I. IN the Appendix of the Defence of Free-thinking in Mathematics, the Author, out of his greatly benevolent and truly Chriſtian Spirit, has compoſed a CATECHISM which he recommends to my Scholars: This CATECHISM with its Introduction I ſhall tranſcribe in his own Words, and fully and diſtinctly anſwer the ſeveral Queſtions contained therein.
II. "THIS Vindicator, indeed, by his diſſembling nine Parts in ten of the Difficulties propoſed in the [Page 4] Analyſt, ſheweth no Inclination to be CATECHISED by me. But his Scholars have a Right to be informed. I therefore, recommend it to them, not to be impoſed on by hard Words and magiſterial Aſſertions, but carefully to pry into his Senſe, and ſiſt his Meaning, and particularly to inſiſt on a diſtinct Anſwer to the following Queſtions."
"LET them ask him, whether he can conceive Velocity without Motion, or Motion without Extenſion, or Extenſion without Magnitude? If he anſwers that he can, let him teach them to do the ſame. If he cannot, let him be asked, how he reconciles the Idea of a Fluxion which he gives (P. 13.) with common Senſe? Again, let him be asked, whether nothing be not the Product of nothing multiply'd by ſomething? And if ſo, when the Difference between the Gnomon and the Sum of the Rectangles vaniſheth, [Page 5] whether the Rectangles themſelves do not alſo vaniſh? that is, when ab is nothing, whether Ab + Ba be not alſo nothing? that is, whether the Momentum of AB be not nothing? Let him then be asked, what his Momentums are good for, when they are thus brought to nothing? Again, I wiſh he were asked to explain the Difference between a Magnitude infinitely ſmall and a Magnitude infinitely diminiſh'd? If he faith there is no Difference, then let him be asked, how he DARES to explain the Method of Fluxions by the Ratio of Magnitudes infinitely diminiſh'd (P.9.) when Sir Iſaac Newton hath expreſly excluded all Conſideration of Quantities infinitely ſmall? If this able Vindicator ſhould ſay that Quantities infinitely diminiſh'd are nothing at all, and conſequently that, according to him, the firſt and laſt Ratios are Proportions between nothings, let him be deſired to make Senſe of this or explain what he means by 'Proportion between nothings. If he ſhould ſay the ultimate Proportions [Page 6] are the Ratios of mere Limits, then let him be asked how the Limits of Lines can be proportioned or divided? After all, who knows but this Gentleman, who hath already complained of me for an uncommon Way of treating Mathematics and Mathematicians, may (as well as the Cantabrigian) cry out Spain and the Inquiſition when he finds himſelf thus cloſely purſued and beſet with Interrogatories? That we may not therefore ſeem too hard on an innocent Man, who probably meant nothing, but was betray'd by following another into Difficulties and Straits that he was not aware of, I ſhall propoſe one ſingle Expedient, by which his Diſciples (whom it moſt concerns) may ſoon ſatisfy themſelves, whether this Vindicator really underſtands what he takes upon him to vindicate. It is in ſhort, that they would ask him to explain the ſecond, third or fourth Fluxions upon his Principles. Be this the Touchſtone of his Vindication. If he can do it, I ſhall own my ſelf much miſtaken: If he cannot, it will be evident [Page 7] that he was much miſtaken in himſelf, when he preſumed to defend Fluxions without ſo much as knowing what they are. So having put the Merits of the Cauſe on this Iſſue, I leave him to be tried by his Scholars."
III. In this CATECHISM I am firſt to be: asked, "Whether I can conceive Velocity without Motion, or Motion without Extenſion, or Extenſion without Magnitude?" Theſe Queſtions are more clearly expreſs'd in the 29th and 30th Queries of his Analyſt, where he asks, "Whether we can form an Idea or Notion of Velocity diſtinct from and excluſive of its Meaſures? Whether Motion can be conceived in a Point of Space? And if Motion cannot, whether Velocity can? And if not, whether a firſt or laſt Velocity can be conceived in a mere Limit, either initial or final, of the deſcribed Space?" I anſwer, I can conceive Velocity and Motion in a Point of Space; that is without any aſſignable Length or Extenſion deſcribed by it, and ſo might he too if he had underſtood and conſider'd [Page 8] the Nature of Motion. For Motion is an Effect of ſome Cauſe acting on the Thing moved; which Effect ſetting aſide all Reſiſtance, will ever be proportional to the whole Action of the generating Cauſe: And therefore if a Cauſe acts continually upon a given Thing without any Interruption, there muſt be a continual Increaſe of its Velocity. The Velocity cannot be the ſame in any two different Points of the Space deſcribed, however near thoſe Points may be to each other. For if it was, there muſt: be a Ceſſation of the Action of the moving Cauſe during the Paſſage of the Thing thro' the Space comprehended between the two Points; which is contrary to the Suppoſition.
