OF FLUXIONS.

IN the method of fluxions geometrical mag⯑nitudes are not preſented to the mind, as compleatly formed at once, but as riſing gradually before the imagination by the mo⯑tion of ſome of their extremes ^*.

NOW, if this line DE be put in motion (ſuppoſe ſo as to keep always parallel to itſelf,) as ſoon as it

HERE it is obvious, that the velocity, wherewith the ſpace augments, is not to be underſtood lite⯑rally the degree of ſwiftneſs, with which either the line FG, or any other line or point appertaining to the curve actually moves; but as this ſpace, while the line FG moves on uniformity, will increaſe more, in the ſame portion of time, at ſome places, than at others; the terms velocity and celerity are [5] applied in a figurative ſenſe to denote the degree, wherewith this augmentation in every part proceeds.

BUT we may diveſt the conſideration of the fluxi⯑on of the ſpace from this figurative phraſe, by cau⯑ſing a point ſo to paſs over any ſtreight line IK, that the length IL meaſured out, while the line DE is moving from A to F, ſhall augment in the ſame proportion with the ſpace AFH. For this line being thus deſcribed faſter or ſlower in the ſame proportion, as the ſpace receives its aug⯑mentation; the velocity or degree of ſwiftneſs, wherewith the point deſcribing this line actually mores, will mark out the degree of celerity, where⯑with the ſpace every where increaſes. And here the line IL will preſerve always the ſame analogy to the ſpace AFH, in ſo much, that, when the line DE is advanced into any other ſituation MNO, if IP be to IL in the proportion of the ſpace AMN to the ſpace AFH, the fluxion of the ſpace at MN will be to the fluxion thereof at FH, as the velocity, wherewith the point deſcribing the line IK moves at [...], to the velocity of the ſame at L. And if any other ſpace QRST be deſcribed along with the former by the like motion, and at the ſame time a line VW, ſo that the portion VX ſhall always have to the length IL the ſame propor⯑tion, as the ſpace QRST bears to the ſpace AFH; the fluxion of this latter ſpace at TS will be to the fluxion of the former at FH, as the velocity, where⯑with the line VW is deſcribed at X, to the velocity, wherewith IK is deſcribed at L. It will hereafter appear, that in all the applications of fluxions to geo⯑metrical problems, where ſpaces are concerned, no⯑thing [6] more is neceſſary, than to determine the ve⯑locity wherewith ſuch lines as theſe are deſcribed ^*.

IN the ſame manner may a ſolid ſpace be concei⯑ved to augment with a continual flux, by the mo⯑tion of ſome plane, whereby it is bounded; and the velocity of its augmentation (which may be eſti⯑mated in like manner) will be the fluxion of that ſolid.

FLUXIONS then in general are the ve⯑locities, with which magnitudes varying by a con⯑tinued motion increaſe or diminiſh; and the mag⯑nitudes themſelves are reciprocally called the fluents of thſe fluxions ^**.

AND as different fluents may be underſtood to be deſcribed together in ſuch manner, as conſtant⯑ly to preſerve ſome one known relation to each other; the doctrine of fluxions teaches, how to aſſign at all times the proportion between the ve⯑locities, wherewith homogeneous magnitudes, vary⯑ing thus together, augment or diminiſh.

THIS doctrine alſo reaches on the other hand, how from the relation known between the fluxions, to diſcover what relation the fluents themſelves, bear to each other.

IT is by means of this proportion only, that fluxions are applied to geometrical, uſes;, for this [7] doctrine never requires any determinate degree of velocity to be aſſigned for the fluxion of any one fluent. And that the proportion between the fluxi⯑ons of magnitudes is aſſignable from the relation known between the magnitudes themſelves, I now proceed to ſhew.

FOR let any other ſituations, that theſe moving points ſhall have at the ſame inſtant of time, be ta⯑ken, either farther advanced from E and F, as at G and H, or ſhort of the ſame, as at I and K; then if EG be denoted by e, CH, the length paſſed over by the point moving on the line CD, while the point in the line AB has paſſed from A [8] to G, will be expreſſed by [...]; and if EI be denoted by e, CK, the length paſſed over by the point moving on the line CD, while the point mo⯑ving in AB has got only to I, will be denoted by [...]: or reducing each of theſe terms into a ſeries, CH will be denoted by [...] and CK by [...]. Hence all the terms of the former ſeries, except the firſt term, viz. [...] will denote FH; and all the latter ſeries, except the firſt term. viz. [...] will denote KF.

WHEN the number n is greater than unite, while the line AB is deſcribed with a uniform mo⯑tion, the point, wherewith CD is deſcribed, moves with a velocity continually accelerated; for if IE be equal to EG, FH will be greater than KF.

IN like manner KF bears to IE a leſs propor⯑tion than that, which the velocity of the point in CD has at F, to the velocity of that in AB. For as the point in CD, in moving from K to F, pro⯑ceeds with a velocity continually accelerated; with the velocity, it has acquired at F, if uniformly con⯑tinued, it would deſcribe in the ſame ſpace of time a line longer than KF.

IN the laſt place I ſay, that no line whatever, that ſhall be greater or leſs than the line repreſen⯑ted by the ſecond term of the foregoing ſeries (viz. the term [...]) will bear to the line denoted by e the ſame proportion, as the velocity, where⯑with the point moves at F, bears to the velocity of the point moving in the line AB; but that the ve⯑locity at F is to that at E as [...] to e, or as [...] to [...].

[10] IF poſſible let the velocity at F bear to the velo⯑city at E a greater ratio than this, ſuppoſe the ratio of p to q.

ON the other hand, if poſſible, let the velocity at F bear to the velocity at E a leſs ratio than that of [...] to e: let this leſſer ratio be that of r to s.

IN the ſeries whereby CK is denoted, e may be taken ſo ſmall, that any one term propoſed ſhall exceed the whole ſum of all the following terms, when added together. Therefore let e be taken ſo ſmall, that the third term [...] ex⯑ceed all the following terms [...], [...], &c. added together. But e may alſo be ſo ſmall, that the ratio of [...] to [...], the double of the third term, ſhall be greater than any ratio, [12] that can be propoſed; and the ratio of [...] to e ſhall come leſs ſhort of the ratio of [...] to e, than any other ratio, that can be named. Therefore let this ratio exceed the ratio of r to s; then the term [...] ex⯑ceeding the whole ſum of all the following terms in the ſeries denoting CK, the whole ſeries [...] or KF, will in every caſe bear to e, or EI a greater ra⯑tio than that of r to s, or of the velocity at F to the velocity at E, which is abſurd. For it has above been ſhewn, that the firſt of theſe ratios is leſs than the laſt.

