_Upon Horizontal Planes._ 1. The greatest range is at 45° nearly. 2. The ranges with different elevations with the same charge, are as the double sines of the angles of elevation. 3. Any angle and its complement give the same range nearly. 4. The times of flight, are as the sines of the angles of elevation. 5. The altitude of the curve, at any elevation, is found by this proportion: as Radius : tangent of angle of elevation :: range : altitude. ----- 4 6. The time of flight at 45° is equal the square root of the range in feet, divided by 4, or more nearly = √(quotient)² of the range in feet, divided by 16.1, or the space passed through in the first second by gravity. Having the first graze with a given elevation and charge, to determine the charge for any other first graze and elevation, multiply the known charge and elevation into the proposed first graze; also the proposed elevation into the known first graze, and divide the first product by the last, for the charge required. _Upon inclined Planes, at 45° Elevation._ _Case 1st. Given the charge and inclination of the plane, to find the range._ Multiply the horizontal range with this given charge, (found in the tables of ranges) by the number found opposite the angle of inclination of the plane, in the first column of multipliers, in the table of amplitudes, under the head _Ascents_, if it be inclined above the horizon; and _Descents_, if below the horizon, for the range required. _Case 2d. Given the range and inclination of the plane, to find the charge._ Multiply the number found in the abovementioned table opposite the angle of inclination of the plane, in the second column of multipliers, under the head _Ascents_, or _Descents_, according as it is above or below the horizon, by the given range; for the range on a horizontal plane at 45°, the charge for which may be found from the tables of ranges. _Upon Inclined Planes, at any Elevation._ There are always two elevations with which any range, (less than the greatest) may be made; and these elevations are always the complements of each other. The greatest range upon a horizontal plane is at 45°; or when the direction bisects the angle formed by the horizontal and vertical plane; also the greatest range upon any plane is made with that direction which bisects the angle between the plane and the zenith; and all other directions which make equal angles with this direction, (on each side of it) will also make equal ranges on the said plane; for the direction that bisects the angle between any plane and the zenith is the same with respect to that plane as the direction at 45° is with respect to the plane of the horizon. _Rules._—1st. The elevation which gives the greatest range on a given ascent, is equal to half the sum of 90° added to the ascent.