HISTORY. The invention of the calculus meth od is generally referred to the latter half of the Seventeenth Century. but the course of its early development really leads much further back. Thus, the 'method of exhaustion.' which, as first applied, consisted in comparing the area bounded by a given curve with the area of an inscribed or circumscribed polygon whose number of sides is continually increased. is related to the present calculus through the doctrine of limits. Simi larly, the surfaces of the sphere. cylinder. and cone were compared with prismatic and pyra midal surface-. By this method -Archimedes cal culated the value of -71-, obtained the areas of the parabola, ellipse. and one of the spirals, and found tile ratio of a spherical surface to the snr face of the circumscribed cylinder. Kepler (105) was the first to improve this method by introducing into geometry the idea of infinity. He considered the circle as composed of an in finite number of triangles (with their vertices at the centre and with their bases on the ein-um ference). and the cone as composed of an infinite number of pyramids. The next advance is due to Cavalieri (q.v.), who effected quadrature he summing the infinitesimal elements into which he divided his areas. and established the properties of the centre of gravity relating to solids of revolution. :so fair the end sought by mathema ticians was the solution of 1114 rticular problems, as the and quadrature of certain curves. Thin, also, Wallis extended the applica tion of Cavalieri's method of indivisible,. Des cartes (It1:37 I increased its power by the intro duction of coordinate geometry (see ANALyrIc t: EOM ETIt Y I and Fermat applied it to maxima and minima (q.v.). lint it remained for Leibnitz and Newton to devise a general notation and to organize existing principles into a comprehensive science. The principles of Newton. which later appeared under the title of Fluxion:: (q.v.). were first published in his Prineipin 1687). The basal idea of his calculus is that of velocity. A line, surface, or solid is conceived as generated respectively by a moving point. line, or surface. The velocity of a moving point, and its compo n•nts along the axes of coordinates for successive intervals, were called fluxion:. The velocity of the moving point was called the fluxion of the arc generated, and the are the fluent of the point's velocity. The velocity of a moving point being regarded as eonstant, the ratio of its component tluxions determined the nature of the path de scribed. In general, the relation between the fiuxions being given, the relations between the coordinates of the point were sought, and con versely. An elementary change in velocity I flux ion) along the X-axis was designated by [x].
or ,r along the by [y], or fi. Leibnitz used the symbol di- instead of a symbolism which has endured, while Newton's fluxional notation disappeared in the first half of the Nineteenth Century. The first publication of Leibnitz's principles appeared in the A•ta Erm/i toram (Leipzig. IGS3). His method differed from Newton's, not only in its symladism, but also in its relation to pure number. The instan taneous changes in any continuously varying L.agnitude. regarded by Leibnitz as taking place
by infinitely small differences, savor less of mechanics than do Newton's components of veloc ity. The basal idea. however, in the two systems is the same, and each calculus consists of two parts—( differential calculus, which investi gates the rules f)tr deducing the relation between the infinitely small different:es of quantities from the relation whieh exists between the quantities themselves; 121 the integral calculus, which treats of the inverse problem, i.e. to determine the relation of the quantities when that of their differenees is k •11.
The intInenee of ealeulus has been so extensive on nearly all branches of mathematics that no attempt will be made in what follows to give other than the most prominent names associated with its development. The theory of inlinitesi mats, whirl' lies at the foundation of differential calculus, has received adequate treatment at the hands of Cauehy. Jordan, and Picard. With the of the general theory of functions are connected the names of Clairant. D'Alembert, Euler, Lagrange, t:auss. Cayley. Cauchy, Riemann, Weierstrass, and Lie: with elliptic and Abelian functions the names of Lan. don. Jakob Bernoulli, Alaelanrin. D'Alembert. Legendry. Cl•bsell, Abel. Jacobi, Eisenstein, and lirioschi; with the theory of the potential. Lagrange, Creep, t:•uss, Dirichlet. Rhona TM Neumann. licitly. and lieltrami; with differen tial equations, the 'Bemoan's. Clairaut.
Euler, Lagrange, Alonge, Cauchy, Clehsch, and Lie: and \vitt' the •aleulus of variations, Jakob Ber111M111. Il lApital. Lagrange. Sams, Cauchy, Hesse. Clebseh, and Weierstrass.
Calculus is essentially a branch of the science of number. It dithers from other branches of this science, as arithmetic. and algebra, by regarding number as continuous, i.e. as being capable of gradual growth and of infinitesimal increase, while they deal with finite and discontinuous number. It differs from ordinary algebra in another respect: in the latter, the values of unknown quantities, and their relat' s with one another. are detected by aid of equations estab lished between these quantities directly; in the calculus, on the other hand, the equations be tween the quantities are obtained by means. of other equations primarily established. not be tween the quantities .themselves, but between certain derivatives from them, or elements of them. This is an artifice of great value, since the relations between the quantities invoh.ed in any problem can, in general. more easily be in ferred from equations between their derivatives than from those between themselves.
CALcuLt•s VARIATIONS. The basis of this calculus is also a method of differentiation, but of a peculiar kind. In ordinary differential cal culus we seek the form whieh f (.r) assumes when s receives an indefinitely small increment, d.r. In the calculus of variations, we seek the laws of the changes attending a slight alteration of the form of the function. or in the transfor mation of one function into another. This cal callus treats the so-called isoperimetrical prob lems, many of which were formerly insoluble. The method has extended appliention in higher physics.