FLUXION
, in the Newtonian analysis, denotes the rate or proportion at which a flowing or varying quantity increases its magnitude or quantity; and it is proportional to the magnitude by which the flowing quantity would be uniformly increased, in a given time, by the generating quantity continuing of the invariable magnitude it has at the moment of time for which the Fluxion is taken: by which it stands contradistinguished from fluent, or the flowing quantity, which is gradually, and indefinitely increasing, after the manner of a space which a body in motion describes.
Mr. Simpson observes, that there is an advantage in thus considering Fluxions, not as mere velocities of increase at a certain point, but as the magnitudes which would be uniformly generated in a given sinite time: the imagination is not here confined to a single point, and the higher orders of Fluxions are rendered much more easy and intelligible. And though Sir Isaac Newton defines Fluxions to be the velocities of motions, yet he has recourse to the moments or increments, generated in equal particles of time, to determine those velocities, which he afterwards directs to expound by finite magnitudes of other kinds.
As to the illustration of this definition, and the rules for finding the Fluxions of all sorts of fluent quantities, see the following article, or the Method of Fluxions.
Method of Fluxions, is the algorithm and analysis of Fluxions, and fluents or flowing quantities.
Most foreigners define this as the method of differences or differentials, being the analysis of indefinitely small quantities. But Newton, and other English authors, call these infinitely small quantities, moments; considering them as the momentary increments of variable quantities; as of a line considered as generated by the flux or motion of a point, or of a surface generated by the flux of a line. Accordingly, the variable quantities are called Fluents, or flowing quantities; and the method of sinding either the Fluxion, or the fluent, the method of Fluxions.
M. Leibnitz considers the same infinitely small quantities as the differences, or differentials, of quantities; and the method of finding those differences, he calls the Differential Calculus.
Besides this difference in the name, there is another in the notation. Newton expresses the Fluxion of a quantity, as of x, by a dot placed over it, thus x.; while Leibnitz expresses his differential of the same <*>, by prefixing the initial letter d, as dx. But, setting aside these circumstances, the two methods are just alike.
The Method of Fluxions is one of the greatest, most subtle, and sublime discoveries of perhaps any age: it opens a new world to our view, and extends our knowledge, as it were, to infinity; carrying us beyond the bounds that seemed to have been prescribed to the human mind, at least infinitely beyond those to which the ancient geometry was confined.
The history of this important discovery, recent as it is, is a little dark, and embroiled. Two of the greatest men of the last age have both of them claimed the invention, Sir I. Newton, and M. Leibnitz; and nothing can be more glorious for the method itself, than the zeal with which the partisans of either side have asserted their title.
To exhibit a just view of this dispute; and of the pretensions of each party, we may here advert to the origin of the discovery, and mark where each claim commenced, and how it was supported.
The principles upon which the Method of Fluxions is founded, or which conducted to it, had been laying, and gradually developing, from the beginning of the last century, by Fermat, Napier, Barrow, Wallis, Slusius, &c, who had methods of drawing tangents, of maxima and minima, of quadratures, &c, in certain particular cases, as of rational quantities, upon nearly the same principles. And it was not wonderful that such a genius as Newton should soon after raise those faint beginnings into a regular and general system of science, which he did about the year 1665, or sooner.
The first time however that the method appeared in print, was in 1684, when M. Leibnitz gave the rules of it in the Leipsic Acts of that year; but without the demonstrations. The two brothers however, John and James Bernoulli, being greatly struck with this new method, applied themselves diligently to it, found out the demonstrations, and applied the calculus with great success.
But before this, M. Leibnitz had proposed his Differential Method, viz, in a letter, dated Jan. 21, 1677, in which he exactly pursues Dr. Barrow's method of tangents, which had been published in 1670: and Newton communicated his method of drawing tangents to Mr. Collins, in a letter dated Dec. 10, 1672; which letter, together with another dated June 13, 1676, were sent to Mr. Leibnitz by Mr. Oldenburgh, in 1676. So that there is a strong presumption that he might avail himself of the information contained in these letters, and other papers transmitted with them, and also in 1675, before the publication of his own letter, containing the first hint of his differential method. Indeed it sufficiently appears that Newton had invented his method before the year 1669, and that he actually made use of it in his Compendium of Analysis and Quadrature of Curves before that time. His attention seems to have been directed this way, even before the time of the plague which happened in London in 1665 and 1666, when he was about 28 years of age.
