EXPLANATION OF THE TABLES. 
225 
The second column and the last but one contain a time scale, cor¬ 
responding to the upper and lower degree; thus 73° 33' 30" corresponds 
to 4h. 54m. 14s. This scale is very convenient for converting arc and 
time, but it is introduced to suit those computations in which the time' 
itself is an argument. 
The parts for each second are given beyond 9°; from 4° to 9°, to each 
10"; but under 4° the variation is too rapid for their insertion, and 
the mean differences are given in the column marked D.* The parts are 
true for the middle term of the argument; thus, the parts from 20° 30' 
to 20° 45' are true for 20° 37i', and approximate for the rest, but the 
inaccuracy in the extreme case corresponds only to ^ of 1". 
It is, of course, the more correct way to take the parts with reference 
to the nearest term, and to apply them accordingly; thus, to find the sine 
of 9° 40' 28", find it for 9° 40' 30", and subtract the parts for 2". 
For greater accuracy proceed by proportion. 
Direct Process. When the given angle is less than 45°, its log. sine, &c. 
are taken from the top; when greater than 45°, from the bottom; thus, 
the log. sine of 28° 17' is 9*675624; the log. sine of 84° 3' is 9*997654. 
In like manner, the log. sine 9*452060 corresponds to the arc 16° 27', the 
cotangent 9*47714 to the arc 73° 18'. 
The log. sine of an angle is the log. cosine of the complement of the 
angle to 99°, whether in excess or defect; so, likewise, the log. cosine is 
the log. sine of the complement; and the like holds of the tangent and 
cotangent, secant and cosecant. 
When the given angle exceeds 90°, find the log. sine, tangent, or 
secant, for the supplement to 180°. But it is generally easier to find the 
log. co-sine, co-tangent, and co-secant, for the excess above 90°. 
Example 1.—The log. sine of 127° 50' is the log. sine of 52° 10', or the 
log. cos. of 37° 50', which is 9*897516. 
Example 2.—The log. cos. of 163° 49' is the log. cos. of 16° 11', or the 
log. sine of 73° 49', which is 9 * 982441. 
* The diff. D., in the early portion (inserted merely for uniformity), is not 
that of two consecutive terms, but corresponds to half the tabular interval on 
both sides of a term. This is done to avoid breaking the continuity of the 
horizontal lines, which must occur when actual diffs. are exhibited, and is 
teasing to the eye. 
VOL. I. 
Q 