IF n be leſs than unite, the point in the line CD moves with a velocity continually decreaſing; and if [...] be a negative number, this point moves back⯑wards. But in all theſe caſes the demonſtration proceeds in like manner:

THUS have we here made appear, that from the relation between the lines AE and CF, the proportion between the velocities, wherewith they are deſcribed, is diſcoverable; for we have ſhewn, that the proportion of [...] to [...] is the true proportion of the velocity, wherewith CF, or [...] augments, to the velocity, wherewith AE, or x is at the ſame time augmented.

THE points moving on the lines AB, CD may either move both the ſame way, or one for⯑wards and the other backwards.

[14] IN the firſt place ſuppoſe them to move the ſame way, advancing forward from A and C; and ſince ſome given line forms with EI a rectangle equal to that under AG and CH, ſuppoſe QT × EI = AG × CH: then, if K, L, M are contempo⯑rary poſitions of the points moving on the lines AB, CD, EF, when advanced forward beyond G, H and I; and N, O, P, three other contemporary po⯑ſitions of the ſame points, before they are arrived at G, H and I; QT × EM will alſo be = AK × CL, and QT × EP = AN × CO; therefore the rect⯑angle under IM (the difference of the lines EI and EM) and QT will be = AK × HL + CH × GK, and IP × QT = AN × HO + CH × GN.

HERE the proportion of the velocity, which the point moving on AB has at G, to that, which the point moving on CD has at H, may either keep always the ſame or continually vary, and one of theſe ve⯑locities, ſuppoſe that of the point moving on the line CD, have to the other a proportion gradually augmenting; that is, if NG and GK are equal, HL ſhall either be equal to OH or greater. Here, ſince IM × QT is = AK × HL + CH × GK, and IP × QT = AN × HO + CH × GN, where CH × GK is = CH × GN and AK × HL in both caſes greater than AN × HO, IM will be greater than IP; in ſo much that in both theſe caſes the velocity of the point, wherewith the line EF is deſcribed, win have to velocity of the point moving on AB a proportion, gradually augmenting. Here therefore the line IM will bear to GK a greater proportion, than the velocity of the point moving on the line EF, when at I, bears to the velocity of the point moving on the [15] line AB, when at G: and the line PI will have a leſs proportion to NG, than the velocity, which the point moving on the line EF, has at I, to the velocity, which the point moving on the line AB has at G.

IF poſſible let the velocity, which the point moving on EF has at I, be to the velocity, which the point moving on AB has at G, as AG × S + CH × R to the rectangle under R and ſome line QV leſs than QT.

[16] TAKE W to GK in the ratio of S to R; then will AG × S + CH × R be to R × QV as AG × W + CH × GK to QV × GK. Here, becauſe the ratio of the velocity of the point moving on the line CD to the velocity of the point moving on AB either remains conſtantly the ſame, or gradually augments, W is either equal to HL or leſs; but when it is leſs, by diminiſhing HL the ratio of W to HL may be⯑come greater than any ratio, that can be propoſed, ſhort of the ratio of equality. The like is true of die ratio of AG to AK by the diminution of GK. Therefore let GK and HL be ſo diminiſh⯑ed, that the ratio of AG × W to AK × HL ſhall be greater than the ratio of QV to QT; then the ratio of AG × W + CH × GK to AK × HL + CH × GK, that is, to QT × IM is greater than the ratio of QV to QT or of QV × IM to QT × IM; therefore AG × W + CH × GK is greater than QV × IM; and the ratio of AG × W + CH × GK to QV × GK is greater than the ratio of QV × IM to QV × GK, or of IM to GK; but the ratio of IM to GK is greater than that of the velocity, which the point moving on EF has at I, to the velocity, which the point moving on AB has at G; therefore the ratio of AG × W + CH × GK to QV × GK, or that of AG × S + CH × R to QV × R, ſtill more exceeds the ratio of the velocity at I to the velocity at G; and con⯑ſequently the ratio of the velocity at I to the velo⯑city at G is not greater than that of AG × S + CH × R to QT × R.

AGAIN, if poſſible let the velocity, which the point moving on EF has at I, be to the velocity, [17] which the point moving on AB has at G, as AG × S + CH × R to the rectangle under R and ſome line QX greater than QT.

If the points deſcribing AB and CD move backwards together, the velocity at I will be the ſame, and the demonſtration will proceed in like manner.

[20] WE have in our demonſtrations only conſidered the fluxions of lines; but by theſe the fluxions of all other quantities are determined. For we have already obſerved, that the fluxions of ſpaces, whe⯑ther ſuperficial or ſolid, are analogous to the velo⯑cities, wherewith lines are deſcribed, that augment in the ſame proportion with ſuch ſpaces.

THUS we have attempted to prove the truth of the rules, Sir Iſaac Newton has laid down, for finding the fluxions of quantities, by demonſtra⯑ting the two caſes, on which all the reſt depend, af⯑ter a method, which from all antiquity has been allowed as genuine, and univerſally acknowledged to be free from the leaſt ſhadow of uncertainty.

WE ſhall hereafter endeavour to make manifeſt, that Sir Iſaac Newton's own demonſtrations are equally juſt with theſe, we have here exhibited. But firſt we ſhall prove, that in all the applica⯑tions of this doctrine to the ſolution of geometrical problems, no other conception concerning fluxions is neceſſary, than what we have here given. And for this end it will be ſufficient to ſhew, how fluxi⯑ons are to be applied to the drawing of tangents to curve lines, and to the menſuration of curvilinear ſpaces.

HERE, I ſay, the line AC being advanced to any ſituation FG, by what has already been writ⯑ten on the nature of fluxions, without any adven⯑titious conſideration whatever, a tangent may be aſſigned to the curve at the point G.

WHEN the point moves on the line AC with an accelerating velocity, the curve DE will be con⯑vex to the abſciſſe DB. Now if two other ſitua⯑tions HI and KL of the line AC be taken, one on each ſide FG, and MGN be drawn parallel to AB; while the line AC is moving from the ſitua⯑tion HI to FG, the point in it will have moved through the length IM, and while the ſame line AC moves from FG to KL, the point in it will have paſſed over the length NL. And ſince the point moves with an accelerated velocity, IM will be leſs, and NL greater than the ſpace, which would have been deſcribed in the ſame time by the velocity, the point has at G.

WHEN the point moves on the line AC with a velocity gradually decreaſing, the curve will be concave towards the abſciſſe; but in this caſe the method of reaſoning will be ſtill the ſame.

IF the curve DE be the conical parabola, the latus rectum being T, and T × FG = DFq, or FG = [...]; the fluxion of DF will be to the fluxion of [...] (that is, the fluxion of FG) as T to 2DF; therefore OF is to FG in the ſame proportion of T to 2DF, or of DF to 2FG, and OF is half DF.

