The Analyst: a Discourse addressed to an Infidel Mathematician

30. It seems therefore upon the whole that we may safely pronounce, the Conclusion cannot be right, if in order thereto any Quantity be made to vanish, or be neglected, except that either one Error is redressed by another; or that secondly, on the same Side of an Equation equal Quantities are destroyed by contrary Signs, so that the Quantity we mean to reject is first annihilated; or lastly, that from the opposite Sides equal Quantities are subducted.

Therefore to get rid of Quantities by the received Principles of Fluxions or of Differences is neither good Geometry nor good Logic. When the Augments vanish, the Velocities also vanish. The Velocities or Fluxions are said to primò and ultimò, as the Augments nascent and evanescent. Take therefore the Ratio of the evanescent Quantities, it is the same with that of the Fluxions. It will therefore answer all Intents as well. Why then are Fluxions introduced?

Is it not to shun or rather to palliate the Use of Quantities infinitely small? But we have no Notion whereby to conceive and measure various Degrees of Velocity, beside Space and Time, or when the Times are given, beside Space alone. We have even no Notion of Velocity prescinded from Time and Space.

When therefore a Point is supposed to move in given Times, we have no Notion of greater or lesser Velocities or of Proportions between Velocities, but only of longer or shorter Lines, and of Proportions between such Lines generated in equal Parts of Time.

31. A Point may be the limit of a Line: A Line may be the limit of a Surface: A Moment may terminate Time. But how can we conceive a Velocity by the help of such Limits?

It necessarily implies both Time and Space, and cannot be conceived without them. And if the Velocities of nascent and evanescent Quantities, i. e. abstracted from Time and Space, may not be comprehended, how can we comprehend and demonstrate their Proportions? Or consider their rationes primæ and ultimæ.

For to consider the Proportion or Ratio of Things implies that such Things have Magnitude: That such their Magnitudes may be measured, and their Relations to each other known. But, as there is no measure of Velocity except Time and Space, the Proportion of Velocities being only compounded of the direct Proportion of the Spaces, and the reciprocal Proportion of the Times; doth it not follow that to talk of investigating, obtaining, and considering the Proportions of Velocities, exclusively of Time and Space, is to talk unintelligibly?

32. But you will say that, in the use and application of Fluxions, Men do not overstrain their Faculties to a precise Conception of the abovementioned Velocities, Increments, Infinitesimals, or any other such like Ideas of a Nature so nice, subtile, and evanescent. And therefore you will perhaps maintain, that Problems may be solved without those inconceivable Suppositions: and that, consequently, the Doctrine of Fluxions, as to the practical Part, stands clear of all such Difficulties.

I answer, that if in the use or application of this Method, those difficult and obscure Points are not attended to, they are nevertheless supposed. They are the Foundations on which the Moderns build, the Principles on which they proceed, in solving Problems and discovering Theorems. It is with the Method of Fluxions as with all other Methods, which presuppose their respective Principles and are grounded thereon. Although the Rules may be practised by Men who neither attend to, nor perhaps know the Principles.

In like manner, therefore, as a Sailor may practically apply certain Rules derived from Astronomy and Geometry, the Principles whereof he doth not understand: And as any ordinary Man may solve divers numerical Questions, by the vulgar Rules and Operations of Arithmetic, which he performs and applies without knowing the Reasons of them:

Even so it cannot be denied that you may apply the Rules of the fluxionary Method: You may compare and reduce particular Cases to general Forms: You may operate and compute and solve Problems thereby, not only without an actual Attention to, or an actual Knowledge of, the Grounds of that Method, and the Principles whereon it depends, and whence it is deduced, but even without having ever considered or comprehended them.

33. But then it must be remembred, that in such Case although you may pass for an Artist, Computist, or Analyst, yet you may not be justly esteemed a Man of Science and Demonstration.

Nor should any Man, in virtue of being conversant in such obscure Analytics, imagine his rational Faculties to be more improved than those of other Men, which have been exercised in a different manner, and on different Subjects; much less erect himself into a Judge and an Oracle, concerning Matters that have no sort of connexion with, or dependence on those Species, Symbols or Signs, in the Management whereof he is so converfant and expert.

As you, who are a skilful Computist or Analyst, may not therefore be deemed skilful in Anatomy: or vice versa, as a Man who can dissect with Art, may, nevertheless, be ignorant in your Art of computing Even so you may both, notwithstanding your peculiar Skill in your respective Arts, be alike unqualified to decide upon Logic, or Metaphysics, or Ethics, or Religion.

