ANALYST;

OR, A

DISCOURSE

Addressed to an

Infidel MATHEMATICIAN.

WHEREIN

It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.

(1734)

Edited by David R. Wilkins

II. *Their
Principles and Methods to be examined with the same freedom, which they assume
with regard to the Principles and Mysteries of Religion. In what Sense and how
far Geometry is to be allowed an Improvement of the Mind. *

III. *Fluxions
the great Object and Employment of the profound Geometricians in the present
Age. What these Fluxions are. *

IV. *Moments
or nascent Increments of flowing Quantities difficult to conceive. Fluxions of
different Orders. Second and third Fluxions obscure Mysteries. *

V. *Differences,
i. e. Increments or Decrements infinitely small, used by foreign
Mathematicians instead of Fluxions or Velocities of nascent and evanescent
Increments. *

VI. *Differences
of various Orders, i. e. Quantities infinitely less than Quantities
infinitely little; and infinitesimal Parts of infinitesimals of infinitesimals,
&c. without end or limit. *

VII. *Mysteries
in faith unjustly objected against by those who admit them in Science.
*

VIII. *Modern
Analysts supposed by themselves to extend their views even beyond infinity:
Deluded by their own Species or Symbols. *

IX. *Method
for finding the Fluxion of a Rectangle of two indeterminate Quantities, shewed
to be illegitimate and false. *

X. *Implicit
Deference of Mathematical-men for the great Author of Fluxions. Their
earnestness rather to go on fast and far, than to set out warily and see their
way distinctly. *

XI. *Momentums
difficult to comprehend. No middle Quantity to be admitted between a finite
Quantity and nothing, without admitting Infinitesimals. *

XII. *The
Fluxion of any Power of a flowing Quantity. Lemma premised in order to examine
the method for finding such Fluxion. *

XIII. *The
rule for the Fluxions of Powers attained by unfair reasoning. *

XIV. *The
aforesaid reasoning farther unfolded, and shew'd to be illogical. *

XV. *No
true Conclusion to be justly drawn by direct consequence from inconsistent
Suppositions. The same Rules of right reason to be observed, whether Men argue
in Symbols or in Words. *

XVI. *An
Hypothesis being destroyed, no consequence of such Hypothesis to be retained.
*

XVII. *Hard
to distinguish between evanescent Increments and infinitesimal Differences.
Fluxions placed in various Lights. The great Author, it seems, not satisfied
with his own Notions. *

XVIII. *Quantities
infinitely small supposed and rejected by Leibnitz and his Followers.
No Quantity, according to them, greater or smaller for the Addition or
Subduction of its Infinitesimal. *

XIX. *Conclusions
to be proved by the Principles, and not Principles by the Conclusions.
*

XX. *The
Geometrical Analyst considered as a Logician; and his Discoveries, not in
themselves, but as derived from such Principles and by such Inferences.
*

XXI. *A
Tangent drawn to the Parabola according to the calculus differentialis.
Truth shewn to be the result of error, and how. *

XXII. *By
virtue of a twofold mistake Analysts arrive at Truth, but not at Science:
ignorant how they come at their own Conclusions. *

XXIII. *The
Conclusion never evident or accurate, in virtue of obscure or inaccurate
Premises. Finite Quantities might be rejected as well as Infinitesimals.
*

XXIV. *The
foregoing Doctrine farther illustrated. *

XXV. *Sundry
Observations thereupon. *

XXVI. *Ordinate
found from the Area by means of evanescent Increments. *

XXVII. *In
the foregoing Case the supposed evanescent Increment is really a finite
Quantity, destroyed by an equal Quantity with an opposite Sign. *

XXVIII. *The
foregoing Case put generally. Algebraical Expressions compared with Geometrical
Quantities. *

XXIX. *Correspondent
Quantities Algebraical and Geometrical equated. The analysis shewed not to
obtain in Infinitesimals, but it must also obtain in finite Quantities.
*

XXX. *The
getting rid of Quantities by the received Principles, whether of Fluxions or of
Differences, neither good Geometry nor good Logic. Fluxions or Velocities, why
introduced. *

XXXI. *Velocities
not to be abstracted from Time and Space: Nor their Proportions to be
investigated or considered exclusively of Time and Space. *

XXXII. *Difficult
and obscure Points constitute the Principles of the modern Analysis, and are the
Foundation on which it is built. *

XXXIII. *The
rational Faculties whether improved by such obscure Analytics. *

XXXIV. *By
what inconceivable Steps finite lines are found proportional to Fluxions.
Mathematical Infidels strain at a Gnat and swallow a Camel. *

XXXV. *Fluxions
or Infinitesimals not to be avoided on the received Principles. Nice
Abstractions and Geometrical Metaphysics. *

XXXVI. *Velocities
of nascent or evanescent Quantities, whether in reality understood and signified
by finite Lines and Species. *

XXXVII. *Signs
or Exponents obvious; but Fluxions themselves not so. *

XXXVIII. *Fluxions,
whether the Velocities with which infinitesimal Differences are generated?
*

XXXIX. *Fluxions
of Fluxions or second Fluxions, whether to be conceived as Velocities of
Velocities, or rather as Velocities of the second nascent Increments?
*

XL. *Fluxions
considered, sometimes in one Sense, sometimes in another: One while in
themselves, another in their Exponents: Hence Confusion and Obscurity.
*

XLI. *Isochronal
Increments, whether finite or nascent, proportional to their respective
Velocities. *

XLII. *Time
supposed to be divided into Moments: Increments generated in those Moments: And
Velocities proportional to those Increments. *

XLIII. *Fluxions,
second, third, fourth, &c. what they are, how obtained, and how
represented. What Idea of Velocity in a Moment of Time and Point of Space.
*

XLIV. *Fluxions
of all Orders inconceivable. *

XLV. *Signs
or Exponents confounded with the Fluxions. *

XLVI. *Series
of Expressions or of Notes easily contrived. Whether a Series, of mere
Velocities, or of mere nascent Increments, corresponding thereunto, be as easily
conceived? *

XLVII. *Celerities
dismissed, and instead thereof Ordinates and Areas introduced. Analogies and
Expressions useful in the modern Quadratures, may yet be useless for enabling us
to conceive Fluxions. No right to apply the Rules without knowledge of the
Principles. *

XLVIII. *Metaphysics
of modern Analysts most incomprehensible. *

XLIX. *Analysts
employ'd about notional shadowy Entities. Their Logics as exceptionable as their
Metaphysics. *

L. *Occasion
of this Address. Conclusion. Queries. *

I. Though I am a Stranger to your Person, yet I am not, Sir, a Stranger to the Reputation you have acquired, in that branch of Learning which hath been your peculiar Study; nor to the Authority that you therefore assume in things foreign to your Profession, nor to the Abuse that you, and too many more of the like Character, are known to make of such undue Authority, to the misleading of unwary Persons in matters of the highest Concernment, and whereof your mathematical Knowledge can by no means qualify you to be a competent Judge. Equity indeed and good Sense would incline one to disregard the Judgment of Men, in Points which they have not considered or examined. But several who make the loudest Claim to those Qualities, do, nevertheless, the very thing they would seem to despise, clothing themselves in the Livery of other Mens Opinions, and putting on a general deference for the Judgment of you, Gentlemen, who are presumed to be of all Men the greatest Masters of Reason, to be most conversant about distinct Ideas, and never to take things on trust, but always clearly to see your way, as Men whose constant Employment is the deducing Truth by the justest inference from the most evident Principles. With this bias on their Minds, they submit to your Decisions where you have no right to decide. And that this is one short way of making Infidels I am credibly informed.

