v = s/t;
and this shows that time is not the measure of velocity, as the one diminishes when the other increases.
This suffices to show that the phrase “movement is measured by time” must be interpreted in a very limited sense, as simply meaning that between movement and time there is a necessary connection, and that, all other things remaining equal, the length of the movement is proportional to the length of the time employed. Yet this does not mean that the length of the movement depends entirely on the time employed, for the same length may be described in different times; but it means that the time employed depends on the material and formal extent of the movement, as above explained; for, according as we take different velocities, different lengths will be described in equal time, and equal lengths in different times. It is not the time that extends the movement, but it is the movement that by its extension extends its own time.
The true measure of movement is its velocity; for the measure of any given quantity is a unit of the same kind, and velocity is the unit of movement. Time, as measured by us, is a number which arises from the mensuration of the movement by its velocity; and therefore time results from the movement as already measured. This shows again that time is not the measure of the extent of the movement. We have seen, also, that time is not the measure of the intensity of the movement. It follows, therefore, that the quantity of movement is not measured by time.
Time, being the ratio of two quantities mathematically homogeneous, is represented by an abstract number. Yet the same time may be expressed by different numbers, according as we measure it by different units, as days, hours, minutes, etc. These numbers, however, are only virtually discrete, as time cannot be discontinued.
Balmes from the equation
v = s/t
deduces the consequence that “the velocity is essentially a relation; for it cannot be otherwise expressed than by the ratio of the space to the time.”[81] We think that this conclusion is faulty. Space and time are not homogeneous quantities; hence the mathematical ratio of space to time is not an abstract but a concrete number, and therefore it represents an absolute quantity. Space divided by time is a length divided into equal parts; hence the quotient—viz., the velocity—represents the length of the movement made in the unit of time. And since Balmes admits that the length of the movement is a quantity having a determinate value, we do not see how he can escape the consequence that velocity, too, is a quantity of the same kind, and not a mere relation. “In the expression of velocity,” says Balmes, “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 cubic 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” (loc. cit.) This passage shows that a length described in space is, according to Balmes, an absolute quantity. And since the mathematical value of velocity represents a length described in space, as we have just proved, it follows that velocity has an absolute value.
But leaving aside all mathematical considerations, we may show that velocity has an absolute value by reference to metaphysical data. What is velocity but the development in extension of the intensity of the momentum impressed on a material point? Now, the intensity of the momentum is an absolute quantity, equal to the quantity of the action by which it is produced. Hence it is evident that, as the action has an absolute value, greater or less, according to circumstances, so also the momentum impressed has an absolute value; and consequently the velocity also, which is nothing else than the momentum itself as developing its intensity into extension, has an absolute value, and is an absolute quantity.