A VINDICATION, &c.

UNDER Pretence of ſome A⯑buſes committed by Mathe⯑maticians, in Virtue of the Au⯑thority they derive from their Profeſſion, the Author of the Minute Philoſopher, in a Libel called the Ana⯑lyſt, has declared 'em Infidels, Makers of Infidels, and Seducers of Mankind in Matters of the higheſt Concernment: This he profeſſes to have done, not from any real Knowlege of his own, but from the credible Information of others; but he has neither produc'd his Informers, nor proved the Accuſation in any one Inſtance; and therefore it is Defamatory.

[4] But they aſſume an Authority, it ſeems, in Things foreign to their Profeſſion, and un⯑dertake to decide in Matters whereof their Knowledge can by no Means qualify them to be competent Judges: And as this Prac⯑tice, if not prevented, may be of dan⯑gerous Conſequence; he has undertaken to enquire into the Object, Principles and Method of Demonſtration, admitted by the Mathematicians of the preſent Age, with the ſame Freedom, he ſays, they pre⯑ſume to treat the Principles and Myſte⯑ries of Religion, to the end, that all Men may ſee what Right they have to lead or what Encouragement others have to fol⯑low them.

And whereas Sir Iſaac Newton has pre⯑ſum'd to interpoſe in Prophecies and Re⯑velations, and to decide in religious Af⯑fairs, it has been thought proper to begin with his Method of Fluxions, and to try what cou'd be done with that Me⯑thod, with the Inventor himſelf, and with his Followers: And what has been done with 'em every intelligent Reader is able to judge.

[5] If this Writer may be credited, the Ob⯑jects about which the Method of Fluxi⯑ons is converſant, are difficult to conceive or imagine diſtinctly; the Notions are moſt abſtracted incomprehenſible Meta⯑phyſics, not to be admitted for the Foun⯑dations of clear and accurate Science; the Principles are obſcure, repugnant, pre⯑carious; the Arguments admitted in Proofs, are fallacious, indirect, illogical; and the Inferences and Concluſions not more juſt, than the Conceptions of the Principles are clear.

How far the Credulous and Injudicious may become infected by this uncommon Way of treating Mathematics and Mathe⯑maticians, is not eaſy to foreſee, and there⯑fore it will be neceſſary to give a ſhort Ac⯑count of the Nature of Fluxions, and of the Objects about which the Method is converſant; and when it ſhall be made apparent, that this Author has not under⯑ſtood the Metaphyſics he wou'd refute; it will not be difficult to defend the Prin⯑ciples and their Demonſtrations, from any Imputations of Fallacy or Repugnancy, [6] which have yet been pointed at by him or any other Writer.

‘In the Method of Fluxions, Sir Iſaac Newton conſiders mathematical Quantities, not as compoſed of the ſmalleſt Parts, but as deſcribed or ge⯑nerated by continual Motion. Lines are deſcribed, and by being deſcribed are generated, not by an Appoſition of Parts, but by the Motion of Points; Surfaces by the Motion of Lines, So⯑lids by the Motion of Surfaces, Angles by the Rotation of their Sides, Times by a continual Flux, and ſo of the reſt. And by conſidering that Quantities, in⯑creaſing in equal Times, and generated by increaſing, become greater or leſs, according as the Velocity with which they increaſe, and are generated, is greater or leſs, he found a Method of deter⯑mining the Quantities themſelves from the Velocities of the Motions, or of the Increments, with which they are generated; calling the Velocities of the Motions or of the Increments Fluxions, and the Quantities generated Fluents.’

[7] The momentaneous Increments or De⯑crements of flowing Quantities, he elſe⯑where calls by the Name of Moments, and conſiders the Increments as added or affirmative Moments, and the Decrements as ſubducted or negative ones: By Mo⯑ments we may underſtand the naſcent or evaneſcent Elements or Principles of finite Magnitudes, but not Particles of any de⯑terminate Size, or Increments actually generated; for all ſuch are Quantities themſelves, generated of Moments.

The Magnitudes of the momentaneous Increments or Decrements of Quantities are not regarded in the Method of Fluxions, but their firſt or laſt Proportions only; that is, the Proportions with which they begin or ceaſe to exiſt: Theſe are not their Proportions immediately before or after they begin or ceaſe to exiſt, but the Proportions with which they begin to exiſt, or with which they vaniſh. If the Lines AC and BE are ſuppoſed to be ge⯑nerated in the ſame Time, by the Mo⯑tions of the Points A and B, to C and E; and if by continuing the Motions of [8] thoſe Points to D and F, they generate DC and EF, ſynchronal Increments of AC and BE;

it is evident that the Points D and F may flow back in the ſame Time to C and E, and by flowing back perpetual⯑ly leſſen the Magnitudes of thoſe Increments till at laſt they vaniſh together, when the Points D and F come to coincide with C and E: Now the ultimate Ratio of thoſe Increments is that Ratio with which they vaniſh and become nothing; or the Ratio with which they ceaſe to be: And the firſt Ratio of them is the Ratio with which they begin to exiſt, at the very firſt ſetting out of the Points from C and E towards D and F.