If the Cauſe acts continually upon the moving Point, but with different Degrees of Strength during the Time of the Motion; the Velocity will not increaſe with the Time, nor with the Square Root of the [Page 10] Space deſcribed; but will ſtill increaſe from the Beginning to the End of the Motion, and not be the ſame in any two different Points of the Space deſcribed, however near to each other. Let the Velocity increaſe with the n Power of the Time, that is, let V be as T n ; and then V will be as [...] or as [...], putting S for the Space deſcrib'd: Whence it appears that the Velocity will not be the ſame in any two different Points of the Space deſcribed: For it muſt vary upon the leaſt Change of the Space S, and conſequently be different in every different Point of AD; which ſhews that this Author has been greatly miſtaken in imagining that there can be no Motion, no Velocity, in a Point of Space.
IV. The next Queſtions in the CATECHISM, depending upon each other, run thus. "Let him be asked, whether nothing be not the Product of nothing multiply'd by ſomething? And if ſo, when the Difference between the Gnomon [Page 11] and Sum of the two Rectangles vaniſheth, whether the Rectangles themſelves do not alſo vaniſh? that is when a b is nothing, whether Ab + Ba be not alſo nothing? that is, whether the Momentum of AB be not nothing? And let him be asked what his Momentums are good for when thus they are reduced to nothing?" As to the firſt of theſe Queſtions I agree with him that nothing is the Product of nothing multiply'd by ſomething; but muſt know what he means by the vaniſhing of the Gnomon and Sum of the two Rectangles in the ſecond, before I give him a direct Anſwer. If by vaniſhing he means that they vaniſh and become nothing as Areas, I grant they do; but abſolutely deny, upon ſuch an Evaneſcence of the Gnomon and Sum of the two Rectangles by the moving back of the Sides of the Gnomon till they come to coincide with thoſe of the Rectangle, that nothing remains. For there ſtill remain the moving Sides, which are now become the Sides of the Rectangle; into which Sides both the Gnomon and the Sum of [Page 12] the two Rectangles are now turned by this retroverted Motion. And as the Gnomon and Sum of the two Rectangles, upon the Evaneſcence of their Areas by this retroverted Motion, are both converted into the two Sides of the Rectangle AB, ſo in the Inſtant of that Converſion, their Motions are exactly the ſame; or the Motion of the Gnomon is the ſame with the Sum of the Motions of the two Rectangles, when they evaneſce, and are converted into the two Sides of the Rectangle AB.
If a Point moves forward to generate a Line, and afterwards the ſame Point moves back again to deſtroy the Line with the very ſame Degrees of Velocity, in all Parts of the Line which it had in thoſe Parts when moving forward to generate it; in the Inſtant the Line vaniſhes as a Length, nothing of a Line will remain; but ſtill the generating Point will remain, together with the Velocity it had at the very Beginning of its Motion. And the Caſe is the very ſame with reſpect to a Rectangle increaſing by the Motion of [Page 13] its Sides: For upon the Evaneſcence of a generated Gnomon, there ſtill remain the Sides of the Rectangle into which the Gnomon by its Evaneſcence is converted, together with the Velocities of thoſe Sides that is, when the Gnomon evaneſces there ſtill remains Ab + Ba.
Hence it appears, that if mathematical Quantities be increaſed in equal Times by Motion, their iſochronal Increments muſt be made to vaniſh by a Retroverſion of the Motion, before we can obtain the Motions with which they vaniſh, or begin to be generated; that is, before we can obtain the Fluxions of the Quantities, the Name given by Sir Iſaac Newton to thoſe Motions. So then, this Author has been much out in ſuppoſing that upon the Evaneſcence of the Gnomon CGK, or of the curvilineal Figure BDFE, the Momentum or Fluxion of the Rectangle CDK, or of the Area ABD, vaniſhes. Conſequently, he has been greatly miſtaken in every one of theſe Queſtions.
