OF FLUXIONS.

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

THUS the line AB may be conceived to be tra­ced out gradu­ally by a point moving on from A to B,

[figure]

either with an equable motion, or with a velocity in any manner varied. And the velocity, or degree of swiftness, with which this point moves in any part of the line, AB, is called the fluxion of this line at that place.

AGAIN, suppose two lines AB and AC to form a space unbounded toward BC; and upon AB a line DE to be erected.

[figure]

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

[Page 4] has passed the point A, a space bounded on all sides will begin to appear between these three lines. For instance, when DE is moved into the situa­tion FG, these three lines will include the space AFH.

[figure]

Here it is evident, that this space will increase fa­ster or slower, ac­cording to the degree of velocity, where­with the line DE shall move. It is also evi­dent, that though the line DE should move with an even pace, the space AFH would not for that reason in­crease equably; but where the line AC was farthest distant from AB, the space AFH would increase fastest. Now the velocity or celerity, wherewith the space AFH at all times increases, is called the fluxion of that space.

HERE it is obvious, that the velocity, wherewith the space augments, is not to be understood lite­rally 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 [Page 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 fluxi­on of the space from this figurative phrase, by cau­sing 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 aug­mentation; the velocity or degree of swiftness, wherewith the point describing this line actually mores, will mark out the degree of celerity, where­with 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 propor­tion, 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, where­with 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 geo­metrical problems, where spaces are concerned, no­thing [Page 6] more is necessary, than to determine the ve­locity wherewith such lines as these are described ^*.

IN the same manner may a solid space be concei­ved to augment with a continual flux, by the mo­tion of some plane, whereby it is bounded; and the velocity of its augmentation (which may be esti­mated in like manner) will be the fluxion of that solid.

FLUXIONS then in general are the ve­locities, with which magnitudes varying by a con­tinued motion increase or diminish; and the mag­nitudes themselves are reciprocally called the fluents of thse fluxions ^**.

AND as different fluents may be understood to be described together in such manner, as constant­ly to preserve some one known relation to each other; the doctrine of fluxions teaches, how to assign at all times the proportion between the ve­locities, wherewith homogeneous magnitudes, vary­ing 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 [Page 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 fluxi­ons of magnitudes is assignable from the relation known between the magnitudes themselves, I now proceed to shew.

IN the first place, let us suppose two lines AB and CD to be de­scribed toge­ther by two points,

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one setting out from A, and the other from C, and to move in such manner, that if AE and CF are lengths described in the same time, CF shall be analogous to some power of AE, that is, if AE be denoted by the letter x, then CF shall always be denoted by [...]; where a represents some given line, and n any num­ber whatever. Here, I say, the proportion be­tween the velocity of the point moving on AB to the velocity of that moving on CD, is at all times assignable.

FOR let any other situations, that these moving points shall have at the same instant of time, be ta­ken, 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 [Page 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 mo­ving 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 mo­tion, 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.

NOW here, I say, that neither the propor­tion of FH to EG, nor the proportion of KF to IE is the proportion of the velocity, which the point moving on CD has at F, to the uniform [Page 9] velocity, wherewith the point moves on the line AB. For, while that point is advanced from E to G,

[figure]

the point mov­ing on CD has passed from F to H, and has moved through that space with a velocity continually accelerated; therefore, if it had moved during the same interval of time with the velocity, it has at F, uniformly continued, it would not have passed over so long a line; con­sequently FH bears a greater proportion to EG, than what the velocity, which the point moving on CD has at F, bears to the velocity of the point moving uniformly on AB.

IN like manner KF bears to IE a less propor­tion 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, pro­ceeds with a velocity continually accelerated; with the velocity, it has acquired at F, if uniformly con­tinued, 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 represen­ted 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, where­with the point moves at F, bears to the velocity of the point moving in the line AB; but that the ve­locity at F is to that at E as [...] to e, or as [...] to [...].

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

IN the series, whereby CH is denoted, the line e can be taken so small, that any term proposed in the series shall exceed all the following terms toge­ther;

[figure]

so that the double of that term shall be greater than the whole collection of that term, and all that follow. Again, by diminishing e, the ratio of the second term in this series to twice the third, that is, of [...] to [...] or the ratio of x to [...], shall be greater than any, that shall be proposed; consequently the line e may be taken so small, that twice the third term, that is [...], shall be greater than all the terms following the second, and also, that the ratio of [...] to e shall less exceed the ratio of [...] to e, than any other ratio, that can be proposed. Therefore let the ratio of [...] to e be less than the ratio of p to q; then, if [...] be also greater [Page 11] than the third and all the following terms of the series, the ratio of the series [...] to e, that is, the ratio of FH to EG shall be less than the ratio of p to q, 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 greater than the last. Therefore the velocity at F cannot bear to the velocity at E any greater proportion than that of [...] to e.