This is all that is heard of the method, till the year 1687, when Newton's admirable Principia came out, | which is almost wholly built on the same calculus. The common opinion then was, that Newton and Leibnitz had each invented it about the same time: and what seemed to confirm it was, that neither of them made any mention of the other; and that, though they agreed in the substance of the thing, yet they differed in their ways of conceiving it, calling it by different names, and using different characters. However, foreigners having first learned the method through the medium of Leibnitz's publication, which spread the method through Europe, those geometricians were insensibly accustomed to look upon him as the sole, or principal inventor, and became ever after strongly prejudiced in favour of his notation and mode of conceiving it.
The two great authors themselves, without any seeming concern, or dispute, as to the property of the invention, enjoyed the glorious prospect of the progresses continually making under their auspices, till the year 1699, when the peace began to be disturbed.
M. Facio, in a treatise on the Line of Swiftest Descent, declared, that he was obliged to own Newton as the first inventor of the differential calculus, and the first by many years; and that he left the world to judge, whether Leibnitz, the second inventor, had taken any thing from him. This precise distinction between first and 2d inventor, with the suspicion it insinuated, raised a controversy between M. Leibnitz, supported by the editors of the Leipfic Acts, and the English mathematicians, who declared for Newton. Sir Isaac himself never appeared on the scene; his glory was become that of the nation; and his adherents, warm in the cause of their country, needed not his assistance to animate them.
Writings succeeded each other but slowly, on either side; probably on account of the distance of places; but the controversy grew still hotter and hotter: till at length M. Leibnitz, in the year 1711, complained to the Royal Society, that Dr. Keil had accused him of publishing the Method of Fluxions invented by Sir I. Newton, under other names and characters. He insisted that nobody knew better than Sir Isaac himself, that he had stolen nothing from him; and required that Dr. Keil should disavow the ill construction which might be put upon his words.
The Society, thus appealed to as a judge, appointed a committee to examine all the old letters, papers, and documents, that had passed among the several mathematicians, relating to the point; who, after a strict examination of all the evidence that could be procured, gave in their report as follows: “That Mr. Leibnitz “was in London in 1673, and kept a correspondence “with Mr. Collins by means of Mr. Oldenburgh, till “Sept. 1676, when he returned from Paris to Hano“ver, by way of London and Amsterdam: that it did “not appear that Mr. Leibnitz knew any thing of the “differential calculus before his letter of the 21st of “June, 1677, which was a year after a copy of a let“ter, written by Newton in the year 1672, had been “sent to Paris to be communicated to him, and above “4 years after Mr. Collins began to communicate that “letter to his correspondents; in which the Method of “Fluxions was sufficiently explained, to let a man of his “sagacity into the whole matter: and that Sir I. New- “ton had even invented his method before the year “1669, and consequently 15 years before M. Leibnitz “had given any thing on the subject in the Leipsic “Acts.” From which they concluded that Dr. Keil had not at all injured M. Leibnitz in what he had said.
The Society printed this their determination, together with all the pieces and materials relating to it, under the title of Commercium Epistolicum de Analysi Promota, 8vo, Lond. 1712. This book was carefully distributed through Europe, to vindicate the title of the English nation to the discovery; for Newton himself, as already hinted, never appeared in the affair: whether it was that he trusted his honour with his compatriots, who were zealous enough in the cause; or whether he felt himself even superior to the glory of it.
M. Leibnitz and his friends however could not shew the same indifference: he was accused of a theft; and the whole Commercium Epistolicum either expresses it in terms, or insinuates it. Soon after the publication therefore, a loose sheet was printed at Paris, in behalf of M. Leibnitz, then at Vienna. It is written with great zeal and spirit; and it boldly maintains that the Method of Fluxions had not preceded the Method of Differences; and even insinuates that it might have arisen from it. The detail of the proofs however, on each side, would be too long, and could not be understood without a large comment, which must enter into the deepest geometry.
