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 ∷ range4 : altitude.
6. The time of flight at 45° is equal the square root of the range in feet, divided by 4, or more nearly = √quotient² 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 multiplyers, 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.