Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/71

on: that a is the Velocity of LM in ſtatu naſcenti, and b, c, d, e, &c. are the Velocities of the Increments MN, NO, OP, &c. in their reſpective naſcent eſtates. You may proceed, and conſider theſe Velocities themſelves as flowing or increaſing Quantities, taking the Velocities of the Velocities, and the Velocities of the Velocities of the Velocities, i. e. the firſt, ſecond, third, &c. Velocities ad infinitum: which ſucceeding Series of Velocities may be thus expreſſed. a. b — a. c — 2b + a. d — 3c + 3b - a &c. which you may call by the names of firſt, ſecond, third, fourth Fluxions. And for an apter Expreſſion you may denote the variable flowing Line KL, KM, KN, &c. by the Letter x; and the firſt Fluxions by ${\dot {x}}$, the ſecond by ${\ddot {x}}$, the third by ${\dot {\ddot {x}}}$, and ſo on ad infinitum.

XXXVII. Nothing is eaſier than to aſſign Names, Signs, or Expreſſions to theſe Fluxions, and it is not difficult to compute and operate by means of ſuch Signs. But it will be found much more difficult, to omit the Signs and yet retain in our

Minds