M. Leibnitz had begun to work upon a Commercium Epistolicum, in opposition to that of the Royal Society; but he died before it was completed.
A second edition of the Commercium Epistolicum was printed at London in 1722; when Newton, in the preface, account, and annotations, which were added to that edition, particularly answered all the objections which Leibnitz and Bernoulli were able to make since the Commercium first appeared in 1712; and from the last edition of the Commercium, with the various original papers contained in it, it evidently appears that Newton had discovered his Method of Fluxions many years before the pretensions of Leibnitz. See also Raphson's History of Fluxions.
There are however, according to the opinion of some, strong presumptions in favour of Leibnitz; i. e. that he was no plagiary: for that Newton was at least the first inventor, is past all dispute; his glory is secure; the reasonable part, even among the foreigners, allow it: and the question is only, whether Leibnitz took it from him, or fell upon the same thing with him; for, in his theory of abstract notions, which he dedicated to the Royal Academy in 1671, before he had seen any thing of Newton's, he already supposed infinitely small quantities, some greater than others; which is one of the great principles of his system.
Before prosecuting farther the history and improvements of this science, it will be proper to premise somewhat of the principles and practice of it, according to the ideas of the inventor.
1. In the doctrine of Fluxions, magnitudes or quantities, of all kinds, are considered, not as made up of a | number of small parts, but as generated by continued motion, by means of which they increase or decrease: as a line by the motion of a point; a surface by the motion of a line; and a solid by the motion of a surface: which is no new principle in geometry; having been used by Euclid and Archimedes. So likewise, time may be considered as represented by a line, increasing uniformly by the motion of a point. And quantities of all kinds whatever, which are capable of increase and decrease, may in like manner be represented by lines, surfaces, or solids, considered as generated by motion.
2. Any quantity, thus generated, and variable, is called a Fluent, or a flowing quantity. And the rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity, at that position or instant: and it is proportional to the magnitude by which the flowing quantity would be uniformly increased, in a given time, with the generating celerity uniformly continued during that time.
3. The small quantities that are actually generated or described, in any small given time, and by any continued motion, either uniform or variable, are called Increments.
4. Hence, if the motion of increase be uniform, by which increments are generated, the increments will in that case be proportional, or equal, to the measures of the Fluxions: but if the motion of increase be accelerated, the increments so generated, in a given finite time, will exceed the Fluxion; and if it be a decreasing motion, the increment so generated, will be less than the Fluxion. But if the time be indefinitely small, so that the motion be considered as uniform for that instant; then these nascent increments will always be proportional or equal to the Fluxions, and may be substituted for them, in any calculation.
5. To illustrate these definitions: Suppose a point m be conceived to move from the position A, and to generate a line AP, with a motion any-how regulated; and suppose the celerity of the point m, at any position P, to be such, as would, if from thence it should become, or continue, uniform, be sufficient to describe, or pass uniformly over, the distance Pp, in the given time allowed for the Fluxion: then will the said line Pp represent the Fluxion of the said fluent or flowing line AP, at that position.
6. Again, suppose the right line mn to move, from the position AB, continually parallel to itself, with any continued motion, so as to generate the fluent, or flowing rectangle ABQP, whilst the point m describes the line AP; also let the distance Pp be taken, as above, to express the Fluxion of the line or base AP; and complete the rectangle PQ qp. Then, like as Pp is the Fluxion of the line AP, so is the small parallelogram Pq the Fluxion of the flowing parallelogram, AQ; both these Fluxions or increments being uniformly described in the same time.
7. In like manner, if the solid AERP be conceived as generated by the plane PQR moving, from the position ABE, always parallel to itself, along the line AD; and if Pp denote the Fluxion of the line AP. Then, like as the parallelogram Pq, or Pp X PQ, expresses the Fluxion of the flowing rectangle AQ, so likewise shall the Fluxion of the variable solid or prism AR be expressed by the prism Pr, or Pp X the plane PR. And in both these last two cases, it appears that the Fluxion of the generated rectangle, or prism, is equal to the product of the generant, whether line or plane, drawn into the Fluxion of the line along which it moves.
