DIFFICULTIES IN THE EXPLANATION OF VELOCITY.
32. Here arises a serious difficulty: if time be nothing absolute, greater or less velocity is inexplicable. This seems to result even from what we have said, that if the relation of movements be not changed, any augmentation or diminution of velocity is impossible; because, if velocity be in necessary relation to time, and time itself be nothing but the relation of mutations, it is inconceivable how time, and consequently how velocity, can be changed without changing the relation of mutations. Thus it would be impossible for the velocity of the whole mechanism of the universe to be changed, just as it would be absurd to say that the stars and every thing that exists may now experience the same changes of velocity. This would destroy the very idea of velocity; at least if taken as something absolute, wherein different grades may be considered.
33. Let us now examine this difficulty, which indeed deserves to be examined, for it seems to contradict our most common ideas.
First of all, we must premise that velocity is not something absolute, but a relation. Physicists and mathematicians express it by a fraction whose numerator is the space run over and whose denominator is the time consumed. Making V the velocity, S the space, and T the time, we shall have V = S/T. This shows the velocity to be essentiallyarelation; for it cannot be otherwise expressed than by the ratio of the space to the time.
34. This mathematic formula expresses the idea we all have of velocity; it expresses in three letters what the unlettered man repeatedly says to himself. The velocity of two horses is ascertained not solely by the space they have passed over, nor solely by the time they have consumed in their career, but by the greater or less space passed over in a given time; or by the longer or shorter time required to pass over a given space.
To deny, then, to velocity an absolute nature, is nothing new; for we all of us make it essentially consist in a relation.
35. In the expression V = S/T two terms enter, space and time. Viewing the former in the real order, abstraction made of that of phenomena, we more easily come to regard it as something fixed; and we comprehend it in a given case without any relation. A foot is at all times a foot; and a yard, a yard. These are quantities existing in reality; and if we refer them to other quantities, it is only to make sure that they are so; not because their reality depends upon the relation. A cubic foot of water is not a cubic foot because the measure so says, but on the contrary, the measure so says because there is a cubit foot. The measure itself is also an absolute quantity; and in general, all extensions are absolute, for otherwise, we should be obliged to seek measure of measure, and so on to infinity. True, to call things large or small depends upon comparison; but this does not change their own quantity. The diameter of the earth, compared with an inch measure, is immense; but it is an almost imperceptible point compared with the distance of the fixed stars; yet this does not prevent the inch measure, the diameter of the earth, and the distance to the fixedstars, from being values in themselves determinate, and independent of each other.[27]
If the denominator in S/T were a quantity of the same kind as space, that is, having determinate values, existing and conceivable by themselves alone, the velocity, although still a relation, might also have determinate values, not indeed, wholly absolute, but only in the supposition that the two terms, S and T, having fixed values, are compared. Thus, if we require a velocity of 4, we have only to take a fixed quantity of space, and another fixed quantity of time, having the relation to each other of 4 to 1; and this is quite easy, when S and T are both absolute quantities. If, in this supposition, an acceleration or delay be required in the whole universe, nothing more would be required than to augment or diminish the time in which each part would have to traverse its respective space. But from the difficulties which we have on the one hand seen presented to the consideration of time as an absolute thing, and from the fact that, on the other hand, no solid proof can be adduced to show such a property to have any foundation, it follows that we know not how to consider velocity as absolute, even in the sense above explained.
36. Hence a consequence not less important than striking, as to the possibility of a universal acceleration or retardation. If we would have an acceleration or retardation of the whole machine of the universe, and should abandon all motion to which we might refer time, should at once change all, not excluding the operations of our own soul, we should have a problem proposed to us that appears insolvable, nothing less than the realization of an impossibility; the relation of many terms would have to be changed without undergoing any change. If velocity be only the relationof space and time, and time only the relation of spaces traversed, it is the same thing to change them all in the same proportion, and not to change them at all; it is to leave every thing as it is.
37. The singularity of such consequences ought not to be a sufficient excuse for abandoning them. We must not forget that we are examining the common ideas of time and velocity in their most transcendental aspect, and that it is by no means astonishing that our mind finds itself, as it leaves its ordinary walks, in an entirely new atmosphere, wherein it seems to discover contradictions. When we examine the ideas of time and velocity, we unwittingly fall into the error of uniting them in the same explanation. We would prescind them; but this we do only with great difficulty, and we often fall into a vicious circle. Hence it is that when, by a great effort, we succeed in really prescinding, the consequences that follow seem contradictory; but this apparent contradiction arises solely from our not having persevered with due firmness in our prescision; and as, in this case, the understanding starts from two different suppositions, whereas it believes that it starts from one alone, the results seem to it contradictory, which in reality they may not be. The same thing occurs in the examination of the idea of space.[28]