A DISCOURSE Concerning the NATURE and CERTAINTY OF Sir ISAAC NEWTON'S METHODS OF FLUXIONS, AND OF PRIME and ULTIMATE RATIOS.
BY
BENJAMIN ROBINS,
F. R. S.
LONDON:
Printed for W. INNYS and R. MANBY, at the West-End of St.
Paul's-Church-yard.
MDCCXXXV.
[Price One Shilling and Six-pence.]
CONTENTS
INTRODUCTION: of the rise of these methods.
Page
1.
Fluxions described, and when they art velocities in a literal sense, when in a figurative, explained,
p.
3.
General definition of fluxions and fluents.
p.
6.
Wherein the doctrine of fluxions consists.
Ibid.
The fluxions of simple powers demonstrated by exhaustions.
p.
7.
The fluxion of a rectangle demonsrated by the same method.
p.
13.
The general method of finding all fluxions observed to depend on these two,
p.
20.
The application of fluxions to the drawing tangents to curve lines.
Ibid.
Their application to the mensuration of curvilinear spaces.
p.
23.
The superior orders of fluxions described.
p.
29.
Proved to exist in nature.
p.
31.
The method of assigning them.
p.
32.
The relation of the other orders of fluxions to the first demonstrated.
p.
34.
Second fluxions applied to the comparing the curvature of curves.
p.
38.
That fluxions do not imply any motion in their fluents, are the velocities only, wherewith the fluents vary in magnitude, and appertain to all subjects capable of such variation.
p.
42.
Transition to the doctrine of prime and ultimate ratios.
P.
43.
A short account of exhaustions.
p.
44.
The analogy betwixt, the method of exhaustions, and the doctrine of prime and ultimate ratios.
p.
47.
When magnitudes are considered as ultimately equal.
p.
48.
When ratios are supposed to become ultimately the same.
Ibid
The ultimate proportion of two quantities assignable, though the quantities themselves have no final magnitude.
p.
49.
What is to be understood by the ultimate ratios of vanishing quantities, and by the prime ratios of quantities at their origine.
p.
50.
This doctrine treated under a more diffusive form of expression.
p.
53.
Ultimate magnitudes defined.
Ibid.
General proposition concerning them.
p.
54.
Ultimate ratios defined.
p.
57.
General proposition concerning ultimate ratios.
Ibid.
How much of this method was known before Sir Isaac Newton.
p.
58.
This doctrine applied to the mensuration of curvilinear spaces.
p.
59.
And to the tangents of curves.
p.
64.
And to the curvature of curves.
p.
65.
That this method is perfectly geometrical and scientific.
p.
68.
Sir Isaac Newton's demonstration of his rule for finding the fluxion of a power illustrated.
p.
69.
The demonstration of his general rule for finding fluxions illustrated.
p.
71.
Conclusion, wherein is explained the meaning of the word momentum, and the perfection shewn of Sir Isaac Newton's demonstration of the momentum of a rectangle; also the essential difference between the doctrine of prime and ultimate ratios, and that of indivisibles set forth.
p.
75.
INTRODUCTION.
FROM many propositions dispersed through the writings of the ancient geometers, and more especially from one whole treatise, it appears, that the process, by which they investigated the solutions of their problems, was for the most part the reverse of the method, whereby they demonstrated those solutions. But what they have delivered upon the tangents of curve lines, and the mensuration of curvilinear spaces, does not fall under this observation; for the analysis, they made use of in these cases, is no where to be met with in their works. In later times, indeed, a method for investigating such kind of problems has been introduced, by considering all curves, as composed of an infinite number of indivisible streight lines, and curvilinear spaces, as composed in the like manner of parallelograms. But this being an obscure and indistinct conception, it was obnoxious to error.
SIR Isaac Newton therefore, to avoid the imperfection, with which this method of indivisibles was justly charg'd, instituted an analysis for these problems upon other principles. Considering magnitudes not under the notion of being increased by a repeated accession of parts, but as generated by a continued motion or flux; he discovered a method to compare together the velocities, wherewith homogeneous magnitudes increase, and thereby has taught an analysis free from all obscurity and indistinctness.
MOREOVER to facilitate the demonstrations for these kinds of problems, he invented a synthetic form of reasoning from the prime and ultimate ratios of the contemporaneous augments, or decrements of those magnitudes, which is much more concise than the method of demonstrating used in these cases by the ancients, yet is equally distinct and conclusive.
OF this analysis, called by Sir Isaac Newton his method of fluxions, and of his doctrine of prime and ultimate ratios, I intend to write in the ensuing discourse. For though Sir Isaac Newton has very distinctly explained both these subjects, the first in his treatise on the Quadrature of curves, and the other in his Mathematical principles of natural philosophy; yet as the author's great brevity has made a more diffusive illustration not altogether unnecessary; I have here endeavoured to consider more at large each of these methods; whereby, I hope, it will appear, they have all the accuracy of the strictest mathematical demonstration.
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
Newt. Introd ad Quad. Curv.
.
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 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 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 more is necessary, than to determine the velocity wherewith such lines as these are described
Page 23.
