to Secant of FGB, so GM the Distance given to GL; and as Radius to Tangent of FGB, so GM to LM. Then LM - MX in Heights , or + MX in Descents ; or else MX - ML, if the Direction be below the Horizontal Line , is the Fall in the Time that the direct Impulse given in G would have carried the Project from G to L = LX = GY; then by Reason of the Parabola , as LX or GY, is to GL or YX, so is GL to the Parameter sought. To find the Velocity of the Impulse : by Prop. 2, and 4, find the Time in Seconds that a Body would fall the Space LX; and by that dividing the Line GL, the Quote will be the Velocity , or Space moved in a Second sought, which is always a mean Proportional between the Parameter , and 16 Feet, 1 Inch.
Prop. X. Problem 2. Having the Parameter, Horizontal Distance, and Height or Descent of an Object, to find the Elevations of the Line of Direction necessary to hit the given Object; that is, having GM, MX, and the greatest Randon equal to half the Parameter; to find the Angles FGB.
Let the Tangent of the Angle sought be = t, the Horizontal Distance GM = b, the Altitude of the Object MX = h, the Parameter = p, and Radius = r, and it will be,
As r to t, so b to tb⁄r = ML and tb⁄r ∓ h
{ in ascents } | {in descents} = LX, and
ptb⁄r ∓ ph = GL quad. = XY quad. ratione Parabolæ; but
bb ∓ ttbb⁄rr = GL quad. 47. 1. Euclid. Wherefore
ptb⁄r ∓ ph = bb ∓ ttbb⁄rr which Equation transposed, is
ttbb ⁄ rr = ptb ⁄ r ∓ ph - bb , divided by bb