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 [...] ex­ceed 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, [Page 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 [...] ex­ceeding 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 ra­tio 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 back­wards. 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.

[Page 13] AGAIN, in the three lines AB, CD, EF, where the points A, C, E are given, let us suppose G, H and I to be three contemporary positions of the points, whereby the three lines AB, CD, EF are respectively described; and let the motion of the point describing the line EF be so regulated with regard to the motion of the other two points, that

[figure]

the rectangle under EI and some given line may be always equal to the rectangle under AG and CH. Here from the velocities, or degrees of swift­ness, wherewith the points describing AB and CD move, the degree of swiftness, wherewith the point describing EF moves, may be determined.

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

[Page 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 contempo­rary 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 po­sitions 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 rect­angle 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 ve­locities, 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 [Page 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.

NOW let R be to S as the velocity, which the point moving on AB has at G, to the velocity, which the point moving on CD has at H; then I say, that the velocity, which the point moving on EF has at I, will be to the velocity, which the point moving on AB has at G, as AG × S + CH × R to QT × R.

[figure]

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.

[Page 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 be­come 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 diminish­ed, 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 con­sequently the ratio of the velocity at I to the velo­city 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, [Page 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.

HERE let Y be to NG as S to R; then will AG × S + CH × R be to R × QX as AG × Y + CH × NG to QX × NG. But Y will be either greater than HO, or equal to it, and when greater, by diminishing HO, the ratio of Y to HO may become less than any ratio, that can be proposed, greater than the ratio of equality. The like is

[figure]

true of the ratio of AG to AN by the diminution of NG. Therefore let NG and HO be so dimi­nished, that the ratio of AG × Y to AN × HO shall be less than the ratio of QX to QT; then the ratio of AG × Y + CH × NG to AN × HO + CH × NG, that is, to QT × IP, is less than the ratio of QX to QT, or of QX × IP to [Page 18] QT × IP. Consequently AG × Y + CH × NG is less than QX × IP, and the ratio of AG × Y + CH × NG to QX × NG is less than the ratio of QX × IP to QX × NG, or of IP to NG. But the ratio of IP to NG is less 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 × Y + CH × NG to QX × NG, or that of AG × S + CH × R to QX × R, is also less than the ratio of the velocity at I to the velocity at G. Consequently, the ratio of the velocity at I to the velocity at G is not less than that of AG × S + CH × R to QT × R.

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.

BUT if one of the points, as that moving on CD, recedes, while the other on AB advances for­ward, take in CD any fix'd point at pleasure Z; then the point on CD in respect of Z moves also forward. Again, take in the line EF, EΓ to AG as CZ to QT; then AG × CZ is = QT × EΓ; and AG × CH being = QT × EI, AG × HZ will be = QT × ΓI; and by the preceeding case AG × S + ZH × R will be to QT × R as the ve­locity, wherewith the point moving on EF sepa­rates from Γ, when at I, to the velocity, which the point moving on AB has at G. But as AG is continually increasing, and EΓ keeps always in the same proportion to AG; the point Γ will it­self be in motion, and the velocity of the point Γ [Page 19] will be to the velocity at G, as the line EΓ to AG, that is, as CZ to QT, or as CZ × R to QT × R; therefore the velocity, wherewith the point moving on EF, when at I, separates from Γ, being to the velocity of the point moving on AB, when at G, as AG × S + ZH × R to QT × R; the absolute velocity, which the point moving on EF has at I, will be to the absolute velocity, which the point moving on AB has at G, as AG × S ∽ CH × R to QT × R; moving backwards, when

[figure]

it separates from Γ swifter than the point Γ itself moves, that is, when AG × S + ZH × R is greater than CZ × R, or AG × S greater than CH × R; and when the point moving on EF, at I separates from Γ with a slower motion, than that wherewith Γ moves, that is, when CZ × R is greater than AG × S + ZH × R, or AG × S less than CH × R, the point moving on EF, at I advances forward.

[Page 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, whe­ther superficial or solid, are analogous to the velo­cities, 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 demonstra­ting the two cases, on which all the rest depend, af­ter 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 applica­tions 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 fluxi­ons are to be applied to the drawing of tangents to curve lines, and to the mensuration of curvilinear spaces.

IF upon the line AB be erected in any angle ano­ther streight line AC, and it be put in motion upon the line AB towards B keeping always pa­rallel [Page 21] to itself, and proceeding on with a uniform velocity: if a point also moves on the line AC with a velocity in any manner regulated;

[figure]

this point will describe within the angle under CAB some third line DE, which will be a curve, unless the point moves in the line AC likewise with a uniform motion.

