MINUTES OF PROCEEDINGS OP 
42 
right and left. If we turn the short telescope so as to diminish the angle 
ABC to say 35,,71, the arrow-head E will point to 71, and if we diminish 
ABC to 35,00, the arrow-head E will again point to a zero. (See Diagram I.) 
Now, let ns suppose that, having laid our instruments in the manner 
above described, we have read 60 at B, and 80 at C; adding these two 
together, we have 140. This being more than a division, we cut off the first 
figure, and read 40. We now know that the sum of the base angles exceeds 
some exact number of divisions by 40 subdivisions, or to use a symbolical 
expression, that 
/3 + y = 100m + 40. 
Now, we may assume that a lies between 20 and 2,00; for with the latter 
of these values a base of 40 would give a range of 456 yards, and with the 
former a range of 4560. Hence m must be either 70 or 71—the latter 
value being rejected when the readings add up to more than 80. Eeadings 
below 80 render possible two different values of m } and consequently two 
apices of different magnitude, and two ranges. The two ranges, however, 
will be so different that it will be impossible to fail to distinguish the true 
from the false; for let us have read n subdivisions, then our two ranges 
will be 
b 
sin (2,00 — n) * 
b 
sin (1,00 — n)' 
But 2,00 — n is more than double 1,00 — n, 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 shew how, by its 
means, the formula 
sin + y) 
may be solved, and the sides of the triangle BAC 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 readings 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 
b 
sin (70,00 + n) 
sin (71,00 + n) 
