PRO 
PRO 
3.04 T it O 
And from any of these, the angle of direction 
mav be found. 
V. To determine the range on an oblique 
plane ; having given the impetus or the velo- 
city, and the angle of direction. 
Let AE he? the oblique plane, at a given angle 
above or below the horizontal plane AH ; AG 
.the direction of the piece ; and AP the altitude 
due to the projectile velocity at A. (rig. fi.) 
Bv the last prop, find the horizontal range 
AH to the given velocity and direction draw 
HE perpendicular to AM, meeting the oblique 
plane in p. ; draw EF parallel to the direction 
.AG, and FI parallel to HE ; so shall the pro- 
* jeetile pass through I, and the range on the ob- 
lique .plane will be A!. This is evident from the 
properties of the parabola: see Conic Sections, 
■where it is proved, that if AH, AT, are any rvyo 
lines terminated at the curve, and IF, HE, are 
■parallel to the axis ; then is EE parallel to the 
tangent AG. (Figs. 6 and 7.) 
lienee, 1. If AO is drawn perpendicular to 
the plane AI, and AP is bisected by the per- 
pendicular STO ; then with the centre O de- 
scribing a circle through A and P, the same will 
also pass through q ; because the angle GAI, 
■formed by the tangent AG and AI, is equal to 
the angle AP q, which will therefore stand upon 
the same arc Aq. 
2 . If there are given the range and velocity, 
or the impetus, the direction will then be easily 
found thus: Take At — fAI ; draw -Lq perpen- 
dicular to AH, meeting the circle described with 
the radius AO in two points q and y; then Ay 
or Ay will be the direction of the piece. And 
hence it appears that there arc two directions, 
which, with the same impetus, give the very 
same range AI, on the oblique plane. And these 
•two directions make equal angles with AI and 
AP, the piano and the perpendicular, because 
the arc Py = the arc Ay. They also make 
equal angles with a line drawn from A through 
S, because the arc Sy = the arc Sy. 
3. Or, if there are given the range AI, and 
the direction Ay, to find the velocity or impe- 
tus. Take A t = ^AI ; and erectly perpendi- 
cular to AH, meeting the line of direction in q ; 
then draw yP, making the angle AyP = the an- 
gle Akq ; so shall AP be the impetus, or alti- 
tude due to the projectile velocity. 
4. The range on an oblique plane, with a 
given elevation, is directly as the rectangle of 
the cosine of the direction of the piece above the 
horizon, and the sine of the direction above the 
oblique plane, and reciprocally as the square 
ef the cosine of the angle of the plane above or 
below the horizon. 
For, put s = sin. A. yAI or APy, 
e = cos. A yAH or sin. PAy, 
C =55 COS. A. IAH or sin. Akd or Aiy or 
AyP. 
Then, in the tri. APy, C : s :: AP : Ay, 
and in the trian. Aiy, C : e :: Ay : A*, 
therefore by compos. C 2 : cs :: AP : At = ^AI, 
so that the oblique range A I = ~ X 4AP. 
Hence the range is the greatest when Ai is 
the greatest, that is, when iy touches the circle 
in the middle point 8 ; and then the line of di- 
rection passes through S, and bisects the angle 
formed by the oblique plane and the vertex. 
Also the ranges are equal at equal angles above 
and below this direction for the maximum. 
5. The greatest height cv or iy of the projec- 
x* 
tile, above the plane, is equal to - 2 X AP. And 
therefore it is as the impetus and square of the 
sine of direction above the plane directly, and 
square of the cosine of the plane’s inclination 
reciprocally. 
For, C (sin. AyP) \ s (sin. APy) ” AP ' Ay, 
and C (sin. Aiy) ] s (sin. iAy) ” Ay * iy, 
therefore by comp. C : * ** AP ' tq. 
G. The time of flight in the curve AH is =± 
2 s ky , . 
— V — » where £ = 1 G fL- feet. And there- 
c S 
fore it is as the velocity and sine of direction 
above the plane directly, and cosine of the 
plane’s inclination reciprocally. For the time 
of describing the curve, is equal to the time of 
. 4 s 1 
falling freely through Gl, cr 4 iy, or — - x AP. 
C' 
Therefore, the time being as the square root of 
• ilj 2s a I* 
the distance, y'g * — c/AP " V * — — / . 
c ’ ' c ' g 
the time of flight. 
7. From the foregoing corollaries may be col- 
lected the following set of theorems, relating to 
projectiles made on any given inclined planes, 
either above or below the horizontal plane ; in 
which the letters denote as before, namely, 
c — cos. of direction above the horizon, 
C — cos. of inclination of the plane, 
s =z sin. of direction above the plane, 
R the range on the oblique plane, 
T the time of flight, 
V the projectile velocity, 
H the greatest height above the plane, 
a the impetus, or alt. due to the velocity V, 
g 1 6-^2 feet. Then 
R = Ax4i= "v = £T. =t i«H. 
C 'S 
X 2 V 2 JR 
4 d-c 2 4c 
V = i/ 4 ag = C V — = <?C T SS — VgH 
T = — V — =_ = V— - V—. 