HERE, I say, the line AC being advanced to any situation FG, by what has already been writ­ten on the nature of fluxions, without any adven­titious 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 con­vex to the abscisse DB. Now if two other situa­tions 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 situa­tion 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.

[Page 22] LET FO be taken to FG in the proportion of the velocity, wherewith the point F moves on the line AB,

[figure]

to the velocity, which the point mov­ing on the line FP has at G, and the streight line OGQ be drawn, cutting HI in R, and KL in S; then FH will be to MR, and FK to NS in the same proportion. Therefore, from what has been said above, MR will be greater than MI, and NS less than NL; so that the line OQ, which unites with the curve at the point G, lies on both sides the point G, on the same side of the curve; that is, it does not cross, or cut the curve (as geometers speak) but touches it only at the point 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 = DF q, 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.

[Page 23] IN like manner by the consideration of these ve­locities only may the mensuration of curvilinear spaces be effected.

SUPPOSE the curvilinear space ABC to be gene­rated by the parallel motion of the line BC upon the line AD with a uniform velocity, within the space comprehended between the streight line AD and

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the curve line AZ; and let the parallelogram AEFB be generated with it by the motion of BF accompanying BC. Suppose another paral­lelogram

[Page 24] GHIK to be generated at the same time by the motion of the line GH equal to AE or BF, insist­ing 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 ve­locity, 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 far­ther and farther from AD; then it is evident, that while the parallelogram EABF augments uniform­ly, the curvilinear space ABC will increase faster and faster; therefore in this case the point K moves with a velocity continually accelerated.

HERE, if possible, suppose the velocity of the point B to bear a less proportion to the velocity of the point K, than the ratio of BF to BC; that is, let the velocity of B be to the velocity of K, as BF to some line M greater than BC. Then it is possible to draw within the curve ACZ towards D a line, as ON, parallel to BC, which, though it exceed BC, shall be less than M; and will the ratio of the velo­city of the point B to the velocity of the Point K, [Page 25] be less than the ratio of BF to NO, or than the ratio of the parallelogram BP to the parallelo­gram BO; therefore still less than the ratio of the parallelogram BP to the space BCON. Farther let the parallelogram KIRQ be taken equal to the space BCON, then will the point K have moved from K to Q in the time, that the point B has

[figure]

moved from B to N. Now the parallelogram BP is to the parallelogram KR as BN to KQ, that is, as the velocity, wherewith the point B passes over BN, to the velocity, wherewith KQ would be described in the same time with a uniform motion. But as the point K moves with a velocity con­tinually [Page 26] accelerated, its velocity at K is less than this uniform velocity now spoken of; therefore the velocity of the point B bears a greater proportion to the velocity of the point K than the parallelo­gram BP bears to the parallelogram KR; that is, than the parallelogram BP bears to the space BCON; but the first of these ratios was before found less than the last; which involves an absurdity. Therefore the velocity of B bears not to the velocity of K a less proportion than that of BF to BC.

[figure]

AGAIN, if possible, let the velocity of B bear to the velocity of K a greater proportion than that [Page 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 cur­vilinear 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 paral­lelogram 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.

[Page 28] HENCE it appears, that nothing more is neces­sary 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 velo­city of the point, wherewith AB is described, shall bear the same proportion at any place B to the ve­locity, 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 sub­ject of Sir Isaac Newton's Treatise upon the Qua­drature of Curves.

FOR example, if ACZ be a conical parabola as before, and Γ × BC = AB q; 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 se­cond part of our present design, but to consider [Page 29] the variations, to which these velocities are sub­ject.

WHEN fluents are not augmented by a uni­form velocity, it is convenient in many problems to consider how these velocities vary This varia­tion Sir Isaac Newton calls the fluxion of the fluxi­on, and also the second fluxion of the fluent; di­stinguishing 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 ^*.

IN the two fluents AE and CF, whose fluxions we compared at page 7, &c. where AE being deno­ted by x, CF was equal to [...], and the fluxion of AE bore to the fluxion of CF the proportion of [...] to [...].

[figure]

Here it is evident, that the antecedent [...] of this proportion being a fix'd quantity, and the consequent [...] a variable one; the fluxion of AE does not bear to the fluxion of CF always the same proportion. If n be the number 2, the fluxion of AE is to the fluxion of CF as a to the [Page 30] variable quantity 2 x; and if n be the number 3, the fluxion of AE to that of CF will be as a2 to 3 x2. Therefore if AE be described with an uni­form velocity, when n is any number greater than unite, CF is so described with a velocity continually accelerating, that when n is = 2, this velocity aug­ments in the same proportion as CF itself increases; and when n is = 3, it augments in the duplicate of that proportion, &c.

