Fluxions. — Chambers's Encyclopaedia

Fluxions. The method of fluxions and fluents was the name given after Newton to that branch of mathematics which with a different notation is known after Leibnitz as the differential and integral calculus. Newton, representing quantities in the manner of Euclid and others by lines, looked upon them not with Leibnitz as made up of very small parts, but as described by a continuous motion. 'From considering,' says Newton in the introduction to his Tractatus de Quadratura Curvarum (1704—the first formal exposition of fluxions published), 'that quantities increasing in equal times and generated by this increasing become greater or less according as the velocity with which they increase and are generated is greater or less, I was in quest of a method of determining quantities from the velocities of the motions or increments with which they are generated; and naming the velocities of the motions or increments fluxions, and the quantities generated fluents, I came little by little in the years 1665 and 1666 upon the method of fluxions.' Instead of referring the rate of change of a dependent variable $y$ directly to the independent variable $x$ , as in the differential calculus, the method of fluxions refers each to time ( $t$ ) considered as a uniformly flowing quantity. Thus, the fluxions of $y$ and $x$ , denoted by $\dot{y}$ and $\dot{x}$ , correspond to $\frac{dy}{dt}$ and $\frac{dx}{dt}$ respectively. The fluent of any quantity, say $y$ , was denoted sometimes by $\int y \dot{x}$ , sometimes by $y'$ . The notation adopted by Newton was on the whole clumsy, and has been abandoned for that of the differential calculus. In the method of fluxions the notions of prime and ultimate ratios take the place that limits hold in the differential calculus. The most logical and complete, as well as the most bulky, treatise that has ever appeared on fluxions is that by Colin Maclaurin (Edin. 2 vols. 4to, 1742; 2d ed. 2 vols. 8vo, 1801). See also CALCULUS, NEWTON (SIR ISAAC).