Hence, if the deſcribing Points move back to C and E, in the ſame Time wherein by moving forward they gene⯑rated the Increments DC and EF; and in returning have every where the ſame Velocities, at certain Diſtances from C and E, which they had at thoſe Diſtances [9] in going forward; the laſt and firſt Ratios of the Increments will be equal, or they will vaniſh, and become nothing, with the very ſame Ratio with which they began to exiſt.

Hence likewiſe it appears, that to ob⯑tain the laſt Ratio of ſynchronal Incre⯑ments, the Magnitudes of thoſe Incre⯑crements muſt be infinitely diminiſh'd. For their laſt Ratio is the Ratio with which they vaniſh or become nothing: But they cannot vaniſh or become nothing, by a conſtant Diminution, till they are infinite⯑ly diminiſh'd; for without an infinite Diminution they muſt have finite or aſ⯑ſignable Magnitudes, and while they have finite or aſſignable Magnitudes they cannot vaniſh.

The ultimate Ratios with which ſyn⯑chronal Increments of Quantities vaniſh, are not the Ratios of finite Increments, but Limits which the Ratios of thoſe In⯑crements attain, by having their Magni⯑tudes infinitely diminiſh'd: The Proporti⯑ons of Quantities which grow leſs and [10] leſs by Motion, and at laſt ceaſe to be, will continually change, and become dif⯑ferent in every ſucceſſive Diminution of the Quantities themſelves: And there are certain determinate Limits to which all ſuch Proportions perpetually tend, and approach nearer than by any aſſignable Difference, but never attain before the Quantities themſelves are infinitely di⯑miniſh'd; or till the inſtant they evaneſce and become nothing. Theſe Limits are the laſt Ratios with which ſuch Quan⯑tities or their Increments vaniſh or ceaſe to exiſt; and they are the firſt Ratios with which Quantities or the Increments of Quantities, begin to ariſe or come into being.

Quantities, and the Ratios of Quanti⯑ties, which conſtantly tend to an Equali⯑ty, by a Diminution of their Difference, and before the End of ſome finite Time approach nearer to an Equality than by any aſſignable Difference, at laſt become equal. For they become equal when the Difference between them vaniſhes or becomes nothing; and it will [11] vaniſh or become nothing by being in⯑finitely diminiſhed: If the Quantities AC and AD perpetually tend to an Equality, either by the Motion of the Point D to C, or by that of C to D; they will be⯑come equal, and their Ratio a Ratio of Equality, when their Difference CD, by a conſtant Diminution, vaniſhes and be⯑comes nothing, which it will do under a Coincidence of the two Points in C or D; and then either AD becomes AC, and ſo AD / AC or AC / AC is a Ratio of Equa⯑lity, or elſe AC becomes AD and AD / AC becomes AD / AD; which is alſo a Ratio of Equality.

The Fluxions of Quantities are very nearly as the Increments of their Fluents generated in the leaſt equal Particles of Time: If CD and EF be Increments of the Fluents AC and BE, deſcribed in the leaſt equal Particles of Time; the Fluxions in the Points C and E will be nearly as the Increments DC and EF. For from the exceeding Smallneſs of the [12] Times it is evident that the Points D and F, muſt be extreamly near to C and E; and by Conſequence however the Velo⯑cities are accelerated or retarded thro' the Spaces CD and EF, they will be very nearly the ſame in D and F as they were in C and E: But Velocities which are ve⯑ry nearly uniform, will be very nearly proportional to the Spaces deſcribed by them in equal Times; and therefore the Velocities in the Points C and E, which are the Fluxions of AC and BE in thoſe Points, will be very nearly as the Incre⯑ments DC and EF, deſcribed in the leaſt equal Particles of Time.

The Fluxions of Quantities are accu⯑rately in the firſt or laſt Proportions of their naſcent or evaneſcent Increments: Thus the Fluxions of AC and BE, in the Points C and E, are in the firſt or laſt Ratio of the Increments CD and EF. For the firſt or laſt Ratio of the Incre⯑ments CD and EF, is the Ratio with which they begin or ceaſe to exiſt: But the Ratio with which they begin or ceaſe to exiſt, is the ſame with the Ratio of the [13] Velocities in C and E, which are the Fluxions in thoſe Points; and conſequent⯑ly the Fluxions in C and E are in the firſt or laſt Ratio of the Increments CD and EF.