HERE therefore we see, that while one quan­tity 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 propor­tion 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 velo­city augment uniformly. So that it appears by this example, that the variation in the velocity, where­with magnitudes increase, may also vary, and this variation is called the third fluxion of the magni­tude.

IN the same manner may the fluxions of the fol­lowing 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 spe­culation; for most, if not all, the bodies, we have [Page 31] any acquaintance with, do actually move with ve­locities 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 augmen­tations of velocity in equal times. In this case therefore the velocity augments uniformly, and the second fluxion of the line described by the falling bo­dy 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 une­qual; 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 conti­nually changes. In the true theory of falling bo­dies, neither this last variation nor any subsequent one can ever be uniform; so that fluxions of every order do here actually exist.

[Page 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 concep­tion of second, third, and following fluxions; and having shewn, that they are applicable to the cir­cumstances, 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 pre­serve 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 propor­tion, as the fluxion, to which it is analogous, va­ries.

[Page 33] SUPPOSE AB to be a fluent described with a varying motion; the second fluxion at any one point C may be compa­red with the second fluxion at any other point D,

[figure]

by causing the line EF to be described by the motion of a point, so as to keep always the same analogy to the first fluxion of the fluent AB. Suppose EG be to EH, as the first fluxion at C to the first fluxion at D; then the second fluxion at C will be to the second fluxion at D, as the first fluxion of the line EF at G, to the first fluxion of the same at H.

IN like manner, if another fluent IK be genera­ted along with the former fluent AB, and also de­scribed with a va­riable motion;

[figure]

the second fluxion of this latter fluent IK at any place L may be compared with the se­cond fluxion at any part of the former fluent AB, by describing the line MN with such a motion, as always to preserve the same analogy to the first fluxion of the fluent IK, as the line EF bore to the first fluxion of AB. Suppose MO to be to EG, as the first fluxion of IK at L to the first fluxion of AB at C; then the second fluxion at L will be to the second fluxion at C, as the velocity, wherewith the line MN is described at O, to the velocity, wherewith the line EF is described at G.

[Page 34] In the same manner if a line be described ana­logous 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.

LET the line AB be described by the motion of the point C moving with a varying velocity, and let a series of lines be adapted to this line AB in such manner, that the point D, moving upon the first line of this series at the same time with the point C, may ever terminate a line ED analogous

[figure]

to the velocity of the point C; the point F at the same time terminating upon the second line of this se­ries a line GF analogous to the velocity of the point D; and HI upon the third line being by the motion [Page 35] of the point I made ever analogous to the velocity of the point F; &c.

IF now another line KL be described by the mo­tion of the point M, and if a series of lines be a­dapted 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 velo­city of the point M always the same proportion at equal distances from A and K, that then the ve­locity 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 ve­locities 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 e­qual 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 va­riation, [Page 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. Con­sequently, since the velocities, wherewith different lines are described, are as the lines themselves di­rectly, 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.

AGAIN, the velocity of D being to the velocity of N, when AC and KM are equal, always in the same duplicate ratio of T to V, and GF being al­ways

[figure]

to RO as the velocity of the point D to the velocity of the point N, the variation, by increase or diminution, of the line GF to the like variation of RO, while AC and KM receive equal aug­mentation, will also be as the velocity of D to the velocity of N, that is in the duplicate ratio of T to V. But the time, in which the line GF receives its variation, will be to the time, in which RO re­ceives its variation, as V to T. Hence the velocity [Page 37] of the point F will be to the velocity of the point O in the triplicate 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 evi­dent, that the velocity of the point D is to the ve­locity 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 pro­portion increased; its second fluxion will increase in the duplicate of that proportion, the third fluxi­on in the triplicate, and the fourth fluxion in the quadruplicate of that same proportion; it is mani­fest, that the terms in any equation, that shall in­volve 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 [Page 38] fluxions; the terms involving a third fluxion will preserve the same proportion to the terms invol­ving the third power of a first, or the product of a first and second, or the product of three first fluxi­ons; 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 or­dinate 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 al­so equal, the curves in those points must be equally [Page 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.

THE curve ABC being given, let it be required to find a circle equally incurvated with this curve at the point B. Suppose EFG to be this circle, in which the tangent FH at the point F makes with the or­dinate FI the same angle, as the tangent BK, drawn to the other curve ABC at the point B, makes with

[figure]

the ordinate BL of that curve. Now if the two ab­scisses AL and EI are described with equal veloci­ties, the first fluxion of the ordinates LB and IF will be equal; and therefore, if the two curves are equally incurvated at the points B and F, the second fluxi­ons of these ordinates will be also equal. If M be [Page 40] the center of the circle EFG, and ME be deno­ted by a and MI by x, IF will be = [...]; and, by the rules for finding fluxions, the first fluxion of IF will be to the fluxion of MI, or of x, as x to [...].

