distance beyond which it cannot pass; for by reduction of that Equation , h will be found = ¼ p - bb ⁄ p in heights , and bb ⁄ p - ¼ p in descents ; from whence it follows, that all the Points h are in the Curve of the Parabola , whose Focus is the Point from whence the Project is cast, and whose Latus rectum , or Parameter ad Axem is = p . Likewise from the same Equation may the least Parameter or Velocity be found capable to reach the Object proposed; for bb = ¼ pp ∓ ph being reduced, ½ p will be = √ bb + hh ± h { in ascents } | {in descents} which is the Horizontal Range at 45 degrees, of a Project cast with the least Velocity that would just reach the Object , and the Elevation requisite will be easily had; for dividing the so found Semi-parameter by the Horizontal distance given b , the Quote into Radius will be the Tangent of the Elevation sought. This Rule may be of good use to all Bombardiers and Gunners , not only that they may use no more Powder than is necessary, to cast their Bombs into the place assigned, but that they may shoot with much more certainty, for that a small Error committed in the Elevation of the Piece , will produce no sensible Difference in the fall of the Shot: For which Reasons the French Engineers in their late Sieges have used Mortar-pieces inclin'd constantly to the Elevation of 45, proportioning their Charge of Pouder according to the distance of the Object they intend to strike on the Horizon.
And this is all that need to be said concerning this Problem of shooting upon Heights and Descents . But if a Geometrical Construction thereof be required; I think I have one that is as easy as can be expected, which I deduce from the foregoing Analytical Solution , viz. t ⁄ r = p ⁄ 2 b