V. But he goes on. "I wiſh he were asked to explain the Difference between a Magnitude infinitely ſmall and a Magnitude infinitely diminiſhed. If he ſaith there is no Difference: Then let [Page 19] him, be further asked, how he DARES to explain the Method of Fluxions by the Ratio of Magnitudes infinitely diminiſh'd, when Sir Iſaac Newton hath expreſly excluded all Conſideration of Quantities infinitely ſmall? If this able Vindicator ſhou'd ſay that Quantities infinitely diminiſh'd are nothing at all, and conſequently that, according to him, the firſt and laſt Ratios are Proportions between nothings, let him be deſired to make Senſe of this, or explain what he means by Proportion between nothings. If he ſhou'd ſay the ultimate Proportions are the Ratios of mere Limits, then let him be asked how the Limits of Lines can be proportioned or divided?"
As all this Part of the CATECHISM relates to the Meaſures of Fluxions by the firſt and laſt Ratios of iſochronal Increments generated and deſtroy'd by Motion, ſo I have taken it together, and ſhall anſwer the whole in one Section.
[Page 20] Neither Sir Iſaac Newton nor I have ſaid, that Fluxions are meaſured by the Proportions of Magnitudes infinitely ſmall, nor by the Proportions of any Magnitudes whatever generated in equal Times; but that they are meaſured by the firſt or laſt Proportions of iſochronal Increments generated or deſtroy'd by Motion; which Proportions are the Ratios with which ſuch Increments begin to exiſt before they have acquired any Magnitude, or with which they ceaſe to exiſt and vaniſh after they have loſt all Magnitude. Theſe Ratios ſubſiſt when the iſochronal Increments have no Magnitude, for as much as the Motions ſubſiſt with which thoſe Increments, juſt now, in this very Inſtant, begin or ceaſe to exiſt; to which Motions theſe Ratios are proportional.
The Motions in C and E, are as the moving Quantities and Velocities taken together; or as two Points and their Velocities, taken together; or as the Velocities, all Points being equal. And the firſt or laſt Ratio of the iſochronal Increments CD and EF, is compounded of the firſt or laſt Ratio of theſe Spaces, and of the Ratio of the moving Quantities. For the Velocities in C and E being in the firſt or laſt Ratios of theſe [Page 22] iſochronal Spaces, the Motions, which are as the moving Quantities and Velocities taken together, will be as the ſame moving Quantities and the firſt or laſt Ratio of the iſochronal Spaces taken together. If Q and q denote the moving Quantities in C and E, V and v their Velocities, S and s the iſochronal Spaces CD and EF, and Ṡ and ṡ the firſt or laſt Ratio of thoſe iſochronal Spaces; then QV will be to qv, as QṠ to qṡ; and in this caſe V will be to v, as Ṡ to ṡ, becauſe Q and q are equal.
Again, the firſt or laſt Ratio of the iſochronal Spaces FD and DH in the augmented Rectangle EGL (See the Figure in Page 14.) has a real Exiſtence; for as much as it is equal to the Ratio of the two Motions of two Points in D, of one towards F, and the other towards H; which Motions, ſubſiſting when the iſochronal Spaces FD and DH are nothing, preſerve the Exiſtence of the firſt or laſt Ratio of theſe Spaces, or keep it from being a Ratio of nothings. If V and v, denote the Velocities in D towards F [Page 23] and H, Q and q the Sides of the Rectangle DK and DC, and Ṡ and ṡ the firſt or laſt Ratio of the iſochronal Spaces FD and DH; then Q v will be to qV, as Qṡ to qṠ; but Qv + qV is the Fluxion or Motion of the Rectangle CDK, as I have ſhewn before; and therefore the Moment or Meaſure of the Fluxion of the Rectangle will be Qṡ + qṠ. This is a full and clear Anſwer to this Part of the CATECHISM, and ſhews that its Author has been greatly miſtaken in ſuppoſing that I explained the Doctrine of Fluxions by the Ratio of Magnitudes infinitely deminiſh'd, or by Proportions between nothings.
