so much as the Force impressed, is more than the Impediment or Resistance.
- Be it as 1 - r to 1; so one to m. (which m is therefore greater than 1.)
- And therefore the effective Force (and consequently the Celerity) as to a first Moment, is to be 1⁄m of what it would be, had there been no Resistance.
- This 1⁄m is also the remaining Force after such first Moment; and this remaining Force is (for the same Reason) to be proportionally abated as to a second Moment; that is, we are to take 1⁄m thereof, that is 1⁄mm of the impressed Force. And for a third Moment (at equal distance of time) 1⁄mmm; for a fourth 1⁄m4; and so onward infinitely.
- Because the length dispatched (in equal times) is proportional to the Celerities; the Lines of Motion (answering to those equal Times) are to be as 1⁄m, 1⁄m2, 1⁄m3, 1⁄m4, &c. of what they would have been, in the same Times, had there been no Resistance.
- This therefore is a Geometrical Progression; and (because of m greater than 1) continually decreasing.
This decreasing Progression infinitely continued (determining in the same Point of Rest, where the Motion is supposed to expire) is yet of a finite Magnitude; and equal to 1 / m - 1, of what it would have been in so much Time, if there had been no Resistance. As is demonstrated in my Algebra, Chap. 95. Prop. 8. For (as I have elsewhere demonstrated) the Sum or Aggregate of a Geometrical Progression is VR - A / R - 1 (supposing V the greatest Term, A the least, and R the common Multiplier.) That is VR / R - 1 - A / R - 1. Now in the present Case, (supposing the Progression infinitely continued) the least Term A , becomes infinitely small, or = 0. And consequently