This may also be shown by reference to the nature of uniform local movement. When a material point describes a line with uniform velocity, its movement being continuous, its duration is continuous; and therefore every flowing instant of its duration is continuous, as no discontinuous parts can ever be reached in the division of continuum. Hence every flowing instant has still the nature of time. This conclusion is mathematically evident from the equation
t = s/v,
for, v being supposed constant, we cannot assume t = 0 unless we also assume s = 0. But this latter assumption would imply rest instead of movement, and therefore it is out of the question. Accordingly, at no instant of the movement can we assume t = 0; or, which is the same, every flowing instant partakes the nature of time.
The same conclusion can be established, even more evidently, by the consideration of accelerated or retarded movements. When a stone is thrown upwards, the velocity of its ascent suffers a continuous diminution till at last it becomes = 0; and at the very instant it becomes = 0 an opposite velocity begins to urge the stone down, and increases continually so long as the stone does not reach the ground or any other obstacle. Now, a continuous increase or decrease of the velocity means that there are not two consecutive moments of time in which the stone moves at exactly the same rate; and hence nothing but an instant corresponds to each successive degree of velocity. But since the duration of the movement is made up of nothing but such instants, it is clear that the succession of such instants constitutes time; and consequently, as time is continuous, those instants, though infinitesimal, are themselves continuous; and thus every flowing instant is really time.
From this it is plain, first, that although the now, as such, is not time, yet its actual flowing is time.
Secondly, it follows that infinitesimals of time, as employed in dynamics, are not mathematical figments, but realities, for time flows only through infinitesimal instants; and therefore to deny the reality of such infinitesimals would be to deny the reality of time.
Thirdly, we gather that the absolute now differs from an actual infinitesimal of time; because the former, as such, is only a term of time, whereas the latter is the flowing of that term from its immediate before to its immediate after. Hence an infinitesimal of time is infinitely less than any designable duration. In fact, its before and its after are so immediately connected with the same absolute now that there is no room for any designable length of duration between them.
Fourthly, whilst the absolute now is no quantity, the infinitesimal of time is a real quantity; for it implies real succession. This quantity, however, is nascent, or in fieri only; for the now, which alone is intercepted between the immediate before and the immediate after, has no formal extension.
Fifthly, the infinitesimal of time corresponds to a movement by which an infinitesimal of space is described. And thus infinitesimals of space, as considered in dynamics, are real quantities. To deny that such infinitesimals are real quantities would be the same, in fact, as to deny the real extension of local movement; for this movement flows and acquires its extension through such infinitesimals only. And the same is true of the infinitesimal actions by which the rate of local movement is continually modified. These latter infinitesimals are evidently real quantities, though infinitely less than any designable quantity. They have an infinitesimal intensity, and they cause an infinitesimal change in the rate of the movement in an infinitesimal of time.
Evolution of time.—The preceding considerations lead us to understand how it is that in any interval of time there is but one absolute now always the same secundum rem, but changing, and therefore manifold secundum rationem. S. Thomas, in his opuscule De Instantibus, c. ii., explains this truth in the following words: “As a point to the line, so is the now to the time. If we imagine a point at rest, we shall not be able to find in it the causality of any line; but if we imagine that point to be in movement, then, although it has no dimensions, and consequently no divisibility in itself, it will nevertheless, from the nature of its movement, mark out a divisible line.… The point, however, does in no way belong to the essence of the line; for one and the same real term, absolutely indivisible, cannot be at the same time in different parts of the same permanent continuum.… Hence the mathematical point which by its movement draws a line is neither the line nor any part of the line; but, remaining one and the same in itself, it acquires different modes of being. These different modes of being, which must be traced to its movement, are really in the line, whilst the point, as such, has no place in it. In the same manner, an instant, which is the measure of a thing movable, and adheres to it permanently, is one and the same as to its absolute reality so long as the substance of the thing remains unimpaired, for the instant is the inseparable measure of its being; but the same instant becomes manifold inasmuch as it is diversified by its modes of being; and it is this its diversity that constitutes the essence of time.”[82]