II. Whereas then it is supposed, that you apprehend more distinctly, consider more closely, infer more justly, conclude more accurately than other Men, and that you are therefore less religious because more judicious, I shall claim the privilege of a Free-Thinker; and take the Liberty to inquire into the Object, Principles, and Method of Demonstration admitted by the Mathematicians of the present Age, with the same freedom that you presume to treat the Principles and Mysteries of Religion; to the end, that all Men may see what right you have to lead, or what Encouragement others have to follow you. It hath been an old remark that Geometry is an excellent Logic. And it must be owned, that when the Definitions are clear; when the Postulata cannot be refused, nor the Axioms denied; when from the distinct Contemplation and Comparison of Figures, their Properties are derived, by a perpetual well-connected chain of Consequences, the Objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical: which habit strengthens and sharpens the Mind, and being transferred to other Subjects, is of general use in the inquiry after Truth. But how far this is the case of our Geometrical Analysts, it may be worth while to consider.

III. The Method of Fluxions is the general Key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature. And as it is that which hath enabled them so remarkably to outgo the Ancients in discovering Theorems and solving Problems, the exercise and application thereof is become the main, if not sole, employment of all those who in this Age pass for profound Geometers. But whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgment, and that of every candid Reader. Lines are supposed to be generated [NOTE: Introd. ad Quadraturam Curvarum.] by the motion of Points, Planes by the motion of Lines, and Solids by the motion of Planes. And whereas Quantities generated in equal times are greater or lesser, according to the greater or lesser Velocity, wherewith they increase and are generated, a Method hath been found to determine Quantities from the Velocities of their generating Motions. And such Velocities are called Fluxions: and the Quantities generated are called flowing Quantities. These Fluxions are said to be nearly as the Increments of the flowing Quantities, generated in the least equal Particles of time; and to be accurately in the first Proportion of the nascent, or in the last of the evanescent, Increments. Sometimes, instead of Velocities, the momentaneous Increments or Decrements of undetermined flowing Quantities are considered, under the Appellation of Moments.

IV. By Moments we are not to understand finite Particles.
These are said not to be Moments, but Quantities generated from Moments, which
last are only the nascent Principles of finite Quantities. It is said, that the
minutest Errors are not to be neglected in Mathematics: that the Fluxions are
Celerities, not proportional to the finite Increments though ever so small; but
only to the Moments or nascent Increments, whereof the Proportion alone, and not
the Magnitude, is considered. And of the aforesaid Fluxions there be other
Fluxions, which Fluxions of Fluxions are called second Fluxions. And the
Fluxions of these second Fluxions are called third Fluxions: and so on, fourth,
fifth, sixth, *&c.* *ad infinitum*. Now as our Sense is
strained and puzzled with the perception of Objects extremely minute, even so
the Imagination, which Faculty derives from Sense, is very much strained and
puzzled to frame clear Ideas of the least Particles of time, or the least
Increments generated therein: and much more so to comprehend the Moments, or
those Increments of the flowing Quantities in *statu nascenti*, in their
very first origin or beginning to exist, before they become finite Particles.
And it seems still more difficult, to conceive the abstracted Velocities of such
nascent imperfect Entities. But the Velocities of the Velocities, the second,
third, fourth, and fifth Velocities, *&c.* exceed, if I mistake not,
all Humane Understanding. The further the Mind analyseth and pursueth these
fugitive Ideas, the more it is lost and bewildered; the Objects, at first
fleeting and minute, soon vanishing out of sight. Certainly in any Sense a
second or third Fluxion seems an obscure Mystery. The incipient Celerity of an
incipient Celerity, the nascent Augment of a nascent Augment, *i. e.* of
a thing which hath no Magnitude: Take it in which light you please, the clear
Conception of it will, if I mistake not, be found impossible, whether it be so
or no I appeal to the trial of every thinking Reader. And if a second Fluxion be
inconceivable, what are we to think of third, fourth, fifth Fluxions, and so
onward without end?

V. The foreign Mathematicians are supposed by some, even of our own, to proceed in a manner, less accurate perhaps and geometrical, yet more intelligible. Instead of flowing Quantities and their Fluxions, they consider the variable finite Quantities, as increasing or diminishing by the continual Addition or Subduction of infinitely small Quantities. Instead of the Velocities wherewith Increments are generated, they consider the Increments or Decrements themselves, which they call Differences, and which are supposed to be infinitely small. The Difference of a Line is an infinitely little Line; of a Plane an infinitely little Plane. They suppose finite Quantities to consist of Parts infinitely little, and Curves to be Polygons, whereof the Sides are infinitely little, which by the Angles they make one with another determine the Curvity of the Line. Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or any the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust.

VI. And yet in the *calculus differentialis*, which
Method serves to all the same Intents and Ends with that of Fluxions, our modern
Analysts are not content to consider only the Differences of finite Quantities:
they also consider the Differences of those Differences, and the Differences of
the Differences of the first Differences. And so on *ad infinitum*. That
is, they consider Quantities infinitely less than the least discernible
Quantity; and others infinitely less than those infinitely small ones; and still
others infinitely less than the preceding Infinitesimals, and so on without end
or limit. Insomuch that we are to admit an infinite succession of
Infinitesimals, each infinitely less than the foregoing, and infinitely greater
than the following. As there are first, second, third, fourth, fifth
*&c.* Fluxions, so there are Differences, first, second, third
fourth, *&c.* in an infinite Progression towards nothing, which you
still approach and never arrive at. And (which is most strange) although you
should take a Million of Millions of these Infinitesimals, each whereof is
supposed infinitely greater than some other real Magnitude, and add them to the
least given Quantity, it shall be never the bigger. For this is one of the
modest *postulata* of our modern Mathematicians, and is a Corner-stone or
Ground-work of their Speculations.