[23] IN like manner by the conſideration of theſe ve⯑locities only may the menſuration of curvilinear ſpaces be effected.

[24] GHIK to be generated at the ſame time by the motion of the line GH equal to AE or BF, inſiſt⯑ing on the line GL in an angle equal to that under CBD; and let the motion of GH be ſo regulated, that the parallelogram GHIK be always equal to the curvilinear ſpace ABC. Then it is evident, by what has been ſaid above in our explanation of the nature of fluxions, that the velocity, wherewith the parallelogram EABF increaſes, is to the ve⯑locity, wherewith the parallelogram GHIK, or wherewith the curvilinear ſpace ABC increaſes; as the velocity, wherewith the point B moves, to the velocity, wherewith the point K moves.

Now I ſay, the velocity of the point B is to the velocity of the point K as BF to BC.

SUPPOSE the curve line ACZ to recede far⯑ther and farther from AD; then it is evident, that while the parallelogram EABF augments uniform⯑ly, the curvilinear ſpace ABC will increaſe faſter and faſter; therefore in this caſe the point K moves with a velocity continually accelerated.

AGAIN, if poſſible, let the velocity of B bear to the velocity of K a greater proportion than that [27] of BF to BC, that is, the proportion of BF to ſome line S leſs than BC; and let the line TV be drawn parallel to CB, and greater than S, and the parallelogram TB be compleated. Here the ratio of the velocity of the point B to the velocity of the Point K will be greater than the ratio of BF to TV, or than the ratio of the parallelogram BW to the parallelogram BT, therefore ſtill greater than the ratio of the parallelogram BW to the cur⯑vilinear ſpace VTCB. Now if the parallelogram XYIK be taken equal to the ſpace VTCB, that the point deſcribing the line GL may have moved from X to K, while VT has moved to BC; ſince the parallelogram BW is to the parallelogram XI as VB to XK, that is, as the velocity, wherewith the point B has paſſed over VB, to the velocity, wherewith XK would be deſcribed in the ſame time with a uniform motion, the velocity of the point B bears a leſs proportion to the velocity of the point K, than the parallelogram BW bears to the paral⯑lelogram XI, becauſe XK is deſcribed with an accelerating velocity: that is, the velocity of the point B bears a leſs proportion to the velocity of the point K, than the parallelogram BW bears to the ſpace VTCB. But the firſt of thoſe ratios was before found greater than the laſt. Therefore the velocity of B does not bear to the velocity of K a greater proportion than that of BF to BC.

IF the curve line ACZ were of any other form, the demonſtration would ſtill proceed in the ſame manner.

[28] HENCE it appears, that nothing more is neceſ⯑ſary towards the menſuration of the curvilinear ſpace ABC, than to find a line GK ſo related to AB, that, while they are deſcribed together, the velo⯑city of the point, wherewith AB is deſcribed, ſhall bear the ſame proportion at any place B to the ve⯑locity, wherewith the point deſcribing the other line GK moves at the correſpondent place K, as ſome given line AE bears to the ordinate BC of the curve ACZ.

THE method of finding ſuch lines is the ſub⯑ject of Sir Iſaac Newton's Treatiſe upon the Qua⯑drature of Curves.

FOR example, if ACZ be a conical parabola as before, and Γ × BC = ABq; taking GK = [...], the parallelogram HK = [...], = ⅓ AB × BC, is equal to the ſpace ABC; for GK being equal to [...], the fluxion of GK or the velocity, wherewith it is deſcribed at K, will be to the fluxion of AB, or the velocity, wherewith B moves, as [...] or BC to GH or AE.

HAVING thus, as we conceive, ſufficiently explained, what relates to the proportions between the velocities, wherewith magnitudes are generated; nothing now remains, before we proceed to the ſe⯑cond part of our preſent deſign, but to conſider [29] the variations, to which theſe velocities are ſub⯑ject.

WHEN fluents are not augmented by a uni⯑form velocity, it is convenient in many problems to conſider how theſe velocities vary This varia⯑tion Sir Iſaac Newton calls the fluxion of the fluxi⯑on, and alſo the ſecond fluxion of the fluent; di⯑ſtinguiſhing the fluxions, we have hitherto treated of, by the name of the firſt fluxions. Theſe ſecond fluxions may alſo vary in different magnitudes of the fluent, and the variation of theſe is called the third fluxion of the fluent. Fourth fluxions are the changes to which the third are ſubject, and ſo on ^*.

HERE therefore we ſee, that while one quan⯑tity flows uniformly, the other is deſcribed with a varying motion; and the variation in this motion is called the ſecond fluxion of this quantity.

IT is evident farther, that in this inſtance, when n is = 2, the variation of the velocity is uniform: for the velocity keeping always in the ſame propor⯑tion to x, while x increaſes uniformly, the velocity muſt alſo increaſe after the ſame manner. But when n is = 3; ſince the velocity is every where as x2, and x2 does not increaſe uniformly; neither will the velo⯑city augment uniformly. So that it appears by this example, that the variation in the velocity, where⯑with magnitudes increaſe, may alſo vary, and this variation is called the third fluxion of the magni⯑tude.

IN the ſame manner may the fluxions of the fol⯑lowing orders be conceived; each order being the variation found in the preceeding one. And the conſideration of velocities thus perpetually varying, and their variation itſelf changing, is a uſeful ſpe⯑culation; for moſt, if not all, the bodies, we have [31] any acquaintance with, do actually move with ve⯑locities thus modified.

A STONE, for inſtance, in its direct fall towards the earth has its velocity perpetually augmented; and in Galileo's Theory of falling Bodies, when the whole deſcent is performed near the ſurface of the earth, it is ſuppoſed to receive equal augmen⯑tations of velocity in equal times. In this caſe therefore the velocity augments uniformly, and the ſecond fluxion of the line deſcribed by the falling bo⯑dy will in all parts of that line be the ſame; ſo that third fluxions cannot take place in this inſtance; ſince the variation of the velocity ſuffers no change, but is every where uniform.

BUT if the ſtone be ſuppoſed to have its gravity at the beginning of its fall leſs than at the ſurface of the earth, the variation of its velocity at firſt will then be leſs than the variation at the end of its motion; or in other words, the ſecond fluxions in the beginning and end of its fall would be une⯑qual; conſequently, third fluxions would here take place, ſince the variation would be ſwifter, as the body in its fall approached the earth.

THE ſtone in this laſt inſtance then not only moves with a velocity perpetually varying, as in the preceeding example, but this variation conti⯑nually changes. In the true theory of falling bo⯑dies, neither this laſt variation nor any ſubſequent one can ever be uniform; ſo that fluxions of every order do here actually exiſt.