The Fluxions of Quantities are only the Velocities with which thoſe Quanti⯑ties begin to be generated or increaſed; or the Velocities with which the gene⯑rating Quantities begin to ſet out; not the Velocities they have after moving thro' Spaces of any finite or aſſignable Mag⯑nitudes: And therefore if two mathema⯑tical Quantities ſet out together, and be⯑gin to move with Velocities which are as a and b, they muſt begin to deſcribe Spaces in the ſame Proportion with a and b; or the Proportion with which the Spaces begin to exiſt or to be deſcribed, muſt be the ſame with that which the Velocities have at the very Beginning of the Motion. For in the very Beginning of the Motion there is neither any Change of Velocity from Acceleration or Retar⯑dation, nor Difference of Time.

[14] Hence it appears that to obtain the Ratios of Fluxions, the correſponding ſynchronal or iſochronal Increments muſt be leſſened in infinitum. For the Mag⯑nitudes of ſynchronal or iſochronal In⯑crements muſt be infinitely diminiſhed and become evaneſcent, in order to obtain their firſt or laſt Ratios, to which Ratios the Ratios of their correſponding Fluxions are equal.

Hence likewiſe it appears that the Mo⯑ments of like Quantities, compared with each other, are in Ratios compounded of the Ratios of the generating Quantities, taken when firſt they begin to move, and of the Velocities with which they ſet out: Or in Ratios compounded of the Ratios of the generating Quantities when firſt they begin to move, and of the firſt Ratios of their ſynchronal naſcent Incre⯑ments. The Moments of Lines there⯑fore are as the generating Points and as the Velocities with which they begin to move taken together: The Moments of Surfaces, which become greater or leſs by carrying of moveable Lines along im⯑moveable [15] ones, are in Ratios compound⯑ed of the Ratios of the moving Lines, and of their firſt Velocities, or firſt Ra⯑tios of the Increments which begin to riſe with thoſe Velocities: And the whole Motion by which Squares or Rectangles begin to alter, either from an Augmenta⯑tion or Diminution of their Sides, is the Sum of the naſcent Motions of thoſe Sides, or the Sum of the naſcent Incre⯑ments ariſing with the firſt Motions of the Sides: For the Proportion of naſcent Increments is the ſame with that of the Motions with which they begin to be generated.

From this ſhort Account of the Na⯑ture of Fluxions, compared with the Ana⯑lyſt, it appears that the Author of that Paper is greatly miſtaken in the Object of 'em; and he is alſo miſtaken in the Principles: For he thinks the Moment or Fluxion of a Rectangle, contain'd un⯑der two indeterminate Quantities A and B, from whence are deduc'd Rules for obtaining the Moments or Fluxions of all other Products or Powers whatever, is [16] no where truly determin'd by Sir Iſaac Newton: But he ought to have read Sir Iſaac with more Care and Attention than he ſeems to have done, before he ſet up to decide and dictate in Matters of this Nature; and he wou'd do well yet to read him with Attention.

If any Rectangle CK be increaſed from an Augmentation of its Sides by Motion, ſo as that DK becomes LG in the ſame Time that DC becomes EG; the Moment of that Rectangle is the Sum of the Rectangles of DK into the Mo⯑ment of DC, and of DC into the Mo⯑ment of DK: That is, putting A and B for the Sides DK and DC, and a and b for their reſpective Moments, the Mo⯑ment of the Rectangle AB will be Ab + Ba.

For the Gnomon CGK in the Inſtant it begins or ceaſes to exiſt is the Mo⯑ment of the Rectangle CK: But the firſt or laſt Ratio of that Gnomon to the Sum of the Rectangles LD and FC is a Ratio of Equality: For the Difference [17] between the Gnomon and the Sum of thoſe Rectangles perpetually leſſens, by a conſtant Diminution of the Increments FD and DH, or by an Ap⯑proach of the Points F and H towards D;

