I. The Velocity of a Body, is the Ratio of the Quantity of Motion to the Quantity of Matter in the Body; and is as the Ratio of the Quantity of the Action of the moving Force to the Weight of the Body; or as the Ratio of the Quantity of that Action to the Density and Magnitude of the Body taken together. That is, putting V for the Velocity of a Body, M for the Quantity of its Motion, F for the Quantity of the Action of the Force which generates that Motion, D for the Density of the Body, B for its Bulk or Magnitude [Page 35] and W for its Weight; V is M / Q, and is as F / W, or as F / DB.

For the Quantity of Motion is the Quantity of Matter and Velocity taken together; that is, M is QV: And con­sequently V is M / Q: But the Motion of a Body is as the Quantity of the Action of the Force which generates it, Effects be­ing always proportional to their adequate Causes, that is, M is as F; and at a given Distance from the Centre of the Earth, the Quantity of Matter in a Body, is as its Weight, or as its Denfity and Magni­tude taken together; that is, Q is as W, or as DB: And therefore V is as F / W, or as F / DB.

The Forces which generate Motion in Bodies may be of different Kinds, as a Blow, Pressure, Weight; all which may [Page 36] be conceived: And Density and Bulk may likewise be conceiv'd: Consequent­ly the two Measures of Velocity, F / W and F / DB, may both be conceived, and yet nei­ther of them includes Time and Space. This Author therefore has been grosly mistaken in asserting that Velocity neces­sarily implies both Time and Space, and cannot be conceiv'd without them.—And that 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.

II. The Velocity of a Body, exists in the Body it self while it conti­nues in Motion.

For the Velocity of a Body, is an Ef­fect of some Cause acting upon it: But this Effect can exist no where except in the Body acted upon; for could it exist [Page 37] any where else, an Effect might exist where there is no Cause to produce it, or in other Words, an Effect might exist without a Cause, which is absurd: And therefore the Velocity of a Body exists in the Body it self while it continues in Motion.

Hence it appears that a Body in Mo­tion, will have a Velocity inherent in it self during the whole Time of its Mo­tion: And consequently there must be a Velocity where-ever the Body is, exclu­sive of Time and Space. If instead of a Body the Thing moved be a Point, its Velocity will exist in a Point, and suc­cessively will exist in every Point of Space through which the Point moves.

Here I may properly take notice of this Author's Objection against my Proof that Velocity can exist in a Point; my Argument for it was conceived in these Words. ‘If a Cause acts continually up­on a given Body or Thing, in order to move it without any Interruption, there must ensue a continual Increase [Page 38] of its Velocity; and consequently no two Points of the Space described, however near to each other, can be assign'd, in which the Velocity is the same: For that wou'd manifestly sup­pose a Cessation of the Action of the moving Cause during the Passage of the Body or Thing thro' the Interval com­prehended between the two Points.’ Now he thinks that from the continual Action of a moving Force, and from the generated Velocity not being the same in any two different Points of the de­scribed Space, it will not follow that Ve­locity can exist in a Point of Space. But in this he is mistaken. For the conti­nual Action of the moving Force neces­sarily preserves a continual Velocity; and if the generated Velocity be not the same in any two different Points of the de­scribed Space, a Velocity must of Conse­quence exist in every Point of that Space.

III. The Velocity of a Body is the Rate or Degree of its Tendency forward.

[Page 39] For the Velocity of a Body, is a Part of its Motion by the first Proposition, and exists in the Body, by the second: But there is nothing existing in a Body moved or translated from one Place to another besides its Quantity of Matter, and the Rate or Degree of its Tendency forward: And therefore the Velocity of a Body is the Rate or Degree of its Ten­dency forward.

IV. The Account of Velocity given in the two preceding Propo­sitions, is agreeable to Sir Isaac New­ton's Notion of Velocity; who con­stantly excludes described Space from his Idea of that Term.