[32] THE ſame is true of the motion of the planets in their elliptic orbs; of the motion of light at the confines of different mediums, and of the motion of all pendulous bodies.

IN ſhort, an uniform unchangeable velocity is not to be met with in any of thoſe bodies, that fall under our cogniſance; for in order to continue ſuch a motion as this, it is neceſſary, that they ſhould not be diſturbed by any force whatever, either of impulſe or reſiſtance; but we know of no ſpaces, in which at leaſt one of theſe cauſes of variation does not operate.

HAVING thus explained the general concep⯑tion of ſecond, third, and following fluxions; and having ſhewn, that they are applicable to the cir⯑cumſtances, which do really occur in all motion, we are acquainted with; we will now endeavour to declare the manner of aſſigning them.

AND in the firſt place ſecond fluxions may be compared together, as follows. Suppoſe any line to be ſo deſcribed by motion, that it always pre⯑ſerve the ſame analogy to the firſt fluxion of any magnitude; then the velocity, wherewith this line is deſcribed, that is, the fluxion of this line, will be analogous to the ſecond fluxion of the aforeſaid magnitude. For it is evident, that this line will perpetually alter in magnitude in the ſame propor⯑tion, as the fluxion, to which it is analogous, va⯑ries.

[34] In the ſame manner if a line be deſcribed ana⯑logous to the ſecond fluxion of any magnitude, the fluxion of this line will expreſs the third fluxion of that magnitude, and ſo of all the other orders of fluxions.

IN the next place the relation, in which the ſeveral orders of fluxions ſtand with regard to each other, will appear by the following propoſition.

IF now another line KL be deſcribed by the mo⯑tion of the point M, and if a ſeries of lines be a⯑dapted to this line KL in the like analogy by the motion of the points N, O, P, ſo that QN be to ED as the velocity of the point M to the velocity of the point C, RO to GF as the velocity of the point N to that of the point D, and SP to HI as the velocity of the point O to that of F; I ſay, that if the velocity of the point C has to the velo⯑city of the point M always the ſame proportion at equal diſtances from A and K, that then the ve⯑locity of D to that of N will be in the duplicate of that proportion; the velocity of F to that of O in the triplicate of that proportion; the velocity of I to that of P in the quadruplicate of that proportion, and ſo on in the ſame order, as far as theſe ſeries of lines are extended.

SUPPOSE the velocity of the point C be always to the velocity of the point M, as the line T to the line V, when theſe points are at equal diſtances from A and K. Then, ſince the times, in which equal lines are deſcribed, are reciprocally as the ve⯑locities of the deſcribing points; the time, in which AC receives any additional increment, will be to the time, in which KM ſhall have received an e⯑qual increment, as V to T.

NOW ED is always to QN in the proportion of T to V. Therefore the variation, by increaſe or diminution that ED ſhall receive to the like va⯑riation, [36] which QN ſhall receive; while the lines AC, KM are augmented by equal increments, will be alſo as T to V. But the time, wherein ED will receive that variation, to the time, wherein QN will receive its variation, will be as V to T. Con⯑ſequently, ſince the velocities, wherewith different lines are deſcribed, are as the lines themſelves di⯑rectly, and as the times of deſcription reciprocally, the velocity of the point D to that of the point N will be in the duplicate ratio of T to V.

AFTER the ſame manner, the velocity of the point I will appear to have to the velocity of the point P the quadruplicate of the ratio of T to V.

BUT from what we have ſaid above, it is evi⯑dent, that the velocity of the point D is to the ve⯑locity of the point N, as the ſecond fluxion of AC to the ſecond fluxion of KM; the velocity of the point F to the velocity of the point O, as the third fluxion of AC to the third fluxion of KM; and the velocity of the point I to the velocity of the point P, as the fourth fluxion of AC to the fourth fluxion of KM. And hence appears the truth of Sir Iſaac Newton's obſervation at the end of the firſt propoſition of his book of Quadratures, that a ſecond fluxion, and the ſecond power of a firſt fluxion, or the product under two firſt fluxions; a third fluxion, and the third power of a firſt, or the product under a firſt and ſecond, and ſo on; are homologous terms in any equation. For, as it appears by this propoſition, that if the velocity, wherewith any fluent is augmented, be in any pro⯑portion increaſed; its ſecond fluxion will increaſe in the duplicate of that proportion, the third fluxi⯑on in the triplicate, and the fourth fluxion in the quadruplicate of that ſame proportion; it is mani⯑feſt, that the terms in any equation, that ſhall in⯑volve a ſecond fluxion, will preſerve always the ſame proportion to the terms involving the ſecond power of a firſt fluxion, or the product of two firſt [38] fluxions; the terms involving a third fluxion will preſerve the ſame proportion to the terms invol⯑ving the third power of a firſt, or the product of a firſt and ſecond, or the product of three firſt fluxi⯑ons; and the terms containing a fourth fluxion will keep the ſame proportion to the terms containing the fourth power of a firſt, the product of a ſecond and the ſecond power of a firſt, the ſecond power of a ſecond, or the product of a firſt and third; &c. however be increaſed or diminiſhed the firſt fluxion, or the velocity, wherewith the fluents augment.

IN the problems concerning curve lines, which relate to the degree of curvature in any point of thoſe curves, or to the variation of their curvature in different parts, theſe ſuperior orders of fluxions are uſeful; for by the inflexion of the curve, whilſt its abſciſſe flows uniformly, the fluxion of the or⯑dinate muſt continually vary, and thereby will be attended with theſe ſuperior orders of fluxions.

FOR example, were it required to compare the different degrees of curvature either of different curves, or of the ſame curve in different parts, and in order thereto a circle ſhould be ſought, whoſe degree of curvature might be the ſame with that of any curve propoſed, in any point, that ſhould be aſſigned; ſuch a circle may be found by the help of ſecond fluxions. When the abſciſſes of two curves flow with equal velocity; where the ordinates have equal firſt fluxions, the tangent; make equal angles with their reſpective ordinates. If now the ſecond fluxions of theſe ordinates are al⯑ſo equal, the curves in thoſe points muſt be equally [39] deflected from their tangents, that is, have equal degrees of curvature. Upon this principle ſuch circles, as have here been mentioned, may be found by the following method.

