Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/20

 beginning of the motion is given alo. Then from the length of the tangent rL, there is given both the velocity proportional to it. and the reitance proportional to the velocity in any place r.

But ince the length 2DP is to the latus rectum of the parabola as the gravity to the reitance in D; and, from the velocity augmented, the reitance is augmented in the ame ratio, but the latus rectum of the parabola is augmented in the duplicate of that ratio; it is plain that the length 2DP is augmented in that imple ratio only; and is therefore always proportional to the velocity; nor will it be augmented or diminihed by the change of the angle CDP, unles the velocity be alo changed.

Hence appears the method of determining the curve DraF, nearly, from the phænomena, and thence collecting the reitance and velocity with which the body is projected. Let two imilar and equal bodies be projected with the ame velocity, from the place D, in different angles CDP, CDp; and let the places F, f, where they fall upon the horizontal plane DC; be known. Then taking any length for DP or Dp, uppoe the reitance in D to be to the gravity in any ratio whatoever, and let that ratio be expounded by any length SM. Then by computation, from that aumed length DP, find the lengths DF, Df ; and from the ratio $$\frac{Ff}{DF}$$, found by calculation, ubduct the ame ratio as found by experiment; and let the difference be expounded by the perpendicular MN. Repeat the ame a econd and a third time, by auming always a new ratio SM of the reitance to the gravity, and collecting a new difference MN. Draw the affirmative differences on one ide of the right line SM and the negative on the other ide; and through the points N, N, N draw a regular curve N N N, cutting