, c X X c s c .? 
And from any of these, the angle of direction 
may be found. 
Of the Path of Projectiles, as depending on the 
Res i dance of the Air. 
For a long time after Galileo, philosophers 
seemed to be satisfied with the parabolic theory 
of projectiles, deeming the effect of the air’s re- 
sistance on the path as of no consequence. In 
process of time, however, as the true philosophy 
began to dawn, they began to suspect that the 
resistance of the medium might have some ef- 
fect upon the projectile curve, and they set 
themselves to consider this subject with some at- 
tention. 
Huygens, supposing that the resistance of the 
air was proportional to the velocity of the mov- 
ing body, concluded that the line described by 
it would be a kind of logarithmic curve. 
But Newton, having clearly proved, that the 
resistance to the body is not proportional to the 
velocity itself, but to the square of it, shews, in 
his Principia, that the line a projectile describes, 
approaches nearer to an hyperbola than a para- 
bola. 
Mr. Robins has shewn that, in some cases, the 
resistance to a cannon-ball amounts to more 
than 20 times the weight of the hall ; and Dr. 
Hutton, having prosecuted this subject far be- 
yond any former example, has sometimes found 
this resistance amount to near 100 times the 
weight of the ball, viz. when it moved with a 
velocity of 2000 feet per second, which is a rate 
of almost 23 miles in a minute. 
Mr. Robins has not only detected the errors 
of the parabolic theory of gunnery, which takes 
no account of the resistance of the air, but 
shews how to compute the real range of Resisted 
bodies. 
There is an odd circumstance which often 
takes f>lace in the motion of bodies projected 
with considerable force, which shews the great 
complication and difficulty of this subject; name- 
ly, that bullets in their flight are not only de- 
pressed beneath their original direction by the 
action of gravity, but are also frequently driven 
to the right or left of that direction by the ac- 
tion of some other force. 
Now if it was true that bullets varied tlieir 
direction by the action of gravity only, then it 
ought to happen that the errors in their flight 
to the right or left of the mark they were aimed 
at, should increase in the proportion of the dis- 
tance of the mark from the piece only. But this 
is contrary to all experience : the same piece 
which will carry its bullet within pn inch of the 
intended mark at 10 yards distance, cannot be 
relied on to 10 inches in 100 yards, much less 
to 30 in 300 yards. 
And this inequality can only arise from the 
track of the bullet being incurvated sideways as 
well as downwards ; for by this means the dis- 
tance between the incurvated line and the line 
of direction, will increase in a much greatef 
ratio than that of the distance ; these lines coin- 
ciding at the mouth of the piece, and afterwards 
separating in the manner of a curve from its 
tangent, if the mouth of the pifeee is considered 
as the point of contact. 
This is put beyond a doubt from the experi- 
ments made by Mr. Robins ; who found also 
that the direction of the shot in the perpendi- 
cular iine was not less uncertain, falling some- 
times 200 yards short of what it did at other 
times, although there was no visible cause of dif- 
ference in making the experiment. See Rifi.e. 
PROJECTION, in mechanics, the act of giv- 
ing a projectile its motion. 
If the direction of the force, by which the 
projectile is put in motion, is perpendicular to. 
the horizon, the projection is said to be perpen- 
dicular ; if parallel to the apparent horizon, ft 
is said to be an horizontal projection; and if it 
makes an oblique angle with the horizon, the 
projection is oblique. In all cases, the angle 
which the line of direction makes with the hori- 
zontal line, is called the angle of elevation of 
the projectile, or of depression when the line of 
direction points below the horizontal line. 
Projection, in perspective, denotes the ap- 
pearance or representation of an object on the 
perspective plane. So, the projection of a point, 
is a point where the optic ray passes from the 
objective point through the plane to the eye ; 
or it is the point where the plane cuts the optic 
ray. And hence it is easy to conceive what i* 
meant by the projection of a line, a plane, or a 
♦olid. 
Projection of the Sphere in Plano , is a repre- 
sentation of the several points or places of the 
surface of the sphere, and of the circles described 
upon it, upon a transparent plane placed be- 
tween the eye and the sphere, or such as they 
appear to the eye placed at a given distance. 
For the laws of this projection, see Perspec- 
tive : the projection of the sphere being only a 
particular case of perspective. 
The chief use of the projection of the sphere, 
is in the construction of planispheres, maps, and 
charts ; which are said to be of tills or that pro- 
jection, according to the several situations of 
the eye, and the perspective plane, with regard 
to the meridians, parallels, and other points or 
places to he represented. 
The most usual projection of maps of the 
world, is that on the plane of the meridian, 
which exhibits a right sphere ; the first meridian 
being the horizon. The next is that on the plane 
of the equator, which has the pole in the centre, 
and the meridians the radii of a circle, & c. and 
this represents a parallel sphere. See Map. 
The projection of the sphere is usually di- 
vided into orthographic and stereographic ; to 
which may be added gnomonic. 
Projection orthographic, is that In which the 
surface of the sphere is drawn upon a plane, 
cutting it in the middle ; the eye being placed 
at an infinite distance vertically te one of the 
hemispheres. And, 
