351 FLUXIONS. 1. The doctrine of fluxions, by many degrees the most important discovery that has been made in abstract science in modern times, comprehends the analysis of quantities considered as variable. It consists of two principal branches, the first of which shows how the relation may be found between the variation in any quantity and the variation of any function of that quantity; and the second shows how, from the variation in the function, the quantity on which the function depends may be discovered. The former of these is by English mathematicians called the direct, and the latter the inverse method of fluxions, but by foreigners they have been generally denominated the differential, and the integral calculus. 2. It is agreed on all hands that either to Sir Isaac Newton, or M. Leibnitz, the honor of discovering this admirable method of investigation belongs. But whether they separately made the discovery, or Leibnitz took advantage of some hints which he might have had from a common friend of Newton and himself, and published as his own what he thus obtained, has never been satisfactorily determined. Certain it is that the method came from the hand of Leibnitz both in its form and metaphysics, in a shape exceedingly different from the manner in which it was explained by Newton; and experience has shown that the Leibnitzian form of the calculus is much better adapted to the higher class of investigations than that of Newton. 3. M. Leibnitz unquestionably was the first person that laid the principles of the method before the public. This he did in the Leipsic acts of 1684; where he gave precepts, but without demonstrations,for performing some elementary operations in the method; and there can be no doubt that, long before that period, he was intimately acquainted with its principles. 4. But though Leibnitz was the first that published any thing on the subject, there can be little doubt that Newton had first made the discovery; for he had made use of it prior to 1669 in his Compendium of Analysis and Quadrature of Curves; and there are traces of this method in matters which must have engaged his attention three or four years before that period. In 1687 his Principia appeared; the most stupendous achievement of human intellect that the world has ever seen, a work entirely founded on the fluxionary calculus. Till about 1699, it appeared to be generally taken for granted that Newton and Leibnitz had separately and independently made the discovery; but about this period Nicholas Facio de Duillier, a Genoese, retired to England, and, it has been said, conceiving that he had been undervalued by Leibnitz, published a little tract on the curve of swiftest descent; and he took occasion to say, that, for the sake of truth and his own conscience, he was obliged to declare Newton the first inventor of the new calculus; and that he left others the task of determining whether Leibnitz the second inventor had borrowed from the English mathematician. 5. Leibnitz, hurt at the remark and the insinuation conveyed in it, answered, however, with great moderation; that he could not believe that M. Facio's remark was made with Newton's approbation; and that he would not enter into any dispute with that great man for whom he had the most profound veneration. That when he published his differential calculus, in 1684, he had been master of it about eight years. He admitted that Newton informed him, about the same time, of his knowing how to draw tangents by a general method which was not impeded by irrational quantities; but, as the information was unaccompanied by any explanation, he could not know whether this method was or was not deduced from the differential calculus; especially as Huygens, who was at that time unacquainted with this calculus, affirmed himself to be in possession of a method of drawing tangents which possessed the same advantages. That the first English work in which he had seen the differential calculus explained was in the preface to Wallis's Algebra, not published till 1693; and that, relying on all circumstances, he appealed entirely to the candor of Newton. 6. Writings succeeded each other at first but slowly; but, as the partizans of each grew more zealous and positive, the controversy grew hotter, till at length, in 1711, M. Leibnitz complained loudly to the Royal Society, of the conduct of Dr. Keil, who had accused him of having published the method of fluxions invented by Newton as his own, merely disguising his piracy by devising other names and characters. The Royal Society accordingly appointed a committee to examine all the writings relative to the question, and in 1712 published these writings with the report of the comm ttee under the title of Commercium epistolicum de Analysi promota. The conclusion of the report is, that Keil had not calumniated Leibnitz. It has been said that in this business Newton did not appear, but left the care of his reputation to his countrymen; but this is a mistake, for in the course of the dispute Newton wrote two very sharp letters against Leibnitz, in which there is evidently some art employed to weaken those strong testimonies of esteem which on previous occasions he had expressed for him, particularly in the celebrated