VII. All these Points, I say, are supposed and believed by certain rigorous Exactors of Evidence in Religion, Men who pretend to believe no further than they can see. That Men, who have been conversant only about clear Points, should with difficulty admit obscure ones might not seem altogether unaccountable. But he who can digest a second or third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity. There is a natural Presumption that Mens Faculties are made alike. It is on this Supposition that they attempt to argue and convince one another. What, therefore, shall appear evidently impossible and repugnant to one, may be presumed the same to another. But with what appearance of Reason shall any Man presume to say, that Mysteries may not be Objects of Faith, at the same time that he himself admits such obscure Mysteries to be the Object of Science?

VIII. It must indeed be acknowledged, the modern
Mathematicians do not consider these Points as Mysteries, but as clearly
conceived and mastered by their comprehensive Minds. They scruple not to say,
that by the help of these new Analytics they can penetrate into Infinity it
self: That they can even extend their Views beyond Infinity: that their Art
comprehends not only Infinite, but Infinite of Infinite (as they express it) or
an Infinity of Infinites. But, notwithstanding all these Assertions and
Pretensions, it may be justly questioned whether, as other Men in other
Inquiries are often deceived by Words or Terms, so they likewise are not
wonderfully deceived and deluded by their own peculiar Signs, Symbols, or
Species. Nothing is easier than to devise Expressions or Notations for Fluxions
and Infinitesimals of the first, second, third, fourth, and subsequent Orders,
proceeding in the same regular form without end or limit . . . . *&c.* or *dx*. *ddx*.
*dddx*. *ddddx*. *&c.* These Expressions indeed are clear
and distinct, and the Mind finds no difficulty in conceiving them to be
continued beyond any assignable Bounds. But if we remove the Veil and look
underneath, if laying aside the Expressions we set ourselves attentively to
consider the things themselves, which are supposed to be expressed or marked
thereby, we shall discover much Emptiness, Darkness, and Confusion; nay, if I
mistake not, direct Impossibilities and Contradictions. Whether this be the case
or no, every thinking Reader is intreated to examine and judge for himself.

IX. Having considered the Object, I proceed to consider the
Principles of this new Analysis by Momentums, Fluxions, or Infinitesimals;
wherein if it shall appear that your capital Points, upon which the rest are
supposed to depend, include Error and false Reasoning; it will then follow that
you, who are at a loss to conduct your selves, cannot with any decency set up
for guides to other Men. The main Point in the method of Fluxions is to obtain
the Fluxion or Momentum of the Rectangle or Product of two indeterminate
Quantities. Inasmuch as from thence are derived Rules for obtaining the Fluxions
of all other Products and Powers; be the Coefficients or the Indexes what they
will, integers or fractions, rational or surd. Now this fundamental Point one
would think should be very clearly made out, considering how much is built upon
it, and that its Influence extends throughout the whole Analysis. But let the
Reader judge. This is given for Demonstration. [NOTE: Naturalis Philosophiæ
principia mathematica, l. 2. lem. 2.] Suppose the Product or Rectangle *AB*
increased by continual Motion: and that the momentaneous Increments of the Sides
*A* and *B* are *a* and *b*. When the Sides *A* and
*B* were deficient, or lesser by one half of their Moments, the Rectangle
was

,

.And as soon as the Sides

or

.From the latter Rectangle subduct the former, and the remaining Difference will be

X. Such reasoning as this for Demonstration, nothing but
the obscurity of the Subject could have encouraged or induced the great Author
of the Fluxionary Method to put upon his Followers, and nothing but an implicit
deference to Authority could move them to admit. The Case indeed is difficult.
There can be nothing done till you have got rid of the Quantity *ab*. In
order to this the Notion of Fluxions is shifted: it is placed in various Lights:
Points which should be as clear as first Principles are puzzled; and Terms which
should be steadily used are ambiguous. But notwithstanding all this address and
skill the point of getting rid of *ab* cannot be obtained by legitimate
reasoning. If a Man by Methods, not geometrical or demonstrative, shall have
satisfied himself of the usefulness of certain Rules; which he afterwards shall
propose to his Disciples for undoubted Truths; which he undertakes to
demonstrate in a subtile manner, and by the help of nice and intricate Notions;
it is not hard to conceive that such his Disciples may, to save themselves the
trouble of thinking, be inclined to confound the usefulness of a Rule with the
certainty of a Truth, and accept the one for the other; especially if they are
Men accustomed rather to compute than to think; earnest rather to go on fast and
far, than solicitous to set out warily and see their way distinctly.

XI. The Points or meer Limits of nascent Lines are
undoubtedly equal, as having no more magnitude one than another, a Limit as such
being no Quantity. If by a Momentum you mean more than the very initial Limit,
it must be either a finite Quantity or an Infinitesimal. But all finite
Quantities are expressly excluded from the Notion of a Momentum. Therefore the
Momentum must be an Infinitesimal. And indeed, though much Artifice hath been
employ'd to escape or avoid the admission of Quantities infinitely small, yet it
seems ineffectual. For ought I see, you can admit no Quantity as a Medium
between a finite Quantity and nothing, without admitting Infinitesimals. An
Increment generated in a finite Particle of Time, is it self a finite Particle;
and cannot therefore be a Momentum. You must therefore take an Infinitesimal
Part of Time wherein to generate your Momentum. It is said, the Magnitude of
Moments is not considered: And yet these same Moments are supposed to be divided
into Parts. This is not easy to conceive, no more than it is why we should take
Quantities less than *A* and *B* in order to obtain the Increment of
*AB*, of which proceeding it must be owned the final Cause or Motive is
very obvious; but it is not so obvious or easy to explain a just and legitimate
Reason for it, or shew it to be Geometrical.

XII. From the foregoing Principle so demonstrated, the general Rule for finding the Fluxion of any Power of a flowing Quantity is derived. [NOTE: Philosophiæ naturalis principia Mathematica, lib. 2. lem. 2.] But, as there seems to have been some inward Scruple or Consciousness of defect in the foregoing Demonstration, and as this finding the Fluxion of a given Power is a Point of primary Importance, it hath therefore been judged proper to demonstrate the same in a different manner independent of the foregoing Demonstration. But whether this other Method be more legitimate and conclusive than the former, I proceed now to examine; and in order thereto shall premise the following Lemma. ``If with a View to demonstrate any Proposition, a certain Point is supposed, by virtue of which certain other Points are attained; and such supposed Point be it self afterwards destroyed or rejected by a contrary Supposition; in that case, all the other Points, attained thereby and consequent thereupon, must also be destroyed and rejected, so as from thence forward to be no more supposed or applied in the Demonstration.'' This is so plain as to need no Proof.

