Balmes thought the contrary, for the following reason: “If the denominator, in the expression of velocity, 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.… 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” (loc. cit.)
This reason proves the contrary of what the author intends to establish. In fact, if the denominator were of the same kind as the numerator, the quotient would be an abstract number, as we know from mathematics; and such a number would exhibit nothing more than the relation of the two homogeneous terms—that is, how many times the one is contained in the other. It is precisely because the denominator is not of the same kind as the numerator that the quotient must be of the same kind as the numerator. And since the numerator represents space, which, according to Balmes, is an absolute quantity, it follows that the quotient—that is, the number by which we express the velocity—exhibits a quantity of the same nature: a conclusion in which all mathematicians agree. When a man walks a mile, with the velocity of one yard per second, he measures the whole mile yard by yard, with his velocity. If the velocity were not a quantity of the same kind with the space measured, how could it measure it?
True it is that velocity, when considered in its metaphysical aspect, is not a length of space, but the intensity of the act by which matter is carried through such a length. Yet, since Balmes argues here from a mathematical equation, we must surmise or presume that he considers velocity as a length measured in space in the unit of time, as mathematicians consider it; for he cannot argue from mathematical expressions with logical consistency, if he puts upon them construction of an unmathematical character. After all, it remains true that the velocity or intensity of the movement is always to be measured by the extension of the movement in the unit of time; and thus it is necessary to admit that velocity exhibits an absolute intensive quantity measured by the extension which it evolves.
We therefore “know how to consider velocity as absolute,” though its mathematical expression is drawn from a relation of space to time. The measure of any quantity is always found by comparing the quantity with some unit of measure; hence all quantity, inasmuch as measured, exhibits itself under a relative form as ratio mensurati ad suam mensuram; and it is only under such a form that it can be expressed in numbers. But this relativity does not constitute the nature of quantity, because it presupposes it, and has the whole reason of its being in the process of mensuration.
We have insisted on this point because the confusion of the absolute value of velocity with its relative mathematical expression would lead us into a labyrinth of difficulties with regard to time. Balmes, having overlooked the distinction between the mathematical expression and the metaphysical character of velocity, comes to the striking consequence that “if the whole machine of the universe, not excluding the operations of our soul, were accelerated or retarded, an impossibility would be realized; for the relation of the terms would have to be changed without undergoing any change. If the velocity be only the relation of space to 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” (loc. cit.) The author is quite mistaken. The very equation
t = s/v,
on which he grounds his argument, suffices to show that if the velocity increases, the time employed in measuring the space s diminishes; and if the velocity diminishes, the time increases. This being the case, it is evident that an acceleration of the movements in the whole machine of the universe would be a real acceleration, since the same movements would be performed in less time; and a retardation would be a real retardation, since the same movements would require more time. We are therefore far from realizing an impossibility when we admit that, in the hypothesis of the author, time would vary in the inverse ratio of the velocity of the universal movement.
Division of time.—Philosophers divide time into real and imaginary. We have already explained this division when speaking of flowing duration. The reality of time evidently depends on the reality of movement; hence any time to which no real movement corresponds is imaginary. Thus if you dream that you are running, the time of your running is imaginary, because your running, too, is imaginary. In such a case the real time corresponds to your real movements—say, to your breathing, pulse, etc.—while the dream continues.
Imaginary time is often called also ideal time, but this last epithet is not correct; for, as time is the duration of local movement, it is in the nature of time to be an object of the imagination. And for this reason the duration of the intellectual movements and operations of pure spirits is called time only by analogy, as we have above stated. However, we are wont to think of such a duration as if it were homogeneous with our own time; for we cannot measure it except by reference to the duration of the movements we witness in the material world.
Time is also divided into past, present, and future. The past corresponds to a movement already made, the future to a movement which will be made, and the present to a movement which is actually going on. But some will ask: Is there really any present time? Does not the now, to which the present is confined, exclude all before and all after, and therefore all succession, without which it is impossible to conceive time? We concede that the now, as such—that is, considered in its absolute reality—is not time, just as a point is not a line; for, as the point has no length, so the now has no extension. Yet, as a point in motion describes a line, so also the now, by its flowing from before to after, extends time. Hence, although the now, as such, is not time, its flowing from before to after is time. If, then, we consider the present as the link of the immediate past with the immediate future—that is, if we consider the now not statically, but dynamically—we shall see at once that its actual flowing from before to after implies succession, and constitutes an infinitesimal interval of time.