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 × EM q to IF c. Therefore in the curve ABC the fluxion of QS to the fluxion of AL will be in the same proportion of e × EM q to IF c. Hence by finding first QS, then its fluxion, from the e­quation expressing the nature of the curve ABC, the proportion of e × EM q to IF c will be given. Con­sequently the proportion of e to IF will be also given, because the ratio of EM q to IF q is the same with the given ratio of HF q to HI q, or of KB q to KL q. And hereby the circle EFG will [Page 41] be given, whose curvature is equal to the curvature of the curve ABC at the point B.

SUPPOSE the curve ABC to be the conical pa­rabola, where AL q shall be equal to γ × LB, γ be­ing the latus rectum of the axis. Here e will be to QS as γ to 2 AL; for that is the ratio of the fluxi­on of AL to the fluxion of BL: therefore QS is

[figure]

= [...] AL, and consequently the fluxion of QS to the fluxion of AL (that is e × EM q to IF c) as 2 e to γ, or as 2 e × EM q to γ × EM q; in so much that IF c is = ½ γ × EM q, and the given ratio of IF q to EM q (namely the ratio of KL q to KB q) is the same with the ratio of ½ γ to IF: that is, IF is equal to half the latus rectum apper­taining [Page 42] to the diameter of the parabola, whose ver­tex is 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, when­ever they are supposed to increase by any continued motion of prolongation, dilatation, expansion or other kind of augmentation, provided such augmen­tation 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 inter­pretation of Sir Isaac Newton's appellation of fluxi­ons, 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 fluxi­ons 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 [Page 43] that fluxion. In short, light, heat, sound, the mo­tion of bodies, the power of gravity, and whatever else is capable of variation, and of having that va­riation 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 know­ledge of these velocities only. But the considera­tion 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 lay­ing them one upon another: thus all the right lined spaces, which in the first book of Euclide are pro­ved 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 there­fore 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.

[Page 45] FOR example, in order to prove the equality be­tween the space comprehended within the circum­ference of a circle, and a triangle, whose base should be equal to the circumference of that circle, and its altitude to the semidiameter, Archimedes takes this method. About the circle he describes a polygon as ABC, and makes it appear, that by multiplying the sides of this polygon, there may at length be

[figure]

described such an one, as shall exceed the circle less than by any difference, that shall be proposed, how minute soever that difference be. By this means it was easy to prove, that the triangle DEF, whose base EF should be equal to the circumference of the circle, and altitude ED equal to the semidia­meter, is not greater than the circle. For were it greater, how small soever be the excess, it were pos­sible to describe about the circle a polygon less than the triangle; but the circumference of the polygon is greater than the circumference of the circle, there­fore the polygon can never be less, but must be always greater than the triangle; for the polygon [Page 46] is equal to a triangle, whose altitude is the semidiame­ter of the circle, and base equal to the circumference of the polygon. It appears therefore impossible for the triangle DEF to be greater than the circle.

THUS far Archimedes makes use of the polygon circumscribing the circle and no farther: but inscri­bing another within the circle he proves, by a simi­lar

[figure]

process of reasoning, that it is impossible for the triangle to be less than the circle; whereby at length it becomes certain, that the triangle DEF is neither greater nor less than the circle, but equal to it.

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 ta­ken, which should not be greater than the circle. But a polygon can be circumscribed about the cir­cle, the circumference of which shall exceed the circumference of the circle by less than any line, that [Page 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 New­ton 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.

[Page 48] IN this method any fix'd quantity, which some varying quantity, by a continual augmentation or diminution, shall prepetually approach, but ne­ver pass, is considered as the quantity, to which the varying quantity will at last or ultimately be­come 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 coin­cide ^**.

[Page 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 per­petually 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 mag­nitude, be brought at last nearer to the proportion of equality, than to any other assignable propor­tion; 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.

THE quadrilateral ABCD bears to the quadri­lateral EBCF the proportion of AB + DC to BE + CF, provided the two lines AE and DF are parallel. Now if the line DF be drawn nearer to AE, this proportion of AB + DC to BE + CF will not remain the same, unless the lines DA, CB, FE produced will meet in the same point; and this proportion, by diminishing the distance be­tween [Page 50] DF and AE may at last be brought nearer to the proportion of AB to BE, than to any other whatever.

[figure]

Therefore the proportion of AB to BE is to be considered as the ultimate proportion of AB+DC to BE+CF, or as the ultimate pro­portion of the quadrilateral ABCD to the quadri­lateral EBCF.