XIII. Now the other Method of obtaining a Rule to find the
Fluxion of any Power is as follows. Let the Quantity *x* flow uniformly,
and be it proposed to find the Fluxion of *x*^{n}. In the
same time that *x* by flowing becomes *x* + *o*, the Power
*x*^{n} becomes , *i. e.* by the Method of infinite Series

and the Increments

are to one another as

Let now the Increments vanish, and their last Proportion will be 1 to

XIV. To make this Point plainer, I shall unfold the
reasoning, and propose it in a fuller light to your View. It amounts therefore
to this, or may in other Words be thus expressed. I suppose that the Quantity
*x* flows, and by flowing is increased, and its Increment I call *o*,
so that by flowing it becomes *x* + *o*. And as *x* increaseth,
it follows that every Power of *x* is likewise increased in a due
Proportion. Therefore as *x* becomes *x* + *o*,
*x*^{n} will become : that is, according to the Method of infinite
Series,

And if from the two augmented Quantities we subduct the Root and the Power respectively, we shall have remaining the two Increments, to wit,

which Increments, being both divided by the common Divisor

which are therefore Exponents of the Ratio of the Increments. Hitherto I have supposed that

XV. Nothing is plainer than that no just Conclusion can be
directly drawn from two inconsistent Suppositions. You may indeed suppose any
thing possible: But afterwards you may not suppose any thing that destroys what
you first supposed. Or if you do, you must begin *de novo*. If therefore
you suppose that the Augments vanish, *i. e.* that there are no Augments,
you are to begin again, and see what follows from such Supposition. But nothing
will follow to your purpose. You cannot by that means ever arrive at your
Conclusion, or succeed in, what is called by the celebrated Author, the
Investigation of the first or last Proportions of nascent and evanescent
Quantities, by instituting the Analysis in finite ones. I repeat it again: You
are at liberty to make any possible Supposition: And you may destroy one
Supposition by another: But then you may not retain the Consequences, or any
part of the Consequences of your first Supposition so destroyed. I admit that
Signs may be made to denote either any thing or nothing: And consequently that
in the original Notation *x* + *o*, *o* might have signified
either an Increment or nothing. But then which of these soever you make it
signify, you must argue consistently with such its Signification, and not
proceed upon a double Meaning: which to do were a manifest Sophism. Whether you
argue in Symbols or in Words, the Rules of right Reason are still the same. Nor
can it be supposed, you will plead a Privilege in Mathematics to be exempt from
them.

XVI. If you assume at first a Quantity increased by
nothing, and in the Expression *x* + *o*, *o* stands for nothing,
upon this Supposition as there is no Increment of the Root, so there will be no
Increment of the Power; and consequently there will be none except the first, of
all those Members of the Series constituting the Power of the Binomial; you will
therefore never come at your Expression of a Fluxion legitimately by such
Method. Hence you are driven into the fallacious way of proceeding to a certain
Point on the Supposition of an Increment, and then at once shifting your
Supposition to that of no Increment. There may seem great Skill in doing this at
a certain Point or Period. Since if this second Supposition had been made before
the common Division by *o*, all had vanished at once, and you must have got
nothing by your Supposition. Whereas by this Artifice of first dividing, and
then changing your Supposition, you retain 1 and *nx*^{n -
1}. But, notwithstanding all this address to cover it, the fallacy is still
the same. For whether it be done sooner or later, when once the second
Supposition or Assumption is made, in the same instant the former Assumption and
all that you got by it is destroyed, and goes out together. And this is
universally true, be the Subject what it will, throughout all the Branches of
humane Knowledge; in any other of which, I believe, Men would hardly admit such
a reasoning as this, which in Mathematics is accepted for Demonstration.

XVII. It may not be amiss to observe, that the Method for
finding the Fluxion of a Rectangle of two flowing Quantities, as it is set forth
in the Treatise of Quadratures, differs from the abovementioned taken from the
second Book of the Principles, and is in effect the same with that used in the
*calculus differentialis*. [NOTE: Analyse des Infiniment Petits, part 1.
prop. 2.] For the supposing a Quantity infinitely diminished and therefore
rejecting it, is in effect the rejecting an Infinitesimal; and indeed it
requires a marvellous sharpness of Discernment, to be able to distinguish
between evanescent Increments and infinitesimal Differences. It may perhaps be
said that the Quantity being infinitely diminished becomes nothing, and so
nothing is rejected. But according to the received Principles it is evident,
that no Geometrical Quantity, can by any division or subdivision whatsoever be
exhausted, or reduced to nothing. Considering the various Arts and Devices used
by the great author of the Fluxionary Method: in how many Lights he placeth his
Fluxions: and in what different ways he attempts to demonstrate the same Point:
one would be inclined to think, he was himself suspicious of the justness of his
own demonstrations; and that he was not enough pleased with any one notion
steadily to adhere to it. Thus much at least is plain, that he owned himself
satisfied concerning certain Points, which nevertheless he could not undertake
to demonstrate to others. [NOTE: *See Letter to* Collins, Nov. 8, 1676.]
Whether this satisfaction arose from tentative Methods or Inductions; which have
often been admitted by Mathematicians (for instance by Dr. *Wallis* in
his Arithmetic of Infinites) is what I shall not pretend to determine. But,
whatever the Case might have been with respect to the Author, it appears that
his Followers have shewn themselves more eager in applying his Method, than
accurate in examining his Principles.

XVIII. It is curious to observe, what subtilty and skill
this great Genius employs to struggle with an insuperable Difficulty; and
through what Labyrinths he endeavours to escape the Doctrine of Infinitesimals;
which as it intrudes upon him whether he will or no, so it is admitted and
embraced by others without the least repugnance. *Leibnitz* and his
followers in their *calculus differentialis* making no manner of scruple,
first to suppose, and secondly to reject Quantities infinitely small: with what
clearness in the Apprehension and justness in the reasoning, any thinking Man,
who is not prejudiced in favour of those things, may easily discern. The Notion
or Idea of an infinitesimal Quantity, as it is an Object simply apprehended by
the Mind, hath been already considered. [NOTE: *Sect*. 5
*and* 6.]
I shall now only observe as to the method of getting rid of such Quantities,
that it is done without the least Ceremony. As in Fluxions the Point of first
importance, and which paves the way to the rest, is to find the Fluxion of a
Product of two indeterminate Quantities, so in the *calculus
differentialis* (which Method is supposed to have been borrowed from the
former with some small Alterations) the main Point is to obtain the difference
of such Product. Now the Rule for this is got by rejecting the Product or
Rectangle of the Differences. And in general it is supposed, that no Quantity is
bigger or lesser for the Addition or Subduction of its Infinitesimal: and that
consequently no error can arise from such rejection of Infinitesimals.