Now ſuppoſe the line NO to be ſo deſcribed, that the fluxion of MI, or of x, ſhall be to the firſt fluxion of IF, as ſome given line e to NP in the line NO, then will NP be = [...]. Suppoſe likewiſe the lines QR to be ſo deſcribed, that the fluxion of AL in the curve ABC ſhall be to the firſt fluxion of LB, as the ſame given line e to QS in the line QR. Here the firſt fluxions of IF and LB being equal, NP and QS are equal. And ſince the ſecond fluxions of IF and LB are equal, the fluxions of NP and QS are alſo equal. But NP was = [...], and by the rules for finding fluxions, the fluxion of NP will be to the fluxion of MI as eaa to [...], that is, as e × EMq to IFc. Therefore in the curve ABC the fluxion of QS to the fluxion of AL will be in the ſame proportion of e × EMq to IFc. Hence by finding firſt QS, then its fluxion, from the e⯑quation expreſſing the nature of the curve ABC, the proportion of e × EMq to IF c will be given. Con⯑ſequently the proportion of e to IF will be alſo given, becauſe the ratio of EMq to IF q is the ſame with the given ratio of HFq to HIq, or of KBq to KLq. And hereby the circle EFG will [41] be given, whoſe curvature is equal to the curvature of the curve ABC at the point B.

THIS is all we think neceſſary towards giving a juſt and clear idea of the nature of fluxions, and for proving the certainty of the deductions made from them. For it muſt now be manifeſt to every reader, that mathematical quantities become the proper object of this doctrine of fluxions, when⯑ever they are ſuppoſed to increaſe by any continued motion of prolongation, dilatation, expanſion or other kind of augmentation, provided ſuch augmen⯑tation be directed by ſome general rule, whence the meaſure of the increaſe of theſe quantities may from time to time be eſtimated. And when different homogeneous magnitudes increaſe after this manner together, one may vary faſter than another. Now the velocity of increaſe in each quantity, is the fluxion of that quantity. This is the true inter⯑pretation of Sir Iſaac Newton's appellation of fluxi⯑ons, Incrementorum velocitates. For this doctrine does not ſuppoſe the fluents themſelves to have any motion. Fluxions are not the velocities, with which the fluents, or even the increments, which thoſe fluents receive, are themſelves moved; but the degrees of velocity, wherewith thoſe increments are generated. Subjects incapable of local motion, ſuch as fluxions themſelves, may alſo have their fluxions. In this we do not aſcribe to theſe fluxi⯑ons any actual motion; for to aſcribe motion, or velocity to what is itſelf only a, velocity, would be wholly unintelligible. The fluxion of another fluxion is only a variation in the velocity, which is [43] that fluxion. In ſhort, light, heat, ſound, the mo⯑tion of bodies, the power of gravity, and whatever elſe is capable of variation, and of having that va⯑riation aſſigned, for this reaſon comes under the preſent doctrine; nothing more being underſtood by the fluxion of any ſubject, than the degree of ſuch its variation.

TO aſſign the velocities of variation or increaſe in different homogeneous quantities, it is neceſſary to compare the degrees of augmentation, which thoſe quantities receive in equal portions of time; and in this doctrine of fluxions no farther uſe is made of ſuch increments: for the application of this doctrine to geometrical problems depends upon the know⯑ledge of theſe velocities only. But the conſidera⯑tion of the increments themſelves may be made ſubſervient to the like uſes upon other principles; the explanation of which leads us to the ſecond part of our deſign.

OF PRIME and ULTIMATE RATIOS.

THE primary method of comparing together the magnitudes of rectilinear ſpaces is by lay⯑ing them one upon another: thus all the right lined ſpaces, which in the firſt book of Euclide are pro⯑ved to be equal, are the ſum or difference of ſuch ſpaces, as would cover one another. This method cannot be applied in comparing curvilinear ſpaces with rectilinear ones; becauſe no part whatever of a curve line can be laid upon a ſtreight line, ſo as wholly to coincide with it. For this purpoſe there⯑fore the ancient geometers made uſe of a method of reaſoning, ſince commonly called the method of exhauſtions; which conſiſts in deſcribing upon the curvilinear ſpace a rectilinear one, which though not equal to the other, yet might differ leſs from it than by any the moſt minute difference whatever, that ſhould be propoſed; and thereby proving, the two ſpaces, they would compare, could be neither greater nor leſs than each other.

HOWEVER, the triangle may be proved not to be leſs than the circle by the circumſcribed polygon alſo. For were it leſs, another triangle DEG, whoſe baſe EG is greater than EF, might be ta⯑ken, which ſhould not be greater than the circle. But a polygon can be circumſcribed about the cir⯑cle, the circumference of which ſhall exceed the circumference of the circle by leſs than any line, that [47] can be named, conſequently by leſs than FG, that is, the circumference of the polygon ſhall be leſs than EG, and the polygon leſs than the triangle DEG; therefore it is impoſſible, that this triangle ſhould not exceed the circle, ſince it is greater than the polygon: conſequently the triangle DEF cannot be leſs than the circle.

THUS the circle and triangle may be proved to be equal by comparing them with one polygon only, and Sir Iſaac Newton has inſtituted upon this principle a briefer method of conception and expreſſion for demonſtrating this ſort of propoſitions, than what was uſed by the ancients; and his ideas are equally diſtinct, and adequate to the ſubject, with theirs, though more complex. It became the firſt writers to chooſe the moſt ſimple form of expreſſion, and the leaſt compounded ideas poſſible. But Sir Iſaac Newton thought, he ſhould oblige the mathematicians by uſing brevity, provided he introduced no modes of conception difficult to be comprehended by thoſe, who are not unſkilled in the ancient methods of writing.

THE conciſe form, into which Sir Iſaac New⯑ton has caſt his demonſtrations, may very poſſibly create a difficulty of apprehenſion in the minds of ſome unexerciſed in theſe ſubjects. But otherwiſe his method of demonſtrating by the prime and ultimate ratios of varying magnitudes is not only juſt, and free from any defect in itſelf; but eaſily to be comprehended, at leaſt by thoſe who have made theſe ſubjects familiar to them by reading the ancients.

[48] IN this method any fix'd quantity, which ſome varying quantity, by a continual augmentation or diminution, ſhall prepetually approach, but ne⯑ver paſs, is conſidered as the quantity, to which the varying quantity will at laſt or ultimately be⯑come equal; provided the varying quantity can be made in its approach to the other to differ from it by leſs than by any quantity how minute ſoever, that can be aſſigned ^*.

UPON this principle the equality between the fore-mentioned circle and triangle DEF is at once deducible. For ſince the polygon circumſcribing the circle approaches to each according to all the conditions above ſet down, this polygon is to be conſidered as ultimately becoming equal to both, and conſequently they muſt be eſteemed equal to each other.

THAT this is a juſt concluſion, is moſt evident. For if either of theſe magnitudes be ſuppoſed leſs than the other, this polygon could not approach to the leaſt within ſome aſſignable diſtance.

