The Analyst: a Discourse addressed to an Infidel Mathematician

37. It is easiest 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 it self and by it self, 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 my self, 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. Some think that an easier Method of conceiving Fluxions is 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.

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. By which means the Analogy of Fluxions may seem better preserved, and the notion rendered more intelligible.

40. 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 is found in the Doctrine of Fluxions.

It may seem therefore, that the Notion might be still mended, and that instead of Fluxions of Fluxions, or 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[14] be divided into the Isochronal Parts Lm, mn, no, oM; and the Increment MN 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 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[15].

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 the 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, fixth 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 - 2b + a. d - 3c + 3b - a. ad infinitum, i. e.

.. ad infinitum.

44. Thus Fluxions may be considered in sundry Lights and Shapes, which seem all equally difficult to conceive.

It is impossible to conceive Velocity without time or space, without either finite length or finite Duration[16], it must seem above the powers of Men to comprehend even the first Fluxions.

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 confounded 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 [17] 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?

46. Divers Series of Quantities and Expressions, Geometrical and Algebraical, may be easily conceived, in Lines, in Surfaces, in Species, to be continued without end or limit.

But it will not be found so easy to conceive a Series, either of mere Velocities or of mere nascent Increments, distinct therefrom and corresponding thereunto.

Some perhaps may be led to think the Author intended a Series of Ordinates, wherein each Ordinate was the Fluxion of the preceding and Fluent of the following, i. e. that the Fluxion of one Ordinate was it self the Ordinate of another Curve; and the Fluxion of this last Ordinate was the Ordinate of yet another Curve; and so on ad infinitum. But who can conceive how the Fluxion (whether Velocity or nascent Increment) of an Ordinate should be it self an Ordinate?

Or more than that each preceding Quantity or Fluent is related to its Subsequent or Fluxion, as the Area of a curvilinear Figure to its Ordinate; agreeably to what the Author remarks, that each preceding Quantity in such Series is as the Area of a curvilinear Figure, whereof the Absciss is z, and the Ordinate is the following Quantity.

47. Upon the whole it appears that the Celerities are dismissed, and instead thereof Areas and Ordinates are introduced. But however expedient such Analogies or such Expressions may be found for facilitating the modern Quadratures, yet we shall not find any light given us thereby into the original real nature of Fluxions; or that we are enabled to frame from thence just Ideas of Fluxions considered in themselves.

In all this the general ultimate drift of the Author is very clear, but his Principles are obscure.

But perhaps those Theories of the great Author are not minutely considered or canvassed by his Disciples; who seem eager, as was before hinted, rather to operate than to know, rather to apply his Rules and his Forms, than to understand his Principles and enter into his Notions. It is nevertheless certain, that in order to follow him in his Quadratures, they must find Fluents from Fluxions; and in order to this, they must know to find Fluxions from Fluents; and in order to find Fluxions, they must first know what Fluxions are.

Otherwise they proceed without Clearness and without Science.

Thus the direct Method precedes the inverse, and the knowledge of the Principles is supposed in both. But as for operating according to Rules, and by the help of general Forms, whereof the original Principles and Reasons are not understood, this is to be esteemed merely technical. Be the Principles therefore ever so abstruse and metaphysical, they must be studied by whoever would comprehend the Doctrine of Fluxions.

Nor can any Geometrician have a right to apply the Rules of the great Author, without first considering his metaphysical Notions whence they were derived.

These how necessary soever in order to Science, which can never be attained without a precise, clear, and accurate Conception of the Principles, are nevertheless by several carelesly passed over; while the Expressions alone are dwelt on and conisdered and treated with great Skill and Management, thence to obtain other Expressions by Methods, suspicious and indirect (to say the least) if considered in themselves, however recommended by Induction and Authority; two Motives which are acknowledged sufficient to beget a rational Faith and moral Persuasion, but nothing higher.

48. You may possibly hope to evade the Force of all that hath been said, and to screen false Principles and inconsistent Reasonings, by a general Pretence that these Objections and Remarks are Metaphysical.

But this is a vain Pretence. For the plain Sense and Truth of what is advanced in the foregoing Remarks, I appeal to the Understanding of every unprejudiced intelligent Reader. To the same I appeal, whether the Points remarked upon are not most incomprehensible Metaphysics.

Metaphysics not of mine, but your own.

I would not be understood to infer, that your Notions are false or vain because they are Metaphysical. Nothing is either true or false for that Reason. Whether a Point be called Metaphysical or no avails little. The Question is whether it be clear or obscure, right or wrong, well or ill-deduced?

49. Although momentaneous Increments, nascent and evanescent Quantities, Fluxions and Infinitesimals of all Degrees, are in truth such shadowy Entities, so difficult to imagine or conceive distinctly, that (to say the least) they cannot be admitted as Principles or Objects of clear and accurate Science: and although this obscurity and incomprehensibility of your Metaphysics had been alone sufficient, to allay your Pretensions to Evidence; yet it hath, if I mistake not, been further shewn, that your Inferences are no more just than your Conceptions are clear, and that your Logics are as exceptionable as your Metaphysics.

Your Conclusions are not attained by just Reasoning from clear Principles; consequently that the Employment of modern Analysts, however useful in mathematical Calculations, and Constructions, doth not habituate and qualify the Mind to apprehend clearly and infer justly; and consequently, that you have no right in Virtue of such Habits, to dictate out of your proper Sphere, beyond which your Judgment is to pass for no more than that of other Men.