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 [Page 4] 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 upon 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 [Page 5] 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 an 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 ^* by the motion of Points, Plains by the motion of Lines, and Solids by the motion of Plains. 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. [Page 7] 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 [Page 8] 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 [Page 9] 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 [Page 10] to be infinitely small. The Diffe­rence of a Line is an infinitely little Line; of a Plain an infinitely little Plain. They suppose finite Quantities to consist of Parts infinitely little, and Curves to be Poly­gones, 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 than 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, [Page 11] 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 [Page 12] 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 x ^.. x ^... x ^⸫. x ^⸬. [...] 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. [Page 14] But if we remove the Veil and look under­neath, if laying aside the Expressions we set our selves 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 [Page 15] 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. ^* 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 [...] i. e. AB − ½ aB − ½ bA + ¼ ab. And as soon as the Sides A and B are increased by the other two halves of their Moments, the Rectangle becomes [...] or AB + ½ aB + ½ bA + ¼ ab. From the latter Rectangle subduct the former, and the remaining diffe­rence will be aB + bA. Therefore the Increment of the Rectangle generated by the intire Increments a and b is aB + bA. [Page 16] Q. E. D. But it is plain that the direct and true Method to obtain the Moment of Increment of the Rectangle AB, is to take the Sides as increased by their whole In­crements, and so multiply them together, A + a by B + b, the Product whereof AB + aB + bA + ab is the augmented Rectangle; whence if we subduct AB, the Remainder aB + bA + ab will be the true Increment of the Rectangle, exceeding that which was obtained by the former illegitimate and indirect Method by the Quantity ab. And this holds universally be the Quantities a and b what they will, big or little, Finite or Infinitesimal, Incre­ments, Moments, or Velocities. Nor will it avail to say that ab is a Quantity ex­ceeding small: Since we are told that in re­bus mathematicis errores quàm minimi non sunt contemnendi. ^* 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 [Page 17] 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 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 expresly 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 [Page 19] 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 ^*. 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 [Page 20] 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 x ⁿ + nox ⁿ−1 + [...] oox ⁿ−2 + &c. and the Incre­ments o and nox ⁿ−1 + [...] oox ⁿ−2 + &c. are one to another as 1 to nx ⁿ−1 + [...] ox ⁿ−2 + &c.. Let now the In­crements vanish, and their last Proportion will be 1 to nx ⁿ−1. But it should seem [Page 21] that this reasoning is not fair or conclusive. For when it is said, let the Increments vanish, i. e. let the Increments be nothing, or let there be no Increments, the former Supposition that the Increments were something, or that there were Increments, is destroyed, and yet a Consequence of that Supposition, i. e. an Expression got by virtue thereof, is retained. Which, by the foregoing Lemma, is a false way of reasoning. Certainly when we suppose the Increments to vanish, we must sup­pose their Proportions, their Expressions, and every thing else derived from the Sup­position of their Existence to vanish with them.

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. [Page 22] Therefore as x becomes x + o, x ⁿ will become [...]: that is, accord­ing to the Method of infinite Series, x ⁿ + nox ⁿ−1 + [...] oox ⁿ−2 + &c. And if from the two augmented Quantities we subduct the Root and the Power respec­tively, we shall have remaining the two Increments, to wit, o and nox ⁿ−1 + [...] oox ⁿ−2 + &c. which Increments, being both divided by the common Divi­sor o, yield the Quotients 1 and nx ⁿ−1 + [...] ox ⁿ−2 + &c. which are there­fore Exponents of the Ratio of the Incre­ments. Hitherto I have supposed that x flows, that x hath a real Increment, that o is something. And I have proceeded all along on that Supposition, without which I should not have been able to have made so much as one single Step. From that Supposition it is that I get at the Incre­ment of x ⁿ, that I am able to compare it with the Increment of x, and that I find the Proportion between the two In­crements. I now beg leave to make a new Supposition contrary to the first, i. e. I will suppose that there is no Increment [Page 23] of x, or that o is nothing; which second Supposition destroys my first, and is in­consistent with it, and therefore with eve­ry thing that supposeth it. I do never­theless beg leave to retain nx ⁿ−1, which is an Expression obtained in virtue of my first Supposition, which necessarily pre­supposeth such Supposition, and which could not be obtained without it: All which seems a most inconsistent way of arguing, and such as would not be allow­ed of in Divinity.