as will be manifeſt on taking the Ratio between the ſaid Gnomon and the Sum of the Rectangles, at ſeveral Diſtances of the Points F and H from D: For what⯑ever be the Magnitudes of a and b, when F and H firſt begin to move back to⯑wards D, the Gnomon CGK and Sum of the Rectangles LD and FC, will be as Ab + Ba + ba and Ab + Ba; when thoſe Points, by moving towards D, have leſſen'd the Increments of DK and DC to ½a and ½b, the Gnomon and Sum of the Rectangles will be as Ab + Ba + ½ba and Ab + Ba; when they have leſſen'd the Increments to ¼a and ¼b, the Gno⯑mon and Sum of the Rectangles will be [18] as Ab + Ba + ¼ab and Ab + Ba; and as Ab + Ba + [...] ab and Ab + Ba, when they have leſſen'd thoſe Increments to [...] a and [...] b: Hence it appears, that un⯑der a conſtant Diminution of the Incre⯑ments a and b, by the Motion of the Points F and H towards D, the Gnomon CGK and the Sum of the Rectangles CF and DL, conſtantly tend to an Equali⯑ty by a continual Diminution of their Difference FH, and that they become equal, and their Ratio becomes a Ratio of Equality, in the Inſtant that Difference vaniſhes and the Points F and H coincide with D; or in other Words the Gnomon and Sum of the Rectangles LD and FC be⯑gin or ceaſe to be under a Ratio of Equali⯑ty: And therefore the Sum of thoſe Rect⯑angles, or Ab + Ba, is the Moment of AB.

Hence, the Gnomon CGK, or Ab + Ba + ab, found by taking the Difference between the Rectangles EL and CK, or by deducting the Rectangle AB from a Rectangle contain'd under the Sides A and B increaſed by their whole Incre⯑ments, is not the Moment or Fluxion [19] of the Rectangle AB, except in the ve⯑ry Inſtant when it begins or ceaſes to exiſt: And this will alſo appear by con⯑ſidering it in another Light. For the Moment of the Rectangle CK, or the Motion with which it firſt begins to al⯑ter, either by increaſing or decreaſing, is the Sum of the naſcent Motions of its Sides; and the naſcent Motions of its Sides, are meaſur'd by their reſpec⯑tive Magnitudes in the very Inſtant they firſt begin to change, and by the Velo⯑cities with which they begin to move taken together; and the Velocities with which the Sides begin to move being in the firſt Ratio of the momentaneous Spaces which ariſe with 'em; it follows that the Sum of the naſcent Motions of the Sides, is the Sum of DK multiply'd into DH in its naſcent State, and of CD multiply'd into DF in its naſcent State: But DH and DF in their naſcent States, are the Moments of DC and DK: And therefore the whole Moment of the Rectangle AB, is Ab + Ba.

[20] In determining the Moments of Quan⯑tities, Sir Iſaac Newton expreſly tells us, that we are only to conſider: the Ratios with which they begin or ceaſe to exiſt; and to obtain thoſe Ratios, it is not neceſſary that the inſochronal In⯑crements ſhou'd have finite Magnitudes. ‘Cave tamen intellexeris particulas finitas, ſays he, Particulae finitae non ſunt Momenta, ſed Quantitates ipſae ex Momentis genitae. Intelligenda ſunt Principia jamjam naſcentia finitarum Magnitudinum. Neque enim ſpectatur in hoc Lemmate mag⯑nitudo Momentorum, ſed prima na⯑ſcentium proportio.’ And in another Place, ‘Fluxiones ſunt quam proximè ut Fluentium Augmenta aequalibus Temporis particulis quam minimis genita, et, ut accurate loquar, ſunt in primâ ratione Augmentorum na⯑ſcentium; exponi autem poſſunt per lineas quaſcunque, quae ſunt ipſis proportionales.’ And again, ‘Siquan⯑do facili rerum conceptui conſulens dixero Quantitates quam minimas, vel evaneſcentes, vel ultimas; cave [21] intelligas quantitates magnitudine determinatas. ſed cogita ſemper di⯑minuendas ſine limite.’

From theſe Paſſages it appears, that the Gnomon CGK in its naſcent or evaneſcent State only, or in the Inſtant it begins or ceaſes to exiſt, is the Moment or Fluxion of the Rectangle CK; and in a naſcent or evaneſcent State, when on⯑ly the Increments of Quantities become their Moments, its Ratio to Ab + Ba, which is the Sum of the Rectangles LD and FC, is a Ratio of Equality. By di⯑miniſhing the Magnitudes of a and b, which are Increments of DK and DC, it is obvious that the Gnomon CGK di⯑miniſhes faſter in Proportion, than the Sum of the Rectangles FC and DL does; and by diminiſhing faſter, it continually approaches to an Equality with that Sum, and attains the Equality only, when their Difference FH becomes evaneſcent, that is, when the Points F and H come to coincide with D; ſo that here is no Ar⯑tifice or falſe Reaſoning uſed, to get rid of HF, or ab, that Term having no [22] Exiſtence at the very Beginning of the Motion, or in the naſcent State of the Augments.