VI. I come now to the laſt Part of the CATECHISM, which ſtands thus. "I ſhall propoſe one ſingle Expedient, by which his Diſciples (whom it moſt concerns) may ſoon ſatisfy themſelves, whether this Vindicator really underſtands what he takes upon him to vindicate. It is in ſhort, that they wou'd ask him to explain the ſecond, third, or fourth Fluxions upon his Principles. Be this [Page 24] the Touchſtone of his Vindication: If he can do it, I ſhall own my ſelf much miſtaken: If he cannot it will be evident that he was much miſtaken in himſelf, when he preſumed to deſend Fluxions without ſo much as knowing what they are. So having put the Merits of the Cauſe on this Iſſue, I leave him to be tried by his Scholars."
I do not wonder that this Author ſhou'd have no clear Ideas or Conceptions of ſecond, third or fourth Fluxions, when he has no clear Conceptions of the common Principles of Motion, nor of the firſt and laſt Ratios of the iſochronal Increments of Quantities generated and deſtroy'd by Motion. For Fluxions, according to Sir Iſaac Newton, are the Motions with which the iſochronal Increments of Quantities begin or ceaſe to exiſt, or the Motions of the generating Quantities in the very Limits or Extremities of the Fluents: Thus the Fluxions of Solids are the Motions of Surfaces; the Fluxions of Surfaces, the Motions of Lines; the Fluxions of Lines, [Page 25] the Motions of Points; and the Fluxions of Points are nothing, for Points in their own Nature are invariable, and therefore incapable of being generated or increaſed by Motion: And if the firſt Fluxions of Quantities be Motions, it follows, that the Mutations of theſe Motions and the Mutations of thoſe Mutations, which are the ſecond and third Fluxions of the Quantities, muſt likewiſe be Motions.
Firſt, ſecond and third Fluxions do really exiſt, and may be clearly and diſtinctly conceiv'd by attending to the Motions of the ſeveral Parts of a Cube, namely, of its Surfaces of their Lines and Points, in the Inſtant it begins to be increas'd by Motion. For if A denotes the Side of a Cube generated by an uniform Motion, whoſe Velocity is expreſs'd by a; the firſt Fluxion of the Cube, according to theſe Principles, will be expreſſed by 3aA2; its ſecond Fluxion, which is the Fluxion of 3aA2, will be expreſſed by 6a2A, or by 6aA × a; its third Fluxion, which is the firſt Fluxion on of 6a2A, will be expreſs'd by 6a3 or by 6a2 × a; and its fourth Fluxion will [Page 26] be nothing: But all theſe Fluxions or Motions do exiſt, and may be clearly and diſtinctly conceived in the Motion of a Cube, at the very End of its Generation, or at the very Beginning of its Augmentation, by Motion; for it begins to be augmented by the Sum of the Motions of three of its Squares comprehending any one of its ſolid Angles, each of which Squares being denoted by A2 and their Velocity outward by a, the Fluxion of the Cube or the Motion with which it begins to increaſe or to be enlarged, will be 3aA2; and this is the firſt Fluxion of the Cube; and the three moving Squares begin to be augmented, in the very ſame Inſtant wherein the Cube begins to enlarge, each by the Sum of the Motions of its two adjoining Sides, and conſequently the Motions of thoſe Sides to augment the cubic Surface, will be expreſs'd by 6aA, but that Surface at the ſame Inſtant of Time moves outward to augment the Solid with a Velocity which is alſo denoted by a, and therefore the whole Motion of the fix moving Sides of the three Squares for increaſing [Page 27] or enlarging the Cube, will be expreſſed by 6aA × a or by 6a2A; and this is the ſecond Fluxion of the Cube; and when the three moving Squares begin thus to increaſe, ſideways and outwards, for the Enlargement of the Cube, their ſix moving Sides begin to be augmented by the Motion of ſix Points; and the common Velocity of thoſe Points in order to increaſe the Sides of the Squares, being the ſame with the Velocity of thoſe Sides to increaſe the Cubic Surface, and with the Velocity of that Surface to augment the Solid; the whole Motion with which thoſe Points begin to enlarge the Cube, will be expreſs'd by 6a3 or by 6a3 p; and this is the third Fluxion of the Cube: Theſe three Kinds of Motion do all neceſſarily exiſt and may be clearly and diſtinctly conceived in the Syſtem of Motion whereby a Cube begins to be augmented: And therefore the firſt, ſecond and third Fluxions of Quantities may be diſtinctly conceived, and fully explained upon the Principles of Sir Iſaac Newton.
