Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/21

INFINITESIMAL CALCULUS 11 the fluxions; (3) when the equation contains the fluents and the fluxions of three or more quantities. The problem of finding the fluent when the fluxion is known is the simplest case of the first class, and is the same as the method of integration of Leibnitz. It was usually styled in Newton s time the method of quad ratures, for it is reducible to the problem of finding the area of a curve, since it can be easily seen that the fluxion of an area is the ordiuate, when the abscissa is taken as the principal fluent. The second class comes under what is now called the solution of differential equations ; this was styled in Newton s time the "inverse method of tangents." Newton s third class is now treated of under the solution of "partial differential equations." The infinitely small parts by which the variable quantities increase in an indefinitely small time were called by Newton the "moments" of the fluent quantities; thus, he represented an infinitely small portion of time, called a moment, by o ; then the moments or infinitely small increments of u, x, y, &c., are represented by uo, xo, yo, &c. ; so that if u, x, y, &c., denote the values of the fluents at any instant, their values at the end of an indefinitely small interval of time are represented by n + uo, x + xo, &c. For instance, let the fluents x, y, be connected by the equation x 3 - ax 2 + axy - y 3 ; then, substituting x + xo for x, and y + yo for y, subtracting the original equation, and dividing by o, we get Sx^x - 2axx + ayx + axy - 3y 2 # + Sxx^o + &c. Hence, regarding o as an evanescent quantity, we obtain, neglect ing the powers of o, Sx^x 2axx + a yx + axy - Sy^y ; consequently x -. y = 3?/ 2 - ax : 3a; 2 - lax + ay. This, as Newton observes, furnishes a ready method of drawing the tangent at any point on a curve. In fact, it is, changing the notation, equivalent to Barrow s method already considered. Newton adds, that in like manner we may neglect, in all cases, the terms multiplied by the second and higher powers of o, and thus find an equation between x, y and their fluxions x, y. A good deal of confusion has arisen from the word fluxion having been commonly employed by the early English writers in the sense of an infinitely small increment. Thus, as is abundantly shown by Professor De Morgan in his tract on the early history of infinitesimals in England (Phil. Mag., 1852), all the early writers on fluxions, up to 1704, except Newton and Cheyne, employed the notation x to represent an infinitely small increment, calling it a fluxion. It is even remarkable that, in the extract from the Commercium Epistolicum which we have given, the words moment and fluxion seem to have been employed as synonymous. It should also be observed that in Newton s earliest papers his method is strictly infinitesimal ; and in the first edition of his Prindpia (1C87) the description of fluxions is founded on infinitely small increments ; so that the original conception of the calculus in England, as well as on the Continent, was based on infinitesimal principles. Objection has frequently been made to Newton s method of fluxions, that it introduced a foreign idea, namely, that of motion, into geometry and analysis. This objection is scarcely well founded, and was indeed answered by Newton when he stated that all his method contemplates is that one of the variables should increase uniformly (xquabili fluxu} as we conceive time to do. Leibnitz, like Newton, supposed any variable magnitude as continually increasing or diminishing, by momentary increments or decrements. These instantaneous changes he regarded as infinitely small differences. Thus the in finitely small difference of a variable u was represented by du. His calculus also, like Newton s, had two parts : (1) the differential calculus, which investigated the rules for deducing the relation between these infinitely small differ ences of quantities from the relation which exists between the quantities themselves ; (2) the integral calculus, which treated of the inverse problem, viz., the determination of the relation of the quantities when that of their differences is known. This corresponds to Newton s inverse method of fluxions, as the differential calculus does to his direct method. It is not necessary to go into further detail hero on Leibnitz s method, as it will be more fully considered subsequently ; in fact, all our treatment of the calculus will be merely a development of this method. The infinitesimal calculus had in the outset its Objec- opponents, such as the Abbe de Catelan, a zealous tors - Cartesian, who declared in his Logistique Universellc, ct Mcthode pour les Tangentes (1G94), that it would be better to extend the principles of the Cartesian geometry than to seek for new methods ; and this was said in the preface of a book composed on the principles, somewhat disguised, of the very calculus of which he was an opponent. It had another adversary in Nieuwentijt, a man who had written some tolerable works on morality and religion, but who had slight pretensions to be regarded as a geometer. Catelan was satisfactorily answered by De 1 Hopital, as was Nieuwentijt by Leibnitz, and afterwards by Bernoulli and Hermann, who proved that this adversary of the cal culus really did not know what he opposed. For instance, Nieuwentijt, while admitting differentials of the first order, rejected all those of higher orders. For such a difference of treatment there is no foundation, for, if we imagine in a circle an infinitely small chord of the first order, the versine is an infinitely small line of the second order. The calculus had a more formidable enemy in Rolle, a skilful algebraist, but a man full of confidence in his own notions, rash in forming his opinions, and jealous of the inventions of others. He attacked the certainty of its principles, and attempted to show that its conclusions were at variance with those obtained by methods previously known, which were acknowledged to be correct. His attack was repelled by Varignon, who completely obviated the objections to the truth of the principles. These dis putes occupied the French Academy a considerable part of the year 1701. The members were chiefly mathematicians advanced in years, who had been long accustomed to other methods, and were therefore not much disposed to receive new doctrines. Some took no part in the dispute, yet were not sorry to perceive a storm raised against a theory for which they had no great liking ; others, more under the influence of their passions and prejudices, declared open war against it. Rolle brought forward objection upon objection ; and, although Varignon answered them in suc cession, yet the former always claimed the victory. In the end the dispute degenerated into a quarrel, and com missioners were appointed to decide on it. These were Gouye, Cassini, and De la Hire. They, however, pro nounced no judgment ; but the public opinion, or at least the opinion of geometers, was in favour of Varignon. The first controversy thus ended, or rather was suspended for want of a decision from the commission ; but Rolle soon renewed hostilities. The defence was next taken up by Saurin. The ground of attack was the indefinite form which the calculus gives for the subtangent of a curve at a point where two branches intersect each other, and which in this case is expressed by the fraction , Saurin s answer was satisfactory ; but Rolle, intrenched in masses of calculation, obstinately maintained the combat. The Academy w : as again appealed to in 1705. The Abb6 Bignon, who con ducted its affairs, undertook to decide the controversy, with the assistance of Gallois and De la Hire, two judges by no means favourable to Saurin. They gave no absolute