This would be true, even admitting that you understood your own Principles and could demonstrate them.

34. If it is said, that Fluxions may be expounded or expressed by finite Lines proportional to them: Which finite Lines, as they may be distinctly conceived and known and reasoned upon, so they may be substituted for the Fluxions, and their mutual Relations or Proportions be considered as the Proportions of Fluxions: By which means the Doctrine becomes clear and useful.

I answer that if, in order to arrive at these finite Lines proportional to the Fluxions, there be certain Steps made use of which are obscure and inconceivable, be those finite Lines themselves ever so clearly conceived, it must nevertheless be acknowledged, that your proceeding is not clear nor your method scientific. For instance, it is supposed that AB being the Absciss, BC the Ordinate,

..

and VCH a Tangent of the Curve AC, Bb or CE the Increment of the Absciss, Ec the Increment of the Ordinate, which produced meets VH in the Point T, and Cc the Increment of the Curve.

The right Line Cc being produced to K, there are formed three small Triangles, the Rectilinear CEc, the Mixtilinear CEc, and the Rectilinear Triangle CET.

These 3 Triangles are different from each other, the Rectilinear CEc being less than the Mixtilinear CEc, whose Sides are the three Increments abovementioned, and this still less than the Triangle CET. It is supposed that the Ordinate bc moves into the place BC, so that the Point c is coincident with the Point C; and the right Line CK, and consequently the Curve Cc, is coincident with the Tangent CH.

In which case the mixtilinear evanescent Triangle CEc will, in its last form, be similar to the Triangle CET:

Its evanescent Sides CE, Ec, and Cc will be porportional to CE, ET, and CT the Sides of the Triangle CET.

Therefore it is concluded, that the Fluxions of the Lines AB, BC, and AC, being in the last Ratio of their evanescent Increments, are proportional to the Sides of the Triangle CET, or, which is all one, of the Triangle VBC similar thereunto. [12]

It is particularly remarked and insisted on by the great Author, that the Points C and c must not be distant one from another, by any the least Interval whatsoever: But that, in order to find the ultimate Proportions of the Lines CE, Ec, and Cc (i. e. the Proportions of the Fluxions or Velocities) expressed by the finite Sides of the Triangle VBC, the Points C and c must be accurately coincident, i. e. one and the same. A Point therefore is considered as a Triangle, or a Triangle is supposed to be formed in a Point.

Which to conceive seems quite impossible. Yet some there are, who, though they shrink at all other Mysteries, make no difficulty of their own, who strain at a Gnat and swallow a Camel.

35. I know not whether it be worth while to observe, that possibly some Men may hope to operate by Symbols and Suppositions, in such sort as to avoid the use of Fluxions, Momentums, and Infinitesimals after the following manner. Suppose x to be one Absciss of a Curve, and z another Absciss of the same Curve.

Suppose also that the respective Areas are xxx and zzz: and that z — x is the Increment of the Absciss, and zzz — xxx the Increment of the Area, without considering how great, or how small those Increments may be. Divide now zzz — xxx by z - x and the Quotient will be zz + zx + xx: and, supposing that z and x are equal, this same Quotient will be 3xx which in that case is the Ordinate, which therefore may be thus obtained independently of Fluxions and Infinitesimals.

But herein is a direct Fallacy: for in the first place, it is supposed that the Abscisses z and x are unequal, without which supposition no one step could have been made; and in the second place, it is supposed they are equal; which is a manifest Inconsistency, and amounts to the same thing that hath been before confidered[13].

There is reason to apprehend, that all Attempts for setting the abstruse and fine Geometry on a right Foundation, and avoiding the Doctrine of Velocities, Momentums, &c. will be found impracticable, till such time as the Object and End of Geometry are better understood, than hitherto they seem to have been.

The great Author of the Method of Fluxions felt this Difficulty, and therefore he gave into those nice Abstractions and Geometrical Metaphysics, without which he saw nothing could be done on the received Principles; and what in the way of Demonstration he hath done with them the Reader will judge.

He used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them.

But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments And what are these same evanescent Increments?

They are neither finite Quantities, nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

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.

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 generated 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 — 2b + a. d — 3c + 3b - a &c. which you may call by the names of 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.