Resultant relative duration—that is, an interval of flowing duration—admits of the same division into real and imaginary. It is real when a real continuous flowing connects the before with the after; in all other suppositions it will be imaginary. It may be remarked that the “real continuous flowing” may be either intrinsic or extrinsic. Thus, if God had created nothing but a simple angel, there would have been no other flowing duration than a continuous succession of intellectual operations connecting the before with the after in the angel himself, and thus his duration would have been measured by a series of intrinsic changes. It is evident that in this case one absolute when suffices to extend the interval of duration; for by its gliding from before to after it acquires opposite formalities through which it can be relatively opposed to itself as the subject and the term of the relation. If, on the contrary, we consider the interval of duration between two distinct beings—say Cæsar and Napoleon—then the real continuous flowing by which such an interval is measured is extrinsic to the terms compared; for the when of Cæsar is distinct from, and does not reach, that of Napoleon; which shows that their respective whens have no intrinsic connection, and that the succession comprised between those whens must have consisted of a series of changes extrinsic to the terms compared. It may seem difficult to conceive how an interval of continuous succession can result between two terms of which the one does not attain to the other; for, as a line in space must be drawn by the movement of a single point, so it seems that a length in duration must be extended by the flowing of a single when from before to after. The truth is that the interval between the whens of two distinct beings is not obtained by comparing the when of the one with that of the other, but by resorting to the when of some other being which has extended its continuous succession from the one to the other. Thus, when Cæsar died, the earth was revolving on its axis, and it continued to revolve without interruption up to the existence of Napoleon, thus extending the duration of its movement from a when corresponding to Cæsar’s death to a when corresponding to Napoleon’s birth; and this duration, wholly extrinsic to Cæsar and Napoleon, measures the interval between them.
As all intervals of duration extend from before to after, there can be no interval between co-existent beings, as is evident. In the same manner as two beings whose ubications coincide cannot be distant in space, so two beings whose whens are simultaneous cannot form an interval of duration.
All real intervals of duration regard the past; for in the past alone can we find a real before and a real after. The present gives no interval, as we have just stated, but only simultaneousness. The future is real only potentially—that is, it will be real, but it is not yet. What has never been, and never will be, is merely imaginary. To this last class belong all the intervals of duration corresponding to those conditional events which did not happen, owing to the non-fulfilment of the conditions on which their reality depended.
As to the measurement of flowing duration a few words will suffice. The when considered absolutely is incapable of measuring an interval of duration, for the reason that the when is unextended, and therefore unproportionate to the mensuration of a continuous interval; for the measure must be of the same kind with the thing to be measured. Just as a continuous line cannot be made up of unextended points, so cannot a continuous interval be made up of indivisible instants; hence, as a line is divisible only into smaller and smaller lines, by which it can be measured, so also an interval of duration is divisible only into smaller and smaller intervals, and is measured by the same. These smaller intervals, being continuous, are themselves divisible and mensurable by other intervals of less duration, and these other intervals are again divisible and mensurable; so that, from the nature of the thing, it is impossible to reach an absolute measure of duration, and we must rest satisfied with a relative one, just as in the case of a line and of any other continuous quantity. The smallest unit or measure of duration commonly used is the second, or sixtieth part of a minute.
But, since continuous quantities are divisible in infinitum, it may be asked, what prevents us from considering a finite interval of duration as containing an infinite multitude of infinitesimal units of duration? If nothing prevents us, then in the infinitesimal unit we shall have the true and absolute measure of duration. We answer that nothing prevents such a conception; but the mensuration of a finite interval by infinitesimal units would never supply us the means of determining the relative lengths of two intervals of duration. For, if every interval is a sum of infinite terms, and is so represented, how can we decide which of those intervals is the greater, since we cannot count the infinite?
Mathematicians, in all dynamical questions, express the conditions of the movement in terms of infinitesimal quantities, and consider every actual instant which connects the before with the after as an infinitesimal interval of duration in the same manner as they consider every shifting ubication as an infinitesimal interval of space. But when they pass from infinitesimal to finite quantities by integration between determinate limits, they do not express the finite intervals in infinitesimal terms, but in terms of a finite unit, viz., a second of time; and this shows that, even in high mathematics, the infinitesimal is not taken as the measure of the finite.
Since infinitesimals are considered as evanescent quantities, the question may be asked whether they are still conceivable as quantities. We have no intention of discussing here the philosophical grounds of infinitesimal calculus, as we may have hereafter a better opportunity of examining such an interesting subject; but, so far as infinitesimals of duration are concerned, we answer that they are still quantities, though they bear no comparison with finite duration. What mathematicians call an infinitesimal of time is nothing else rigorously than the flowing of an actual “when” from before to after. The “when” as such is no quantity, but its flowing is. However narrow the compass within which it may be reduced, the flowing implies a relation between before and after; hence every instant of successive duration, inasmuch as it actually links its immediate before with its immediate after, partakes of the nature of successive duration, and therefore of continuous quantity. Nor does it matter that infinitesimals are called evanescent quantities. They indeed vanish, as compared with finite quantities; but the very fact of their vanishing proves that they are still something when they are in the act of vanishing. Sir Isaac Newton, after saying in his Principia that he intends to reduce the demonstration of a series of propositions to the first and last sums and ratios of nascent and evanescent quantities, propounds and solves this very difficulty as follows: “Perhaps it may be objected that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and, when they are vanished, is none. But by the same argument it may be alleged that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, the velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, not afterwards, but with which they vanish. In like manner, the first ratio of nascent quantities is that with which they begin to be.” From this answer, which is so clear and so deep, it is manifest that infinitesimals are real quantities. Whence we infer that every instant of duration which actually flows from before to after marks out a real infinitesimal interval of duration that might serve as a unit of measure for the mensuration of all finite intervals of succession, were it not that we cannot reckon up to infinity. Nevertheless, it does not follow that an infinitesimal duration is an absolute unit of duration; for it is still continuous, even in its infinite smallness; and accordingly it is still divisible and mensurable by other units of a lower standard. Thus it is clear that the measurement of flowing duration, and indeed of all other continuous quantity, cannot be made except by some arbitrary and conventional unit.