THE 110YAL AliTILLEliY INSTITUTION. 
43 
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 whieh 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 circum¬ 
ference. 
Each logarithmic graduation in this scale is marked with the number of 
subdivisions in the angle to whose sine it belongs. 
As sin 71,77* 18 = *01, log sin 71,77*18 = — 2, and it is at 77*18 that 
the zero point of the lower logarithmic rim will fall, an arrow-head is 
marked here, and the word “ tape ” written beneath it. 
Erom 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. 
Eor by the formula, 
b 
1 sin (/3 + y) ’ 
. *. log r — log 1) = — log sin (J3 + y). 
But log r — log b is the distance from the point on the upper ring gra¬ 
duated r to that graduated b; and — log sin (/? + y). is the distance from 
the arrow-head to the point on the upper rim of the body graduated 
(/? + 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 gradua¬ 
tions r and (/3 + y) must also coincide. 
We have already stated that for readings less than 80 two values of 
(/? + 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. 
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