8. Hitherto the generant, or generating line or plane, has been considered as of a constant or invariable magnitude; in which case the fluent, or quantity generated, is a parallelogram, or a prism, the former being described by the motion of a line, and the latter by the motion of a plane. In like manner are other figures, whether plane or solid, conceived to be described, by the motion of a variable magnitude, whether it be a line or a plane. Thus, let a variable line PQ be carried with a parallel motion along AP, or whilst a point P is carried along, and describes, the line AP, suppose another point Q to be carried by a motion perpendicular to the former, and to describe the line PQ: let pq be another position of PQ, indefinitely near to the former; and draw Qr parallel to AP. Now in this case there are several fluents or flowing quantities, with their respective Fluxions: viz, the line or fluent AP, the Fluxion of which is Pp, or Qr; the line or fluent PQ, the Fluxion of which is qr; the curve, or oblique line AQ, described by the oblique motion of the point, the Fluxion of which is Qq; and lastly the surface APQ, described by the variable line PQ, and the Fluxion of which is the rectangle PQrp, or PQ X Pp. And in the same manner may any solid be conceived to be described, by the motion of a variable plane parallel to itself, substituting the variable plane for the variable line; in which case, the Fluxion of the solid, at any position, is represented by the va- | riable plane, at that position, drawn into the Fluxion of the line along which it is carried.
9. Hence then it follows generally, that the Fluxion of any figure, whether plane or solid, at any position, is equal to the section of it, at that position, drawn into the Fluxion of the axis, or line along which the variable section is supposed to be perpendicularly carried; i. e. the Fluxion of the figure AQP, is equal the plane PQ X Pp when that figure is a solid, or to the ordinate PQ X Pp when the figure is a surface.
10. It also follows, from the same premises, that, in any curve, or oblique line, AQ, whose absciss is AP, and ordinate is PQ, the Fluxions of these three form a small right-angled plane triangle Q qr; for Qr = Pp is the Fluxion of the absciss AP, qr the Fluxion of the ordinate PQ, and Qq the Fluxion of the curve or right line AQ. And consequently that, in any curve, the square of the Fluxion of the curve, is equal to the sum of the squares of the Fluxions of the absciss and ordinate, when these two lines are at right angles to each other.
11. From the premises it also appears, that contemporaneous fluents, or quantities that flow or increase together, which are always in a constant ratio to each other, have their Fluxions also in the same constant ratio at every position. For, let AP and BQ be two contemporaneous fluents, described in the same time by the motion of the points P and Q, the contemporaneous positions being P, Q, and p, q; and let AP be to BQ, or Ap to Bq, in the constant ratio of n to 1.
| Then is | , |
| and | ; |
| therefore by subtraction, | ; |
| that is, the Fluxion Pp : Fluxion | Qq :: n : 1, |
| the same as Fluent AP : Fluent | BQ :: n : 1; |
But if the ratio of the fluents be variable, so will that of the Fluxions be also, though not in the same variable ratio with the former, at every position.
12. To apply the foregoing principles to the determination of the Fluxions of algebraic quantities, by means of which those of all other kinds are determined, it will be necessary first to premise the notation used in this science, with some observations. As, first, that the final letters of the alphabet z, y, x, w, &c, are used to denote variable or flowing quantities; and the initial letters a, b, c, d, &c, constant or invariable ones: Thus, the variable base AP of the flowing rectangular figure ABQP, at art. 6, may be represented by x; and the invariable altitude PQ, by a : also the variable base or absciss AP, of the figures in art. 8, may be represented by x; the variable ordinate PQ, by y; and the variable curve or line AQ, by z.
Secondly, that the Fluxion of a quantity denoted by a single letter, is represented by the same letter with a point over it: Thus the Fluxion of x is expressed by x., that of y by y., and that of z by z.. As to the Fluxions of constant or invariable quantities, as of a, b, c, &c, they are equal to 0 or nothing, because they do not slow, or change their magnitude.
Thirdly, that the increments of variable or flowing quantities, are also denoted by the same letters with a small (′) over them: So the increments of x, y, z, are x′, y′, z′.
13. From these notations, and the foregoing principles, the quantities and their Fluxions, there considered, will be denoted as below.
