HORIZONTAL range, of a piece of ord nance, is the distance at which a ball falls on, or strikes a horizontal plane, whatever be the angle of elevation or direction of the"piec'e. When the piece is pointed pa rallel- to the horizon, the range is then called the point-blank, or point-blank range; The greatest horizontal range, in the parabolic, theory, or in a vacuum, is that made with the piece elevated to 45 degrees, and is equal to double the height from which a heavy body must freely fall, to acquire the velocity with which the shot is discharged. Thus, a shot being discharged with the velocity of v feet per second ; because gravity "generates the velocity '2 g, or 32* feet, in the first second of time, by falling org feet, and because the spaces descend ed are as the squares Of ;he velocities, v" therefore as 4,e;: — the space a 4g body must descend to acquire the velocity v of the shot, or the space due to the locity v ; consequently the double of this, v' or v' is the greatest horizontal range with the velocity v, or at an eleva tion of 45 degrees, which is nearly half the square of a quarter of the velocity.
In other elevations, the horizontal range is as the sine of double the angle of ele vation: so that, any other elevation being e, it will be, v" v' As radius 1 : sin. 2 the range at the elevation e, with the ve locity v. But in a resisting medium, like the atmosphere, the actual ranges fall far short of the above theorems, 'in so much that with the great velocities the actual or real ranges may be less than the tenth part of the potential ranges; so that some balls, which actually range but a mile or two, would in vacuo range or thirty miles. And hence also it happens, that the elevation of the piece; to shoot farthest in the resisting medium, is always below 45°, and gradually the more below it as the velocity is greater, so that the greater velocities with which balls are discharged from cannon with gunpowder require an elevation of the gun equal to but about 30°, or even less. And the less the size of the balls is too, the less must this angle of elevation be, to shoot the farthest with a given velocity. See GUN