AS i OE 
GUNNERY. 
ving thus ascertained the general nature of the 
rr ith of a projectile, we must now examine its motion 
forces, we shall consider the velocity of pro- 
as having been generated by falling through 
determinate height. 
ual Be ee oe betty, draw 
} meeting the in 
_ 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 pede with the same velocity, 
in the time of falling through Bt. Therefore BR is to 
Bras the time of falling through VB to that of falling 
through Bé. But, since BT is equal to VB, Br is to 
BR as the time of falling through B ¢ to the time of fall- 
ing through BT. Therefore BR is to Br as the time 
of falling through VB to that of falling through Bz. 
But, since BT is equal to VB, Br isto BR as the time 
of falling through B ¢ tothe time of falling through BT. 
Therefore we have Bit: BT=B7?: BR*. But, in the 
la, we have Bé: BT =# C*: TS, = B?: TS’, 
. ‘ore TS is equal to BR, or to twice VB or BT. 
herefore TS? =4 BT?, = 4BT x BV, =BT x 4. BV. 
But, ina parabola, the =— of any ordinate TS is 
equal to the rectangle of absciss BT and the para- 
meter of that diameter. Therefore 4 VB is the para« 
meter of the diameter BT, and VB is the fourth part of 
that parameter. 
If, therefore, the horizontal line VZ be drawn, it is 
the directrix of the parabola described by a body pro- 
jected from B in any direction, with the velocity ac- 
quired 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. 
Gor. 2. Hence also we learn that the velocities in 
any two points, such as B and D, are proportional to 
the portions vy and D¢ of the tangents through those 
points which are intercepted by the same diameters. 
us, vy is a portion of the t By, intercepted 
by the diameters DD’ and EE’, which also intercept a 
portion of the tangent Dt. For these portions of tan- 
are in byte ee a ratio of the lines VB and 
tb. Now the velocities acquired by falling through 
VB and ZD.are in this subduplicate ratio of the spaces 
h 
_» Such is the Galilean Theory of the parabolic motion 
of projectiles ; a doctrine valuable for its intrinsic excel- 
lence, and which will always be respectable among 
pl phers, as the first example of a problem in the 
i department of mechanical philosophy. 
; es are now to consider ae as the pen of the 
art ery. But it ma affirmed, at setting out, 
that then. shesey is of very. Bittle 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 methods of 
inting the piece after each di When the gun 
1s within $00 yards of the object, the gunner points it 
: ° & 
578 
straight on it, or rather a little above, to compensate Parabolic 
for the small deflection which obtains, even at this small Theory of 
distance. Sometimes the piece is elevated at a small S&*ey- 
angle, andl the shot, discharged with a very moderate 
velocity, drops on the ground, and bounds along, de- 
stroying 407 mone cme. But, in all these cases, the 
gunner is directed entirely by practice, and it cannot be 
said that the parabolic t 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 maga- 
zines, to destroy troops by bursting among them, &c. 
Such objects being generally under cover of the works 
of a place, cannot be hit by a direct shot, and therefore 
the shells are thrown with such elevated directions, 
that t over the works, and produce their effect, 
These shells are of great weight, sometimes exceeding 
200 Ib. The mortar from which they are discharged 
must be exceedingly strong, that it may resist the explo~ 
sion of the powder able to impel this vast mass to a 
t distance. They are therefore most unwieldy ; and 
it is found most convenient to have them almost soli 
and un ble in their position. The shell is thrown 
to the intended distance by employing a proper quanti- 
ty of powder. This is found incomparably easier than 
to vary the elevation of the mortar. 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 elevation had been cho- 
sen. 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 ordi occa-~ 
sions, 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 ad- 
vantage to him. We shall therefore consider the chief 
problems that the artillerist has to resolve, but with the 
utmost brevity ; and the reader will soon see, that more 
minute 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 by 
the symbol f. Thus, in Plate CCLXXXVI. and Fig. prarr 
2,3,4, FA is the height through which the body is ccuxxxvn 
to fall, in order to acquire the velocity with Fig-2, 3,4 
which it is projected from A. 
The distance AB between the piece of ordnance and 
the object, is called the ampirrupe, and also the range 
=r 
Let the angle EAB contained between the vertical 
and the direction of the object be called the anoxz or 
POSITION = p. 
And let the angle DAB contained between that di- 
rection and the axis of the piece, be called the direction 
of the mortar = d, and let z express the zenith distance 
or angle EAD, contained between the axis of the mor- 
tar and the vertical line AE. 
The leading problem, from which almost all the 
others may be derived, is the following. 
Let a shell be thrown from A, (Fig. 3, 4) with the Fig, 3, 4 
velocity required by falling throug vertical FA, so 
as to hit an object B. Required the direction AD of 
the projection. 
Let AH be a horizontal line, and AB the line of po- 
sition of the object. In the vertical AF, take AE = 
4 AF, and on EKA describe an arch of a circle EDd A, 
which shall touch the line of position AB, Draw through 