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 be­tween AB and BE is for this reason called the ultimate proportion of the two quadrilaterals, be­cause it is the proportion, which those quadrila­terals can never actually have to each other, but the limit of that proportion.

THE quadrilaterals may be continually dimini­shed, either by dividing BC in any known propor­tion in G drawing HGI parallel to AE, by di­viding again BG in the like manner, and by con­tinuing this division without end; or else the line DF may be supposed to advance towards AE with an uninterrupted motion, 'till the quadrila­terals quite disappear, or vanish. And under this latter notion these quadrilaterals may very proper­ly [Page 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 quadrila­teral 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 ulti­ma quantitatum evanescentium; and I have ren­dered the Latin participle evanescens, by the En­glish 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 quadri­laterals ABCD, BEFC become vanishing quan­tities 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 propor­tions to each other; so the ratio between AB and BE is not to be called merely Ratio harum quan­titatum evanescentium, but Ultima ratio ^*.

[Page 52] SHOULD we suppose the line DF first to coin­cide with the line AE, and then recede from it, by that means giving birth to the quadrilaterals;

[figure]

un­der this conception the ra­tio of AB to BE may very justly be considered as the ratio, wherewith the quadrilaterals by this motion commence; and this ratio may also properly be called the first or prime ratio of these quadrilaterals at their origine.

HERE I have attempted to explain in like man­ner 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 concep­tion of the quadrilaterals they are to be called nas­cantes, not only at the very instant of their first pro­duction, 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 mo­tion: 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 ^*.

[Page 53] WE see here the same ratio may be called some­times 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 exami­nation 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 demonstra­tions 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, accord­ing to this definition, is to be called the ultimate magnitude of the polygon circumscribing it; be­cause 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 [Page 54] soever; and yet the polygon can never become ei­ther equal to the circle or less.

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

AGAIN the triangle DEF is the ultimate mag­nitude of the triangle DEG; because the base EG, being always equal to the circumference of the po­lygon,

[figure]

will constantly be greater than the base EF, equal to the circumference of the circle only, and yet EG may be made to approach EF nearer than by any difference, that can be named.

UPON this definition we may ground the fol­lowing proposition; That, when varying magni­tudes [Page 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.

AS A is a varying magnitude continually ap­proaching to C, but can never become equal to it, A may be either always greater or always less than C.

[figure]

In the first place suppose it greater. When A is greater than C, in approaching to C it is con­tinually diminished; therefore B keeping always in the same pro­portion to A, B in approaching to its limit D is also continually diminished.

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 ul­timate magnitude of A in the sense of the preceed­ing 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. There­fore, since C is to E as A to B, B will always ex­ceed E; consequently can never approach to D so near as by the excess of E above D: which is ab­surd. For since D is supposed the ultimate mag­nitude [Page 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 ul­timate 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.

[Page 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 pro­portion than to any other, that can be assigned, but can never pass it: this determined proportion is then called the ultimate proportion, or the ulti­mate 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 pro­position: 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.

[Page 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, com­prehend 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 trea­tise 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 subjoin­ed to our first definition; from which he demon­strates with brevity and elegance his propositions con­cerning the mensuration and center of gravity of the sphere, spheroid, parabolical and hyperbolical co­noids. This author writ before the doctrine of in­divisibles 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 [Page 59] of his demonstrations at the same time very judi­ciously exposing the inconclusiveness of the reason­ings from indivisibles. However, the consideration of the limits of varying proportions, when the quan­tities expressing those proportions have themselves no limits, which renders this method of prime and ultimate ratios much more extensive, we owe in­tirely to Sir Isaac Newton. That this method, as thus compleated, is applicable not only to the sub­jects treated by the ancients in the method of ex­haustions, 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 pro­blems 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 ex­ample.

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 [Page 60] abscisse then will the curvilinear space ADB be equal to the parallelogram EG.

IN the curve let the abscisse AD be divided into any number of equal parts AI, IL, LN, ND, and let the ordinates IK, LM, NO be drawn, and also in the parallelogram EG the correspondent