XIX. And yet it should seem that, whatever errors are admitted in the Premises, proportional errors ought to be apprehended in the Conclusion, be they finite or infinitesimal: and that therefore the of Geometry requires nothing should be neglected or rejected. In answer to this you will perhaps say, that the Conclusions are accurately true, and that therefore the Principles and Methods from whence they are derived must be so too. But this inverted way of demonstrating your Principles by your Conclusions, as it would be peculiar to you Gentlemen, so it is contrary to the Rules of Logic. The truth of the Conclusion will not prove either the Form or the Matter of a Syllogism to be true: inasmuch as the Illation might have been wrong or the Premises false, and the Conclusion nevertheless true, though not in virtue of such Illation or of such Premises. I say that in every other Science Men prove their Conclusions by their Principles, and not their Principles by the Conclusions. But if in yours you should allow your selves this unnatural way of proceeding, the Consequence would be that you must take up with Induction, and bid adieu to Demonstration. And if you submit to this, your Authority will no longer lead the way in Points of Reason and Science.

XX. I have no Controversy about your Conclusions, but only
about your Logic and Method. How you demonstrate? What Objects you are
conversant with, and whether you conceive them clearly? What Principles you
proceed upon; how sound they may be; and how you apply them? It must be
remembred that I am not concerned about the truth of your Theorems, but only
about the way of coming at them; whether it be legitimate or illegitimate, clear
or obscure, scientific or tentative. To prevent all possibility of your
mistaking me, I beg leave to repeat and insist, that I consider the Geometrical
Analyst as a Logician, *i. e.* so far forth as he reasons and argues; and
his Mathematical Conclusions, not in themselves, but in their Premises; not as
true or false, useful or insignificant, but as derived from such Principles, and
by such Inferences. And forasmuch as it may perhaps seem an unaccountable
Paradox, that Mathematicians should deduce true Propositions from false
Principles, be right in the Conclusion, and yet err in the Premises; I shall
endeavour particularly to explain why this may come to pass, and shew how Error
may bring forth Truth, though it cannot bring forth Science.

XXI. In order therefore to clear up this Point, we will suppose for instance that a Tangent is to be drawn to a Parabola, and examine the progress of this Affair, as it is performed by infinitesimal Differences.

Let

But herein there is an error arising from the aforementioned false supposition, whence the value of

There was therefore an error of defect in making

But if you multiply

truly. There was therefore an error of excess in making

which followed from the erroneous Rule of Differences. And the measure of this second error is

Therefore the two errors being equal and contrary destroy each other; the first error of defect being corrected by a second error of excess.

XXII. If you had committed only one error, you would not
have come at a true Solution of the Problem. But by virtue of a twofold mistake
you arrive, though not at Science, yet at Truth. For Science it cannot be
called, when you proceed blindfold, and arrive at the Truth not knowing how or
by what means. To demonstrate that *z* is equal to

let

Likewise from the Nature of the Parabola

and because

will be equal to

which being reduced gives

XXIII. Now I observe in the first place, that the
Conclusion comes out right, not because the rejected Square of *dy* was
infinitely small; but because this error was compensated by another contrary and
equal error. I observe in the second place, that whatever is rejected, be it
every so small, if it be real, and consequently makes a real error in the
Premises, it will produce a proportional real error in the Conclusion. Your
Theorems therefore cannot be accurately true, nor your Problems accurately
solved, in virtue of Premises, which themselves are not accurate, it being a
rule in Logic that *Conclusio sequitur partem debiliorem*. Therefore I
observe in the third place, that when the Conclusion is evident and the Premises
obscure, or the Conclusion accurate and the Premises inaccurate, we may safely
pronounce that such Conclusion is neither evident nor accurate, in virtue of
those obscure inaccurate Premises or Principles; but in virtue of some other
Principles which perhaps the Demonstrator himself never knew or thought of. I
observe in the last place, that in case the Differences are supposed finite
Quantities ever so great, the Conclusion will nevertheless come out the same:
inasmuch as the rejected Quantities are legitimately thrown out, not for their
smallness, but for another reason, to wit, because of contrary errors, which
destroying each other do upon the whole cause that nothing is really, though
something is apparently thrown out. And this Reason holds equally, with respect
to Quantities finite as well as infinitesimal, great as well as small, a Foot or
a Yard long as well as the minutest Increment.

XXIV. For the fuller illustration of this Point, I shall consider it in another light, and proceeding in finite Quantities to the Conclusion, I shall only then make use of one Infinitesimal.

Suppose the straight Line

wherein if for

And supposing

which is the true value of the Subtangent. And since this was obtained by one only error,

XXV. Upon the whole I observe, *First*, that
*v* can never be nothing so long as there is a secant. *Secondly*,
that the same Line cannot be both tangent and secant. *Thirdly*, that
when *v* and *NO* [NOTE: *See the foregoing Figure.*]
vanisheth, *PS* and *SR* do also vanish, and with them the
proportionality of the similar Triangles. Consequently the whole Expression,
which was obtained by means thereof and grounded thereupon, vanisheth when
*v* vanisheth. *Fourthly*, that the Method for finding Secants or
the Expression of Secants, be it ever so general, cannot in common sense extend
any further than to all Secants whatsoever: and, as it necessarily supposeth
similar Triangles, it cannot be supposed to take place where there are not
similar Triangles. *Fifthly*, that the Subsecant will always be less than
the Subtangent, and can never coincide with it; which Coincidence to suppose
would be absurd; for it would be supposing, the same Line at the same time to
cut and not to cut another given Line, which is a manifest Contradiction, such
as subverts the Hypothesis and gives a Demonstration of its Falshood.
*Sixthly*, if this be not admitted, I demand a Reason why any other
apagogical Demonstration, or Demonstration *ad absurdum* should be
admitted in Geometry rather than this: Or that some real Difference be assigned
between this and others as such. *Seventhly*, I observe that it is
sophistical to suppose *NO* or *RP*, *PS*, and *SR* to be
finite real Lines in order to form the Triangle *RPS*, in order to obtain
Proportions by similar Triangles; and afterwards to suppose there are no such
Lines, nor consequently similar Triangles, and nevertheless to retain the
Consequence of the first Supposition, after such Supposition hath been destroyed
by a contrary one. *Eighthly*, That although, in the present case, by
inconsistent Suppositions Truth may be obtained, yet that such Truth is not
demonstrated: That such Method is not conformable to the Rules of Logic and
right Reason: That, however useful it may be, it must be considered only as a
Presumption, as a Knack, an Art, rather an Artifice, but not a scientific
Demonstration.

XXVI. The Doctrine premised may be farther illustrated by the following simple and easy Case, wherein I shall proceed by evanescent Increments.

