and 'tis this, having made the right Angle GDF, (_Tab. 5. Fig. 3._) make DF = _½p_, or greatest Range, and GD = _b_ the Horizontal Distance, and DB = _h_ the perpendicular heighth of the Object; to be laid upwards from D, if the Object be above the Horizon, or downwards if below it. Parallel to GD draw FA, and make it equal to GB the Hypothenusal Distance of the Object; and with the Center A and Radius FB = _½p ± h_, sweep an Arch, which shall if the thing be possible, intersect the indeterminate Perpendicular DF in two Points K and L, to which draw the Lines, GL, GK; I say, the Angles DGK, DGL, are the Elevations requisite to strike the Object B. _Demonstration._ The Square of FK or FL, is equal to FB_q_ - GB_q_: or (_½p ± h_)² - _bb_ - _hh_ or _¼pp ± ph - bb_, and therefore √(_¼pp ± ph - bb_) is = FK = FL, and by Consequence DK, DL = _½p_ ± √(_½pp ± ph - bb_). And as DG: DK and DL :: Radius: Tangents sought, which coincides with our Algebraical Expression thereof. _Prop._ XI. To determine the Force or _Velocity_ of a _Project_, in every Point of the _Curve_ it describes. To do this we need no other _Præcognita_, but only the third Proposition, _viz._ That the _Velocity_ of _falling Bodies_, is double to that which in the same time, would have described the Space _fallen_ by an equable Motion: For the _Velocity_ of a Project, is compounded of the constant equal _Velocity_ of the impressed Motion, and the _Velocity_ of the _Fall_, under a given _Angle_, _viz._ the Complement of the _Elevation_: For Instance, in _Fig. 2._ in the time wherein a Project would move from G to L, 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.