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 [Page 24] 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 [Page 25] 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 [Page 26] 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 ^*. 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 [Page 27] 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 ^*. 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 [Page 28] 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 ^*. 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 [Page 29] 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. [Page 30] 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 the 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 coversant with, and whether you conceive them clearly? [Page 31] 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 AB be a Curve, the Abscisse AP = x, the ordinate PB = y, the Difference of the Abscisse PM = dx, the Difference of the Ordinate RN = dy. Now by supposing the Curve to be a Polygon, and consequent­ly BN, the Increment or Difference of the Curve, to be a straight Line coincident [Page 33] with the Tangent, and the differential Triangle BRN to be similiar to the tri­angle TPB the Subtangent PT is found a fourth Proportional to RN ∶ RB ∶ PB: that is to dy ∶ dx ∶ y. Hence the Subtangent will be [...]. But herein there is an error arising from the forementioned false sup­position, whence the value of PT comes out greater than the Truth: for in reality it is not the Triangle RNB but RLB, which is similar to PBT, and therefore (in­stead of RN) RL should have been the first term of the Proportion, i. e. RN + NL, i. e. dy + z: whence the true expression for the Subtangent should have been [...]. There was therefore an error of defect in making dy the divisor: which error was equal to z, i. e. NL the Line comprehend­ed between the Curve and the Tangent. Now by the nature of the Curve yy = px, supposing p to be the Parameter, whence by the rule of Differences 2 ydy = pdx and dy = [...]. But if you multiply y + dy by it self, and retain the whole Product without rejecting the Square of the Diffe­rence, [Page 34] it will then come out, by substitu­ting the augmented Quantities in the E­quation of the Curve, that dy = [...] truly. There was therefore an error of excess in making dy = [...], which followed from the erroneous Rule of Differences. And the measure of this second error is [...] = z. Therefore the two errors being equal and contrary destroy each other; the first er­ror of desect 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 BR or dx be m and RN or dy be n. By the thirty third Proposition of the first Book of the Conics of Apollonius, and from similar [Page 35] Triangles, as 2 x to y so is m to n + z = [...]. Likewise from the Nature of the Parabola yy + 2 yn + nn = xp + mp, and 2 yn + nn = mp: wherefore [...] = m: and because yy = px, [...] will be equal to x. Therefore substituting these values instead of m and x we shall have n + z = [...]: i. e. n + z = [...]: which being reduced gives z = [...] Q. E. D.