For, as I remember, whenever he uses the Word Velocity in his Principles of Philosophy, he speaks of the Velocity of a Body existing in some one certain Place; by which it appears, that he confines Ve­locity to the Place of the Body: But there is no Space described by a moving [Page 40] Body existing in one and the same Place: And therefore he excludes all Space de­scribed from his Idea of Velocity. And he expresly does it in his Doctrine of Fluxions. For he calls Velocities Flux­ions; and by Fluxions declares, that he does not understand any Increments ge­nerated, and consequently not any Spaces described: Whence it necessarily follows, that Velocities, in his Doctrine of Flux­ions, do not imply any Spaces described. Therefore the Proposition is true.

The Velocity of a Body, is an Effect communicated to the Body by the Ac­tion of some moving Force or Cause, and is retain'd in it by the Inertness of Mat­ter, till destroy'd by the Action of some contrary Force or Resistance; and by be­ing retain'd in it, becomes a Cause of the Body's going forward, or of its being translated continually into a new Place: So that the continual Translation of a Body into a new Place, is really an Effect of Velocity, which Velocity may be conceived to exist in the Body prior to [Page 41] the Progress of that Body, as all Causes may be conceiv'd to exist prior to the Effects produc'd by them.

V. If the Velocity of a Body or Thing moved be given, or increase with the Time of the Motion, it will be measur'd by the Space described apply'd to the Time of its Descrip­tion: And if it increases any how either regularly or irregularly, it will be measur'd by the first Ratio of the Space to be described in a given Particle of Time. If S de­notes the Space described in the Time T, and S the first Ratio of the Space to be described in a given Particle of Time, then V will be as S / T when the Velocity is either given, or increases with the Time of the Motion; and as S when it increases any how regularly or irregularly.

[Page 42] Case 1. If the Velocity be given, that is, if the Body constantly goes on at the same Rate, the Space described, reckon­ing from the Beginning of the Motion, will be as the Time of its Description; and consequently the Ratio of the Space described to the Time of its Description; that is, S / T will be a given Quantity: But one given Quantity may be the Measure of another: And therefore V will be measur'd by S / T.

Case 2. If the Velocity increases with the Time of the Motion, the Space de­scrib'd must increase with the Square of the Time; whence S / T will be as T, that is, as V: And consequently in this Case also V will be measur'd by S / T.

Case 3. If the Velocity increases any how either regularly or irregularly, the Space to be described in a given Particle of Time, will begin to exist, no Part of [Page 43] it being yet described, with the Ratio of the Velocity: But the Ratio with which that Space begins to exist or to be de­scribed, is its first Ratio: And therefore the Velocity will be measured by the first Ratio of the Space to be described in a given Particle of Time. Consequently V in all Cases will be measur'd by S.

The two first Cases of this Proposi­tion are particular ones, and obtain only when the Force which generates the Mo­tion acts either by one single Impulse, or continually with the same Degree of Strength during the whole Time of Mo­tion: The last is a general one, and ob­tains in all Cases whatever: This general Measure of Velocity Sir Isaac Newton uses in his Doctrine of Fluxions: It con­tinually exists in the final Limits or Ex­tremities of Quantities actually generated by Motion, and just beginning to be in­creas'd by a Continuance of that Mo­tion, tho' as yet no Parts of their iso­chronal Increments are described; for those Increments must begin to exist in the Ra­tio of the Velocities, which the genera­ting [Page 44] Quantities have in the final Limits of the Quantities generated, which are the initial Limits of their Increments. If the Ratio of the Velocities existing in any two Points be 4/3 or 3/2, the first Ratio of iso­chronal Increments commencing from those Points, will be 4/3 or 3/2.

Having shewn what Sir Isaac Newton means by the Word Velocity, and given an Account of its Measures; I will now, by way of Inference, shew the Weakness of this Author's Objections against the se­veral Parts of the foregoing Answer; and in doing of this I shall be less methodical, because I intend to pursue him in the order of his Reasons.