[figure]

lines PQ, RS and TV. In the curve compleat the parallelograms IW, LX, NY, and in the parallelogram EG make the parallelogram PZ e­qual to the parallelogram IW, the parallelogram [Page 61] RΓ equal to LX, and the parallelogram TΔ e­qual to NY: then the whole figure IKWMXOYD will be equal to the whole Figure PZΓΔH. But in the curve, by increasing the number and di­minishing the breadth of these parallelograms, the figure IKWMXOYD will approach nearer and nearer in magnitude to the curvilinear space ADB; in so much that their difference may be reduced to less than any space, that shall be assigned; there­fore the curvilinear space ADB is the ultimate magnitude of the figure IKWMXOYD. Far­ther, since the parallelogram EG is in all parts so related to the curve, that the ultimate ratio of every portion, as LN, of the abscisse AD to RT, the correspondent portion of EH, is the same with the ratio of EF or RS, to LM; the ultimate ratio of the parallelogram LX, or its equal RΓ, to the parallelogram RV, is the ratio of equality. This is also true of all the other correspondent pa­rallelograms; therefore, the ultimate ratio of the figure PZΓΔH to the parallelogram PG is the ratio of equality; that is, the figure PZΓΔH, by increasing the number of its parallelograms, can be brought nearer to the parallelogram PG than by any difference whatever, that may be proposed. Moreover, by increasing of the number of ordi­nates in the curve, the residuary portion AI of the abscisse can be reduced to less than any magni­tude, that shall be proposed; whereby the paral­lelogram EQ, corresponding to this portion of the abscisse, may be also reduced to less than any magnitude, that can be proposed; and the paral­lelogram PG be brought to differ less from EG than by any assignable magnitude. Since therefore the [Page 62] figure PZΓΔH can be brought nearer to the pa­rallelogram PG than by any difference, that can be assigned; the Figure PZΓΔH can be brought also

[figure]

nearer to the parallelogram EG than by any differ­ence, that can be assigned. Consequently the pa­rallelogram EG is the ultimate magnitude of the figure PZΓΔH. Therefore the figures PZΓΔH and IK WMXOYD being equal varying mag­nitudes, and the ultimate magnitudes of equal varying magnitudes being equal, the curvilinear space ADB is equal to the parallelogram EG.

[Page 63] SUPPOSE the curve ABC were a cubical para­bola convex to the abscisse, that is, suppose Θ a given line, and Θ q × 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 Θ q × EF to AL c + ¼ AL q × LN + AL × LN q + ¼ LN c. Now it is evident, that the ratio of LN to RT can never be so great as the ratio of Θ q × EF to AL c; 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 propo­sed. Consequently by the preceeding definition of what is to be understood by an ultimate ratio, the ratio of Θ q × EF to AL c is the ultimate ratio of LN to RT. But AL c being = Θ q × LM, Θ q × EF is to AL c as EF to LM. Therefore the ratio of EF to LM is the ultimate ratio of LN to RT. Consequently, by the preceeding ge­neral proposition, the parallelogram EG is equal to the curvilinear space ADB. And this paral­lelogram is equal to ¼ AD × DB.

[Page 64] AGAIN this method is equally useful in deter­mining the situation of the tangents to curve lines.

IN the curve ABC, whose abscisse is AD, let EB be a tangent at the point B. Let BF be the ordinate at the same point B, and GH another or­dinate paral­lel to it, which shall meet the tan­gent in I, and the line BK, parallel to the abscisse AD, in K.

[figure]

Here the ra­tio of HK, the difference of the ordinates, to BK can never be the same with the ratio of BF to FE, unless by the figure of the curve the tangent chance to cut it in some point remote from B; this ratio of BF to FE being the same with that of IK to KB. But it is farther evident, that the nearer GH is to FB, the ratio of KH to KB will approach so much the nearer to the ratio of IK to KB; and the angle, which the curve BC makes with the tangent BI being less than any right-lined an­gle, it is manifest, that GH may be made to ap­proach towards FB, 'till the ratio of HK to KB, shall at last approach nearer to the ratio of IK to KB, or of BF to FE, than to any other ratio whatever, that shall be proposed; that is, the ra­tio of BF to FE is the ultimate ratio of HK to [Page 65] KB. Therefore, if from the properties of the curve ABC the ratio of HK to KB be determined, and from thence their ultimate ratio assigned; this ratio thus assigned will be the ratio of BF to FE; because all the ultimate ratios of the same variable ratio are the same with each other.

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 + FG q to Z q. Consequently the ratio of HK to BK can never be so small as the ratio of 3AF q to Z q; but by diminishing BK it may be brought nearer to that ratio, than to any other whatever; that is, the ratio of 3AF q to Z q is the ultimate ratio of HK to KB. Therefore, if BF bear to FE the ratio of 3AF q to Z q, 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 3AF q to Z q, IK will be = [...], and HK being = [...], HI will [Page 66] be = [...]. Now by this line HI may the curvature of curve lines be compared.