In all the foregoing figures, put
| the variable or flowing line | AP = x, |
| in art. 6, the constant line | PQ= a, |
| in art. 8, the variable ordinate | PQ= y, |
| the variable curve or right line | AQ= z; |
14. The principles and notation being now laid down, we may proceed to the practice and rules of this doctrine, which consists of two principal parts, called the direct and inverse method of Fluxions; viz, the direct method, which consists in finding the Fluxion of any proposed fluent or flowing quantity; and the inverse method, which consists in finding the fluent of any proposed Fluxion. As to the former of these two problems, it can always be determined, and that in finite algebraic terms; but the latter, or finding of fluents, only in some certain cases, except by means of infinite series.—First then, of
15. To find the Fluxion of the product or rectangle of two variable quantities; let ARQP = xy be the flowing or variable rectangle, generated by two lines RQ and PQ moving always perpendicular to each other, from the positions AP and AR; denoting the one by x, and the other by y; and suppose x and y to be so related, that the curve AQ always passes through their intersection Q, or the opposite angle of the rectangle.
Now this rectangle consists of the two trilineal spaces APQ, ARQ, of which the Fluxion of the former is PQ X Pp or x.y, and that of the latter is RQ X Rr or xy., by art. 8; therefore the sum of the two, x.y + xy., is the Fluxion of the whole rectangle xy or ARQP.
The same otherwise.—Let the sides of the rectangle, x and y, by flowing, become x + x′ and y + y′: then the product of the two, or xy + x′y + xy′ + yy′ will be the new or contemporaneous value of the flowing rectangle PR or xy; subtract the one value from the other, | and the remainder xy′ + x′y + x′y′, will be the increment generated in the same time as x′ or y′; of which the last term x′y′ is nothing, or indefinitely small in respect of the other two terms, because x′ and y′ are indefinitely small in respect of x and y; which term being therefore omitted, there remains xy′ + x′y for the value of that increment: and hence, by substituting x. and y. for x′ and y′, to which they are proportional, there arises xy. + x.y for the value of the Fluxion of xy; the same as before.
17. Hence may be derived the Fluxions of all powers and products, and of all other forms of algebraic quantities whatever. And first for the continual products, of any number of quantities, as xyz, or wxyz, or vwxyz, &c. For xyz put q or pz, so that p = xy, and xyz = pz = q. Now, taking the Fluxion of q = pz, by the last article, it is ; but p = xy, and so by the same article; substituting therefore these values of p and p. instead of them, in the value of q., this becomes , the Fluxion of xyz required; which is therefore equal to the sum of the products arising from the Fluxion of each letter or quantity multiplied by the product of the other two.
Again, to determine the Fluxion of wxyz, the continual product of four variable quantities; put this product, viz, wxyz or qw = r, where q = xyz as above; then, taking the Fluxion by the last article, ; and this, by substituting for q and q. their values as above, becomes , the Fluxion of wxyz as required; consisting of the Fluxion of each quantity drawn into the products of the other three.
In the very same manner it is found that the Fluxion of vwxyz is v.wxyz + vw.xyz + vwx.yz + vwxy.z + vwxyz.; and so on, for any number of quantities whatever; in which it is always found that there as many terms as there are variable quantities in the proposed fluent, and that these terms consift of the Fluxion of each variable quantity multiplied by the product of all the rest of the quantities.
18. From hence is easily derived the Fluxion of any power of a variable quantity, as of x2, or x3, or x4, &c. For, in the rectangle or product xy, if x = y, then is the product xy = xx or x2, and also its Fluxion , the Fluxion of x2.
Again, if all the three x, y, z be equal; then is the product of the three xyz = xxx or x3; and its Fluxion , the Fluxion of x3.
And in the same manner it will appear that the Fluxion of x4 is = 4x3x., that of x5 is = 5x4x., that of xn is = nxn - 1x.; where n is any positive whole number. That is, the Fluxion of any positive integral power, is equal to the exponent of the power (n), multiplied by the next less power of the same quantity (xn - 1), and by the Fluxion of the root (x).