First then, in Sect. 3. he takes notice of my Freedom in calling his Analyst a Libel: I am sorry my free Manner should offend him; but I must continue to call it a Libel, till he produces Proofs to the contrary. He tells me, I well know a bad Vindication is the bitterest Libel. I cannot say I know it; but I know that defaming one of the greatest and best [Page 45] Men, who had nothing at heart besides the Promotion of true Philosophy and true Religion; is a most bitter Libel: A bad Vindication may proceed from a good Mind and an honest Intention; but De­famation and Detraction can arise from neither.

In Sect. 4, 5. he finds fault with my Proof that Velocity can exist in a Point of Space. But I have shewn the Justness of that Proof under the second Proposi­tion, to which I refer him. That Proof may perhaps appear more evident to some, by considering the Motion of a Body in falling from a State of Rest by the Force of its own Weight. For setting aside the Resistance of the Air, and supposing the Weight of the Body to be the same at all near Distances from the Surface of the Earth; its Velocity, from the continued Action of that Weight, will increase con­tinually as the Time of falling increases, or as the square Root of the Length de­scribed increases, reckoning that Length from the Beginning of the Motion; and therefore Velocity will exist and be dif­ferent [Page 46] in every different Point of the de­scribed Space: Both these are necessary and inevitable Consequences of the continued Action of the Weight of the Body during the Time of its falling; and of the Ve­locity's being proportional to the square Root of the Space described.

In Sect. 6, 7. he charges me with gi­ving an Account of Motion different from Sir Isaac Newton, ‘ who distin­guisheth two Sorts of Motion, abso­lute and relative. The former he de­fineth to be a Translation from abso­lute Place to absolute Place, the lat­ter from one relative Place to ano­ther. Mine, which exists in a Point, which may be conceiv'd without Space describ'd, he says , is plainly neither of these Sorts of Motion, but some third Kind, which he is at a loss to comprehend. But he can clearly com­prehend that, if we admit Motion without Space, then Sir Isaac New­ton 's Account of it must be wrong: For Place by which he defines Motion is, according to him, a Part of [Page 47] Space.’ Now in Answer to this I say that all Motion, which can neither be ge­nerated nor changed but by Forces im­press'd on a Body, necessarily exists in the Body it self while it continues to change its Place, and is the Quantity of Matter moved and its Velocity or Degree of Tendency forward taken together: The continual Translation of a Body therefore into a new Place is, as I have before ob­served, an Effect of this Tendency for­ward in the Body, and not the Tenden­cy itself; consequently Space describ'd is an Effect of Velocity, and not the Ve­locity itself. Velocity, according to Sir Isaac Newton, does not necessarily im­ply any Space described: And therefore, consider'd as an Effect existing in the Body, can be nothing but the Rate or Degree of its Tendency forward, as I have proved it to be in Prop. III. Con­sequently I have not given a different Account of Motion from Sir Isaac New­ton, but an Account every way consistent with his Principles.

[Page 48] In Sect. 8. I find the following Pas­sage transcribed from the Analyst. ‘ Ve­locity necessarily implies both Time and Space, and cannot be conceiv'd without them. And if the Velocities of nascent and evanescent Quantities, that is, abstracted from Time and Space, may not be comprehended, how can we comprehend and demonstrate their Proportions? or consider their rationes primae & ultimae. For to con­sider the Proportion or Ratio of Things, implieth that such Things have Magnitude: That such their Magnitudes may be measur'd, 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?’ This Passage I have fully [Page 49] answer'd, in having proved that there are other Measures of Velocity besides Time and Space. I have given two general Measures of it in the first Proposition, and one in the fifth; all of which may be clearly conceiv'd, and yet not one of them includes or implies either Space described or Time; and I know no general Measure of it which does. I agree with him that to consider the Proportion or Ratio of Things implieth that such Things have Magnitude, but ‘to consider the first or last Proportions of Quantities does not at all imply that such Quantities, have Magnitudes. These are not the Pro­portions of first or last Quantities, or of any generated Magnitudes whatever, but the Proportions of the Velocities with which Quantities begin or cease to have Magnitudes.’