LET the streight line AB touch the curve CBD in the point B; CE being the abscisse of the curve, and BF the ordinate at B. Take any other point G in the curve, and through the points G, B de­scribe the circle BGH, that shall touch the line AB

[figure]

in B; lastly, draw IKGL parallel to FB. Here are two angles formed at the point B with the circle, one by the line BK, the other by the curve; and the pro­portion of the first of these angles to the second will be different in different distances of the point G from the point B. And by the approach of G to B the angle between the circle and curve will be dimini­shed, even so much as at length to bear a less pro­portion to the angle between the circle and tangent, than any, that can be proposed. That is, by the [Page 67] approach of the point G to B the angle between the tangent and circle may be brought nearer to the an­gle between the tangent and the curve, than by any difference how minute soever homogeneous to those angles; therefore the magnitude of the circle being continually varied by the gradual approach of G to B, and the angle between the tangent and circle thereby also varied; the angle between the tangent and curve is the ultimate magnitude of these angles. That is, the ultimate of these circles determines the degree of curvature of the curve CBD at the point B. But in the circle the rectangle under LKG is equal to the square of BK. And whereas the mag­nitude of KL will perpetually vary by the ap­proach of the point G towards B; if BM taken in FB produced be the ultimate magnitude of KL, the circle described through M and B to touch the tangent AK in B will be the circle, by which the cur­vature of the curve CBD in B is to be estimated.

SUPPOSE the curve CBD to be the cubical pa­rabola as before, where Z q × FB is = CF c, then KG will be = [...]. Hence LK (= [...]) is = [...]. But it is evident, that in a given situation of the tangent AB the ratio of BK q to FI q 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 [Page 68] than by any difference, that can be proposed. Therefore, by our former definition of ultimate magnitudes, [...] is the ultimate mag­nitude 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 need­ful for illustrating the doctrine of prime and ulti­mate 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 con­ception 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 pro­ceed to consider the demonstrations, which Sir Isaac Newton has himself given, upon the principles of this method, of his precepts for assigning the fluxi­ons of flowing quantities.

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

SIR Isaac Newton has comprehended his di­rections for computing the fluxions of quan­ties 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.

FOR determining the fluxion of a simple power suppose the line AB to be de­noted by x, and another line CD to be de­noted by [...], or by considering a as unite, CD will be denoted by xn.

[figure]

[Page 70] SUPPOSE the points B and D to move in equal spaces of time into two other positions E and F; then DF will be to BE in the ratio of the velocity, wherewith DF would be described with an uniform motion, to the velocity, wherewith BE will be descri­bed in the same time with an uniform motion. But if the point de­scribing the line AB moves uni­formly; the ve­locity, where­with the line CD is described, will not be uniform.

[figure]

Therefore the space DF is not described with a uniform velocity; in so much that the velocity, wherewith DF would be uniformly described, is never the same with the velocity at the point D. But by diminishing the magnitude of DF, the uni­form velocity, wherewith DF would be described, may be made to approach at pleasure to the velo­city at the point D. Therefore the velocity at the point D is the ultimate magnitude of the velocity, wherewith DF would be uniformly described. Consequently the ratio of the velocity at D to the velocity at B is the ultimate ratio of the velocity, wherewith DF would be uniformly described, to the velocity, wherewith BE is uniformly described. But DF being to BE as the velocity, wherewith DF would be uniformly described, to that, where­with BE is uniformly described, the ultimate ratio of DF to BE is also the ultimate ratio of the first of these velocities to the last; because all the ulti­mate ratios of the same varying ratio are the same with each other. Therefore the ratio of the velo­city [Page 71] at D to the velocity at B, that is, of the fluxi­on of CD to the fluxion of AB, is the same with the ultimate ratio of DF to BE.

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 ulti­mate 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; there­fore 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 Qua­dratures 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 contem­poraneous [Page 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, where­with the augment of z would be uniformly descri­bed 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 exhi­bits the relation between the three quanties x, y, z, as far as the same can be expressed by a single equa­tion; so this second equation, with the assistance of the first, will express the relation between the aug­ments of these quantities. But the first of these equa­tions may be taken out of the latter; whence will arise this third equation [...] [Page 73] [...]; which also expresses the relation between the several increments; and like­wise if o be a given quantity, this equation will e­qually express the relation between the velocities, wherewith these several increments are generated respectively by a uniform motion. And this equa­tion being divided by o will be reduced to more simple terms, and yet will equally express the rela­tion 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 e­quation 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 [Page 74] equation represents the relation of the fluxions of the quantities x, y, z.

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

IN the beginning of this discourse we have en­deavoured 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 de­monstrations, we have not only shewn, that his doctrine of fluxions is an unerring guide in the so­lution of geometrical problems, but also that he himself had fully proved the certainty of this me­thod. For accomplishing this last part of our un­dertaking it was necessary to explain at large ano­ther method of reasoning established by him, no less worthy consideration; since as the first inabled him to investigate the geometrical problems, where­by 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.