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 ever 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, [Page 36] 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 MQ cuts the Curve AT in the Points R and S. Suppose LR a Tangent at the Point R, AN the Abscisse, NR and OS Ordinates. Let AN be produced to O, and RP be drawn parallel to NO. Suppose AN = x, NR = y, NO = v, PS = z, the subsecant MN = S. Let the Equation y = xx express the nature of the Curve: and supposing y and x increased by their finite Increments, we get y + z = xx + 2 xv + vv: whence the former [Page 38] Equation being subducted there remains z = 2 xv + vv. And by reason of similar Triangles PS ∶ PR ∷ NR ∶ NM, i. e. z ∶ v ∷ y ∶ s = [...], wherein if for y and z we substitute their values, we get [...] = s = [...]. And supposing NO to be infinitely diminished, the subsecant NM will in that case coincide with the subtan­gent NL, and v as an Infinitesimal may be rejected, whence it follows that S = NL = [...]; which is the true va­lue of the Subtangent. And since this was obtained by one only error, i. e. by once rejecting one only Infinitesimal, it should seem, contrary to what hath been said, that an infinitesimal Quantity or Difference may be neglected or thrown away, and the Conclusion nevertheless be accurately true, although there was no double mistake or rectifying of one error by another, as in the first Case. But if this Point be through­ly considered, we shall find there is even here a double mistake, and that one com­pensates or rectifies the other. For in the [Page 39] first place, it was supposed, that when NO is infinitely diminished or becomes an Infinitesimal, then the Subsecant NM be­comes equal to the Subtangent NL. But this is a plain mistake, for it is evident, that as a Secant cannot be a Tangent, so a Subsecant cannot be a Subtangent. Be the Difference ever so small, yet still there is a Difference. And if NO be infinitely small, there will even then be an infinitely small Difference between NM and NL. There­fore NM or S was too little for your sup­position, (when you supposed it equal to NL) and this error was compensated by a second error in throwing out v, which last error made s bigger than its true va­lue, and in lieu thereof gave the value of the Subtangent. This is the true State of the Case, however it may be disguised. And to this in reality it amounts, and is at bottom the same thing, if we should pretend to find the Subtangent by hav­ing first found, from the Equation of the Curve and similar Triangles, a ge­neral Expression for all Subsecants, and then reducing the Subtangent under this general Rule, by considering it as the [Page 40] Subsecant when v vanishes or becomes nothing.

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 or NO ^* 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 [Page 41] 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, [Page 42] as a Knack, an Art or 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 AB = x,

BC = y, BD = o, and that xx is equal to the Area ABC: It is proposed to find the Ordinate y or BC. When x by flowing becomes x + o, then xx becomes xx + 2 xo + oo: And the Area ABC becomes ADH, and the Increment of xx will be equal to BDHC the Increment of the [Page 43] Area, i. e. to BCFD + CFH. And if we suppose the curvilinear Space CFH to be qoo, then 2 xo + oo = yo + qoo which divided by o gives 2 x + o = y + qo. And, supposing o to vanish, 2 x = y, in which Case ACH will be a straight Line, and the Areas ABC, CFH, Triangles. Now with regard to this Reasoning, it hath been already remarked ^*, that it is not le­gitimate or logical to suppose o to vanish, i. e. to be nothing, i. e. that there is no Increment, unless we reject at the same time with the Increment it self every Con­sequence of such Increment, i. e. what­soever could not be obtained but by sup­posing such Increment. It must never­theless be acknowledged, that the Problem is rightly solved, and the Conclusion true, to which we are led by this Method. It will therefore be asked, how comes it to pass that the throwing out o is attended with no Error in the Conclusion? I an­swer, the true reason hereof is plainly this: Because q being Unite, qo is equal to o: And therefore 2 x + o − qo = y = 2 x, [Page 44] the equal Quantities qo and o being de­stroyed by contrary Signs.

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 [Page 45] 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 ^* 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 [Page 46] and illogical. But if we fairly de­lineate the Area and its Increment, and divide the latter into two Parts BCFD and CFH ^*, 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, nox ⁿ−1 + [...] oox ⁿ−2 + &c. = BDFC + CFH. And only the first Members, on each Side of the Equation being retained, nox ⁿ−1 = BDFC: And dividing both Sides by o or BD, we shall get nx ⁿ−1 = BC. Admitting, therefore, that the curvilinear Space CFH is equal to the rejectaneous Quantity [...] oox ⁿ−2 + &c.. 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 BD, whether that of an infinitesimal Difference or a finite Increment ever so great. It is there­fore plain, that the supposing the rejectaneous [Page 47] algebraical Quantity to be an infinitely small or evanescent Quantity, and there­fore to be neglected, must have produced an Error, had it not been for the curvi­linear Spaces being equal thereto, and at the same time subducted from the other Part or Side of the Equation agreeably to the Axiom, If from Equals you subduct Equals, the Remainders will be equal. For those Quantities which by the Analysts are said to be neglected, or made to vanish, are in reality subducted. If therefore the Conclusion be true, it is absolutely neces­sary that the finite Space CFH be equal to the Remainder of the Increment expressed by [...] oox ⁿ−2 &c. 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 [Page 48] of the Increment of the Absciss nor of any Power thereof, another which includes the Expression of the Increment it self, and a 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, [Page 49] 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 the 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 [Page 50] 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.

