Page:The New International Encyclopædia 1st ed. v. 04.djvu/34

* CALCULUS. 20 divided his areas, and established the projierties of the centre of jjravity relating to solids of revolution. So far the end sought liy niathenia- tieians was the solution of particular prohlenis. as the rectification and (piadraturo of certain curves. Thus, also, Wallis extended the ai)i)lica- tion of Cavalieri's method of indivisibles. Des- cartes (16.37) increased its power by the intro- duction of coordinate geometry (sec Analytic tiKOMETBY). and Terniat applied it to maxima and minima (q.v. ). But it remained for Leibnitz and Newton to ilevise a general notation and to organize existing princi|)les into a comprehensive science. The principles of Xewton. which later api)eared under the title of fluxions (q.v.), were first published in his I'rincipia (1087). 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- nents along" the axes of coordinates for successive intervals, were called Huxions. The velocity of the moving point was called the fluxion of the arc generated, and the arc the lluent of the point's velocity. The velocity of a moving point being regarded as constant, "the ratio of its component fluxions determined the nature of the path de- scribed. In general, the relation between the fluxions 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 .r'. [j-]. or x; along the '-axis by i/'. [^].or f/. Leibnitz used the symbol dx instead of x', a symbolism which has" endured, while Xewton's fluxional notation disappeared in the first halt of the Nineteenth Century. The first publication of Leibnitz's principles appeared in the Acta Enidi- torum (Leipzig. ltiS4). His method differed from Newton's, not only in its symbolism, but also in its relation to pure numlicr. The instan- taneous changes in any continuously varying ii.agnilude, regarded by Leibnitz as taking place by infinitely snuUI differences, savor less of mechanics than do Newton's components of veloc- ity. The basal idea, however, in the two systems is the same, aiul each calculus consists of two parts — (1) differential calculus, which investi- gates the rules for deducing the relation between the inhnitely small differences of quantities from the relation which exists between the quantities themselves; (2) the integral calculus, which treats of the inverse problem, i.e. to determine the relation of the quantities when that of their differences is known. The influence of calculus 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 develoi)ment. The theory of infinitesi- mals, which lies at the foundaticm of dilVcrential calculus, has received adeipiate treatment at the hands of Gauss. Cauchy, .Tordan, and Picard. With the develoi)ment of the general theory of functions are connected the names of Clairaut. D'Alembert, Kuler, Lagrange, Gauss, Cayley, Cauchy, Riemanii, Weierstrass, and Lie: with elliptic and Abelian functions the names of Lan- don, Jakob Hcrnoulli, Maclaurin. D'Alembert. Legendre, Clcbsch. Abel. .Jacohi, Kisenstcin, and Brioschi; with the theory of the potential. Lagrange, Green. Gauss. Diriehlet. Riemann, Neumann, Heine, and Beltrami; with differen- CALCULUS. tial equations, the Bernoullis, Riecati, Clairaut, Kuler, Lagrange, .Monge, Cauchy, Clebsch, Boole, and Lie; ami with the calculus of variation-, .lakob Bernoulli, I'llOpital. Lagrange. Sarrus, Cauchy, Hesse, Clebsch, and Weierstrass, Calculus is essentially a branch of the science of number. It differs from other branches of this science, as arithmetic and algebra, by regarding luiniber as continuous, i.e. as being capable of gradual growth and of infinitcsinuil increase, while they deal with finite and discontinuous luiniber. It differs from ordinary algebra in another respect : in the latter, the values of luiknown (piantities, and their relations with one another, are detected by aid of equations estab- lislied l)etween these quantities directly; in the calculu-, on the other hand, the equations be- tween the (|uantities are obtained by means of other equations primarily established, not be- tween the quantitie's themselves, but between certain derivatives from them, or elements of them. This is an artifice of great value, since the relations between the quantities involved in any problem can, in general, more easily be in- ferred from equations between their derivatives than from those between themselves. Calculus of Vaultioxs. 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 which f(x) assumes when X receives an indefinitely small increment. (Ix. In the calculus of variations, we seek the laws of the changes attending a slight alteration of the fonn of the function, or in the transfor- mation of one function into another. This cal- culus treats the so-called isoperimetrical prob- lems, many of which were formerly insoluble. The method has extended application in higher physics. Calculus of Fixite Diffkrexcks is a calculus concerned with the changes of functions due to finite changes in the variables involved, hence without the assumption of continuity. E. A, and 2 are important symbols, E denoting the opera- tion of increasing the independent varialilc of a function by unity. A the corresponding incre- ment of the function, and S the siunmation of all valvies of the function for integral values of the variable from unity to any desireil numl«'.r. Calculus of Opekatioxs, as the term indi- cates, is a systematic method of treating prob- lems by operating algebraically upon symbols of operation. If the symbols o and V prefixed to a quantity represent operations of the same class, the law of operations is distributive (.see DisTRinuTlvE Law) when 0(j + i/) = 0'" + <^'''. the law is commutative (see Associative Law) when 4,^(x) =-(oc). If <>"' represents the repetition of the operation 0hi times, the law of indices in ordinary algebra is expressed by 0/« { " (.V) J = *'"+" (-v) = .^"+"' (-V) = " "' (v) } These laws are true for diflferentiation, for dx(u+v)=dx[ii) +rf.v(i'), (distributive law) d.rl'lyi'i) ! = <)'} <*-*• (") } ' (eomiuutative law) ,j»',;«i,/l = f?™+"(i. (index law). Calculus of PROBAniLtTiES. A systematic analvtic treatment of the doctrine of probability (q.v.) by use of the differential and integral cal- culus.