THE ROYAL ARTILLERY INSTITUTION. 
181 
then our two ranges will be 
sin (70,00 + n) sin (2,00— n)’ 
b 
B = 
b 
b 
sin (71,00 + %) sin (1,00— n) 
“But 2,00 —n is more than double 1,00—%, and small angles vary 
almost as their sines ; therefore, the greater of the two ranges will at least 
be double the less. 
“We now proceed to describe the calculating roller, and show how, by 
its means, the formula 
b 
sin (P + y) 
may be solved, and the sides of the triangle BA C obtained without actual 
computation. 
“ The roller is a built-up cylinder, consisting of a body and two rings, 
these rings being free to rotate round the axis of the cylinder. The 
lower rim of the body and the upper rim of the lower ring are marked 
from left to right with 100 equal graduations, corresponding to the 
number of subdivisions in a division; by means of these rims the read¬ 
ings at B and C may be added together, and if their sum exceed 100, the 
first figure will be cut off. At the zero point of the lower ring is marked 
the word “ breech.” This is brought opposite the reading of the first 
angle-finder (60), when the reading (80) of the second angle-finder on 
the lower ring will come under (40)—the sum of the two readings less 
100—on the body. 
“ This will be seen in Diagram II., where the two rims are represented 
as if in one plane, in order that all parts of them may be visible at once. 
“ On the lower edge of the upper ring is graduated, from left to right, a 
scale, giving the differences of logarithms of all numbers between 400 
and 4000, each logarithm being marked with the number to which it 
belongs. The size of the graduations is so proportioned to the size of 
the roller that log 4000 —log 400 or unity is exactly one circumference 
of the circle, and the points 4000 and 400 consequently coincide; while, 
as all powers of ten have logarithms free from fractions, the point at 
which 1000 is marked will be the true zero point of the logarithmic 
scale. 
“ On the upper edge of the body is graduated from right to left a scale 
giving the differences of logarithms for the sines of all angles between 
71,80 and 70,00, which are the greatest and least values the sum of the 
base angles can have. As the sines of very small angles are proportional 
to the angles themselves, 
sin 2,00=10 sin 20 ; 
sin 70,00 = 10 sin 71,80. 
Hence the difference between their logarithms will be unity, or one cir¬ 
cumference. 
“ Each logarithmic graduation in this scale is marked with the number 
of subdivisions in the angle to whose sine it belongs. 