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 [Page 51] 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 [Page 52] 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 [Page 53] 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 [Page 54] 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 [Page 55] 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. It is evident these three 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, [Page 56] 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: And its evanescent Sides CE, Ec, and Cc will be porpor­tional to CE, ET, and CT the Sides of the Triangle CET. And 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. ^* It it 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 [Page 57] 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.

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 [Page 58] 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 consi­dered ^*. 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 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. 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 [Page 59] 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 [Page 60] 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 [Page 61] on: that a is the Velocity of LM in statu nascenti, and b, c, d, e, &c. are the Velocities of the Increments MN, NO, OP, &c. in their respective nascent estates. You may proceed, and consider these Ve­locities themselves as flowing or increasing Quantities, taking the Velocities of the Velocities, and the Velocities of the Ve­locities of the Velocities, i. e. the first, second, third, &c. Velocities ad infinitum: which succeeding Series of Velocities may be thus expressed. a. b − a. c − 2 b + a. d − 3 c + 3 b − a &c. which you may call by the names of 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 x ^., the second by x ^.., the third by x ^⸫, and so on ad infini­tum.

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 [Page 62] 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, [Page 63] 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 [Page 64] 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. By 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. [Page 65] 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 ^* be divided into the Isochronal Parts Lm, mn, no, oM; and the Increment MN into the Parts Mp, pq, qr, rN Isochronal to the for­mer; as the whole Increments LM, MN are proportional to the Sums of their de­scribing Velocities, even so the homolo­gous Particles Lm, Mp are also propor­tional to the respective accelerated Veloci­ties with which they are described. And as the Velocity with which Mp is gene­rated, exceeds that with which Lm was generated, even so the Particle Mp ex­ceeds [Page 66] 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 ^*.

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, [Page 67] 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 [Page 68] 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. x ^.. x ^... x ^⸫. x ^⸬. 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 ^†, 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 [Page 69] 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 z″. z′. z. z ^.. z ^⸬. z ^⸫. ^* 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, [Page 70] 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 Abfciss 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 [Page 72] Science. Thus the direct Method precedes the inverse, and the knowledge of the Principles is supposed in both. But as for operating according to Rules, and by the help of general Forms, whereof the ori­ginal Principles and Reasons are not un­derstood, this is to be esteemed merely technical. Be the Principles therefore ever so abstruse and metaphysical, they must be studied by whoever would comprehend the Doctrine of Fluxions. Nor can any Geometrician have a right to apply the Rules of the great Author, without first considering his metaphysical Notions whence they were derived. These how necessary soever in order to Science, which can never be attained without a precise, clear, and accurate Conception of the Principles, are nevertheless by several carelesly passed over; while the Expres­sions alone are dwelt on and 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 [Page 73] 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 [Page 75] 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 [Page 76] 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?

[Page 77] 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, [Page 78] concerning what they neither do nor can conceive? And whether the Doc­trine of Forces be not a sufficient Proof of this? ^*

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?

[Page 79] 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 [Page 80] 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 parti­ticular 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 [Page 81] 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 [Page 82] 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?

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 [Page 83] 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 [Page 84] 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? Or whether a pa­rabolical Area can in fact be measured more accurately than a Circular?

[Page 85] 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 Science of the Principles? And whether a Man can have Science 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?

[Page 86] 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 likliest 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 [Page 87] 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 [Page 88] 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 [Page 89] 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 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?

[Page 90] 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 [Page 91] Extensions, it will not follow that un­limited Approximations compleatly an­swer the Intention of Geometry?

Qu. 54. Whether the same things which are now done by Infinities 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?

[Page 92] Qu. 57. Whether from this, and other concurring Causes, the Minds of specula­tive Men have not been born 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?

[Page 93] 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, 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?

[Page 94] 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?