single constant which would have a single value absolutely invariable; they would be led without any doubt to regard it as an essential constant.
A word in passing to forestall an objection: the inhabitants of this imaginary world could neither observe nor define the area-constant as we do, since the absolute longitudes escape them; that would not preclude their being quickly led to notice a certain constant which would introduce itself naturally into their equations and which would be nothing but what we call the area-constant.
But then see what would happen. If the area-constant is regarded as essential, as depending upon a law of nature, to calculate the distances of the planets at any instant it will suffice to know the initial values of these distances and those of their first derivatives. From this new point of view, the distances will be determined by differential equations of the second order.
Yet would the mind of these astronomers be completely satisfied? I do not believe so; first, they would soon perceive that in differentiating their equations and thus raising their order, these equations became much simpler. And above all they would be struck by the difficulty which comes from symmetry. It would be necessary to assume different laws, according as the aggregate of the planets presented the figure of a certain polyhedron or of the symmetric polyhedron, and one would escape from this consequence only by regarding the area-constant as accidental.
I have taken a very special example, since I have supposed astronomers who did not at all consider terrestrial mechanics, and whose view was limited to the solar system. Our universe is more extended than theirs, as we have fixed stars, but still it too is limited, and so we might reason on the totality of our universe as the astronomers on their solar system.
Thus we see that finally we should be led to conclude that the equations which define distances are of an order superior to the second. Why should we be shocked at that, why do we find it perfectly natural for the series of phenomena to depend upon the initial values of the first derivatives of these distances, while we hesitate to admit that they may depend on the initial values of the second derivatives? This can only be because of the habits of mind created in us by the constant study of the generalized principle of inertia and its consequences.
The values of the distances at any instant depend upon their initial values, upon those of their first derivatives and also upon something else. What is this something else?
If we will not admit that this may be simply one of the second derivatives, we have only the choice of hypotheses. Either it may be supposed, as is ordinarily done, that this something else is the absolute orientation of the universe in space, or the rapidity with which this orientation varies; and this supposition may be correct; it is certainly the most convenient solution for geometry; it is not the most satisfactory for the philosopher, because this orientation does not exist.
Or it may be supposed that this something else is the position or the velocity of some invisible body; this has been done by certain persons who have even called it the body alpha, although we are doomed never to know anything of this body but its name. This is an artifice entirely analogous to that of which I spoke at the end of the paragraph devoted to my reflections on the principle of inertia.
But, after all, the difficulty is artificial. Provided the future indications of our instruments can depend only on the indications they have given us or would have given us formerly, this is all that is necessary. Now as to this we may rest easy.
CHAPTER VIII
Energy and Thermodynamics
Energetics.—The difficulties inherent in the classic mechanics have led certain minds to prefer a new system they call energetics.
Energetics took its rise as an outcome of the discovery of the principle of the conservation of energy. Helmholtz gave it its final form.
It begins by defining two quantities which play the fundamental rôle in this theory. They are kinetic energy, or vis viva, and potential energy.
All the changes which bodies in nature can undergo are regulated by two experimental laws:
1º The sum of kinetic energy and potential energy is constant. This is the principle of the conservation of energy.
2º If a system of bodies is at A at the time t0 and at B at the time t1, it always goes from the first situation to the second in such a way that the mean value of the difference between the two sorts of energy, in the interval of time which separates the two epochs t0 and t1, may be as small as possible.
This is Hamilton's principle, which is one of the forms of the principle of least action.
The energetic theory has the following advantages over the classic theory:
1º It is less incomplete; that is to say, Hamilton's principle and that of the conservation of energy teach us more than the fundamental principles of the classic theory, and exclude certain motions not realized in nature and which would be compatible with the classic theory:
2º It saves us the hypothesis of atoms, which it was almost impossible to avoid with the classic theory.
But it raises in its turn new difficulties:
The definitions of the two sorts of energy would raise difficulties almost as great as those of force and mass in the first system. Yet they may be gotten over more easily, at least in the simplest cases.
Suppose an isolated system formed of a certain number of material points; suppose these points subjected to forces depending only on their relative position and their mutual distances, and independent of their velocities. In virtue of the principle of the conservation of energy, a function of forces must exist.
In this simple case the enunciation of the principle of the conservation of energy is of extreme simplicity. A certain quantity, accessible to experiment, must remain constant. This quantity is the sum of two terms; the first depends only on the position of the material points and is independent of their velocities; the second is proportional to the square of these velocities. This resolution can take place only in a single way.
The first of these terms, which I shall call U, will be the potential energy; the second, which I shall call T, will be the kinetic energy.
It is true that if T + U is a constant, so is any function of T + U,
Φ (T + U).
But this function Φ (T + U) will not be the sum of two terms the one independent of the velocities, the other proportional to the square of these velocities. Among the functions which remain constant there is only one which enjoys this property, that is T + U (or a linear function of T + U, which comes to the same thing, since this linear function may always be reduced to T + U by change of unit and of origin). This then is what we shall call energy; the first term we shall call potential energy and the second kinetic energy. The definition of the two sorts of energy can therefore be carried through without any ambiguity.
It