19. Next, for the Fluxion of any fraction, as x<*>/y of one variable quantity divided by another; put the proposed fraction x/y = q; then multiplying by the denominator, x = qy; and, taking the Fluxions, ; and, by division, (by substituting the value of q, or the Fluxion of x./y, as required. That is, the Fluxion of any fraction, is equal to the Fluxion of the numerator drawn into the denominator, minus the Fluxion of the denominator drawn into the numerator, and the remainder divided by the square of the denominator.
20. Hence too is easily derived the Fluxion of any negative integer power of a variable quantity, as of x- n or 1/xn, which is the same thing. For here the numerator of the fraction is 1, whose Fluxion is nothing; and therefore, by the last article, the Fluxion of such a fraction, or negative power, is barely equal to minus the Fluxion of the denominator, divided by the square of the said denominator. That is, the Fluxion of ; which is the same rule as before for integral powers.
Or, the same thing is otherwise derived immediately from the Fluxion of a rectangle or product, thus: put the proposed fraction, or quotient, 1/xn = q; then is qxn = 1; and, taking the Fluxions, ; hence , and (dividing by xn), (by substituting 1/xn for q), - nx./(xn+1) or -nx-n-1x.; the same as before.
21. Much in the same manner is obtained the Fluxion of any surd, or fractional power of a fluent quantity, as of xm/n or √nxm. For, putting the proposed quantity xm/n = q, then, raising each to the n power, xm = qn; take the Fluxions, ; divide by nqn-1, : which is still the same rule as before, for finding the Fluxion of any power of a sluent quantity, and which is therefore general, whether the exponent be positive or negative, or integral or fractional. |
22. For the Fluxions of Logarithms: Let A be the principal vertex of an hyperbola, having its asymptotes CD, CP, with the ordinates DA, BA, PQ, &c, parallel to them. Then, from the nature of the hyperbola, and of logarithms, it is known that any space ABPQ is the log. of the ratio of CB to CP, to the modulus ABCD. Now put 1 = CB or BA the side of the square or rhombus DB; m = the modulus, or area of DB, or sine of the angle C to the radius 1; also the absciss CP = x, and the ordinate PQ = y. Then, by the nature of the hyperbola, CP X PQ is always equal to DB, that is, xy = m; hence y = m/x, and the fluxion of the space, or x.y is mx./x = PQ qp the fluxion of the log. of x, to the modulus m. And in the ordinary hyp. logarithms the modulus m being 1, therefore x./x is the fluxion of the hyp. log. of x; which is therefore equal to the Fluxion of the quantity, divided by the quantity itself. And the same might be brought out in several other ways, independent of the figure of the hyperbola.
23. By means of the Fluxions of logarithms, are determined those of exponential quantities, i. e. quantities which have their exponent also a flowing or variable quantity. These exponentials are of two kinds, viz, when the root is a constant quantity, as ex; and when the root is variable, as yx.
In the former case, put the proposed exponential ex = z, a single variable quantity; then take the logarithm of each, so shall log. of ; take the fluxions of these, so shall ; hence , the fluxion of the proposed exponential ex; and which therefore is equal to the said proposed quantity, drawn into the fluxion of the exponent, and also into the log. of the root.
24. Also in the 2d case, put the exponential yx=z; then the logarithms give log. , and the fluxions give ; hence (by substituting yx for z) , is the fluxion of the proposed exponential yx; which therefore consists of two terms, of which the one is the fluxion of the proposed quantity considering the exponent only as constant, and the other is the fluxion of the same quantity considering the root as constant.
Of Second, Third, &c Fluxions.—Having explained the manner of considering and determining the first fluxions of flowing or variable quantities; it remains now to consider those of the higher orders, as 2d, 3d, 4th, &c, fluxions.
25. If the rate or celerity with which any flowing quantity changes its magnitude, be constant, or the same, at every position; then is the fluxion of it also constantly the same. But if the variation of magnitude be continually changing, either increasing or decreasing; then will there be a certain degree of fluxion peculiar to every point or position; and the rate of variation or change in the fluxion, is called the Fluxion of the Fluxion, or the second Fluxion of the given fluent quantity. In like manner, the variation or fluxion of this 2d fluxion is called the third Fluxion of the first proposed fluent quantity; and so on.