The next two Sections relate to the Moment of a Rectangle. If the Rect­angle CDK be generated by the Motions of two right Lines, and from a Continu­ance of their Motions be increased by the Gnomon CGK; and if A and B are [Page 50] put for the Sides DK and DC, and a / b for the first Ratio of their isochronal In­crements DF / DH, which first Ratio is equal to the Ratio of the Velocities in D to­wards

[figure]

F and H; the Moment or Muta­tion of the Rectangle CDK, will be Ab+Ba, as I have fully proved in the foregoing Answer. In the Augmentation of a Rectangle no Motions exist but the Motions of its Sides, which Motions, according to Sir Isaac Newton, con­stitute its Moment or Mutation: And if the Sides of the Rectangle EGL, flow back till they coincide with those of the Rectangle CDK, and the gene­rated [Page 51] Gnomon CGK vanishes; I have proved that the Motion subsisting in the Instant of its Evanescence, which is also the Moment of the Rectangle CDK, is the Sum of the Motions of the Sides of that Rectangle, or Ab+Ba; and he ought either to allow it or prove the con­trary. To say that a and b must denote the Magnitudes of the Increments DF and DH, and not their first Ratio, or the Ratio of the Velocities in D towards F and H, is to assert a manifest Falshood. For the Velocities during the Passage of the moving Quantities from D to F and H, may be so changed that the Magni­tudes of the Increments DF and DH, shall have a Proportion very different from that of the Velocities in D towards F and H: And therefore DF and DH, how small soever, cannot measure the Fluxions of DK and DC; and conse­quently cannot be the Things expressed by a and b.

In the next place he charges me with explaining Fluxions by the Ratio of [Page 52] Magnitudes infinitely diminish'd, al­though I had expresly told him, that they are not measured or expounded by the Proportions of any generated Magnitudes whatever; but by the first or last Propor­tions of isochronal Increments generated or destroyed by Motion. The Passage which occasion'd this senseless Notion, is contain'd in the 9th Page of the Vindi­cation, and stands in these Words. ‘To obtain the last Ratio of synchronal In­crements, the Magnitudes of those In­crements must be infinitely diminish'd: For their last Ratio is the Ratio with which they vanish and become no­thing.’ To which may be added a like Passage in the 14th Page of the same Vindication, in these Words. ‘The Magnitudes of synchronal Increments must be infinitely diminish'd and be­come evanescent in order to obtain their first or last Ratios, to which Ra­tios the Ratios of their corresponding Fluxions are equal.’ Now a fair and ingenuous Reader, who had nothing at heart but the establishing of Truth and [Page 53] Science, would easily have collected from either of these Passages, that I did not explain or expound Fluxions by the Ra­tio of Magnitudes infinitely diminish'd, but by the first or last Ratios of Incre­ments generated or destroyed in equal Times; that is, by the Ratios of the Velocities with which those Increments begin or cease to exist.

In the 20th Page of the foregoing An­swer, I have asserted, that ‘the first or last Proportions of isochronal Incre­ments, subsist when the Increments themselves have no Magnitudes; for­asmuch 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.’ In Answer to this the Author of the Reasons asks his Rea­der, ‘ Whether the isochronal Incre­ments themselves subsist when they have no Magnitude? Whether there can be an Increment where there is no Increase, or Increase where [Page 54] there is no Magnitude? Whether if Magnitudes be not generated by Mo­tion in equal Times, what else is ge­nerated therein, or what else is it that Mr. Walton calls isochronal?’ To the two first of these Questions I re­ply in the negative, and to the last in the affirmative; and would fain know what he infers from thence. I did not say that isochronal Increments subsist when they have no Magnitudes, but that the first and last Ratios of such Incre­ments subsist when the Increments them­selves have no Magnitudes; and gave him the Reason why they subsist, which ac­cording to his usual Candour he has taken no notice of.

