182 
MINUTES OF PROCEEDINGS OF 
“As sin 71,7718=*01, log sin 71,7718=—-2, and it is at 7718 that 
the zero point of the lower logarithmic rim will fall, an arrow-head is 
marked here, and the word “ tape ” written beneath it. 
“ From the figures on the lower rim of the body to the corresponding 
figures of the upper rim, diagonal lines are drawn. These in no way 
affect the principle of the instrument, and serve only to guide the eye. 
“We will now describe the use of the roller. 
“ As soon as the number of yards in the base is known, the upper ring 
is turned round until that number comes opposite the arrow-head 
marked “tape.” The two readings of the angle-finders are then added 
together, as already described, by means of the lower ring. The 
number representing their sum on the upper rim of the body will be 
opposite the range on the upper ring. 
“ For, by the formula, 
_ b 
r ~ sin (/3 + y)’ 
.log r — log b — — log sin (/? + y). 
“ But log r — log b is the distance from the point on the upper ring, 
graduated r to that graduated b ; and — log sin ifi -t- y) is the distance 
from the arrow-head to the point on the upper rim of the body 
graduated (fi + y); the difference in sign corresponding to the contrary 
directions in which the two logarithmic scales are graduated. Therefore 
these two distances must be equal. 
“ But the graduation b coincides with the arrow-head; therefore the 
graduations r and (/3 + y) must also coincide. 
“We have already stated that, for readings less than 80, two values of 
(fS + y) are possible, and that we have consequently two ranges given; 
but that one being more than double the other, they were easily 
distinguished. A reference to Diagram II., fig. 2, will show this to be 
the case. 
“ We now proceed to the method by which the range is ascertained with 
one finder only. 
“ The guns are dressed so as to be in line with some object to the right 
or left, and laid on the object aimed at. The angle-finder is then placed 
first on one gun and then on the other, and the short telescope is each 
time laid on the object to the flank, the long one on the object aimed at. 
We thus read the angles /3 and 3 (Diagram I.) 
Let /3= 100 m+p. 
£=100 n + q. 
Then 
/? + y = 72,00— a, 
= 72,00—(3—/?), 
= 72,00-{(100 n +<?)—-(100 m + p)}, 
= (72 , + m—n). 100 — (q— p). 
Hence we have the rule— 
“ From the gun-number of the gun nearest the square object take the 
gun-number of the gun furthest from the square object, and subtract the 
difference from 100. 
“ In working without a tape, we have to lay one angle-finder first on the 
breech and then on the muzzle of the other gun, then, taking the triangle 