And these orders of fluxions are denoted by the fluent letter or quantity, with the corresponding number of points over it; viz, 2 points for the 2d fluxion, 3 for the 3d fluxion, 4 for the 4th fluxion, and so on. So the different orders of the fluxions of x, are x., x.., x∴ x...., &c; where each is the fluxion of the one next before it.
26. This description of the higher orders of fluxions may be illustrated by the three figures at the 8th article; where, if x denote the absciss AP, and y the ordinate PQ; and if the ordinate PQ or y flow along the absciss AP or x, with an uniform motion; then the fluxion of x, viz x. = Pp or Qr is a constant quantity, or x..=o, in all the figures. Also, in fig. 1, in which AQ is a right line, y. is = rq, or the fluxion of PQ, is a constant quantity, or y..=o; for, the angle Q, = the angle A, being constant, Qr is to rq, or x. to y., in a constant ratio. But in the 2d figure, rq, or the fluxion of PQ, continually increases more and more; and in fig. 3 it continually decreases more and more; and therefore in both these cases y has a 2d fluxion, being positive in fig. 2, but negative in fig. 3: and so on for the other orders of fluxions.
27. Thus, if for instance, the nature of the curve be such, that x3 is everywhere equal to a2y; then, taking the fluxions, it is a2y. = 3x2x.; and, considering x. always as a constant quantity, and taking always the fluxions, the equations of the several orders of fluxions will be as below; viz, the 1st fluxions a2y. = 3x2x., the 2d fluxions a2y.. = 6xx.2, the 3d fluxions a2y∴ = 6x.3, the 4th fluxions a2y.... = o, and all the higher fluxions = o or nothing.
Also the higher orders of fluxions are found in the same manner as the lower ones. Thus, The 1st flux. of y3 is 3y2y.;
28. In the foregoing articles, it has been supposed that the fluents increase; or that their fluxions are positive; but it often happens that some fluents decrease, and that therefore their fluxions are negative: and whenever this is the case, the sign of the fluxion must be changed, or made contrary to that of the fluent. So, of the rectangle xy, when both x and y increase together, the fluxion is x.y + xy.: but if one of them, as y, decrease, while the other, x, increases; then the fluxion of y being -y., the fluxion of xy will in that case be x.y - xy.. This may be illustrated by the annexed rec- | tangle APQR = xy, supposed to be generated by the motion of the line PQ from A towards C, and by the motion of the line RQ from B towards A: For, by the motion of PQ, from A towards C, the rectangle is increased, and its fluxion is + x.y; but by the motion of RQ, from B towards A, the rectangle is decreased, and the fluxion of the decrease is xy.; therefore, taking the fluxion of the decrease from that of the increase, the fluxion of the rectangle xy, when x increases and y decreases, is x.y-xy..
For the Inverse Method, or the finding of fluents, see Fluent. And for the several applications of this science to Maxima and Minima, the drawing of TANCENTS, &c, see the respective articles.
An idea of the principles of Fluxions being now delivered, as above, we may next consider somewhat of the chief writings and improvements that have been made by divers authors, since the first discovery of them: indeed some of the chief improvements may be learned by consulting the preface to Dr. Waring's Meditationes Analyticæ.
The inventor himself brought the doctrine of Fluxions to a considerable degree of perfection; as may be seen by many specimens of this science, given by him; particularly in his Principia, in his Tract on Quadratures, and in his Treatise on Fluxions, published by Mr. Colson; from all which it will appear, that he not only laid down the whole theory of this method, both direct and inverse; but also applied it in practice, to the solution of many of the most useful and important problems in mathematics and philosophy.
Various improvements however have been made by many illustrious authors on this science; particularly by John Bernoulli, who treated of the fluents belonging to the fluxions of exponential expressions; James Bernoulli, Craig, Cheyne, Cotes, Manfredi, Riccati, Taylor, Fagnanus, Clairaut, D'Alembert, Euler, Condorcet, Walmesley, Le Grange, Emerson, Simpson, Landen, Waring, &c. There are several other treatises also on the principles of Fluxions, by Hayes, Newyentyt, L'Hôpital, Hodson, Rowe, &c, &c, delivering the elements of this science in an easy and familiar way.
