it descends from L to X, and by the third Proposition has acquired a Velocity , which in that time would have carried it by an equable Motion from L to Z, or twice the Descent LX; and drawing the Line GZ, I say, the Velocity in the Point X, compounded of the Velocities GL and LZ under the Angle GLZ, is to the Velocity impress'd in the Point G, as GZ is to GL; this follows from our second Axiom , and by the 20 and 21 Prop. lib. 1. conic. Midorgii , XO parallel and equal to GZ shall touch the Parabola in the Point X. So that the Velocities in the several Points, are as the lengths of the Tangents to the Parabola in those Points, intercepted between any two Diameters : And these again are as the Secants of the Angles , which those Tangents continued make with the Horizontal Line GB. From what is here laid down, may the comparative Force of a Shot in any two Points of the Curve , be either Geometrically or Arithmetically discover'd.
Corollary.
From hence it follows, that the force of a Shot is always least at U, or the Vertex of the Parabola, and that at equal distances therefrom, as at T and X, G and B its force is always equal, and that the least force in U is to that in G and B, as Radius to the Secant of the Angle of Elevation FGB.
These Propositions considered, there is no question relating to Projects, which, by the help of them, may not easily be Solved; and tho' it be true that most of them are to be met withal, in Galileus, Torricellius and others, who have taken them from those Authors, yet their Books being Foreign, and not easy to come by, and their Demonstrations long and difficult, I thought it not amiss to give the whole Doctrine here in English, with such short Analytical Proof of my own, as might be sufficient to evince