RATIOS alſo may ſo vary, as to be confined after the ſame manner to ſome determined limit, and ſuch limit of any ratio is here conſidered as that, with which the varying ratio will ultimately coin⯑cide ^**.

[49] FROM any ratio's having ſuch a limit, it does not follow, that the variable quantities exhibiting that ratio have any final magnitude, or even limit, which they cannot paſs.

FOR ſuppoſe two magnitudes, B and B + A, whoſe difference ſhall be A, are each of them per⯑petually increaſing by equal degrees. It is evident, that if A remains unchanged, the proportion of B + A to B is a proportion, that tends nearer and nearer to the proportion of equality, as B becomes larger; it is alſo evident, that the proportion of B + A to B may, by taking B of a ſufficient mag⯑nitude, be brought at laſt nearer to the proportion of equality, than to any other aſſignable propor⯑tion; and conſequently the ratio of equality is to be conſidered as the ultimate ratio of B + A to B. The ultimate proportion then of theſe quantities is here aſſigned, though the quantities themſelves have no final magnitude.

THE ſame holds true in decreaſing quantities.

HERE theſe quadrilaterals can never bear one to the other the proportion between AB and BE, nor have either of them any final magnitude, or even ſo much as a limit, but by the diminution of the diſtance between DF and AE they diminiſh continually without end: and the proportion be⯑tween AB and BE is for this reaſon called the ultimate proportion of the two quadrilaterals, be⯑cauſe it is the proportion, which thoſe quadrila⯑terals can never actually have to each other, but the limit of that proportion.

THE quadrilaterals may be continually dimini⯑ſhed, either by dividing BC in any known propor⯑tion in G drawing HGI parallel to AE, by di⯑viding again BG in the like manner, and by con⯑tinuing this diviſion without end; or elſe the line DF may be ſuppoſed to advance towards AE with an uninterrupted motion, 'till the quadrila⯑terals quite diſappear, or vaniſh. And under this latter notion theſe quadrilaterals may very proper⯑ly [51] be called vaniſhing quantities, ſince they are now conſidered, as never having any ſtable magnitude, but decreaſing by a continued motion, 'till they come to nothing. And ſince the ratio of the quadrila⯑teral ABCD to the quadrilateral BEFC, while the quadrilaterals diminiſh, approaches to that of AB to BE in ſuch manner, that this ratio of AB to BE is the neareſt limit, that can be aſſigned to the other; it is by no means a forced conception to conſider the ratio of AB to BE under the notion of the ratio, wherewith the quadrilaterals vaniſh; and this ratio may properly be called the ultimate ratio of two vaniſhing quantities.

THE reader will perceive, I am endeavouring to explain Sir Iſaac Newton's expreſſion Ratio ulti⯑ma quantitatum evaneſcentium; and I have ren⯑dered the Latin participle evaneſcens, by the En⯑gliſh one vaniſhing, and not by the word evaneſcent, which having the form of a noun adjective, does not ſo certainly imply that motion, which ought here to be kept carefully in mind. The quadri⯑laterals ABCD, BEFC become vaniſhing quan⯑tities from the time, we firſt aſcribe to them this perpetual diminution; that is, from that time they are quantities going to vaniſh. And as during their diminution they have continually different propor⯑tions to each other; ſo the ratio between AB and BE is not to be called merely Ratio harum quan⯑titatum evaneſcentium, but Ultima ratio ^*.

HERE I have attempted to explain in like man⯑ner the phraſe Ratio prima quantitatum naſcentium; but no Engliſh participle occuring to me, whereby to render the word naſcens, I have been obliged to uſe circumlocution. Under the preſent concep⯑tion of the quadrilaterals they are to be called naſ⯑cantes, not only at the very inſtant of their firſt pro⯑duction, but according to the ſenſe, in which ſuch participles are uſed in common ſpeech, after the ſame manner, as when we ſay of a body, which has lain at reſt, that it is beginning to move, though it may have been ſome little time in mo⯑tion: on this account we muſt not uſe the ſimple expreſſion Ratio quantitatum naſcentium; for by this we ſhall not ſpecify any particular ratio; but to denote the ratio between AB and BE we muſt call it Ratio prima quadrilaterûm naſcentium ^*.

[53] WE ſee here the ſame ratio may be called ſome⯑times the prime, at other times the ultimate ratio of the ſame varying quantities, as theſe quantities are conſidered either under the notion of vaniſhing, or of being produced before the imagination by an uninterrupted motion. The doctrine under exami⯑nation receives its name from both theſe ways of epxpreſſion.

THUS we have given a general idea of the manner of conception, upon which this doctrine is built. But as in the former part of this diſcourſe we confirmed the doctrine of fluxions by demonſtra⯑tions of the moſt circumſtantial kind; ſo here, to remove all diſtruſt concerning the concluſiveneſs of this method of reaſoning, we ſhall draw out its firſt principles into a more diffuſive form.

FOR this purpoſe, we ſhall in the firſt place define an ultimate magnitude to be the limit, to which a varying magnitude can approach within any degree of nearneſs whatever, though it can never be made abſolutely equal to it.

THUS the circle diſcourſed of above, accord⯑ing to this definition, is to be called the ultimate magnitude of the polygon circumſcribing it; be⯑cauſe this polygon, by increasing the number of its ſides, can be made to differ from the circle, leſs than by any ſpace, that can be propoſed how ſmall [54] ſoever; and yet the polygon can never become ei⯑ther equal to the circle or leſs.

IN like manner the circle will be the ultimate magnitude of the polygon inſcribed, with this dif⯑ference only, that as in the firſt caſe the vary⯑ing magnitude is always greater, here it will be leſs than the ultimate magnitude, which is its limit.

UPON this definition we may ground the fol⯑lowing propoſition; That, when varying magni⯑tudes [55] keep conſtantly the ſame proportion to each other, their ultimate magnitudes are in the ſame proportion.

LET A and B be two varying magnitudes, which keep conſtantly in the ſame proportion to each other; and let C be the ultimate magnitude of A, and D the ultimate magnitude of B. I ſay that C is to D in the ſame proportion.

NOW, if poſſible, let the ratio of C to D be greater than that of A to B, that is, ſuppoſe C to have to ſome magnitude E, greater than D, the ſame proportion as A has to B. Since C is the ul⯑timate magnitude of A in the ſenſe of the preceed⯑ing definition, A can be made to approach nearer to C than by any difference, that can be propoſed, but can never become equal to it, or leſs. There⯑fore, ſince C is to E as A to B, B will always ex⯑ceed E; conſequently can never approach to D ſo near as by the exceſs of E above D: which is ab⯑ſurd. For ſince D is ſuppoſed the ultimate mag⯑nitude [56] of B, it can be approached by B nearer than by any aſſignable difference.

