It is plain that BR is the space which would be uni formly described with the velocity of projection in the time of falling through VB. Also B r is the space that would be uniformly described, with the same velocity, in the time of falling through B t. Therefore BR is to B ; as the time of falling through VB to that of falling through B t. But, since BT is equal to VII, B r is to BR as the time of falling through B t to the time of falling through BT. Therefore BR is to Br as the time of falling through VB to that of falling through Bt. But, since BT is equal to VB, B r is to BR as the time of falling through B t to the time of falling through BT. Therefore we have BT = : But, in the parabola, we have B t: BT=t : TS', = : Therefore TS is equal to BR, or to twice VB or BT. Therefore =4 = 4 BT X BV, = BTx 4 By. But, in a parabola, the square of any ordinate TS is equal to the rectangle of the absciss BT and the parameter of that diameter. There fore 4 VB is the parameter of the diameter BT, and VB is the fourth part of that parameter.
If, therefore the horizontal line VZ be drawn, it is the di rectrix of the parabola described by a body projected from B in any direction, with the velocity acquired by falling from V.
Cor. 1. As this is true for any other point, C, D, &c. it follows that the velocity in any point of the path is that which a heavy body would acquire by falling from the directrix to that point.
Cor. 2. Hence also we learn that the velocities in any two points, such as B and D, are proportional to the por tions v p and D t of the tangents through those points which are intercepted by the same diameters. Thus, vy is a portion of the tangent B y, intercepted by the diameters DD' and EE', which also intercept a portion of the tan gent D t. For these portions of tangents are in the sub duplicate ratio of the lines VB and ZD. Now the velo cities acquired by falling through VB and ZD are in this subduplicate ratio of the spaces fallen through.
Such is the Galilean Theory of the parabolic motion of projectiles ; a doctrine valuable for its intrinsic excellence, and which ti ill always be respectable among philosophers, as the first example of a problem in the higher depart ment of mechanical philosophy.
We are now to consider it as the foundation of the art of gunnery. But it may be affirmed, at setting out, that the theory is of very little use for directing the practice of cannonading. Here it is necessary to ap proach as near as possible to the object, and the hurry of service allows no time for geometrical of pointing the piece after each discharge. \Vhen the gun is within 300 yards of the object, the gunner points it straight on it, or rather a little above, to compensate for the small deflection which obtains, even at this small dis tance. Sometimes the piece is elevated at a small angle,
and the shot, discharged with a very moderate velocity, drops on the ground, and bounds along, destroying the enemy's troops. But, in all these cases, the gunner is di rected entirely by practice, and it cannot be said that the parabolic theory is of any service to him.
Its principal use is for directing the bombardier in the throwing of shells. With these it is proposed to destroy buildings, to break through the roofs of magazines, to de stroy troops by bursting among them, &c. Such objects being generally under cover of the works of a place, can not be hit by a direct shot, and therefore the shells arc thrown with such elevated directions, that they get over the works, and produce their effect. These shells are of great weight, sometimes exceeding 200 lb. The mortar from which they are discharged must be exceedingly strong, that it may resist the explosion of the powder able to impel this vast mass to a great distance. They are therefore most unwieldy ; and it is found most convenient to have them almost solid, and unchangeable in their po sition. The shell is thrown to the intended distance by employing a proper quantity of powder. This is found incomparably easier than to vary the elevation of the mor tar. We shall also find, that when a proper elevation has been selected, a small deviation from it, unavoidable in such service, is much less detrimental than if another ele vation had been chosen. Mortars, therefore, are frequently cast in one piece with their bed or carriage, having an elevation that is not far from being the best on all ordinary occasions, and the rest is done by repeated trials with dif ferent charges of powder.
Still, however, in this practice, the parabolic motion must be understood, that the bombardier may avail him self of any occasional circumstance that may be of advan tage to him. We shall therefore consider the chief pro blems that the artillerist has to resolve, but with the ut most brevity ; and the reader will soon see, that more mi nute discussion would be of very little service.
The velocity of projection is measured by the fall that is necessary for acquiring it. It has generally been called the force, or IMPETUS ; we shall distinguish it be the symbol f. Thus, in Plate CCLXXXVI. and Fig. 2, 3, 4, FA is the height through which the body is supposed to fall, in order to acquire the velocity with which it is pro jected from A.