scholia to prop. 7. book 2, of the Principia, in which Newton says, In a correspondence in which I was engaged with the very learned geometrician Mr. Leibnitz ten years ago, having informed him that I was acquainted with a method of determining the maxima and minima, drawing tangents, and doing other similar things which succeeded equally in rational equations ar. radical quantities, and having concealed this method by transposing the letters of the words, which signified, an equation containing any number of flowing quantities being given, to find the fluxions, and inversely; that celebrated gentleman answered that he had found a similar method; and this which he communicated to me, differed from mine only in the enunciation and rotation.' To this the edition of 1714 adds and in the idea of the generation of the quantities.' This shows clearly that Newton at that time believed that the discovery of Leibnitz was independent of his own. In the edition of the Principia which was published in 1726, ten years after the death of Newton, the above scholium was omitted. It appears too, that the Royal Society was sensible that in hastening the publication of the documents that made against Leibnitz, without waiting for those which he promised in his defence, it might be accused of partiality, for it declared soon after that it had no intention of passing judgment in the case, but left the world at liberty to discuss it, and give its opinion. 7. On the whole, while we think that the Commercium Epistolicum has made it plain that Newton was the first inventor, yet we are bound in candor to state that we do not think that Leibnitz was indebted for what he discovered on the subject to the previous inventions of his illustrious contemporary. 8. The method of fluxions was brought to a considerable degree of perfection by the labors of its inventors, and the Messrs. Bernouilli; but none of the great men of that day perhaps foresaw the improvements which a century would make in this new instrument of investigation, that had just been put into their hands. Maclaurin, Simpson, Lander, Waring, and Emerson, among our own countrymen, have all contributed to the improvement of some parts of the analysis. They were all of the school of Newton. But the lead in improvement has been taken by our continental neighbours of the Leibnitzian school. Euler, D'Alembert, Arbogort, and, above all, La Grange, have immensely extended the bounds of the method. 9. At present, among the leading mathematicians of this country, the logarithm of Newton has been generally abandoned for that of Leibnitz; and the labors and the talents of Woodhouse, Herschel, Babbage, Lardner, Airy, and a host of other enthusiastic cultivators of science, induce us to hope that Britain will soon, as in by-gone days, be foremost in the ranks, as well of science as of art. SECT. II. DEFINITIONS, PRINCIPLES, AND 10. In the fluxionary calculus, quantities of all kinds are considered as generated by 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, a solid by the motion of a surface, and an angle by the rotation of one of the lines which contain it; time in all cases flowing uniformly and since, when we consider magnitude only, without regarding position, figure, and other affections, all quantities may be represent 11. Any variable quantity thus generated is called by English mathematicians a fluent, or flowing quantity; and by the continental mathmaticians an integral; and the rate of increase or decrease of the variable quantity at any instant, is in this country called the flurion, and on the continent the differential of that quantity. 12. To illustrate these definitions, suppose a point m be conceived to move from the posi tion A, and to generate a line AP, by 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 custinue uniform, be sufficient to cause the point to pass uniformly over the distance Pp in the time allowed for the fluxion, then will the said line Pp represent the fluxion of the fluent, or flowing line AP, at that position. B Р Q 9 C m PR D 13. Again, suppose the right line m n, to move from the position A B, continually parallel to itself, with any continued motion, so as to generate the fluent, or A flowing rectangle A BQP, whilst the point m describes the line AP; also let the distance Pp be taken, as before, to represent the fluxion of the line, or base AB, and complete the rectangle PQqp, then, like as Pp is the fluxion of the line AP, so is Pg the fluxion of the flowing parallelogram AQ; for if the line Pp be supposed to be generated with a uniform celerity in a given time, the parallelogram Pq will also be generated with a uniform celerity in the same time. E R B Q A D 14. In like manner, if the solid AERP be conceived to be generated by the plane PQR, moving from the position AB E, always parallel to itself, along the line AD, and if Pp denote the fluxion of the line AP: Then, like as the rectangle PQqp expresses the fluxion of the flowing rectangle ABQP, so also shall the fluxion of the variable solid, or prism ABERQP, be expressed by the prism PQRrqp. And in both of these last two cases, it appears that the fluxion of the generated rectangle, or prism, is equal to the product of the generating line, or plane, drawn into the fluxion of the line along which it moves. 