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