Piece; that is, how far differing Charges will cast the same Shot out of it; which may most conveniently be engraven on the outside thereof, as a standing Direction to all Gunners, who shall from thence forward have occasion to use that Piece: And were this Matter well ascertained, it might be worth the while to make all Mortars of the like Diameter as near as may he, alike in length of Chase, Weight, Chamber, and all other Circumstances. This Discovery that the utmost Range on an inclined Plane, is, when the _Axis_ of the Piece makes equal Angles with the Perpendicular and the Object; compared with what I have demonstrated of the same Problem in the aforesaid Discourse does lead to and discover two very ready Theorems; the one, to find the greatest Horizontal Range at 45° Elevation, by any Shot made upon any inclined Plane, with any Elevation of the Piece whatsoever: And the other to find the Elevations proper to strike a given Object, with any Force greater than what suffices to reach it with the aforesaid middle Elevation. Both which being performed by one single Proportion, may be very serviceable to such as are concerned in the Practice of Gunnery, but are unwilling to trouble themselves with tedious and difficult Rules. The two Propositions are these. _PROP. I._ A Shot being made on an inclined Plane, having the Horizontal Distance of the Object it strikes, with the Elevation of the Piece, and the Angle at the Gun between the Object and the Perpendicular; to find the greatest Horizontal Range of that Piece, laden with the same Charge; that is, half the _Latus rectum_ of all the _Parabolæ_ made with the same _Impetus_. _RULE._ Take half the Distance of the Object from the _Nadir_, and take the Difference of the given Elevation from that half; the Versed Sine of twice that Difference subtract from the Versed Sine of the Distance of the Object from the _Zenith_: Then shall the Difference of those Versed Sines be to the Sine of the Distance of the Object from the _Zenith_, as the Horizontal Distance of the Object strook, to the greatest Horizontal Range at 45°. _PROP. II._ Having the greatest Horizontal Range of a Gun, the Horizontal Distance and Angle of Inclination of an Object to the Perpendicular, to find the two Elevations necessary to strike that Object. _RULE._ Halve the Distance of the Object from the _Nadir_; this half is always equal to the half Sum of the two Elevations we seek. Then say, _As the greatest Horizontal Range is to the Horizontal Distance of the Object: So is the Sine of the Angle of Inclination or Distance of the Object from the Perpendicular, to a fourth Proportional; which fourth being subtracted from the Versed Sine of the Distance of the Object from the _Zenith_, leaves the Versed Sine of the Difference of the Elevations sought; which Elevations are therefore had by adding and subtracting the half Difference to and from the aforesaid half Sum._