If the Lines AC and BE be generated by the Motions of two Points setting out from A and B, and be increas'd in equal Times from a Continuance of their Motions by the Increments CD and EF; these Increments will begin to exist with the Velocities which the moving Points [Page 55] have in their initial Limits at C and E;

[figure]

and the Ratio of those Velo­cities will be the Ratio with which the Increments begin to exist, that is, their first Ratio; con­sequently the first Ratio of CD and EF subsists when the Incre­ments themselves do not subsist, that is, before they have acquired any Magnitude whatever. The first Ratio of the Increments CD and EF, is not the Ratio of those Increments, nor of any Parts of them; it is not their Ratio con­sidered as existing and having finite Mag­nitudes, but the Ratio with which they begin to exist and have Magnitudes. If the Velocities in C and E be as 3 and 2, the first Ratio of the isochronal Incre­ments CD and EF, will be the Ratio of 3 to 2, and may be expounded by any two Lines which are as 3 to 2; but those Lines are not the Increments themselves nor any Parts of them. For the Velo­cities may perpetually vary in the Pro­gress of the moving Points from C and E, to D and F; and their Variation may [Page 56] be such that CD and EF, however small, may be in a very different Pro­portion from that of 3 to 2, as I have observed above: So that the Fluxions of Lines cannot be measured or ex­pounded by any Proportions of their iso­chronal Increments, except their first or last Proportions.

In order to assist the Understanding of his Correspondent he says, that in the Genesis of a Rectangle by Motion, I have suppos'd two Points to exist at the same Time in one Point, and to be moved different Ways without stirring from that Point. This he is pleas'd to call a Riddle, and says that I have the Con­science to call it a full and clear An­swer to Part of his CATECHISM. I do not indeed profess much Skill in making or explaining of Riddles, but I can see it will not be very difficult to clear this Af­fair. Let any two Lines KF and CH, intersecting each other in D, be describ'd by the Motions of two Points, setting out from K and C; ( See the Figure in p. 50.) and let the Velocities of the mo­ving [Page 57] Points be such, that they arrive at D exactly at the same Instant of Time: Upon their Arrival in D, it is plain the two Points will exist in the Point D; which he may venture to allow, since mathe­matical Points have no Magnitude: So far the Riddle is clear. And since the two Points can exist in D, as they cer­tainly may without any Absurdity, they must preserve their respective Velocities or Tendencies forward to F and H, and go on in the Directions DF and DH, by Virtue of those Velocities or Tendencies, there being nothing to change or destroy them: But they cannot go on in those Directions without stirring from the Point D: And therefore he has been mistaken in saying I supposed two Points to be moved different Ways in the same Point, without stirring from that Point. Far­ther, from the Genesis of the Rectangle CDK, it is obvious that the common In­tersection of the moving Lines IB and IA, in which the two Points are sup­posed to exist at the same Time, will continually go forward with the Lines [Page 58] themselves; but it cannot continually go forward with the Lines themselves unless it continually goes into a new Place. Surely this Author's Friends must be a­shamed of him for calling this a Riddle, which is plain to any one who has in the least considered these Things.

In Sect. 14. he honestly confesses he does not understand Sir Isaac 's Doctrine so far as to frame a positive Idea of his Fluxions; and yet from certain nega­tive Conceptions thereof he takes upon him to say that by the Genesis of a Cube I have not explain'd the Nature of se­cond, third and fourth Fluxions in a Way agreeable to that Doctrine. But in this he is mistaken. For (by Lem. 2. L. 2. Princip. Newt.) if A flows with an uni­form Velocity express'd by a, 3aA ² is the Moment of A ³, 6a ²A the Moment of 3aA ², 6a ³ the Moment of 6a ²A, and o the Moment of 6a ³: But according to Sir Isaac the Moment of the Moment of A ³, is the second Moment of A ³; the Moment of that second Moment, is its [Page 59] third Moment; and the Moment of the third Moment, is its fourth Moment: And therefore 3aA ², 6a ² A, 6a ³ and o, are the first, second, third, and fourth Moments of A ³. Farther, I have shewn that all these Moments or Motions exist and may be clearly and distinctly conceiv'd, in the Instant that a Cube generated by an uni­form Motion, begins to be increas'd by a Continuance of that Motion: And there­fore in the System of Motion whereby a Cube begins to be augmented, I have gi­ven an Explanation of first, second and third Fluxions every way agreeable to Sir Isaac's Doctrine: And he ought either to allow it or shew the contrary.

