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. Note here, that in _Descents_, if the _Tangent_ exceed the _Quote_, as it does when _ph_ is more than _bb_, the _direction_ of the lower _Elevation_ will be below the _Horizon_, and if _ph_ = _bb_, it must be directed _Horizontal_, and the _Tangent_ of the upper _Elevation_ will be _pr/b_: Note likewise, that if _4bb + 4ph_ in _Ascents_, or _4bb - 4ph_ in _Descents_, be equal to _pp_, there is but one _Elevation_ that can hit the _Object_, and its _Tangent_ is _pr/2b_. And if _4bb + 4ph_ in _Ascents_, or _4bb - 4ph_ in _descents_, do exceed _pp_, the _Object_ is without the reach of a _Project_ cast with that _Velocity_, and so the thing impossible. From this _Equation_ _4bb ∓ 4ph_ = _pp_ are determined the utmost limits of the reach of any _Project_, and the Figure assigned, wherein are all the _heights_ upon each _Horizontal distance_ beyond which it cannot pass; for by reduction of that _Equation_, _h_ will be found = _¼p_ - _bb/p_ in _heights_, and _bb/p_ - _¼p_ 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_ = _¼pp_ ∓ _ph_ being reduced, _½p_ 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/2b_ ± √((_¼pp ± ph - bb_)/_bb_),