Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/20

10 INFINITESIMAL CALCULUS few copies printed being distributed as presents. In 1715 an elaborate account of the contents of this report was published by order of the Royal Society in their Transac tions. The manuscript of almost the whole of this account has in recent years been found in Newton s own handwrit ing. (Brewster s Life of Newton, vol. ii. p. 75.) In 1722 what is usually considered the second edition was published. The latest and most important edition is that of M. Biot and M. Lefort, published in Paris in 185G, in which many additional letters and documents necessary for an impartial appreciation of the question are added. It would occupy too large a share of our space to detail the long and bitter controversy to which the Commerciiim Epistolicum gave rise. It suffices to state that from the time of its publication until long after the death of Leibnitz 1 (November 14, 171G), and of Newton (March 28, 1727), this controversy was carried on, first between Newton and Leibnitz, and afterwards by their respective admirers. The feeling which induces men to exalt their own nation at the expense of their neighbours contributed im mensely to increase the bitterness of the dispute. It is the less necessary nowadays to enter into the merits of this great quarrel, inasmuch as it has long been agreed on, by all mathematicians who have examined into the controversy, that Newton and Leibnitz are both justly entitled to be regarded as independent discoverers of the principles of the calculus, and that, while Newton was certainly master of the method of fluxions before Leibnitz discovered his method, yet Leibnitz had several years priority of publication. The dispute seems, however, to have had a very injurious effect on the progress of mathematics in England ; for, partly owing to the natural veneration for the lofty genius of Newton, but mainly, it would appear, in consequence of the strong national prejudice produced by the bitterness of the above-mentioned controversy, British mathematicians, for considerably more than a century, failed to perceive the great superiority of the notation originated by Leibnitz to that which Newton introduced. And thus, while the Bernoullis, Euler, D Alembert, Clairaut, Lagrange, Laplace, Legendre, and a number of other eminent Contin ental mathematicians were rapidly extending knowledge, by employing the infinitesimal calculus in all branches of mathematics, pure and applied, and producing a number of great treatises in every department, in England com paratively little progress was made. In fact it was not until 1815 when three Cambridge graduates, who each afterwards rose to great distinction, Sir John Herschel, Babbage, and Peacock, published a trans lation of Lacroix s smaller treatise on the calculus that the algorithm universally adopted by Continental writers was introduced into the studies of the British universities. The great superiority of Leibnitz s system of notation was soon acknowledged, and thus an immense impetus given to the study of mathematics, in all its branches. Ever since that time the method of fluxions, 2 as a distinct method, has become almost obsolete ; and it is now strange to read Newton s own assertion in the preface to the Commercium -Epistolicum, in which he claims that the method of fluxions 1 That Leibnitz at the time of his death was occupied with a reply justifying his title to an independent discovery of the calculus, has been brought to light in recent years by Dr Gerhardt s publication (1846) of the manuscript entitled Ilistoria ct Origo Calculi Di/erentialis a G. G. LeilmUio. In his letter from Vienna, 25th August 1714, to Chamber - layne, Leibnitz expressed his purpose, on his return to Hanover, to pub lish an impartial Commercium Epistolicum. This, however, remained for others to accomplish. 2 That the fluxional notation in combination with that of differen tials lias its advantages is exhibited iu many physical works; we may instance Thomson ami Tait s Treatise on Natural Philosophy. is more elegant, more natural, more geometrical, more useful, more certain, and incomparably more universal, than that of Leibnitz. We next proceed to give a brief account- of the notation and principles of the method of fluxions, as that which was first discovered. The idea of a fluxion, as its name indicates, originated from that of motion, and all geometrical magnitudes were considered by Newton as capable of generation by con tinuous motion. Thus lines are conceived as generated by the motion of points, surfaces by that of lines, solids by surfaces, &c. Again, if we conceive a moving point as describing a curve, and the curve referred to coordinate axes, then the velocity of the moving point can be decom posed into two others, one parallel to the axis of x, the other to that of y ; these velocities are called the " fluxions " of x and y respectively, and the velocity of the point is the fluxion of the arc. Reciprocally, the arc is called the "fluent" of the velocity with which it is described ; and the ordinates x and ?/ are the fluents of their velocities re spectively. Again, if the velocity of the moving point be regarded as constant, the fluxions of the abscissa and orcli- nate of any point on the curve (except in the case of a right line) will be variable ; and their ratio at each instant will depend on the nature of the curve, i.e., on the relation between the coordinates. Reciprocally, the relation be tween the coordinates depends necessarily on that which exists at each instant between their fluxions. Hence we may seek to determine the relation between the fluxions, when we know that which exists between the coordinates, i.e., the equation of the curve ; and reciprocally we may seek to discover the relation between the coordinates when we know that between their fluxions, cither alone or com bined with the coordinates themselves. The first part of the problem is called the " method of fluxions," and the second the " inverse method of fluxions." Again, in the same case, not only do the coordinates x and y change, but also the subtangent, normal, radius of curva ture, <fec. ; that is to say, each of these quantities increases or decreases more or less rapidly, as well as the coordinates themselves. All these quantities, accordingly, have fluxions, whose ratios are also determined by the motion of the point. Consequently these quantities may in like manner be regarded as " fluents." Similar remarks apply to areas and surfaces regarded as fluents. Newton observes that he does not consider the time formally (formaliter), but supposes that one of the proposed quantities increases equably (xqucibili fluxu), to which the others are referred (tanquam ad tempus). This fluent may be chosen at pleasure, and is what we now are accustomed to call the independent variable. Again, if any quantities, regarded as fluents, be represented by letters, such as u, x, y, z, &c., the corre sponding fluxions are represented by u, x, y, 2, &c., respectively. Next, if u, x, y, z be regarded as variable or fluent quantities, their fluxions are represented by ii, x, y z, and are the fluxions of the fluxions of u, x, y, &c., i.e., the second fluxions. If one of these, x for instance, be taken as the " principal fluxion," then a; is a constant, and consequently x = 0. In like manner we may have third fluxions, as well as those of higher orders. Again, u, x, y, &c., may be regarded as themselves the fluxions of other quantities called their fluents. These quantities were represented by Newton, sometimes by u , x , y , &c., in other places by [?<], [#], &c. ; and from them it may be desired to proceed to the fluents. Newton remarks that this second general problem in volves three cases : (1) when the. equation contains the fluxions of two quantities and but one of their fluents ; (2) when the equation involves both the fluents as well as both