Our conviction then of the objective reality of matter (at least from the point of view of the Natural Philosopher) is based upon the experimental truth that we can neither increase nor diminish its quantity, in fact on what we may conveniently for our present purpose call the Conservation of Matter.
95. Here let us pause for a moment to compare together this view of matter and the definition of the laws of the universe, which we have already given. The laws of the universe we defined ([Art. 54]) to be the laws according to which the beings in the universe are trammelled by the Governor thereof as regards time, space, and sensation. Now, it may be asked, is this definition consistent with a belief in the objective reality of matter? Our reply is, that to our minds the two are in perfect accordance.
We do not here intend to enter into any metaphysical discussion. It is enough for us to say that our practical working certainty of the reality of matter depends upon the facts, firstly, that it offers resistance to our imagination and our will, and, secondly, that in particular it offers absolute resistance to all attempts to change its quantity. We shall soon see that experiment teaches us that both properties belong to something else.
96. Returning from this digression let us therefore assume that the objective reality of the external universe has been proved, and that this reality is strongly impressed upon us in virtue of that principle which we have called the conservation of matter.
But as soon as we grant this, we are obliged by our reason, however little our senses may incline us to it, or rather however much they may dispose us against it, to allow objective reality to whatever else may be found to be in the same sense conserved. (We have here italicised these four words for a reason which will afterwards appear.) This is a question which deserves and must secure careful consideration.
97. In abstract dynamics several things are said and mathematically proved by deductions from experiment to be conserved, but one only of these in the strict sense in which we have spoken of the conservation of matter. We will examine them briefly, and our non-mathematical readers must pardon us if we make use of certain technical expressions belonging to the domain of mathematical physics.
[It is absolutely essential that the reader should have clear notions on these points, for there is widespread confusion and error as to the meaning even of so simple and elementary a term as ‘force.’ He will often find it used indifferently in either of two senses which have no connection whatever with one another; and unless he completely gets over this abuse of language he need not hope to be able to follow the present portion of our preliminary argument. Force proper is a pull, push, weight, pressure, etc., and can be measured, in the vernacular of engineers, as equivalent to so many pounds weight; but the unjustifiable use of the word applies it to work done by a force, so many pounds raised so many feet, i.e. force overcome through a space. Two such things are of different kinds, and cannot possibly be compared together. They differ in fact in precisely the same way as length or breadth differs from superficial area, i.e. as a linear foot differs from a square foot! And the modern abuse of the word is more outrageous, alike to science and to common sense, than would be the attempt to assign the height of a mountain in acres! For the absurdity does not end even here. We have, as yet, absolutely no proof whatever that force proper has objective existence. In all probability there is no such thing as force (which is suggested to us by the impressions of our muscular sense), any more than there is such a thing as Sound, or Light, which are mere names for physical impressions produced upon special nerves by the energy of undulatory motions of certain media. The term, however, is a very convenient one for the rate of transference or transformation of energy per unit of length in a given direction.]
(1.) Conservation of Momentum.—What is understood by this is a mere direct consequence of Newton’s first interpretation of his Third Law of Motion, viz., that Action and Reaction are equal and opposite. In this first interpretation Newton tells us to consider actions and reactions as forces proper, or (their equivalents) quantities of motion. This is the term employed by Newton; but we now designate it momentum, and measure it by the product of the mass and the velocity of a body. Stated in its simplest form, this law asserts that the momentum of a system of bodies, measured in any direction whatever, is not altered by their mutual action, whether that action be of the nature of traction, attraction, repulsion, or impact. And we see at once from this third law of motion that it must be so, because the change of momentum, in any direction, of any one part of the system, per unit of time, is the measure of the force acting on that part in that direction. Whatever momentum in this particular direction is gained by one member of the system must have been lost by other members, but not from their whole momentum, merely from the part of it in this direction. It thus appears that the (algebraic) sum of the momenta generated by the mutual actions of the system is zero.
These momenta are in fact directed magnitudes (like the forces of which they are the measure), and are therefore capable of cancelling one another when their numerical amounts are equal and their directions are opposite. In this sense the conservation is of the same nature as that of the imagined electric or magnetic fluids, where no portion whatever of one kind can be produced without the simultaneous appearance of an equal quantity of the other, a quantity just capable of neutralising it. This is obviously not in any sense analogous to the Conservation of Matter of which we have just spoken.
As an illustration take a loaded cannon. Before firing, neither cannon nor ball had momentum. After firing, the ball has a certain momentum, the cannon (in virtue of its recoil) an equal and opposite momentum. If we could exactly reverse the motions of the cannon and ball just as they separate, the impact between them would just reduce each to rest, and no momentum would be left. Considered separately after the discharge, each has momentum, but in the complete system of cannon and ball there is no momentum—there being equal quantities of positive and negative, in the same line. In fact momentum cannot be produced or destroyed in any system as a whole. This is the Conservation referred to. It is as if a man always when he received a sum of money fell to the same amount in debt—the state of his affairs, as shown by his books, would of course not be altered.