AFTER the ſame manner, neither can the ratio of D to C be greater than that of B to A.

IF the varying magnitude A be leſs than C, it follows, in like manner, that neither the ratio of C to D can be leſs than that of A to B, nor the ratio of D to C leſs than that of B to A.

IT is an evident corollary from this propoſition, that the ultimate magnitudes of the ſame or equal varying magnitudes are equal.

NOW from this propoſition the fore-mentioned equality between the circle and triangle DEF will again readily appear. For the circle being the ul⯑timate magnitude of the polygon, and the triangle DEF the ultimate magnitude of the triangle DEG; ſince the polygon and the triangle DEG are equal, by this propoſition the circle and triangle DEF will be alſo equal.

IF the preceeding propoſition be admitted, as a genuine deduction from the definition, upon which it is grounded; this demonſtration of the equality of the circle and triangle cannot be excepted to. For no objection can lie againſt the definition itſelf, as no inference is there deduced, but only the ſenſe explained of the term defined in it.

[57] THE other part of this method, which concerns varying ratios, may be put into the ſame form by defining ultimate ratios as follows.

IF there be two quantities, that are (one or both) continually varying, either by being continually augmented, or continually diminiſhed; and if the proportion, they bear to each other, does by this means perpetually vary, but in ſuch a manner, that it conſtantly approaches nearer and nearer to ſome determined proportion, and can alſo be brought at laſt in its approach nearer to this determined pro⯑portion than to any other, that can be aſſigned, but can never paſs it: this determined proportion is then called the ultimate proportion, or the ulti⯑mate ratio of thoſe varying quantities.

TO this definition of the ſenſe, in which the term ultimate ratio, or ultimate proportion is to be underſtood, we muſt ſubjoin the following pro⯑poſition: That all the ultimate ratios of the ſame varying ratio are the ſame with each other.

SUPPOSE the ratio of A to B continually varies by the variation of one or both of the terms A and B. If the ratio of C to D be the ultimate ratio of A to B, and the ratio of E to F be likewiſe the ultimate ratio of the ſame; I ſay, the ratio of C to D is the ſame with the ratio of E to F.

[58] IF poſſible, let the ratio of E to F differ from that of C to D. Since the ratio of C to D is the ultimate ratio of A to B, the ratio of A to B, in its approach to that of C to D, can be brought at laſt nearer to it, than to any other whatever. Therefore if the ratio of E to F differ from that of C to D, the ratio of A to B will either paſs that of E to F, or can never approach ſo near to it, as to the ratio of C to D: in ſo much that the ratio of E to F cannot be the ultimate ratio of A to B, in the ſenſe of this definition.

THE two definitions here ſet down, together with the general propoſitions annexed to them, com⯑prehend the whole foundation of this method, we are now explaining.

WE find in former writers ſome attempts toward ſo much of this method, as depends upon the firſt definition. Lucas Valerius in a moſt excellent trea⯑tiſe on the Center of gravity of ſolid bodies, has given a propoſition nothing different, but in the form of the expreſſion, from that we have ſubjoin⯑ed to our firſt definition; from which he demon⯑ſtrates with brevity and elegance his propoſitions con⯑cerning the menſuration and center of gravity of the ſphere, ſpheroid, parabolical and hyperbolical co⯑noids. This author writ before the doctrine of in⯑diviſibles was propoſed to the world. And ſince, Andrew Tacquet, in his treatiſe on the Cylindrical and annular ſolids, has made the ſame propoſition, though ſomething differently expreſſed, the baſis [59] of his demonſtrations at the ſame time very judi⯑ciouſly expoſing the inconcluſiveneſs of the reaſon⯑ings from indiviſibles. However, the conſideration of the limits of varying proportions, when the quan⯑tities expreſſing thoſe proportions have themſelves no limits, which renders this method of prime and ultimate ratios much more extenſive, we owe in⯑tirely to Sir Iſaac Newton. That this method, as thus compleated, is applicable not only to the ſub⯑jects treated by the ancients in the method of ex⯑hauſtions, but to many others alſo of the greateſt importance, appears from our author's immortal treatiſe on the Mathematical principles of natural philoſophy.

HOWEVER, we ſhall farther illuſtrate this doctrine in applying it to the ſame general pro⯑blems as thoſe, whereby the uſe of fluxions was above exemplified.

WE have already given one inſtance of its uſe in determining the dimenſions of curvilinear ſpaces; we ſhall now ſet forth the ſame by a more general ex⯑ample.

LET ABC be a curve line, its abſciſſe AD, and an ordinate DB. If the parallelogram EFGH, formed upon the given line EF under the ſame angle, as the ordinates of the curve make with its abſciſſe, be in all parts ſo related to the curve, that the ultimate ratio of any portion of the abſciſſe AD to the correſpondent portion of the line EH, ſhall be that of the given line EF to the ordinate of the curve at the beginning of that portion of the [60] abſciſſe then will the curvilinear ſpace ADB be equal to the parallelogram EG.

[63] SUPPOSE the curve ABC were a cubical para⯑bola convex to the abſciſſe, that is, ſuppoſe Θ a given line, and Θq × LM = AL c. If EH be = [...] × EF, then the parallelogram EG will be equal to the ſpace ADB.

As EH is = [...], ER will be = [...] and ET = [...], conſequently RT = [...]. Therefore the parallelogram EG is here ſo related in all parts to the curve, that LN is to RT as Θq × EF to ALc + ¼ ALq × LN + AL × LNq + ¼ LNc. Now it is evident, that the ratio of LN to RT can never be ſo great as the ratio of Θq × EF to ALc; but yet, by diminiſhing LN, the ratio of LN to RT may at laſt be brought nearer to this ratio than to any other whatever, that ſhould be propo⯑ſed. Conſequently by the preceeding definition of what is to be underſtood by an ultimate ratio, the ratio of Θq × EF to ALc is the ultimate ratio of LN to RT. But ALc being = Θq × LM, Θq × EF is to ALc as EF to LM. Therefore the ratio of EF to LM is the ultimate ratio of LN to RT. Conſequently, by the preceeding ge⯑neral propoſition, the parallelogram EG is equal to the curvilinear ſpace ADB. And this paral⯑lelogram is equal to ¼ AD × DB.

[64] AGAIN this method is equally uſeful in deter⯑mining the ſituation of the tangents to curve lines.