After Sir Iſaac had ſo expreſly told us what he meant by Moments and Fluxions, and by naſcent or evaneſcent Quantities, one wou'd imagine it impoſ⯑ſible to have miſtaken and miſrepreſented him in the Manner this Author has done. He ſeems indeed to have been lead, or rather to have been deceived, by an Opi⯑nion that there can be no firſt or laſt Ratios of mathematical Quantities or of their iſochronal Increments generated or deſtroy'd by Motion; imagining that no ſuch Quantities, by any Diviſion or Di⯑minution whatever, can be exhauſted or reduc'd to nothing: But if Lines, Sur⯑faces and Solids can be generated or aug⯑mented by the Motion of Points, Lines, and Surfaces, they may likewiſe be de⯑ſtroy'd or diminiſh'd by the Motion of the ſame Points, Lines and Surfaces, in returning to the Places from whence they firſt ſet out. While a generating Quantity moves back thro' the ſame [23] Space it before deſcribed in moving for⯑ward, the Quantity generated, or its Augment, continually leſſens; and by perſevering in a State of decreaſing, it muſt in ſome finite Time vaniſh and be⯑come nothing; and therefore mathema⯑tical Quantities, by a conſtant Diminu⯑tion, may be reduc'd to nothing: And ſuch as are thus generated or deſtroy'd in equal Times by Motion, or which a⯑riſe and vaniſh together, will ariſe or va⯑niſh under certain Ratios, which are their firſt or laſt Ratios; or the Ratios with which they begin or ceaſe to be: But it may be neceſſary to perſue this Caſe a little farther, and ſee whether Sir Iſaac Newton's Demonſtration of it cannot be defended, and proved to be geometrical.

‘Suppoſe any Rectangle AB aug⯑mented by continual Motion; and the momentaneous Increments of its Sides A and B to be denoted by a and b; the Moment of the generated Rect⯑angle will be meaſured by Ab + Ba.’

‘[24] For when the Sides A and B want⯑ed half of their Moments, the Rect⯑angle was [...] or AB − ½Ab − ½Ba + ¼ab: And as ſoon as the Sides A and B are augmented by the other halves of their Moments, it be⯑comes [...], or AB + ½Ab + ½Ba + ¼ab: From this Rectangle deduct the former, and there will re⯑main Ab + Ba: Therefore the Incre⯑ment of the Rectangle AB, generated with a and b the whole Increments of the Sides, is Ab + Ba.’

Now, in determining the Moment of a Rectangle, there is nothing to be con⯑ſidered, when it firſt begins to be aug⯑mented by the Motions of its Sides, but the Sides themſelves and the Velocities with which they begin to move; or the Sides and the firſt Ratio of the Spaces deſcribed by them. And therefore the true Moment of the Rectangle AB, or the Law according to which it begins to be augmented, on the Principles of Sir Iſaac Newton, will only be the Sum of [25] the Rectangles Ab and Ba; for the Sides A and B begin to move with Ve⯑locities which are as b and a: But this Moment Ab + Ba, is manifeſtly equal to the Difference between the Rectangles [...] and [...]; and therefore Sir Iſaac's determination of it is geometrical.

From the foregoing Principle ſo de⯑monſtrated, the general Rule for finding the Moment or Fluxion of any Power of a flowing Quantity, is eaſily deduc'd: It is eaſy, from hence, to infer that the Moment or Fluxion of Aⁿ is as nAⁿ−1, or that the Fluxion of xⁿ is as nxⁿ−1: But becauſe this is alſo determined in a manner ſeemingly different, by Sir Iſaac, in his Introduction to the Quadrature of Curves, the Author of the Analyſt ob⯑ſerves, ‘That there ſeems to have been ſome inward Scruple or Conſciouſneſs of Defect in the foregoing Demon⯑ſtration.’ And he repeats the ſame Reflection in another Place, adding with⯑al, ‘That Sir Iſaac was not enough pleaſed with any one Notion ſteadily [26] to adhere to it:’ But Reflections of this Nature deſerve no Regard unleſs it be allowable, by way of Return, to ob⯑ſerve that the Perſon who makes 'em has very often been guilty of like Prac⯑tices himſelf.^*

The Proof given in the Introduction to the Quadratures, is ſaid to be a moſt inconſiſtent way of arguing; as pro⯑ceeding to a certain Point of the Demon⯑ſtration upon Suppoſition of an Incre⯑ment, and then in a fallacious Manner, ſhifting the Suppoſition to that of no In⯑crement; and to ſhew the Inconſiſteny with greater Force, a Lemma is pre⯑miſed by Way of Axiom; as if ſome ve⯑ry obvious and natural Application of an apparent Truth, wou'd at once over⯑turn the Whole of Sir Iſaac's Demon⯑ſtration: But that Lemma, however true in it ſelf, is no Way pertinent to the Caſe for which it was intended; and [27] therefore ſuch Inferences as are made in Virtue of it, with relation to the Point in diſpute, are illegitimate, and inconſiſtent with the Rules of true reaſoning.