Suppose

XXVII. As on the one hand it were absurd to get rid of
*o* by saying, let me contradict my self: Let me subvert my own Hypothesis:
Let me take it for granted that there is no Increment, at the same time that I
retain a Quantity, which I could never have got at but by assuming an Increment:
So on the other hand it would be equally wrong to imagine, that in a geometrical
Demonstration we may be allowed to admit any Error, though ever so small, or
that it is possible, in the nature of Things, an accurate Conclusion should be
derived from inaccurate Principles. Therefore *o* cannot be thrown out as
an Infinitesimal, or upon the Principle that Infinitesimals may be safely
neglected. But only because it is destroyed by an equal Quantity with a negative
Sign, whence *o* - *qo* is equal to nothing. And as it is illegitimate
to reduce an Equation, by subducting from one Side a Quantity when it is not to
be destroyed, or when an equal Quantity is not subducted from the other Side of
the Equation: So it must be allowed a very logical and just Method of arguing,
to conclude that if from Equals either nothing or equal Quantities are
subducted, they shall still remain equal. And this is a true Reason why no Error
is at last produced by the rejecting of *o*. Which therefore must not be
ascribed to the Doctrine of Differences, or Infinitesimals, or evanescent
Quantities, or Momentums, or Fluxions.

XXVIII. Suppose the Case to be general, and that
*x*^{n} is equal to the Area *ABC*, whence by the
Method of Fluxions the Ordinate is found *nx*^{n - 1} which
we admit for true, and shall inquire how it is arrived at. Now if we are content
to come at the Conclusion in a summary way, by supposing that the Ratio of the
Fluxions of *x* and *x*^{n} are found [NOTE:
*Sect*. 13.]
to be 1 and *nx*^{n - 1}, and that the Ordinate of the Area
is considered as its Fluxion; we shall not so clearly see our way, or perceive
how the truth comes out, that Method as we have shewed before being obscure and
illogical. But if we fairly delineate the Area and its Increment, and divide the
latter into two Parts *BCFD* and *CFH*, [NOTE: *See the Figure in
Sect*. 26.]
and proceed regularly by Equations between the algebraical and geometrical
Quantities, the reason of the thing will plainly appear. For as
*x*^{n} is equal to the Area *ABC*, so is the Increment
of *x*^{n} equal to the Increment of the Area, *i.
e.* to *BDHC*; that is, to say,

And only the first Members, on each Side of the Equation being retained,

and that when this is rejected on one Side, that is rejected on the other, the Reasoning becomes just and the Conclusion true. And it is all one whatever Magnitude you allow to

equal I say to the finite Remainder of a finite Increment.

XXIX. Therefore, be the Power what you please, there will arise on one Side an algebraical Expression, on the other a geometrical Quantity, each of which naturally divides it self into three Members: The algebraical or fluxionary Expression, into one which includes neither the Expression of the Increment of the Absciss nor of any Power thereof, another which includes the Expression of the Increment it self, and the third including the Expression of the Powers of the Increment. The geometrical Quantity also or whole increased Area consists of three Parts or Members, the first of which is the given Area, the second a Rectangle under the Ordinate and the Increment of the Absciss, and the third a curvilinear Space. And, comparing the homologous or correspondent Members on both Sides, we find that as the first Member of the Expression is the Expression of the given Area, so the second Member of the Expression will express the Rectangle or second Member of the geometrical Quantity; and the third, containing the Powers of the Increment, will express the curvilinear Space, or third Member of the geometrical Quantity. This hint may, perhaps, be further extended and applied to good purpose, by those who have leisure and curiosity for such Matters. The use I make of it is to shew, that the Analysis cannot obtain in Augments or Differences, but it must also obtain in finite Quantities, be they ever so great, as was before observed.

XXX. 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 opposite Sides equal Quantities are subducted.
And 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 be
*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, besides Space and Time, or when the Times
are given, besides 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 and shorter Lines, and of Proportions between
such Lines generated in equal Parts of Time.

XXXI. 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 primae* and *ultimae*? 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?

XXXII. 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.

XXXIII. 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 conversant 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. And this would be true, even admitting that you understood your own
Principles and could demonstrate them.

XXXIV. 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

XXXV. 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 3*xx* 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 such
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 considered. [NOTE: *Sect*. 15.]
And there is indeed 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 the 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 in to 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. It must, indeed, be acknowledged, that 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?

XXXVI. 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

XXXVII. 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 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.

XXXVIII. 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.

XXXIX. 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.

XL. 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 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.

XLI. For the more distinct Conception of all which it may
be considered, that if the finite Increment *LM* [NOTE: *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 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.
[NOTE: *Sect*. 36.]

XLII. 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*.

XLIII. 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, 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*.

XLIV. Thus Fluxions may be considered 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, [NOTE: *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.

XLV. 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 [NOTE: 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?

XLVI. 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.

XLVII. 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 obtained 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 considered 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.

XLVIII. 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. And 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?

XLIX. 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. It should seem therefore upon the whole, that 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.

L. Of a long Time I have suspected, that these modern Analytics were not scientifical, and gave some Hints thereof to the Public about twenty five Years ago. Since which time, I have been diverted by other Occupations, and imagined I might employ my self better than in deducing and laying together my Thoughts on so nice a Subject. And though of late I have been called upon to make good my Suggestions; yet, as the Person, who made this Call, doth not appear to think maturely enough to understand, either those Metaphysics which he would refute, or Mathematics which he would patronize, I should have spared my self the trouble of writing for his Conviction. Nor should I now have troubled you or my self with this Address, after so long an Intermission of these Studies; were it not to prevent, so far as I am able, your imposing on your self and others in Matters of much higher Moment and Concern. And to the end that you may more clearly comprehend the Force and Design of the foregoing Remarks, and pursue them still further in your own Meditations, I shall subjoin the following Queries.

*Query 1.* Whether the Object of Geometry be not
the Proportions of assignable Extensions? And whether, there be any need of
considering Quantities either infinitely great or infinitely small?

*Qu. 2.* Whether the end of Geometry be not to
measure assignable finite Extension? And whether this practical View did not
first put Men on the study of Geometry?

*Qu. 3.* Whether the mistaking the Object and End
of Geometry hath not created needless Difficulties, and wrong Pursuits in that
Science?

*Qu. 4.* Whether Men may properly be said to
proceed in a scientific Method, without clearly conceiving the Object they are
conversant about, the End proposed, and the Method by which it is pursued?

*Qu. 5.* Whether it doth not suffice, that every
assignable number of Parts may be contained in some assignable Magnitude? And
whether it be not unnecessary, as well as absurd, to suppose that finite
Extension is infinitely divisible?

*Qu. 6.* Whether the Diagrams in a Geometrical
Demonstration are not to be considered, as Signs of all possible finite Figures,
of all sensible and imaginable Extensions or Magnitudes of the same kind?

*Qu. 7.* Whether it be possible to free Geometry
from insuperable Difficulties and Absurdities, so long as either the abstract
general Idea of Extension, or absolute external Extension be supposed its true
Object?

*Qu. 8.* Whether the Notions of absolute Time,
absolute Place, and absolute Motion be not most abstractedly Metaphysical?
Whether it be possible for us to measure, compute, or know them?

