Solution. As Radius 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 is tt⁄rr = pt⁄br ∓ ph⁄bb - 1.
this Equation shews the Question to have 2 Answers, and the Roots thereof are t⁄r = p⁄2b ∓ √ pp ∓ 4ph / 4bb - 1; from which I derive the following Rule.
Divide half the Parameter by the Horizontal distance, and keep the Quote; viz. p⁄2b then say, as square of the distance given to the half Parameter, so half Parameter ∓ double height | descent to the square of a Secant = pp ∓ 4ph / 4bb. The Tangent answering to that Secant, will be √ pp ∓ 4ph / 4bb - 1 or Square of Radius, so then the sum and difference of the afore-found Quote, and this Tangent will be the Roots of the Equation, and the Tangents of the Elevations sought.