CCLXXXVI. 
574 
Parabolic the object the vertical line BD, cutting the circle in D 
Theory of and d, and join ADand Ad. I say that AD or Adare 
Gunnery. the directions required. Join ED and Ed. 
For, because AB touches the circle in A, the angle 
ADE is equal to the exterior angle EA a, or DBA, and 
the alternate angles EAD, ADB are equal. The trian- 
gles ADB and EAD are therefore similar, and DB: DA 
— DA: AE, and DA? = DBx EA. Therefore B is in 
a parabola, of which the vertical AI is a diameter, AD 
a tangent in A, and AE the parameter of that diameter. 
If, therefore, the body be projected from A in the di- 
rection AD, with the velocity acquired by falling 
through FA, the fourth part of this parameter, it will 
describe a parabola AVB which —s through B. 
By the same reasoning, it is demonstrated that the 
body will hit the mark B, if projected in the direction 
Ad with the same velocity, describing the parabola A vB. 
From this very simple construction, we may draw 
several very instructive corollaries. 
Cor. 1. When the vertical line passing through B 
cuts the circle EDA, it always cuts it in two points D 
and d, giving two directions AD and Ad, either of 
which will solve the problem. 
Cor. 2. But if the vertical through 6 only touch the 
circle, as it touches it in one point only, it gives but one 
direction, along which the body must be projected to 
hit the mark b. This direction is AG. 
Cor. 8. The direction AG evidently bisects the angle 
EAB, and the directions AD and Ad are equidistant 
from the middle direction AG. 
Cor. 4, If the vertical passing through B do not meet 
the circle described on AE, according to the conditions 
specified, the object is too remote to be struck by a bo« 
dy projected from A with the prema cio by fall- 
ing from F. There is no direction that will enable it 
to go so far onthe line AB. The distance A is the 
_ greatest possible with this velocity, and it is attained by 
taking the elevation AG which bisects the angle EAB. 
We may therefore'call Ab the maximum range on the 
line AB, and AG the middle direction. 
Cor. 5. The distances on a given line of position to 
which a body will be projected in a given direction 
AD, are proportional to the squares of the velocities of 
prevention: For the figure being similar, the range AB 
the same proportion to AF, the fall necessary for 
acquiring the velocity. Now the falls are in the du- 
plicate ratio of the velocities required by falling. There- 
fore, &c. 
The converse of this problem is solved with the same 
facility of construction. 
Let a body be projected in the direction AD, with 
the velocity acquired by falling through FA, it is re- 
as to find to what distance it will reach on the line 
Poate 
Fig. 3, 4. 
Describe, as before, on AE, = 4 AF, the circle EDA, 
touching AB, and cutting AD inD. Through D draw 
the vertical DB, cutting ABin B. Then B isthe point 
to which the projectile will reach. The proof is too 
evident to need discussion. 
_ Lastly, suppose the object B to be given, and also the 
line of direction AD (which is a very common case, 
> cog. Sa our mortars are often’ so fixed in their beds 
that their elevation ean be very little altered) it is re- 
uired to determine the velocity that must be given ‘to 
the projectile. 
_ Draw through the object the vertical BD, meeting the 
direction in D. Draw the vertical AE, and make it a 
third proportional to DB and DA, that is, make AE = 
GUNNERY. 
< : ast / 
ae and take FA = ©. ‘Then FA is the fall which 11 
will generate the velocity required for the projection. 
The Ratisretzation of ‘he is also Mee ede ‘| Pam 
Notwithstanding the great simplicity of the con~ ct 
struction of tame prole ae esas in 
cal solutions for practice with equal simplicity, except - 
when the line a pouition is horizontal, as x Fig. 2. 
This indeed is the most general case, and there are few 
situations so abrupt as to deviate very far from this 
case, the eens height of a fortress commonly bearing 
but a small proportion to the distance of the mortar. 
When AB is a horizontal plane, as in Fig. 2. the arch 
EDA is a semicircle. . : 
Ad is equal to AC, 
E 
numeri- 1S: 
In this case the maximum range 
the radius of the circle, and equal to twice the height 
FA necessary for acquiring the velocity of the projec« 
tion. 
This a, range is obtained by elevating the mor- 
tar 45 degrees from the horizon. sf ; 
The ranges, with different directions, are r= 
tional to the sines of twice the angles of elgvataon: 
For, drawing GC, DL, dl, b stamens to EA, and 
drawing the radii CD and Cd, we have CG equal to 
the range Ad and /d, to the range AB. Now 
CG is the sine of the angle ACG, which is double of 
GAB, and /d isthe sine of AC d, which is double of 
AEd, which is equal'to the elevation d AB; and the ~ 
same is true of all other elevations. We may orate 
ony this analogy as radius to the sine of twice 
angle of elevation, so is twice the height necessary for 
acquiring the velocity to the range of the projection on 
a horizontal plane. 
The height to which the projectile rises above the 
horizontal plane is as the square of the sine of elevation. 
For OV the axis of the parabola is 1th of DB or LA ;— 
and FA, the height to which the projectile would rise 
straight upward, is }th of EA. Now EA: LA=EA?: | 
AD?=rad.?: sin.* D,=rad.?: sin.? elevation. There« 
fore FA: VO=rad.! : sin. elevation ; also VO:v O= 
sin.? DAB: sin.?, d AB, &c. 
The times of the flights are as the sines of the ele« 
vation. For the velocities in the directions AD, Ad, 
being the same, the times of describing AD and Ad 
uniformly will be as AD and Ad. Now AD and Ad 
are as the sines of the angles AED and AE d, which are 
equal to the angles DAB and d AB. Now the times of 
describing AD and A d uniformly with the velocity of 
projection are the same with the times of describing the 
parabolas AVB and A vB. 
When the object to be struck is on an inclined plane 
AB, ascending, as in Fig. 3, the arch EDA is less than’ 
a semicircle ; and when it is on a descending plane, as 
in Fig. 4. EDA is greater than a semicircle. This con~ 
siderably embarrasses the process for obtaining the di« 
rection, when the impetus and the object are given) or 
conversely. It has been much canvassed by the many 
authors who deliver theories of gunnery, and the para« 
bola affords many very pretty methods of solving the 
problem. Dr Halley's, in the Philosophical Tran 
actions, No. 179. is peculiarly ‘elegant. Mr Thomas 
Simpson’s also, in No. 486. is extremely ingenious and 
comprehensive, and has been reduced to a very — 
py res by Frisius in his Cosm —— But er 
of these methods shew so distinctly the connection be- 
tween the different circumstances of the motions, or 
keep the general principle so much in view, as the one 
here given ; and all the arithmetical ae mae which 
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