*Qu. 9.* Whether Mathematicians do not engage
themselves in Disputes and Paradoxes, concerning what they neither do nor can
conceive? And whether the Doctrine of Forces be not a sufficient Proof of this?
[NOTE: See a *Latin* treatise, *De Motu*, published at
*London*, in the year 1721.]

*Qu. 10.* Whether in Geometry it may not suffice
to consider assignable finite Magnitude, without concerning our selves with
Infinity? And whether it would not be righter to measure large Polygons having
finite Sides, instead of Curves, than to suppose Curves are Polygons of
infinitesimal Sides, a Supposition neither true nor conceivable?

*Qu. 11.* Whether many Points, which are not
readily assented to, are not nevertheless true? And whether those in the two
following Queries may not be of that Number?

*Qu. 12.* Whether it be possible, that we should
have had an Idea or Notion of Extension prior to Motion? Or whether if a Man had
never perceived Motion, he would ever have known or conceived one thing to be
distant from another?

*Qu. 13.* Whether Geometrical Quantity hath
coexistent Parts? And whether all Quantity be not in a flux as well as Time and
Motion?

*Qu. 14.* Whether Extension can be supposed an
Attribute of a Being immutable and eternal?

*Qu. 15.* Whether to decline examining the
Principles, and unravelling the Methods used in Mathematics, would not shew a
bigotry in Mathematicians?

*Qu. 16.* Whether certain Maxims do not pass
current among Analysts, which are shocking to good Sense? And whether the common
Assumption that a finite Quantity divided by nothing is infinite be not of this
Number?

*Qu. 17.* Whether the considering Geometrical
Diagrams absolutely or in themselves, rather than as Representatives of all
assignable Magnitudes or Figures of the same kind, be not a principal Cause of
the supposing finite Extension infinitely divisible; and of all the Difficulties
and Absurdities consequent thereupon?

*Qu. 18.* Whether from Geometrical Propositions
being general, and the Lines in Diagrams being therefore general Substitutes or
Representatives, it doth not follow that we may not limit or consider the number
of Parts, into which such particular Lines are divisible?

*Qu. 19.* When it is said or implied, that such a
certain Line delineated on Paper contains more than any assignable number of
Parts, whether any more in truth ought to be understood, than that it is a Sign
indifferently representing all finite Lines, be they ever so great. In which
relative Capacity it contains, *i. e.* stands for more than any
assignable number of Parts? And whether it be not altogether absurd to suppose a
finite Line, considered in it self or in its own positive Nature, should contain
an infinite number of Parts?

*Qu. 20.* Whether all Arguments for the infinite
Divisibility of finite Extension do not suppose and imply, either general
abstract Ideas or absolute external Extension to be the Object of Geometry? And,
therefore, whether, along with those Suppositions, such Arguments also do not
cease and vanish?

*Qu. 21.* Whether the supposed infinite
Divisibility of finite Extension hath not been a Snare to Mathematicians, and a
Thorn in their Sides? And whether a Quantity infinitely diminished and a
Quantity infinitely small are not the same thing?

*Qu. 22.* Whether it be necessary to consider
Velocities of nascent or evanescent Quantities, or Moments, or Infinitesimals?
And whether the introducing of Things so inconceivable be not a reproach to
Mathematics?

*Qu. 23.* Whether Inconsistencies can be Truths?
Whether Points repugnant and absurd are to be admitted upon any Subject, or in
any Science? And whether the use of Infinites ought to be allowed, as a
sufficient Pretext and Apology, for the admitting of such Points in Geometry?

*Qu. 24.* Whether a Quantity be not properly said
to be known, when we know its Proportion to given Quantities? And whether this
Proportion can be known, but by Expressions or Exponents, either Geometrical,
Algebraical, or Arithmetical? And whether Expressions in Lines or Species can be
useful but so far forth as they are reducible to Numbers?

*Qu. 25.* Whether the finding out proper
Expressions or Notations of Quantity be not the most general Character and
Tendency of the Mathematics? And Arithmetical Operation that which limits and
defines their Use?

*Qu. 26.* Whether Mathematicians have sufficiently
considered the Analogy and Use of Signs? And how far the specific limited Nature
of things corresponds thereto?

*Qu. 27.* Whether because, in stating a general
Case of pure Algebra, we are at full liberty to make a Character denote, either
a positive or a negative Quantity, or nothing at all, we may therefore in a
geometrical Case, limited by Hypotheses and Reasonings from particular
Properties and Relations of Figures, claim the same Licence?

*Qu. 28.* Whether the Shifting of the Hypothesis,
or (as we may call it) the *fallacia Suppositionis* be not a Sophism,
that far and wide infects the modern Reasonings, both in the mechanical
Philosophy and in the abstruse and fine Geometry?

*Qu. 29.* Whether we can form an Idea or Notion of
Velocity distinct from and exclusive of its Measures, as we can of Heat distinct
from and exclusive of the Degrees on the Thermometer, by which it is measured?
And whether this be not supposed in the Reasonings of modern Analysts?

*Qu. 30.* Whether Motion can be conceived in a
Point of Space? And if Motion cannot, whether Velocity can? And if not, whether
a first or last Velocity can be conceived in a mere Limit, either initial or
final, of the described Space?

*Qu. 31.* Where there are no Increments, whether
there can be any *Ratio* of Increments? Whether Nothings can be
considered as proportional to real Quantities? Or whether to talk of their
Proportions be not to talk Nonsense? Also in what Sense we are to understand the
Proportion of a Surface to a Line, of an Area to an Ordinate? And whether
Species or Numbers, though properly expressing Quantities which are not
homogeneous, may yet be said to express their Proportion to each other?

*Qu. 32.* Whether if all assignable Circles may be
squared, the Circle is not, to all intents and purposes, squared as well as the
Parabola? Of whether a parabolical Area can in fact be measured more accurately
than a Circular?

*Qu. 33.* Whether it would not be righter to
approximate fairly, than to endeavour at Accuracy by Sophisms?

*Qu. 34.* Whether it would not be more decent to
proceed by Trials and Inductions, than to pretend to demonstrate by false
Principles?

*Qu. 35.* Whether there be not a way of arriving
at Truth, although the Principles are not scientific, nor the Reasoning just?
And whether such a way ought to be called a Knack or a Science?

*Qu. 36.* Whether there can be Science of the
Conclusion, where there is not Evidence of the Principles? And whether a Man can
have Evidence of the Principles, without understanding them? And therefore,
whether the Mathematicians of the present Age act like Men of Science, in taking
so much more pains to apply their Principles, than to understand them?

*Qu. 37.* Whether the greatest Genius wrestling
with false Principles may not be foiled? And whether accurate Quadratures can be
obtained without new *Postulata* or Assumptions? And if not, whether
those which are intelligible and consistent ought not to be preferred to the
contrary? *See* Sect. XXVIII *and* XXIX.