Nothing is more plain and obvious, than that Quantities which begin to exiſt together under certain Proportions, and with certain Velocities; may become evaneſcent and ceaſe to exiſt, under the ſame Proportions and with the ſame Velo⯑cities; and this is all Sir Iſaac ſuppoſes in that Determination of the Fluxion of xⁿ; and it is not very obvious, that the Lemma which this Author has hit upon, is ap⯑plicable to Caſes of ſuch a Nature.

That the Reader may ſee how ſtrict⯑ly Sir Iſaac Newton has kept to the ſame Principle in this Determination, how ſteadily he adheres to the ſame Method, and how ill the Author of the Analyſt has proved his Imputations; it will be ne⯑ceſſary to perſue this Point, and conſi⯑der the Proof it ſelf.

[28] Let it be required to find the Fluxi⯑on of xⁿ, ſuppoſing x to increaſe uni⯑formly.

Suppoſe x in any finite Particle of Time, to become greater than before, by a finite Increment, whoſe Magnitude is expreſs'd by o. Then, in the ſame Time that x, by flowing becomes x + o, the n Power of x will become xⁿ + noxⁿ−1 + [...] o2 xⁿ−2 + &c.. Conſequently the Magnitudes of the ſynchronal Increments of x and of xⁿ, are to each other as 1 and nxⁿ−1 + [...] oxⁿ−2 + &c.. Now, let the Increments decreaſe by flowing back, in like Manner as they increas'd before by flowing forward, and continually grow leſs and leſs till they vaniſh; and their ultimate Ratio, that is, the Ratio with which they become evaneſcent, will be expreſs'd by 1 and nxⁿ−1: But the Fluxi⯑ons of Quantities are in the laſt Ratio of their evaneſcent Arguments; and by Conſequence the Fluxion of x is to that of xⁿ, as 1 to nxⁿ−1.

[29] In this Computation, Sir Iſaac endea⯑vours to collect the Proportion with which the iſochronal Increments of x and of xⁿ, begin or ceaſe to exiſt: Their Proportion obtain'd on Suppoſition that o is ſomething, is allowed to be the ſame with that of 1 and nxⁿ−1 + [...] oxⁿ−2 + &c.. And it muſt be acknowleg'd that this Ratio has a Limite dependent on the Magnitude of o, which Limite it cannot attain before the Increments are in⯑finitely diminiſh'd and become evane⯑ſcent; and when, by an infinite Diminu⯑tion, they become evaneſcent, no other Terms of their Ratio will be af⯑fected, ſo as to vaniſh with 'em, but ſuch as are govern'd or regulated by them: In the Inſtant therefore that o va⯑niſhes, [...] oxⁿ−2 + &c. and all enſuing Terms of the Series abſolutely vaniſh together; but the Terms 1 and nxⁿ−1 remain in⯑variable under all poſſible Changes of the Increments, from any finite Degrees of Magnitude whatever, even till they be⯑come [30] evaneſcent: They therefore expreſs the laſt Ratio, under which the iſochro⯑nal Increments of x and xⁿ vaniſh, or the Proportion of the Velocities with which thoſe Increments ceaſe to exiſt: Sir Iſaac Newton then rightly retain'd 'em for the Meaſures of the Ratio of the Fluxi⯑ons of x and xⁿ, tho' got in Virtue of his firſt Suppoſition; and the Fallacy, the Inconſiſtency, lies on the Side of this Author; who wou'd have them re⯑jected on the Authority of a Lemma not to the purpoſe.

To make this Point ſtill more plain and obvious, I ſhall propoſe the reaſoning in a ſtronger Light: It amounts there⯑fore to this, or may in other Words, be thus expreſſed: If x be ſuppos'd to flow uniformly, the Fluxions of x and xⁿ, will be as 1 and nxⁿ−1. For in the ſame Time that x by flowing, becomes x + o, xⁿ will become [...], which by the Method of infinite Series, is equal to xⁿ + noxⁿ−• + [...] o2 xⁿ−2 + &c.. Conſe⯑quently [31] the Increments of x and xⁿ, ge⯑nerated in the ſame Time, are o and noxⁿ−1 + [...] o2 xⁿ−2 + &c.. But the na⯑ſcent or evaneſcent Increment of xⁿ, is as its Fluxion; and in either of theſe States the Ratio of noxⁿ−1 + [...] o2 xⁿ−2 + &c. to noxⁿ−1 is a Ratio of Equality: For as the Magnitude of o becomes leſs and leſs, the Quantities noxⁿ−1 + [...] o2 xⁿ−2 + &c.. and noxⁿ−1 conſtantly tend to an Equali⯑ty, by a continual Diminution of their Difference; and they become equal, and their Ratio becomes a Ratio of Equality, when their Difference vaniſhes; that is, in the Inſtant o becomes evaneſcent, or in the Inſtant that the Increment of xⁿ firſt begins to exiſt: For as they vaniſh toge⯑ther under a Ratio of Equality, ſo they begin to exiſt together under the ſame Ratio; and therefore in the naſcent or evaneſcent State of o, the Fluxions of x and xⁿ, are as o and noxⁿ−1, which are manifeſtly to each other as 1 and nxⁿ−1.

