It may seem therefore that the notion might be still mended, and that instead of fluxions of fluxions, of fluxions of fluxions of fluxions, and instead of second, third, or fourth, &c. fluxions of a given quantity, it might be more consistent and less liable to exception to say, the fluxion of the first nascent increment, i.e. the second fluxion; the fluxion of the second nascent increment i.e. the third fluxion; the fluxion of the third nascent increment, i.e. the fourth fluxion - which fluxions are conceived respectively proportional, each to the nascent principle of the increment succeeding that whereof it is the fluxion.
41. For the more distinct conception of all which it may be considered that if the finite increment LM [See the foregoing scheme in sect. 36.] be divided into the isochronal parts Lm , mn , no , oM ; and the increment MN divided into the parts Mp , pq , qr , rN isochronal to the former; as the whole increments LM , MN are proportional to the sums of their describing velocities, even so the homologous particles Lm , Mp are also proportional to the respective accelerated velocities with which they are described. And, as the velocity with which Mp is generated, exceeds that with which Lm was generated, even so the particle Mp exceeds the particle Lm . And in general, as the isochronal velocities describing the particles of MN exceed the isochronal velocities describing the particles of LM , even so the particles of the former exceed the correspondent particles of the latter.
And so this will hold, be the said particles ever so small. MN therefore will exceed LM if they are both taken in their nascent states: and that excess will be proportional to the excess of the velocity b above the velocity a . Hence we may see that this last account of fluxions comes, in the upshot, to the same thing with the first. [Sect.
36.]
42. But, notwithstanding what hath been said, it must still be acknowledged that the finite particles Lm or Mp , though taken ever so small, are not proportional to the velocities a and b ; but each to a series of velocities changing every moment, or which is the same thing, to an accelerated velocity, by which it is generated during a certain minute particle of time: that the nascent beginnings or evanescent endings of finite quantities, which are produced in moments or infinitely small parts of time, are alone proportional to given velocities:
that therefore, in order to conceive the first fluxions, we must conceive time divided into moments, increments generated in those moments, and velocities proportional to those increments: that, in order to conceive second and third fluxions, we must suppose that the nascent principles or momentaneous increments have themselves also other momentaneous increments, which are proportional to their respective generating velocities: that the velocities of these second momentaneous increments are second fluxions: those of their nascent momentaneous increments third fluxions. And so on ad infinitum .
43. By subducting the increment generated in the first moment from that generated in the second, we get the increment of an increment. And by subducting the velocity generating in the first moment from that generating in the second, we get a fluxion of a fluxion. In like manner, by subducting the difference of the velocities generating in the two first moments from the excess of the velocity in the third above that in the second moment, we obtain the third fluxion. And after the same analogy we may proceed to fourth, fifth, sixth fluxions &c. And if we call the velocities of the first, second, third, fourth moments, a , b , c , d , the series of fluxions will be as above, a , b - a , c - 2 b + a , d - 3 c + 3 b - a , ad infinitum , i.e. , , , , ad infinitum .
44. Thus fluxions may be conceived in sundry lights and shapes, which seem all equally difficult to conceive. And, indeed, as it is impossible to conceive velocity without time or space, without either finite length or finite duration, [Sect. 31] it must seem above the powers of men to comprehend even the first fluxions. And if the first are incomprehensible, what shall we say of the second and third fluxions, &c.? He who can conceive the beginning of a beginning, or the end of an end, somewhat before the first or after the last, may be perhaps sharpsighted enough to conceive these things. But most men will, I believe, find it impossible to understand them in any sense whatever.
45. One would think that men could not speak too exactly on so nice a subject. And yet, as was before hinted, we may often observe that the exponents of fluxions, or notes representing fluxions are compounded with the fluxions themselves. Is not this the case when, just after the fluxions of flowing quantities were said to be the celerities of their increasing, and the second fluxions to be the mutations of the first fluxions or celerities, we are told that [`De Quadratura Curvarum.'] represents a series of quantities whereof each subsequent quantity is the fluxion of the preceding: and each foregoing is a fluent quantity having the following one for its fluxion?