15. Hitherto the generating line or plane has been considered as of a constant, or invariable magnitude; in which case the fluent, or quantity generated, is a rectangle or a prism, the former being described by the motion of a line, and the latter by the motion of a plane. So in like manner are other figures, whether plane or solid, conceived to be described by the motion of a sented by dr, that of y by dy, &c. In this A Q PP variable magnitude, whether it be a line or a plane. Thus, let a variable line PQ, fig. 4 and 5, be carried with a parallel motion along the line AP, so that while the point P generates the line AP, the line PQ may generate the curvilinear area APQ, by this means the point Q will generate the curve line AQ. Here, therefore, there are several fluents, or flowing quantities, namely, the absciss, or base AP, the ordinate PQ, the curvilinear area APQ, and the curve line AQ. article we shall adopt the English notation, as, in the principal works on the application of fluxions that have hitherto been published in this country, it has been employed. SECT. III. TO FIND THE FLUXION OF ANY PROPOSED VARIABLE QUANTITY. 16. In order to exhibit the fluxions of these able values of and y', the product of ry is A quantities, let Pp, as before, be the fluxion of the base, complete the rectangle PQrp, this rectangle will be the fluxion of the curvilinear area APQ. For if the generating line were supposed to become invariable at the position PQ, it is evident that while the line Pp, was described by the point P with a uniform celerity, the parallelogram Pr would also be generated by the line PQ uniformly, and with the very celerity with which the area APQ was increasing at the position PQ. Next, suppose that the variable line PQ increases uniformly, after leaving the position PQ, with the very degree of celerity of increase it had when in that position; it is evident that the point Q will now describe a straight line Qs, which will be a tan gent to the curve at Q; this line will also be generated with a uniform celerity, viz. the very celerity with which the generating point was moving in the curve at the position Q. Hence it appears, that like as Pp is the fluxion of the base, or absciss AP, and Pr the fluxion of the area APQ, so is the liners the fluxion of the ordinate, or generating line PQ, and Qs the fluxion of the curve line AQ. 19. The fluxion of r is , and that of y is y, therefore the fluxion of +y is i+y; and the fluxion of the product of any two variable similarly the fluxion of r - -y is ry. To find quantities as r and y; let a be increased by any small quantity a', and y by y', then the quantities become + and y+y, whose product is ry + xy+yx+x'y'; which exceeds ry, the product of the two proposed quantities, by ry+ yx'y', the quantity by which, with any assignincreased. In the same manner if r and y were diminished severally by r' and y', their product would be diminished by ry' + yx' — x'y', a quantity which is equal to the former when r and y' are indefinitely small, or when for a', y', we substitute their fluxions i, y, in which case the fluxion of the product is simply ry+y. If for ry z we were to substitute w z, we should as above find its fluxion w+z w = x y z + z (y+y) = 1 y ż + x z ÿ + y zi. Hence the fluxion of the product of any number of variable quantities is obtained by multiplying the fluxion of each by the product of the others, and adding the results together. = r y = y y y-y. Hence from y3 y the fluxion of the dividend multiplied by the divisor, subtract the fluxion of the divisor multiplied by the dividend, and divide the remainder by the square of the divisor, and the result will be the fluxion of the quotient. By considering the divisor and dividend as the denominator and numerator of a fraction, the fluxion of the fraction may be found in the same manner. 21. To find the fluxion of any power of a variable quantity, as a. Let r be increased by the indefinitely small part, then if x + be raised to the nth power, it becomes x + nx n xn- 22. &c.; whence the increase of the nth power arising from an increase of in the root is nr "— 1 *, &c. If i had been negative, then the diminution in the nth power, arising from a diminution of in the root, would have been n r i-n 17. In the doctrine of fluxions, the initial let- - 1 ¿ + n · ters of the alphabet, a, b, c, d, &c., are commonly used to denote constant, or invariable quantities; and the last letters z, r, y, w, &c., to denote variable, or flowing quantities. Thus, the variable line AP, fig. 3, may be represented by r, and the constant line PQ by a; also in fig. 4, the variable absciss AP may be represented by r, the ordinate PQ by y, and the curve line AQ by . 