SUPPOSE the curve ABC again to be a cubical parabola, where BF is = [...], and GH = [...]. Here HK will be = [...]; therefore HK is to FG, or BK, as 3 AF × AG + FGq to Zq. Conſequently the ratio of HK to BK can never be ſo ſmall as the ratio of 3AFq to Zq; but by diminiſhing BK it may be brought nearer to that ratio, than to any other whatever; that is, the ratio of 3AFq to Zq is the ultimate ratio of HK to KB. Therefore, if BF bear to FE the ratio of 3AFq to Zq, the line BE will touch the curve in B: and EF will be equal to ⅓ AF.

AFTER the ſituation of the tangent has been thus determined, the magnitude of HI, the part of the ordinate intercepted between the tangent and the curve, will be known. For example, in this inſtance ſince BF is to FE, that is IK to FG, as 3AFq to Zq, IK will be = [...], and HK being = [...], HI will [66] be = [...]. Now by this line HI may the curvature of curve lines be compared.

SUPPOSE the curve CBD to be the cubical pa⯑rabola as before, where Zq × FB is = CFc, then KG will be = [...]. Hence LK (= [...]) is = [...]. But it is evident, that in a given ſituation of the tangent AB the ratio of BKq to FIq is given; therefore LK will be reciprocally as 3CF + FI, and will continually increaſe, as the point G approaches to the point B, but can never be ſo great, as to equal [...]; yet by the near approach of G to B, LK may be brought nearer to this quantity [68] than by any difference, that can be propoſed. Therefore, by our former definition of ultimate magnitudes, [...] is the ultimate mag⯑nitude of LK. Conſequently, if BM be taken equal to this [...], the circle deſcribed through M is that required.

WE have now gone through all, we think need⯑ful for illuſtrating the doctrine of prime and ulti⯑mate ratios; and by the definitions, we have given of ultimate magnitudes and proportions, compared with the inſtances, we have ſubjoined, of the application of this doctrine to geometrical problems, we hope our readers cannot fail of forming ſo diſtinct a con⯑ception of this method of reaſoning, that it ſhall appear to them equally geometrical and ſcientific with the moſt unexceptionable demonſtration.

THEREFORE we ſhall in the next place pro⯑ceed to conſider the demonſtrations, which Sir Iſaac Newton has himſelf given, upon the principles of this method, of his precepts for aſſigning the fluxi⯑ons of flowing quantities.

OF Sir ISAAC NEWTON'S METHOD Of demonſtrating his Rules for finding FLUXIONS.

SIR Iſaac Newton has comprehended his di⯑rections for computing the fluxions of quan⯑ties in two propoſitions; one in his Introduction to his treatiſe on the Quadrature of curves; the other is the firſt propoſition of the book itſelf. In the firſt he aſſigns the fluxion of a ſimple power, the latter is univerſal for all quantities whatever.

IF now the augment BE be denoted by o, the augment DF will be denoted by [...]. And here it is obvious, that all the terms after the firſt taken together may be made leſs than any aſſignable part of the firſt. Conſequently the proportion of the firſt term [...] to the whole augment may be made to approach within any degree whatever of the proportion of equality; and therefore the ulti⯑mate proportion of [...] to o, or of DF to BE, is that of [...] only to o, or the proportion of [...] to 1.

AND we have already proved, that the proportion of the velocity at D to the velocity at B is the ſame with the ultimate proportion of DF to BE; there⯑fore the velocity at D is to the velocity at B, or the fluxion of xn to the fluxion of x, as [...] to 1.

IN the firſt propoſition of the treatiſe of Qua⯑dratures the author propoſes the relation betwixt three varying quantities x, y, and z to be expreſſed by this equation [...]. Suppoſe theſe qnantities to be augmented by any contem⯑poraneous [72] increments great or ſmall. Let us alſo ſuppoſe ſome quantity o to be deſcribed at the ſame time by ſome known velocity, and let that velocity be denoted by m; the velocity, wherewith the augment of x would be uniformly deſcribed in that time be denoted by ẋ; the velocity, wherewith the augment of y would be uniformly deſcribed in the ſame time by ẏ; and laſtly the velocity, where⯑with the augment of z would be uniformly deſcri⯑bed in the ſame time by ż. Then [...], [...], and [...] will expreſs the contemporaneous increments of x, y, and z reſpectively. Now when x is become [...], y is become [...] and z become [...]; the former equation will become [...]. Here, as the firſt of theſe equations exhi⯑bits the relation between the three quanties x, y, z, as far as the ſame can be expreſſed by a ſingle equa⯑tion; ſo this ſecond equation, with the aſſiſtance of the firſt, will expreſs the relation between the aug⯑ments of theſe quantities. But the firſt of theſe equa⯑tions may be taken out of the latter; whence will ariſe this third equation [...] [73] [...]; which alſo expreſſes the relation between the ſeveral increments; and like⯑wiſe if o be a given quantity, this equation will e⯑qually expreſs the relation between the velocities, wherewith theſe ſeveral increments are generated reſpectively by a uniform motion. And this equa⯑tion being divided by o will be reduced to more ſimple terms, and yet will equally expreſs the rela⯑tion of theſe velocities; and then the equation will become [...]. Now let us form an equation out of the terms of this, from which the quantity o is abſent. This e⯑quation will be [...]; and this equation multiplied by m becomes [...]. It is evident, that this equation does not expreſs the relation of the forementioned velocities; yet by the diminution of o this equation may come within any degree of expreſſing that relation. Therefore, by what has been ſo often inculcated, this equation will expreſs the ultimate relation of theſe velocities. But the fluxions of the quantities x, y, z are the ultimate magnitudes of theſe velocities; ſo that the ultimate relation of theſe velocities is the relation of the fluxions of theſe quantities. Conſequently this laſt [74] equation repreſents the relation of the fluxions of the quantities x, y, z.

IT is now preſumed, we have removed all dif⯑ficulty from the demonſtrations, which Sir Iſaac Newton has himſelf given, of his rules for finding fluxions.

IN the beginning of this diſcourſe we have en⯑deavoured at ſuch a deſcription of fluxions, as might not fail of giving a diſtinct and clear conception of them. We then confirmed the fundamental rules for comparing fluxions together by demonſtrations of the moſt formal and unexceptionable kind. And now having juſtified Sir Iſaac Newton's own de⯑monſtrations, we have not only ſhewn, that his doctrine of fluxions is an unerring guide in the ſo⯑lution of geometrical problems, but alſo that he himſelf had fully proved the certainty of this me⯑thod. For accompliſhing this laſt part of our un⯑dertaking it was neceſſary to explain at large ano⯑ther method of reaſoning eſtabliſhed by him, no leſs worthy conſideration; ſince as the firſt inabled him to inveſtigate the geometrical problems, where⯑by he was conducted in thoſe remote ſearches into nature, which have been the ſubject of univerſal admiration, ſo to the latter method is owing the ſurprizing brevity, wherewith he has demonſtrated thoſe diſcoveries.