[32] Hence it appears, that this Method of finding the Fluxion of xⁿ, upon a Sup⯑poſition that x flows uniformly, is the very ſame with that of finding the Fluxi⯑on of a Rectangle, as it is deſcribed in the ſecond Book of the mathematical Prin⯑ciples: For, as ab the Difference be⯑tween Ab + Ba + ab and Ab + Ba grows leſs and leſs perpetually, by diminiſhing the ſynchronal Increments of the Sides of the Rectangle, and at laſt evaneſces, and in the Inſtant of its Evaneſcence, the Gnomon Ab + Ba + ab becomes equal to the Sum of the Rectangles Ab and Ba; ſo [...] o2 xⁿ−2 + &c. the Dif⯑ference between noxⁿ−1 + [...] o2 xⁿ−2 + &c. and noxⁿ−1 grows leſs and leſs per⯑petually, by diminiſhing the Increment o, and at laſt evaneſces, and in the In⯑ſtant of its Evaneſcence noxⁿ−1 + [...] o2 xⁿ−2 + &c. becomes equal to noxⁿ−1: And as the Gnomon Ab + Ba + ab is not the Moment or Fluxion of the Rectangle AB, but in the Inſtant of its [33] becoming equal to Ab + Ba, ſo noxⁿ−1 + [...] o2 xⁿ−2 + &c.. is not the Moment or Fluxion of xⁿ, but in the Inſtant of its becoming equal to noxⁿ−1.

The Author of the Analyſt therefore, is greatly miſtaken, in thinking Sir Iſaac found the Fluxion of xⁿ, by a Method different from that he uſed in finding the Fluxion of a Rectangle, con⯑tain'd under two flowing Quantities: He ſteadily adheres to one and the ſame Method; namely, that of taking the firſt or laſt Ratios of Quantities, or of their iſochronal Increments, for the Mea⯑ſures of the Ratios of their Fluxions; and uſes no illegitimate Artifice to ob⯑tain theſe firſt or laſt Ratios; unleſs it be accounted illegitimate to ſuppoſe that mathematical Quantities can be gene⯑rated and deſtroyed by Motion.

It is pretended, ‘That the Method for finding the Fluxion of a Rectangle of two flowing Quantities, as it is ſet forth in the Treatiſe of Quadra⯑tures, [34] differs from that found in 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 di⯑miniſh'd and therefore rejecting it, is in Effect the rejecting an Infiniteſi⯑mal.’ But if this Author deduces the Rule from the firſt Propoſition in the Treatiſe of Quadratures, and conſiders it ever ſo little, he will find it the very ſame with that ſet down in the ſecond Book of the Principles: And it is doubt⯑leſs in Effect too the ſame with that uſed in the differential Calculus, ſo far as dif⯑ferent Methods can effect the ſame Thing, but no farther: For Quantities are not rejected in the Method of Fluxions, as in the differential Calculus, on Ac⯑count of their exceeding Smallneſs.

‘But according to the received Prin⯑ciples it is evident, ſays he, that no geometrical Quantity, by being infi⯑nitely diminiſh'd can ever be exhauſt⯑ed or become nothing.’ Now, on the received Principles of Fluxions, this is [35] a direct Abſurdity. For theſe Principles ſuppoſe that mathematical Quantities can be generated by Motion, which he has not yet thought proper to contradict; and conſequently they may alſo by Mo⯑tion be deſtroy'd: For Quantities, and the Augments of Quantities, which in ſome finite Time are produc'd by Mo⯑tion, may perpetually grow leſs and leſs by reverting that Motion; and by con⯑ſtantly growing leſs and leſs, they may come to be infinitely diminiſhed, or to be leſs than any aſſignable Quantities; and from being leſs than any aſſignable Quantities, the Motion ſtill perſevering, they muſt at laſt vaniſh and become no⯑thing; otherwiſe it might be contended that a Body ſetting out from any Place, and, in any finite Time, deſcribing a cer⯑tain length, cou'd never by moving back and returning in the ſame Line, arrive at the Place from whence it firſt ſet out.

