180 
MINUTES OF PROCEEDINGS OF 
“ 'Now, as a is very small—for it cannot exceed 5°, as an isosceles tri¬ 
angle with base 40 yds. and apex 5° will have sides only 456 yds. in 
length—we shall, as will hereafter he proved, incur hut slight error from 
making sin -■ a or cos ^ equal to one, and writing our formula 
<w A £ 
- h 
1 “sin (/ 3 + y)' 
a We shall, hereafter, see that from this formula is derived the principle 
on which the calculating roller is constructed. 
f< 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 gradua¬ 
tions is 0, the subdivisional 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 BOA, 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 BB 
and AB will coincide, and therefore the angle ABC coinciding with the 
angle BBC, will be a right angle, or 36,00 (thirty-six divisions no sub¬ 
divisions), A zero must, therefore, be marked on the index plate, imme¬ 
diately above the axis of the long telescope, to serve as a starting point, 
from which the plate is graduated 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 us 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-f-y=100 m + 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. Readings 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, 
