GUNNERY. 
the sine of twice the angle * A F ; half the 
impetus being radius. 
Whence, at the directions of 15° or 75°, 
the amplitude is equal to the impetus ; for 
from what has been said, half the impetus 
being radius, a fourth part of the amplitude 
is the sine of twice the angle of elevation ; 
but the sine of twice 15°, that is, the sine of 
30°, is always equal to half the radius ; or in 
this case a fourth part of the impetus is equal 
to a fourth part of the amplitude. From 
this and the preceding proposition there are 
two easy practical methods for finding the 
impetus of any piece of ordnance. The 
fourth part of the amplitude is a mean pro- 
portional between the impetus at the curve’s 
principal vertex and its altitude. For M N : 
Ns::N»:NA = sF = tD. 
The altitudes are as the versed sines of 
double the angles of elevation, the impetus 
remaining the same. For making half the 
impetus radius, AN the altitude is the 
versed sine of the angle ACs = twice angle 
jAF. And also, radius : tangent angle 
elevation :: one-fourth amplitude : altitude; 
that is, R : tangent angle s A/: : A f:fs = 
D v. 
Projections on Ascents and Descents, fig. 8, 9. 
If the mark can be hit only with one di- 
rection A G, the impetus in ascents will be 
equal to the sum of half the inclined plane 
and half the perpendicular height, and in 
descents it will be equal to their difference ; 
but if the mark can be reached with two 
directions, the impetus will be greater than 
that sum or difference. For when A G is 
the line of direction, the angle g G A being 
= MAG = GAj; G g = A g, and gz 
added to or subtracted from both makes 
G z half the impetus equal to the sum or dif- 
ference of A g a fourth part of the inclined 
plane, and g z a fourth part of the perpen- 
dicular height. In any other direction F P 
is greater than F o — A F ; and Ff added 
to or subtracted from both, makes f P 
half the impetus greater than the sum or 
difference of A F a fourth part of the 
inclined plane, and Ff a fourth part of 
the perpendicular height. Whence if in 
ascents the impetus be equal to the sum of 
half the inclined plane and half the per- 
pendicular height, or if in descents it be 
equal to their difference, the mark can be 
reached only with one direction ; if the im- 
petus is greater than that sum or difference, 
it may be hit with two directions ; and if 
the impetus is less, the mark can be hit with 
itone at all. 
Prob. II. The angles of elevation, the 
horizontal distance, tpul perpendicular 
height being given, to find the impetus. 
Fig. 8, 9. 
From these data you have the angle of ob- 
liquity, and length of the inclined plane ; 
then as 
As : A M : : S. angle A M s : 8, angle 
A s M : : S. angle s A F : S. angle M A F,and 
AF : As: :S. angle M As: S. angle M A F; 
whence by the ratio of equality, A F : 
A M : : S. angle s A F x S. angle M A s : S', 
angle M AF x S. angle M A F, which gives 
this rule. 
Add the logarithm of A F to twice the lo- 
garithmic sine of the angle M A F ; from 
their sum subtract the logarithmic sines 
of the angles s A F and M A s, and the re- 
remainder will give the logarithm of A M 
the impetus. 
When the impetus and angles of eleva- 
tion are given, and the length of the in- 
clined plane is required, this is the rule. 
Add the logarithm of A M to the logarithmic 
sines of the angles s A F and M As ; from 
their sum subtract twice the logarithmic 
sine of angle M A F, and the remainder will 
give the logarithm of A F the fourth part 
of the length of the inclined plane. 
If the angle of elevation (AH and its 
amplitude A B (fig. 11,) and any other 
angle of elevation t A H is given ; to find 
the amplitude A b for that other angle, the 
impetus A M and angle of obliquity D A H 
remaining the same. 
Describe the circle A G M, take A F a 
fourth part of A B, and A/ a fourth part 
of A b : from the points F, f, draw the 
lines F s and fp parallel to A M, and 
cutting the circle in the points s, p ; then 
A F : A M : : S. angle s A F x S. angle 
M A s : S. angle M A F x S. angle M A F; 
and A M : Af : : S. a ngle M A F xS. angle 
M A F : S. anglep Af x S. angle p AM; 
whence by the ratio of equality. 
A F : Af: : S. angle s A F x S. angle 
M A s : S. angle p Afx S. angle p AM, 
which gives this rule. 
Add the logarithm of A F to the loga- 
rithmic sines of the angles pAf, p AM; 
from their sum subtract the logarithmic 
sines of the angles s A F, s A M, and the re- 
mainder will give the logarithm of Af, a 
fourth part of the amplitude required. 
Prob. III. To fiqd the force or velocity 
of a ball or projectile at any point of the 
curve, having the perpendicular height of 
that point, and the impetus at the point of 