Upon the whole then it appears, that the Method of Fluxions, as deſcrib'd by Sir Iſaac Newton in his Introduction to the Quadrature of Curves, and in the [36] ſecond Book of his mathematical Prin⯑ciples, is not that wretched un-ſcientifi⯑cal Knack ſet forth in the Analyſt; but a Method founded upon obvious, rational, accurate and demonſtrative Principles: It likewiſe appears, that the Concluſions do not ariſe from illegitimate tentative Ways or Inductions, but follow from ſuch Premiſes, and by ſuch Arguments, as are moſt conformable to the Rules of Logic and right Reaſon: All the Skill and Dexterity therefore by this Author ſhewn, in the Inveſtigation of contrary Errors correcting each other, are vain and impertinent. He has miſtaken the Doctrine of Fluxions, and by not right⯑ly diſtinguiſhing its Principles from thoſe of the differential Calculus, has impoſed a falſe Meaſure of Moments upon his Readers, and arguing from that falſe Mea⯑ſure, has unjuſtly charg'd Sir Iſaac with Errors ariſing from it; and, to mend the Matter, has inſtituted Computations to ſhew how thoſe Errors redreſs one ano⯑ther, and how Mathematicians by Means of Errors bring forth Truth and Science.

[37] The Diſpute between the Followers of Sir Iſaac Newton, and the Author of the Analyſt, is not about the Principles of the differential Calculus, but about thoſe of Fluxions; and it is whether theſe Principles in themſelves are clear or ob⯑ſcure, and whether the Inferences from them are juſt or unjuſt, true or falſe⯑ſcientific or otherwiſe: We are not con⯑cerned about Infiniteſimals or minute Dif⯑ferences, but about the Ratios with which mathematical Quantities begin or ceaſe to exiſt by Motion; and to conſider the firſt or laſt Proportions of Quantities does not imply that ſuch Quantities have any fi⯑nite Magnitudes: They are not the Pro⯑portions of firſt or laſt Quantities, but Li⯑mits of Ratios; which Limits, the Ratios of Quantities attain only by an infinite Diminution of their Magnitudes, by which infinite Diminution of their Mag⯑nitudes they become evaneſcent and ceaſe to exiſt. If therefore Quantities may ceaſe to exiſt by Motion, and if the Ratios with which they become evane⯑ſcent be truly determin'd, it will follow that there are no Errors, how⯑ever [38] ſmall, admitted in the Principles of Fluxions; and if no Errors be admitted in the Principles; there can be none in the Concluſions, nor any to be account⯑ed for in the Arguments by which thoſe Concluſions are deduc'd from their Pre⯑miſes: The Hints therefore, which this Author has condeſcended to give the Mathematicians for aſcertaining the Truth of their Concluſions, by means of con⯑trary Errors deſtroying each other, will probably be left to be further extended and apply'd by himſelf, to all the good Purpoſes he pleaſes to extend and apply them; as having more Leiſure, and a Science more tranſcendental^*, and per⯑haps a much greater Curioſity for ſuch Matters, than they have.

It has been obſerv'd before, that Fluxi⯑ons may be expounded by any Lines which are proportional to them; and ſo the Analyſis may be inſtituted, by con⯑ſidering [39] the mutual Relations or Propor⯑tions of finite Quantities, as the Pro⯑portions of Fluxions themſelves. To this it is objected, ‘That if, in order to arrive at theſe finite Lines propor⯑tional to Fluxions, there be certain Steps made uſe of which are obſcure and in⯑conceivable, it muſt be acknowleged, that the Proceeding is not clear, nor the Method ſcientific.’ But there may be many Steps obſcure and incon⯑ceivable to Perſons, who are unacquaint⯑ed with Sir Iſaac Newton's Method of firſt and laſt Ratios, with his Doctrine of Fluxions, and with his Principles of Moti⯑on; and yet thoſe Steps may appear very different to others who have duly conſi⯑der'd 'em: And therefore, till it be made apparent from geometrical Principles that the fluxional Triangle, which evaneſces upon the returning of the Ordinate of any Curve to the Place from whence it firſt ſet out, cannot in its laſt Form, that is, in the Form it has at the Inſtant it becomes evaneſcent, be ſimilar to a Triangle contain'd between the Tangent, the Abſciſs extended and the Ordinate of the [40] ſame Curve; or till it be proved that no Triangle, which is capable of becoming evaneſcent by a Diminution of its Sides from Motion, can be ſimilar in its laſt Form to any plain Triangle whatſoever; we ſhall continue to expound Fluxions by ſuch Right Lines as are proportional to them; and do aſſert, that the Proceeding is clear, and the Method ſcientific.