THE ROYAL ARTILLERY INSTITUTION. 
41 
Now, as we may fairly assume our range over 500 yds., AB and AC are 
very large in proportion to BC; and as the point A is directly opposite 
some point in the base BC, the length of the sides BA, CA, will be much 
the same for any particular magnitude of the angle BAC, whether the 
triangle BAC be isosceles or not; and we may therefore treat the triangle 
BAC as an isosceles triangle of known base and known angles, of which it 
is required to find the sides. (The error resulting from this assumption, 
will hereafter be proved too small to matter in practice). 
Our formula will be 
sin a 
a 
. cos a 
sin(/3 + y) 
r = range = AB = AC, 
3 = base = BC, 
a = ABAC. 
Now, as a is very small—for it cannot exceed 5°, as an isosceles triangle 
with base 40 yds. and apex 5° will have sides only 456 yds. in length—we shall, 
as will hereafter be proved, incur but slight error from making sin ~~ or 
cos ~ equal to one, and writing our formula 
b 
t =--, 
sin (/3 + y) 
"We shall hereafter see that from this formula is derived the principle 
on which the calculating roller is constructed. 
We now proceed to read the base angles by means of the index plates of 
the angle-finders. 
On these plates the circle is divided into 144 divisions (of 2° 30' each), 
which are again subdivided into 100 subdivisions (of 1' 30" each). The 
actual portion of the circle graduated on the index plate is only about eight 
divisions, or 20°. Tenths of divisions only are graduated on the plate, but 
the angles may, by means of a vernier on the steel limb, be read to as little 
as half a subdivision. At each of the divisional graduations is 0, the sub- 
divisional graduations being marked from 1 to 99—the numbers running 
from left to right on left-handed instruments, from right to left on right- 
handed ones. The instruments will consequently show, not the exact 
magnitude of the angles ABC and BCA, but merely their excess over the 
next lowest round number of divisions. Thus, if the steel limb EB, be 
immediately over the long telescope, the lines EB and AB will coincide, 
and therefore the angle ABC coinciding with the angle EBC, will be a 
right angle, or 36,00 (thirty-six divisions no subdivisions). A zero must 
therefore be marked on the index plate, immediately above the axis of the 
long telescope, to serve as a starting point, from which the plate is graduated 
