36. Men too often impose on themselves and others as if they conceived and understood things expressed by signs, when in truth they have no idea, save only of the very signs themselves. And there are some grounds to apprehend that this may be the present case. The velocities of evanescent or nascent quantities are supposed to be expressed, both by finite lines of a determinate magnitude, and by algebraical notes or signs: but I suspect that many who, perhaps never having examined the matter take it for granted, would, upon a narrow scrutiny, find it impossible to frame any idea or notion whatsoever of those velocities, exclusive of such finite quantities and signs. Suppose the line KP described by the motion of a point continually accelerated, and that in equal particles of time the unequal parts KL , LM , MN , NO , &c. are generated. Suppose also that a , b , c , d , e , &c. denote the velocities of the generating point, at the several periods of the parts or increments so generated. It is easy to observe that these increments are each proportional to the sum of the velocities with which it is described: that, consequently, the several sums of the velocities, generated in equal parts of time, may be set forth by the respective lines KL , LM , MN , &c.
generated in the same times. It is likewise an easy matter to say, that the last velocity generated in the first particle of time may be expressed by the symbol a , the last in the second by b , the last in the third by c , and so on: that a is the velocity of LM in statu nascenti , and b , c , d , e , &c.
are the velocities of the increments MN , NO , OP , &c.
in their respective nascent estates. You may proceed and consider these velocities themselves as flowing or increasing quantities, taking the velocities of the velocities, and the velocities of the velocities of the velocities, i.e. the first, second, third &c. velocities ad infinitum :
which succeeding series of velocities may be thus expressed, a , b - a , c - 2 b + a , d - 3 c + 3 b - a &c. which you may call by the names of the first, second, third, fourth fluxions. And for an apter expression you may denote the variable flowing line KL , KM , KN , &c. by the letter x ; and the first fluxions by , the second by , the third by , and so on ad infinitum .
37. Nothing is easier than to assign names, signs, or expressions to these fluxions; and it is not difficult to compute and operate by means of such signs. But it will be found much more difficult to omit the signs and yet retain in our minds the things which we suppose to be signified by them. To consider the exponents, whether geometrical, or algebraical, or fluxionary, is no difficult matter. But to form a precise idea of a third velocity for instance, in itself and by itself, Hoc opus, hic labor . Nor indeed is it an easy point to form a clear and distinct idea of any velocity at all, exclusive of and prescinding from all length of time and space; as also from all notes, signs, or symbols whatsoever. This, if I may be allowed to judge of others by myself, is impossible. To me it seems evident that measures and signs are absolutely necessary in order to conceive or reason about velocities; and that consequently, when we think to conceive the velocities simply and in themselves, we are deluded by vain abstractions.
38. It may perhaps be thought by some an easier method of conceiving fluxions to suppose them the velocities wherewith the infinitesimal differences are generated. So that the first fluxions shall be the velocities of the first differences, the second the velocities of the second differences, the third fluxions the velocities of the third differences, and so on ad infinitum . But, not to mention the insurmountable difficulty of admitting or conceiving infinitesimals, and infinitesimals of infinitesimals, &c., it is evident that this notion of fluxions would not consist with the great author's view; who held that the minutest quantity ought not to be neglected, that therefore the doctrine of infinitesimal differences was not to be admitted in geometry, and who plainly appears to have introduced the use of velocities or fluxions, on purpose to exclude or do without them.
39. To others it may possibly seem that we should form a juster idea of fluxions by assuming the finite, unequal, isochronal increments KL , LM , MN , &c., and considering them in statu nascenti , also their increments in statu nascenti , and the nascent increments of those increments, and so on, supposing the first nascent increments to be proportional to the first fluxions or velocities, the nascent increments of those increments to be proportional to the second fluxions, the third nascent increments to be proportional to the third fluxions, and so onwards. And, as the first fluxions are the velocities of the first nascent increments, so the second fluxions may be conceived to be the velocities of the second nascent increments, rather than the velocities of velocities. But which means the analogy of fluxions may seem better preserved, and the notion rendered more intelligible.
40. And indeed it should seem that in the way of obtaining the second or third fluxion of an equation the given fluxions were considered rather as increments than velocities. But the considering them sometimes in one sense, sometimes in another, one while in themselves, another in their exponents, seems to have occasioned no small share of that confusion and obscurity which are found in the doctrine of fluxions.