.
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
M
vel incrementorum velocitates nominando fluxiones, & quantitates genitas nominando fluentes. Newton. Introd. ad Quadr. Curv.
.
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 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.
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 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.
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
.
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.
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, 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.
THE points moving on the lines AB, CD may either move both the same way, or one forwards and the other backwards.
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 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.
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, 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.
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.
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.
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.
IN like manner by the consideration of these velocities only may the mensuration of curvilinear spaces be effected.
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.
AGAIN, if possible, let the velocity of B bear to the velocity of K a greater proportion than that 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.
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 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
Fluxionum (scilicet primarum) fluxiones seu mutationes magis aut minus celeres fluxiones secundas nominare licet, &c. Newt. Quadr. Curv. in Princip.
.
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 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.
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.
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.
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, 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.
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 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 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
× 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 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, 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 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.
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 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.
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
Princ. Philos. Lib. I. Lern. 1.
.
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
Ibid.
.
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.
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 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
Vid. Princ. Philos. pag. 37. 38.
.
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
Vid. Ibid.
.
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 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.
UPON this definition we may ground the following proposition; That, when varying magnitudes 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 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 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.
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.
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 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 abscisse then will the curvilinear space ADB be equal to the parallelogram EG.
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.
AGAIN this method is equally useful in determining 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 + 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 be =
. Now by this line HI may the curvature of curve lines be compared.
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 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.
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 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
; 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 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.
CONCLUSION.
THUS we have at length finished the whole of our design, and shall therefore put a period to this discourse with the explanation of the term momentum frequently used by Sir Isaac Newton, though we have yet had no occasion to mention it.
AND in this I shall be the more particular, because Sir Isaac Newton's definition of momenta, That they are the momentaneous increments or decrements of varying quantities, may possibly be thought obscure. Therefore I shall give a fuller delineation of them, and such a one, as shall agree to the general sense of his description, and never fail to make the use of this term, in every proposition, where it occurs, clearly to be understood.
IN determining the ultimate ratios between the contemporaneous differences of quantities, it is often previously required to consider each of these differences apart, in order to discover, how much of those differences is necessary for expressing that ultimate ratio. In this case Sir Isaac Newton distinguishes, by the name of momentum, so much of any difference, as constitutes the term used in expressing this ultimate ratio.
IN like manner, if A and B denote varying quantities, and their contemporaneous increments be represented by
a
and
b
; the rectangle under any given line M and
a
is the contemporaneous increment of the rectangle under M and A, and A ×
b
+ B ×
a
+
a
×
b
is the like increment of the rectangle under A, B. And here the whole increment M ×
a
represents the momentum of the rectangle under M, A; but A ×
b
+ B ×
a
only, and not the whole increment A ×
b
+ B ×
a
+
a
×
b,
is called the momentum of the rectangle under A, B; because so much only of this latter increment is required for determining the ultimate ratio of the increment of M × A to the increment of A × B, this ratio being the same with the ultimate ratio of M ×
a
to A ×
b
+ B ×
a
; for the ultimate ratio of A ×
b
+ B ×
a
to A ×
b
+ B ×
a
+
a
×
b
is the ratio of equality. Consequently the ultimate ratio of M ×
a
to A ×
b
+ B ×
a
differs not from the ultimate ratio of M ×
a
to A ×
b
+ B ×
a
+
a
×
b.
THESE momenta equally relate to the decrements of quantities, as to their increments, and the ultimate ratio of increments, and of decrements at the same place is the same; therefore the momentum of any quantity may be determined, either by considering the increment, or the decrement of that quantity, or even by considering both together. And in determining the momentum of the rectangle under A and B▪ Sir Isaac Newton has taken the last of these methods; because by this means the superfluous rectangle is sooner disengaged from the demonstration.
HERE it must always be remembred, that the only use, which ought ever to be made of these momenta, is to compare them one with another, and for no other purpose than to determine the ultimate or prime proportion between the several increments or decrements, from whence they are deduced
Neque spect.
magnitudo momentorum, sed prima nascentium proportio. Newt. Princ. Phil. Lib. II. Lem. 2.
. Herein the method of prime and ultimate ratios essentially differs from that of indivisibles; for in that method these momenta are considered absolutely as parts, whereof their respective quantities are actually composed. But though these momenta have no final magnitude, which would be necessary to make them parts capable of compounding a whole by accumulation; yet their ultimate ratios are as truly assignable as the ratios between any quantities whatever. Therefore none of the objections made against the doctrine of indivisibles are of the least weight against this method: but while we attend carefully to the observation here laid down, we shall be as secure against error, and the mind will receive as full satisfaction, as in any the most unexceptionable demonstration of Euclide.
WE shall make no apology for the length of this discourse: for as we can scarce suspect, after what has been above written, that our readers will be at any loss to remove of themselves, whatever little difficulties may have arisen in this subject from the brevity of Sir Isaac Newton's expressions; so our time cannot be thought misemployed, if we shall at all have contributed, by a more diffusive phrase, to the easier understanding these extensive, and celebrated inventions.
FINIS,
ERRATA.
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