FLIMIONS, in mathematics. The method of fluxions invented by Newton was intimate ly connected with the notion of velocity uniform and variable; and extended that notion, derived from the consideration of a moving point,'to every species of magnitude and quantity. It proposed to determine, in all cases, the rate of increase or. decrease of a magnitude or quantity whose value depends on that of another, which itself varies in value at a uniform and given rate. If x and y represent two such quantities, and y = F (a) represent the law of their dependence, and if x be supposed to be the velocity with which x increases, and y that with which y changes value. Newton undertook by his method to express j in teims of x and of x, or to find y = (x). x. The quantities x and y, which in modern language we call the variables, he called flowing quantities or fluents, and x,'y, which we should represent by dx and dy, and call differentials, he called the F. of x and y. See CALCULUS. To illustrate his notation suppose y = an, it may be
shown that y = nan 'a. Regarding now y as a quantity depending on x and x, and sup posing x to increase uniformly, in which case x is constant, and (a) its fluxion zero, we observe that y may have a fluxion, for it depends on the value assumed by x, when a further changes. We find (y) = n . (n-1). Thus, second fluction or velocity of y, or (y), Newton wrote y. If a had a second fluction, or did not change uniformly, then that fluction he wrote x. The third Ruction of y he wrote y; and so on, pointing as many points over the fluent as there are units in the order of the fluxion. For the fluent, he had no special symbol. Instead of = xa, according to the modern notation, he wrote I putting the expression in an inclosure. For the principles 'on which Leibnitz founded his calculus and its notation, see CALCULUS.