The CATECHISM.

"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.

[Page 9] If the Thing moved be a Point, and the Space described by this Point setting out from a State of Rest at A, be AD;

and if the Point be acted upon con­tinually by a moving Cause during the Time of its going from A to D, and the Cause act continually with the same Degree of Strength during that Time; the Velocity will con­tinually increase with the Time of the Motion, or with the Square Root of the Space described; that is, the Velocities in the Points C and D, will be in the same Proportion as the Times of describing AC and AD, or as the Square Roots of AC and AD. These are the Laws of Motion observed by Bodies falling from small Heights to the Surface of the Earth, setting aside the Resistance of the Air, and supposing the Bodies to have exactly the same Weights during the Times of their Fall.

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 Tⁿ; 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.

To make this more evident to Sense; let the Point which generated the Line AD,

move back again from D to C with the very same Degrees of Ve­locity in every Part of the Incre­ment CD which it had in that part in moving forward from C to D; and when the Increment CD eva­nesces and becomes nothing as a Line by being converted into the moving Point in C; there will remain that that Point together with its Motion in C; for neither the Point nor its Motion in C is destroy'd by the Evanescence of the Line CD.

[Page 14] If the Rectangle CDK be generated by the Motions of two indefinite right Lines IB and IA, of IB down IA and of IA along IB, and from a Continuance of the Motions of these Lines, be in­creased by the Gnomon CGK; and if

the generating Lines return from L and E to K and C, with the very same Degrees of Velocity in the several Points of LK and EC which they had in those Points in moving from K and C to L and E; it is evident, when the Gnomon CGK or the Increment of the Rectangle CDK evanesces and becomes nothing as an Area, by being converted into the Sides DK and DC; that there will remain those Sides and their Velocities in D; neither the Sides nor the Velocities of those [Page 15] Sides being destroy'd by the Evanescence of the Area of the Gnomon. But per­haps he will say, that when the Gnomon evanesces it turns into something more than the Sum of the Sides DK and DC: For the Rectangles LD and FC turn into those Sides, and the Rectangle FH turns into a Point at D; and therefore the Mo­tion of the Gnomon in the Instant it eva­nesces, will become the Sum of the Mo­tions of the Sides DK and DC, and of the Point in D; that is, putting A and B for DK and DC, and a and b for the Velocities in D towards F and H, and p for a Point; it will be equal to Ab + Ba + ap + bp: Let it be so, and yet a+b × p will be nothing when compared with Ab + Ba, nor make the least Augmentation to that Sum when added to it. For let a be to b as 3 to 2, and Ab + Ba + ap + bp will be 2A + 3B + 5p: But 5p or five Points added to a Line, make no­thing but the Line itself; a Point being no Part of a Line; and consequently not in the least increasing its Length when added to it: And therefore the Motion of the Gnomon when it evanesces and [Page 16] turns into the Sides of the Rectangle CDK, will be Ab + Ba, as Sir Isaac Newton makes it.

If the Curvilineal Area ABD be gene­rated by the Motion of any indefinite Line AH down AG, While a Point set­ting out from A moves in the Line AH from A towards H; and if from a Con­tinuance of these two Motions that Area be increased by the Augment BDFE,

consisting of the Rectangle BC and of the Triangle CFD; the Motion with which the Increment BDFE begins to be generated, is the Sum of the two Mo­tions towards C and H in the Point D; which Sum, putting b for the Velocity towards C, a for the Velocity towards H [Page 17] and p for a Point, will be BD×b + ap. Now suppose the Ordinate EF to move back to BD, and the Point F to move back in the Line HE to C, and that in moving back they have the same Degrees of Ve­locity in every Point of EB and EH, which they had before in moving for­ward; it is manifest, that when the Aug­ment BDFE vanishes and becomes no­thing as an Area, by being converted into the Ordinate BD, and into a Point at the Extremity of that Ordinate; there will remain BD and the Point in D, together with their Velocities in that Point; for the Motions of that Line and Point, are not destroy'd by the Evanescence of the Area BDFE, In the Instant EF coin­cides with BD, and the Point F with D, the Area BDFE will be converted into BD + p, and its Motion will become BD × b + pa, as before. But pa, being only so many Points as a contains Units, is real­ly and truly nothing with respect to BD × b; and consequently does not at all increase it when added to it. And there­fore the Fluxion of the curvilineal Area ABD, will be BD × b; or as BD if b [Page 18] be given, or the Line AH move down AG with a Velocity which is exactly the same in every Part of GA.

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.

For Example, the first or last Ratio of the Increments CD and EF generated in equal Times, has a real Existence; for as much [Page 21] as it is equal to the Ratio of the Motions in C and E,

which are the Mo­tions subsisting in those Points when the ischronal Increments just begin or cease to exist. The Existence of the Motions preserves the Existence of these Ratios of the Increments CD and EF. If the Ratio of the Motions in C and E, be that of 4 to 3; the first or last Ratio of the isochronal Increments CD and EF, will likewise be that of 4 to 3, even tho' no Part of those Incre­ments has any Existence.

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.