[Page 28] As firſt, ſecond and third Fluxions are explained by the ſeveral Motions neceſſarily exiſting in the very Inſtant a Cube begins to be augmented, ſo they may likewiſe be explain'd and distinctly comprehended, by conſidering the naſcent or evaneſcent Increments of the ſeveral Parts of a Cube, generated by Motion; provided always that by naſcent or evaneſcent Increments be underſtood not generated Increments of any Magnitude whatever, but only ſuch Quantities or Magnitudes as are proportional to and Conſequently Meaſures of the Motions with which thoſe iſochronal Increments begin or ceaſe to exiſt. For if a, which before denoted Velocity, be now put for the firſt or laſt Ratio of the Space deſcribed by that Velocity in a given Time; 3aA2 will denote the naſcent Increment of the Cube generated by the Motion of three of its Squares comprehending any one of its ſolid Angles; and 6aA will expreſs the Sum of the naſcent Increments of the three moving Squares, which Sum multiply'd into a will be the Increment of the naſcent Solid 3aA2; conſequently [Page 29] 6a2A will be the ſecond naſcent Increment of the Cube; and the naſcent Increment of 6Aa, or of ſix Rectangles each denoted by Aa, will be 6a2, which multiply'd into a gives 6a3 for the naſcent Increment of 6a2A, and therefore 6a3 is the third naſcent Increment of the Cube: All this may be clearly conceiv'd and made evident to Senſe by the Figure in Page 14. For let CDK repreſent one of the three moving Squares comprehending any of the ſolid Angles of a Cube increaſing by Motion, and then three times CDK multiply'd into a, or 3aA2, will expound the firſt naſcent Increment of the Cube; and 3FC + 3LD or 6Aa (which is the naſcent Increment of the three moving Squares) multiply'd into a, will be the ſecond naſcent Increment of the Cube; and, the Rectangle FH or a2 being the naſcent Increment of the Rectangle FC or LD, and 2FH or 2a2 the naſcent Increment of FC + LD or of 2Aa, and 6a2 the naſcent Increment of 6Aa, it follows that 6a2 multiply'd into a, or that 6a3, will be the naſcent Increment of 6Aa2 and conſequently [Page 30] the third naſcent Increment of the Cube. And theſe three diſtinct Orders of Increments, all begin to exiſt together, in the very Inſtant the Cube begins to be augmented by Motion.
This may ſerve as an Anſwer to the laſt Part of the CATECHISM, concerning the Author's Touchſtone of my Vindication: Whether he will own himſelf miſtaken I know not; but I think he ought after his unjuſt and ſhameful Treatment of Sir Iſaac Newton; who in the Introduction to his Quadrature of Curves, in the ſecond Lemma of the ſecond Book, and in the Scholium to the firſt Section of the firſt Book of his Principles of Philoſophy, has deliver'd his Doctrine of Fluxions in ſo clear and diſtinct a Manner, without the leaſt Inconſiſtency in Terms or Arguments, that one wou'd have thought it impoſſible for any Perſon not to have underſtood him, particularly for this Author, who ſays, he had long and maturely conſidered thoſe Principles, and taken as much pains as [Page 31] any Man living to underſtand and make Senſe of them.
I have now done with the CATECHISM; but beg leave before I conclude this Paper, in order to prevent my being CATECHISED any more by this Author, to give the World a ſhort Account of ſome Part of my Faith in Religion. I believe that there is one ſupreme, incorporeal, everliving, intelligent and omnipreſent Being, called GOD, who made and governs the World. I believe that GOD is endued with infinite Power, Knowledge, Wiſdom and Goodneſs; and that to deny or limit any one of theſe Attributes, is in Effect to deny a GOD. I believe, that to ſay GOD cannot create Spirits with a Power, inherent in themſelves and reſulting from their own Frame and Make, of perceiving and knowing. Things of a quite different Nature from their own, by Ideas and Senſations; is in Effect to deny a GOD; for as much as by this Principle his Almighty Power is denied. And laſtly, I believe that this Supreme Being has revealed his Will to Mankind [Page 32] by Moſes, the Prophets, Jeſus Chriſt and the Apoſtles; and that the Doctrine by them deliver'd is therefore divine, and cannot be altered by any Power or Authority upon Earth, nor even by an Angel from Heaven, who is pronounced accurſed ſhou'd he preach any other Goſpel than what is delivered.