He tells his Reader (Sect. 17.) that in saying there can be no fourth Fluxion of a Cube, I make it my Business di­rectly to overthrow Sir Isaac Newton 's Doctrine. For if there can be no fourth Fluxion of a Cube, there can be no se­cond Fluxion of a Line, and a fortiori, no third, fourth or fifth Fluxion. Here it must be observ'd that one Circumstance [Page 60] which I particularly suppos'd, has not been attended to; and by not attending to it, the Reader is made to believe that, according to the Account I have given of Fluxions, there can be no second, third or fourth Fluxion of a Line. Now, in the 25th Page of the foregoing Answer, I suppos'd the Cube to be generated by an UN'FORM Motion; in which Case the Ve­locity express'd by a, will be a given Quantity: But there can be no kind of Mutation or Change of a Velocity which is given; and consequently no second Fluxion of a Line, nor any fourth Flux­ion of a Cube, which increases with such a Velocity. In this therefore I have not endeavoured directly to overthrow Sir Isaac 's Doctrine; I have not, to the great Relief of the learned World, destroy'd an indefinite Rank of Fluxions of diffe­rent Orders; nor have I given an Account of them any way inconsistent with Sir Isaac's Doctrine, as this Author has false­ly asserted.

[Page 61] He says, Sect. 18. that I give up Sir Isaac 's Doctrine of Fluxions, and in­stead thereof humorously substitute what all the World knew before he was born, to wit, the three Dimensions of a Cube and the Genesis thereof by Motion. But this is a Mistake. For I neither give up the Doctrine of Fluxions, nor substitute the Genesis of a Cube by Motion instead of it: I did not introduce that Genesis in order to set aside the Doctrine of Flux­ions; but to exhibit a System of Motion which might explain and illustrate first, second and third Fluxions, so as to make their Existence conceivable to the mean­est Capacity; and thereby to expose the Weakness and Ignorance of this Writer: And I have succeeded in both; although to his discerning Eye it may seem to be a disguising and betraying and giving up the Doctrine. According to that Doc­trine, the Fluxions and Moments of ge­nerated Quantities, are the Velocities and Motions with which the isochronal Incre­ments of those Quantities begin or cease to exist; and though the Generation of [Page 62] a Cube by Motion, might be known be­fore ever Sir Isaac thought of his Flux­ions; yet it was not known, before he publish'd the Principles of that Doctrine, that three distinct Orders of Fluxions or Moments, depending upon each other, existed in and might be clearly and di­stinctly conceiv'd, by attending to that well known and obvious System of Mo­tion, which exists in the Instant a gene­rated Cube begins to be augmented.

In Sect. 20. He intreats me to ex­plain whether Sir Isaac 's Momentum be a finite Quantity, or an Infinitesimal, or a mere Limit. I tell him, that Sir Isaac's Momentum is a FINITE QUANTITY; it is a Product contained under the mo­ving Quantity and its Velocity, or un­der the moving Quantity and first Ratio of the Space described by it in a given Particle of Time; the Velocity being measur'd by the first Ratio of that Space. Now the Velocity of the moving Quan­tity, and first Ratio of the Space de­scribed by it in a given Particle of Time, [Page 63] being both of them finite Quantities, may both be express'd by one and the same Line of a finite Magnitude; but that Line does not exist in the Quantity gene­rated or augmented by Motion. The moving Quantity exists both in the Flu­ent and in the Increment of that Fluent, being the final Limit of the former and the initial Limit of the latter: But the Line which expresses the Velocity, or first Ratio of the Space described by the moving Quantity in a given Particle of Time, does not exist either in the Fluent or in its Increment. By Moments there­fore he is not to understand generated In­crements of Fluents, but certain FINITE PRODUCTS OR QUANTITIES of a very dif­ferent Nature from generated Increments, expressing only the Motions with which those Increments begin or cease to exist.