ACTION, in Mechanics, is, properly speaking, the motion which one body either produces, or tends to pro duce, in another body; though it has been generally employed to denote the effect which one power exerts against another power.
The Principle of least AcTiox, was a name given by Alaupertuis to the law, that when any bodies, acting upon one another, suffer any change in their motion, the mass, multiplied by the space and the velocity, or the quantity oC action which nature employs to effect this change, is the least possible. This principle was attacked, as false, by Koenig, a professor at the Hague, who also maintained, that Leibnitz had described the same principle in 1707, in a letter to Herman. This attack gave rise to a long and keen dispute, in which the Academy of Berlin interfered in behalf of their pre sident. The principle of least action was extended by Euler, who proved, " that, in the trajectoris described by means of central forces, the product of the integral of the velocity, and the element of the curve, is either a minimum or a maximum." This new law, which Euler
recognised only in the case of insulated bodies, was still farther generalized by La Grange, who found, " that the sum of the products of the masses by the integrals of the velocities, multiplied by the elements of the spaces, is always a minimum or a maximum." This principle has been employed by La Grange with great success, in the solution of many difficult dynamical problems. Sec Mem. Acad. Par. 1744 ; 1749, p. 531, 8vo. p. 771; 1752, p. 503, 8vo. p. 765. Mein. Acad. Berlin, 1746, p. 267; 1750; 1752; 1753, p. 310. Act. Lips. Mart. 1751. Eu ler's Traite des Isoptrimetries Lausanne, 1744. La Grange's Mecanique Analytiqzte, 1788, p. 189. Com ment. Bonon. tom. vi. Nov. Comment. Petrop. torn. xx. p. 239. See MECHANICS. (70