*Qu. 38.* Whether tedious Calculations in Algebra
and Fluxions be the likeliest Method to improve the Mind? And whether Mens being
accustomed to reason altogether about Mathematical Signs and Figures, doth not
make them at a loss how to reason without them?

*Qu. 39.* Whether, whatever readiness Analysts
acquire in stating a Problem, or finding apt Expressions for Mathematical
Quantities, the same doth necessarily infer a proportionable ability in
conceiving and expressing other Matters?

*Qu. 40.* Whether it be not a general Case or
Rule, that one and the same Coefficient dividing equal Products gives equal
Quotients? And yet whether such Coefficient can be interpreted by *o* or
nothing? Or whether any one will say, that if the Equation 2 × *o* = 5 ×
*o*, be divided by *o*, the Quotients on both Sides are equal? Whether
therefore a Case may not be general with respect to all Quantities, and yet not
extend to Nothings, or include the Case of Nothing? And whether the bringing
Nothing under the notion of Quantity may not have betrayed Men into false
Reasoning?

*Qu. 41.* Whether in the most general Reasonings
about Equalities and Proportions, Men may not demonstrate as well as in
Geometry? Whether in such Demonstrations, they are not obliged to the same
strict Reasoning as in Geometry? And whether such their Reasonings are not
deduced from the same Axioms with those in Geometry? Whether therefore Algebra
be not as truly a Science as Geometry?

*Qu. 42.* Whether Men may not reason in Species as
well as in Words? Whether the same Rules of Logic do not obtain in both Cases?
And whether we have not a right to expect and demand the same Evidence in both?

*Qu. 43.* Whether an Algebraist, Fluxionist,
Geometrician, or Demonstrator of any kind can expect indulgence for obscure
Principles or incorrect Reasonings? And whether an Algebraical Note or Species
can at the end of a Process be interpreted in a Sense, which could not have been
substituted for it at the beginning? Or whether any particular Supposition can
come under a general Case which doth not consist with the reasoning thereof?

*Qu. 44.* Whether the Difference between a mere
Computer and a Man of Science be not, that the one computes on Principles
clearly conceived, and by Rules evidently demonstrated, whereas the other doth
not?

*Qu. 45.* Whether, although Geometry be a Science,
and Algebra allowed to be a Science, and the Analytical a most excellent Method,
in the Application nevertheless of the Analysis to Geometry, Men may not have
admitted false Principles and wrong Methods of Reasoning?

*Qu. 46.* Whether, although Algebraical Reasonings
are admitted to be ever so just, when confined to Signs or Species as general
Representatives of Quantity, you may not nevertheless fall into Error, if, when
you limit them to stand for particular things, you do not limit your self to
reason consistently with the Nature of such particular things? And whether such
Error ought to be imputed to pure Algebra?

*Qu. 47.* Whether the View of modern
Mathematicians doth not rather seem to be the coming at an Expression by
Artifice, than at the coming at Science by Demonstration?

*Qu. 48.* Whether there may not be sound
Metaphysics as well as unsound? Sound as well as unsound Logic? And whether the
modern Analytics may not be brought under one of these Denominations, and which?

*Qu. 49.* Whether there be not really a
*Philosophia prima*, a certain transcendental Science superior to and
more extensive than Mathematics, which it might behove our modern Analysts
rather to learn than despise?

*Qu. 50.* Whether ever since the recovery of
Mathematical Learning, there have not been perpetual Disputes and Controversies
among the Mathematicians? And whether this doth not disparage the Evidence of
their Methods?

*Qu. 51.* Whether any thing but Metaphysics and
Logic can open the Eyes of Mathematicians and extricate them out of their
Difficulties?

*Qu. 52.* Whether upon the received Principles a
Quantity can by any Division or Subdivision, though carried ever so far, be
reduced to nothing?

*Qu. 53.* Whether if the end of Geometry be
Practice, and this Practice be Measuring, and we measure only assignable
Extensions, it will not follow that unlimited Approximations completely answer
the Intention of Geometry?

*Qu. 54.* Whether the same things which are now
done by Infinites may not be done by finite Quantities? And whether this would
not be a great Relief to the Imaginations and Understandings of Mathematical
Men?

*Qu. 55.* Whether those Philomathematical
Physicians, Anatomists, and Dealers in the Animal Oeconomy, who admit the
Doctrine of Fluxions with an implicit Faith, can with a good grace insult other
Men for believing what they do not comprehend?

*Qu. 56.* Whether the Corpuscularian,
Experimental, and Mathematical Philosophy so much cultivated in the last Age,
hath not too much engrossed Mens Attention; some part whereof it might have
usefully employed?

*Qu. 57.* Whether from this, and other concurring
Causes, the Minds of speculative Men have not been borne downward, to the
debasing and stupifying of the higher Faculties? And whether we may not hence
account for that prevailing Narrowness and Bigotry among many who pass for Men
of Science, their Incapacity for things Moral, Intellectual, or Theological,
their Proneness to measure all Truths by Sense and Experience of animal Life?

*Qu. 58.* Whether it be really an Effect of
Thinking, that the same Men admire the great Author for his Fluxions, and deride
him for his Religion?

*Qu. 59.* If certain Philosophical Virtuosi of the
present Age have no Religion, whether it can be said to be for want of Faith?

*Qu. 60.* Whether it be not a juster way of
reasoning, to recommend Points of Faith from their Effects, than to demonstrate
Mathematical Principles by their Conclusions?

*Qu. 61.* Whether it be not less exceptionable to
admit Points above Reason than contrary to Reason?

*Qu. 62.* Whether Mysteries may not with better
right be allowed of in Divine Faith, than in Humane Science?

*Qu. 63.* Whether such Mathematicians as cry out
against Mysteries, have ever examined their own Principles?

*Qu. 64.* Whether Mathematicians, who are so
delicate in religious Points, are strictly scrupulous in their own Science?
Whether they do not submit to Authority, take things upon Trust, and believe
Points inconceivable? Whether they have not their Mysteries, and what is more,
their Repugnancies and Contradictions?

*Qu. 65.* Whether it might not become Men, who are
puzzled and perplexed about their own Principles, to judge warily, candidly, and
modestly concerning other Matters?

*Qu. 66.* Whether the modern Analytics do not
furnish a strong *argumentum ad hominem* against the Philomathematical
Infidels of these Times?

*Qu. 67.* Whether it follows from the
abovementioned Remarks, that accurate and just Reasoning is the peculiar
Character of the present Age? And whether the modern Growth of Infidelity can be
ascribed to a Distinction so truly valuable?

FINIS.

Links:

D.R. Wilkins(

School of Mathematics

Trinity College, Dublin