1897-98.] Dr J. Burgess on Log. Sines , etc., of Small Arcs. 267 
The method I have found most useful in all cases is nearly as 
simple as this, and does not require the use of the secant. It is 
represented by 
log. sin. {a + x) = log. sin. a + log. (a + x)~ log. a ; 
log. tan. (a + x) — log. tan. a + log. (a + x) - log. a. 
Where a and a + x may he stated in degrees, minutes or seconds, 
and decimals of the same denomination: Thus, taking Vega’s 
example, to find the arc of which 5 ’6271 691 is the log. sine, we 
have — 
log. sin. (a + a?) . . . 5*6271691 
log. sin. 8" = a . ar. co. 4-4113351-10 
log. 8" .... 0*9030900 
a + a" = 8"-7416, log. . . 0-9415942 
Conversely to find the log. tan. of 2° 42' 56"-44 ( = 97 7 6"*44) — 
log. tan. 2° 42' 56" . . 8-6760614 
log. 9776"-44 . . . 3-9901807 
log. 9776" , . ar. co. 6-0098388 — 10 
log. tan. 2° 42' 56"-44 . 8*6760809 
But, while the usual tables give the sines and tangents for each 
second up to about 2°, few of them give the values for seconds 
beyond this, and some not even for the first 2°. We may, how- 
ever, apply this method with a correction that will enable us to 
find the values for any arc in the quadrant. 
For the sine we readily find the difference between 5° + 60" 
computed by the above method and the true value for log. sin. 
5° 1' to be - *0000036,828. Similarly the errors of the method 
are — 
at 5° 
- 36,828 
at 30° 
-224,692 
10° 
73,708 
35° 
-263,940 
15° 
- 110,815 
40° 
- 304,073 
20° 
- 148,264 
45° 
-345,255 
25° 
-186,179 
etc. 
To above 20°, these differences are represented, within less than 
