572 
Parabolic polygonal figure ABCDEFG, all in one vertical plane, 
Theory of an 
Gunnery. 
PLATE 
in every instant or point, such as E, will be found 
in the vertical line KE, drawn from the point at which 
it would have arrived in that instant by the primitive 
ccLxxxvi. projection. 
Fig. I. 
Now, let the interval between these impulses be di- 
minished, and their number be increased without. end. 
It is evident that this polygonal motion will ultimately 
coincide with the motion in a path of continued curva- 
tion, by the continual and unvaried action of gravity. 
The line described by the body has evidently the 
following properties. 
1st, If a number of equidistant vertical lines BB’, 
HCC’, IDD’ ,KEE’, &c. be drawn, cutting the curve in 
B; C, D, E, &c.; and if the chords AB, BC, CD, DE, 
&c. drawn through the points of intersection, be produ- 
ced till they cut the verticals in H, N, O, P, &c. the in~ 
tercepted portions HC, ND, OE, PF, &c. areall equal. 
2d, The curve is a parabola, in which. the verticals 
BB’, CC’, &c. are diameters. The property mentioned 
in the last paragraph belongs exclusively to the para- 
bola. As the circle is the curve of uniform deflection 
in the direction of the radius, so.the parabola is the 
curve of uniform deflection in the direction of the dia- 
meter. That the curve in which the chords drawn 
through the intersection of equidistant verticals cut off 
equal —_e of these verticals is a parabola, is easily 
proved in a variety of ways, Since Bé, Cc, Dd, Ee, 
are all equal, and the verticals are equidistant, Bed E 
must be a straight line. So must CdeF; BE must 
be parallel to CD, and CF to DE. Therefore BF and 
CE are parallel, and are bisected in. m and o by the 
vertical DD’. Also, if FC be produced till it meet the 
next vertical in 7, 7 B is equal to D m, All this is very 
plain. Hence 
2B, or Dm:dm= BE:mF,=mF:0E; 
but dm:Do= mF:0E; 
therefore D m: Do = m F2; 9 E?; 
and D, E, F are in a parabola, of which D m is a dia- 
meter, and o E, m F are semiordinates.. We should 
prove, in the same manner, that BG is parallel to CF, 
and AG to BF,.and D m: DD’ = m F*: D’G:2, and the 
points D, F, G, in the same parabola. 
Thus we have demonstrated, that the equal and par- 
allel impulse of gravity produces a motion in a parabola 
whose diameters are perpendicular to the. horizon. 
This was the great discovery of Galileo, and the finest 
example of his genius. His discoveries in the heavens 
have indeed attracted more notice, and he is oftener 
spoken of as the first person who shewed the mountains 
in the moon, the phases of Venus, the satellites of Ju- 
piter, &c. But in all these he was obliged to his tele- 
scope ; and another person who had common curiosity 
would have seen the same things, But, in the present 
discovery, every step was an effort of judgment and 
reasoning, and the whole investigation was altogether 
novel. - No attempt had been made, since the first 
dawn of mechanical science, to explain a curvilineal 
motion of any kind; and even the lon of the compo- 
sition of motion, though faintly seen by the ancients, | 
had never been applied to any use (except by Stevinus) 
till this sagacious philosopher saw its immense import 
ance, and brought it into constant service. 
_ The process employed by Galileo in this investiga~ 
tion, and which has been copied by almost all the 
writers on the subject, is considerably different from 
the one now gone through. Galileo supposes the heavy 
body to fall in the vertical BB’ with a uniformly. acce« 
lerated motion, describing spaces as the squares of the 
times, He supposes this motion to be compounded with 
GUNNERY. 
the uniform motion in the direction of the tangent BR. _P. 
Then, supposing that B¢ and BT are fallen thror 
while Br and BR are described by the motion of pro- © 
jection, it follows, that because Br is to BR as the time p, 
peepliel ams BiCr, BTSR, we have Bz: Bret 3 
Ss, ood the evinth B, C, S are in a parabola, whose 
diameter is BT, and has BRatangentinB. 
No doubt, the result of these suppositions agrees per= 
fectly with the phenomena, and gives a very easy and 
elegant solution of the question. But, in the first place, _ 
it is more difficult, or takes more di , to. prove 
this continued composition of motion (almost peculiar 
to the case) than to demonstrate the parabolic figure: 
and, secondly, it is not a just narration of the fact of 
the procedure of nature. There is no composition of 
such motions as are here supposed. When the body is 
at C, there is not a motion in the direction aor to 
Br, compounding itself with a motion in. the vertical, 
having the velocity which the falling body would have 
as it passes through the point. ¢. e body is really 
moving in the direction CS of the tangent to the para- 
bola, and it there receives the same infinitesimal impulse 
of gravity that it received at B, Its deflection, there« 
fore, from the line of its motion, does not make any fi- 
nite angle with that motion. Therefore, although Ga- 
lileo’s cossedtonion does very well for a mere mathe« 
matical process, like the navigator’s calculation of the 
ship’s place by tables of difference of latitude and de- 
— it by no means answers the purpose of the ora 
osophical investigation of a natural phenomenon. The 
method we have followed is a bare narration of the 
facts ; considering the motion of the body in every in- 
stant as it really is, and stating the force then really af- 
fecting its motion. 
We have not scrupled to make use of the method 
employed by Newton in the demonstration of his fun« 
damental proposition on curvilineal motions, first con- 
ceiving the action of gravity to be subsultory, and the 
motion to be polygonal, and then inferring a similar 
result from the uninterrupted action of gravity. But if 
any person is so fastidious as to object to this, (as John 
Bernoulli has done to Newton’s method,) he may re~ 
mark, that the motion BJ, which we com with 
BH, in order to produce the motion BC, is just double 
of thes Bt, through which the body falls during 
the motion along BH. Therefore the figure will be 
such, that the curvilineal deflection will be one half of 
B 4, or of HC, and the tangent to the curve, whatever 
it is, will bisect HC.- Then, during the next moment, 
since the deflective action of gravity is supposed the 
same, the body will be as much deflected from its path 
in C, that is, from the new tangent CS, whatever direc- 
tion that tangent may haye, as it was in the preceding 
moment, This gives us s D equal to r C, and this ob- 
tains throughout, Without entering on any discussion 
on the progress of the deflection in the different points 
of the arch BC or CD, it is enough for our purpose to 
shew that the curve described is such that when equi- 
distant verticals are drawn, and tangents drawn mgs 
their intersections with the curve, the portions of | 
verticals cut off by the nts are everywhere equal. 
This also is a property of the exclusively. 
That BCD is a parabola, of which BT is a diameter, 
and BR a tangent, is easily seen. For, drawing Dx 
_parallel to BR, it is plain thatu N=2rC, and ND=2sD, 
=2rC. Therefore vD=4rC, and Bu=4Bi, and 
Bt: Bu=iC?:uD*. And we should prove, in the 
same manner, that y E=9r C, &e, 
