reversible process, another quant.i.ty of heat Q of the absolute[55] temperature TaCA is lowered to the absolute temperature TaCC.” Here Q' is dependent only on Q, TaCA, TaCC, but is independent of the substances used and of the character of the process, so far as that is unaccompanied by loss. Owing to this last fact, it is sufficient to find the relation which obtains for some one well-known physical substance, say a gas, and some definite simple process. The relation found will be the one that holds generally. We get, thus, Q'/(Q' + Q) = (TaCA-TaCC)/TaCA (1) that is, the quotient of the available heat Q' transformed into work divided by the sum of the transformed and transferred heats (the total sum used), the so-called economical coefficient of the process, is, (TaCA-TaCC)/TaCA.

IV. THE CONCEPTIONS OF HEAT.

When a cold body is put in contact with a warm body it is observed that the first body is warmed and that the second body is cooled. We may say that the first body is warmed at the expense of the second body. This suggests the notion of a thing, or heat-substance, which pa.s.ses from the one body to the other. If two ma.s.ses of water m, m', of unequal temperatures, be put together, it will be found, upon the rapid equalisation of the temperatures, that the respective changes of temperatures u and u' are inversely proportional to the ma.s.ses and of opposite signs, so that the algebraical sum of the products is, mu + m'u' = 0.

Black called the products mu, m'u', which are decisive for our knowledge of the process, quant.i.ties of heat. We may form a very clear picture of these products by conceiving them with Black as measures of the quant.i.ties of some substance. But the essential thing is not this picture but the constancy of the sum of these products in simple processes of conduction. If a quant.i.ty of heat disappears at one point, an equally large quant.i.ty will make its appearance at some other point. The retention of this idea leads to the discovery of specific heat. Black, finally, perceives that also something else may appear for a vanished quant.i.ty of heat, namely: the fusion or vaporisation of a definite quant.i.ty of matter. He adheres here still to this favorite view, though with some freedom, and considers the vanished quant.i.ty of heat as still present, but as latent.

The generally accepted notion of a caloric, or heat-stuff, was strongly shaken by the work of Mayer and Joule. If the quant.i.ty of heat can be increased and diminished, people said, heat cannot be a substance, but must be a motion. The subordinate part of this statement has become much more popular than all the rest of the doctrine of energy. But we may convince ourselves that the motional conception of heat is now as unessential as was formerly its conception as a substance. Both ideas were favored or impeded solely by accidental historical circ.u.mstances. It does not follow that heat is not a substance from the fact that a mechanical equivalent exists for quant.i.ty of heat. We will make this clear by the following question which bright students have sometimes put to me. Is there a mechanical equivalent of electricity as there is a mechanical equivalent of heat? Yes, and no. There is no mechanical equivalent of quant.i.ty of electricity as there is an equivalent of quant.i.ty of heat, because the same quant.i.ty of electricity has a very different capacity for work, according to the circ.u.mstances in which it is placed; but there is a mechanical equivalent of electrical energy.