18. The fluxion of a variable quantity is represented by the same letter, with a point over it. Thus the fluxion of r is represented by, the fluxion of y by y, &c. The continental mathematicians represent the fluxion, or differential of any quantity, by prefixing d to the quantity. Thus the fiuxion, or differential, of r is repre VOL. IX. n ད "13,&c.; which in the nascent state 2 2 motion of the point B, and let its rate of increase or diminution be represented by the distance of D from a given point C. Then, if the velocity of B be not uniform, CD will be a variable line; and its rate of increase or decrease will be its fluxion, or the second fluxion of AB. And if the motion of B be such that EF, which by the variations in its length may represent the rate of variation in CD, is also variable; then EF will have its second fluxion GH, which is the third fluxion of AB. Let u= log. r; then ra", whence, by ar-+y, &c. ticle 22, ir log. a × ù, or i= ; or the x log.a fluxion of a logarithm is equal to the fluxion of the quantity divided by the product of that quantity, and the hyperbolic logarithm of the base of the system. If log. a1, as in hyperbolic logarithms, then the fluxion of log. x == 27. Having given in the preceding sections the methods of determining the fluxion of the most usual forms of flowing quantities, we proceed to consider the much more difficult process by which the flowing quantity may be determined from its fluent. There is indeed no method by which the 24. To find the fluxion of r", the root and the fluent can in all cases be deduced from its fluxion; exponent being both variable. Letua, then log. u = v. log. x, hence i. log. x + 1 iv น all that can be done is in general to discover whether the given fluxion agrees, can be made to agree, or to have a known relation to a fluxion, whence i vro, log. + which has been deduced from a known quantity by the direct method, and thence to deduce the fluent of the given fluxion. SECT. IV. OF THE DIFFERENT ORDERS of FLUXIONS. Thus we know that the fluxion of " is n I i, and conversely therefore the fluent of s 25. If the proportion between the fluxion of a root, and the fluxion of any algebraic quantity in r i is r". The fluent of r3y is 2 x y i+js, which that root may be involved, be constant, the therefore the fluent of 2 a x y z + ajr2 is cry. fluxion of that quantity is also constant; otherwise 28. The principal rules for finding fluents, dethe fluxion itself will be a variable quantity, and duced from the various forms of fluxions investiconsequently that fluxion will itself have a flux- gated in the preceding section, are the following: ion, or rate of increase corresponding to its value, at any given instant. This Auxion is called the second fluxion of the original quantity; and this second fluxion may also be still a variable quantity, and consequently have its fluxion, which is called the third fluxion of the proposed magnitude. These orders of fluxions are denominated by the same fluent letter, with a number of points over it corresponding to its order; thus of x, the first fluxion is i, the second, the third, &c. 26. For the sake of illustration, let AB represent any variable quantity generated by the 29. If there is only one fluxional quantity, and no variable quantity, the fluent is found by merely substituting the flowing quantity for its fluxion. Thus the fluent of a t is a r, that of a2+ a2 is √ a2 + x2. multiplied by the fluxion of the root, divide by 30. When any power of a flowing quantity is the fluxion of the root; add 1 to the exponent of the power; divide by the exponent so increased, and the quotient will be the fluent of the proposed flowing quantity. For example, the fluent of 3 aż is 3+ i 6 I For instance, taking 2 ai a2 i, we found a2 — x2 = a + c · a— r. Put therefore multiple of that under the vinculum, put a single 2, whence ri= 2 2' = A B a + x B= 1, whence & = a2); whence F = 23 2 (z — a2) — 4 2o 2 —↓ hyp. log. 1 = a2, the fluent of which is 2-2=3 23(2)(by substitution and reduction) fî · a2 + x2 § · x2 — } a2. ૐ 3 32. When there are several terms involving two or more variable quantities, having the fluxion of each multiplied by the other quantity or quantities, take the fluent of each term as if there were only one variable quantity in it; then, if the fluent of each term be the same, that quantity is the required fluent of the whole. Example.-Required the fluent of x y z + x y z + rys 2. The fluent of each term being ry z, considering in the first, y in the second, and in the third, as the variable quantities, ry z is therefore the required fluent. 33. As the fluxion of A+B · a— A Bax 1 and 2, assume x2 = A+ Ba 2 A+ 3 B 5 +6 then A + B = 0, and 2 A + 3 B = and consequently u hyp. log. x-3 x- 3 X 1, x- 2' have y 2 and add plying both numerator and denominator by r we is the hyp. log of r. In the same manner √ a2 + x2. From this, ar √ a2 + x2 whose fluent is * 2 if the fluent of ax hyp. log. x2) be deducted, the remainder a2 hyp. log. (x + √ a2 + a3) is the required fluent. 37. Sometimes fluents may be found by expanding the terms of their fluxions in a series, when no other method is applicable; and in many instances this method of finding fluents is of the highest importance. An example or two will explain the principles of the method. Let it be required to find the fluent of i√ a2 by means of series. √ a2 expanded into a series by the binomial theorem, or the method of indeterminate r1 co-efficients, is a— 8 a3 2 a 35. When the fluxion is a rational fraction, its &c.; hence i√ a3 denominator may be decomposed into its factors; and, by means of indeterminate inefficients, the fluxion may be decomposed into others of a simpler form. 2 A 2 5 18 16 a 128 a |