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 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 conta n 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 P 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 position 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. 13. 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 A flowing rectangle A m PR D 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 PQgp, then, like as Pp is the fluxion of the line AP, so is Pq 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. 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 A D 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 invaria le 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, sented by dr, that of y by dy, &c. In this 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. is y, 19. The fluxion of r is i, and that of therefore the fluxion of r+y is i + ÿ; and similarly the fluxion of x-y is -y. To find quantities as r and y; let x be increased by any the fluxion of the product of any two variable small quantity x', and y by y', then the quantities become + and y+y, whose product is xy + xy + y x+x'y'; which exceeds product of the two proposed quantities, by x y + yx'y', the quantity by which, with any assign ry, the 16. In order to exhibit the fluxions of these able values of a' and y', the product of xy 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 Q 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. 17. In the doctrine of fluxions, the initial let ters 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, 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 fuxion, or differential, of a is repreVOL. IX. increased. In the same manner if I and y were diminished severally by r' and y', their product would be diminished by ry' + y x' — x'y', a quantity which is equal to the former when r and y' are indefinitely small, or when for r', y', we substitute their fluxions i, y, in which case the fluxion of the product is simply x y + yi. If for ryz we were to substitute w z, we should as above find its fluxion w + z w = x y ż + z (y+y) = x y ŝ + 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. 20. To find the fluxion of the quotient of two variable quantities, as w, then r= wy, whence, by article 19, i wy+yw, or w= y = y put x y y-y. Hence from y3 -w 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 2, then if x + i be raised to the nth power, it becomes + n x SECT. IV. OF THE DIFFERENT ORDERS OF FLUXIONS. 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 x, the first fluxion is i, the second r, the third 2, &c. 26. For the sake of illustration, let AB represent any variable quantity generated by the A 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. Example. The fluxion of x is 3 x i, and the is 6 r * + 3 a 2, the second fluxThe fluxion of this quantity again is 6 × × 2 x x + 6 x x x + 3 x 2 7. + 3 x2 ï. fluxion 3 If y = n x x2 + n x x + ÿ x, &c. SECT. V. OF THE INVERSE METHOD OF FLUXIONS, OR THE METHODS OF DEDUCING THE FLUENTS FROM THEIR FLUXIONS. ; if 2= ij, then 2 i ï = ż ÿ 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 fluent can in all cases be deduced from its fluxion; 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, which has been deduced from a known quantity by the direct method, and thence to deduce the fluent of the given fluxion. Thus we know that the fluxion of r n-1 n , and conversely therefore the fluent of n i is ". The fluent of xy is 2 x y + j x2, therefore the fluent of 2 a x y z + aj x2 is a x2y. 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 no variable quantity, the fluent is found by merely substituting the flowing quantity for its fluxion. Thus the fluent of a r is a r, that of a2 + x2 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 i is 3 ti 6 x or of i a- I a B 2 ai For instance, taking , we found a2 But multiple of that under the vinculum, put a single a + x a-x a2 — x2 = a + cax. Put therefore = A a + x B + then A+B 2 andA B1, whence &= a2 A+Ba- A Bax a 4 x (-) (by substitution and reduction) ··α2 + x22 · x2 - } a2. 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 ży z + xj z + xy i2. The fluent of each term being a yz, considering in the first, y in the second, and in the third, as the variable quantities, x y z is therefore the required fluent. 1 then A + B = 0, and 2 A + 3 B = 1, whence A1 and B= 2' i 1, and i= x-3 and consequently u hyp. log. i X- 2 36. The fluents of many expressions may be derived from the fluents of others; thus, we have is the seen above that the fluent of √ x2 + a2 hyp. log. of r+ √x2+ a2, let it be required to find from this the fluent of i y - rÿ , therefore iy the fluent of is y 2 ; similarly the flu Multi√ a2 + x2 plying both numerator and denominator by r we a2 x x x3x , and and add √ a2x2 + x2 √ a2x2+x a2 x i + x3 i √ a2x2 + x1 whose fluent is r 2 a2 + x2. , or 34. As the fluxion of hyp. log. a is the fluent ofis the hyp. log of r. In the same manner we find that the fluent of log. of x + √x2+a2; for the fluxion of x + From this, ar √ a2 + x2 if the fluent of a hyp. log. +) be deducted, the remainder Let it be required to find the fluent of i√ a2 a by means of series. √ a2 2 expanded into a series by the binomial theorem, or the method of indeterminate co-efficients, is a 2 a 8 a3 16 as 5 28 128 a 35. When the fluxion is a rational fraction, its &c.; hence i√ a2 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 6 a 40 a3 112 as 5 x 1152 a7' &c. 38. There have been collected by several authors a great many forms of fluxions, with thei Again, let it be required to find the fluent of corresponding fluents. They may often save expanded into a by means of series. 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: |