and 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 bastening 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 Commereium 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 NOTATION. 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 717 11. Any variable quantity thus generated is called by English mathematicians a fluent, or flowing quantity; and by the continental mathematicians 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 continue 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. Р 13. Again, suppose the right line m n, to m from the position m Q 9 C PR AB, continually pa- B rallel 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. R 7 B Q 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 PQgp 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 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 PP line AP, the line PQ may generate the curvilinear area AP Q, 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. 19. The fluxion of r is, and that of y is y, therefore the fluxion of +y is iÿ; and similarly the fluxion of x-y is i―y. To find quantities as r and y; let a be increased by any the fluxion of the product of any two variable small quantity a', and y by y', then the quantities become r+r and y+y, whose product is xy + xy + y x + 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 x 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 a′ and y' are indefinitely small, or when for x', y', we substitute their fluxions r, y, in which case the fluxion of the product is simply ry+yi. If for ry z we were to substitute w z, we should as above find its fluxion w+z w = x y ż + z (y+y)= x y ś + x z ÿ + y z . 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. 16. In order to exhibit the fluxions of these able values of and y', the product of ry is 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 lines the fluxion of the ordinate, or generating line PQ, and Qs the fluxion of the curve line AQ. 17. In the doctrine of fluxions, the initial letters of the alphabet, a, b, c, d, &c., are commonly used to denote constant, or invariable quantities; and the last letters z, x, 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 a, the ordinate PQ by y, and the curve line AQ by z. 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 fluxion, or differential, of x is repre VOL. IX. y y y = yiry. Hence from y2 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 r. Let r be increased by the indefinitely small part, then if x + i be raised to the nth power, it becomes " + nx n- 1 i + n rn-2;2, &c.; whence the i n- 1 2 increase of the nth power arising from an increase of in the root is nx *, &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 " — ' n- 1 * -n of —¿,&c.; which in the nascent state would agree with the preceding, that is, when all the terms after the first were zero. Hence, generally, the fluxion of x, nx— ' ¿ That is, multiply the fluxion of the root by the index of the power, and the product by the next lower power of the root for the fluxion of the proposed power. 2 A Let u log. r; then ra", whence, by article 22, log. a X ù, or i= ; or the r 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. a=1, as in hyperbolic logarithms, then motion of the point B, and let its rate of increase :. i, then ynn-1r x2 + n x x ; if 2 = ÿ, then 2 i = i ÿ 2 SECT. V.-OF THE INVERSE METHOD OF FLUX- 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 ", the root and the fluent can in all cases be deduced from its fluxion; exponent being both variable. the fluxion of log. x == Let u=r", then log. u — v. log. x, hence“ 25. If the proportion between the fluxion of a root, and the fluxion of any algebraic quantity in which that root may be involved, be constant, the fluxion of that quantity is also constant; otherwise the fluxion itself will be a variable quantity, and consequently that fluxion will itself have a fluxion, or rate of increase corresponding to its value, at any given instant. This fluxion 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 ☛, the first fluxion is, the second, the third, &c. all that can be done is in general to discover Thus we know that the fluxion of x is n x n-1 n 1 n i is . The fluent of r3y is 2 x y +jr, therefore the fluent of 2 a x y z + ajr2 is a 2y. 28. The principal rules for finding fluents, deduced from the various forms of fluxions investigated in the preceding section, are the following: 29. If there is only one fluxional quantity, and multiplied by the fluxion of the root, divide by For example, the fluent of 3 i is 2 a For instance, taking , we found 2 a multiple of that under the vinculum, put a single variable letter for the compound root, and put its povers and fluxion for those of the same value in the given quantity, and it will then be reduced in the last article that u = hyp. log.+s But to a form in which the last rule may be applied. Example.-Let the given fluxion be F (a2 + r2)3ï3 i, where 3, the index of the quantity without the vinculum, increased by 1 makes 4, double the index of r under the vinculum. Put 2a2+x2, then x2-z-a2; and 2 x i— a2); whence F = } 23 2 (z — a2) = { a2 — x2 = a + x · a A = a + x -x. Put therefore then A+B 2 and'A Again, let i= A+Ba i a2 — x3 Ba a 2 x2 B=0, or A = 1 and + a + x 1 ( 3)(by substitution and reduction) fo・a2 + x2 Example.-Required the fluent of y z + xj z + xy & 2. The fluent of each term being xyz, considering in the first, y in the second, and in the third, as the variable quantities, x y z is therefore the required fluent. 33. As the fluxion of is A+Br2 A+ 3 B; B + then A + B = 0, and 2 A + 3 B = − 5 +6 whence A1 and B= i 2' - 1, and &= x-3 3 x- 2 and consequently u➡hyp. log. 1, 36. The fluents of many expressions may be derived from the fluents of others; thus, we have seen above that the fluent of is the √ x2 + a2 hyp. log. of r + √x2+ a2, let it be required to find from this the fluent of Multi√ a2 + x2 plying both numerator and denominator by r we a2 x x have and add and √ a2x2 + a2 x x + x3 i √ a2x2 + x^ √ a2x2 + the fluent we have whose fluent is r if the fluent of a3× hyp. log. √ x2± a2 is the hyp.a2 x2 + x• √ a2 + 22 log. of x + √ x2±a2; for the fluxion of x + √ x2± a2 is x + (x + a2 + 2) be deducted, the remainder 35. When the fluxion is a rational fraction, its &c.; hence √ 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 2 a 6 a 40 a3 112 as 5 29 &c. 1152 a" 38. There have been collected by several authors a great many forms of fluxions, with theù Again, let it be required to find the fluent of corresponding fluents. They may often save a2 i by means of series. &c. much labor in finding the fluents of complicated expressions, when a fluent is to be found from a fluxionwhich either agrees with, or has an assignable relation to, a fluxion on those collections. They serve in this case much the same purpose to the analyst, that logarithmic tables do to the computer. We give the following as the forms which are of most frequent occurrence in practice, and refer for the most extensive collection with which we are acquainted, to a work on the subject by Meyer Hirsch, which has been lately translated into English: |