1897 - 98 .] Dr J. Burgess' gw Log. Sines , etc., of Small Arcs. 265 
Note on Finding the Logarithmic Sines and Tangents 
of Small Arcs. By J. Burgess, C.I.E., LL.D. 
(Read July 18, 1898.) 
In geodetical and astronomical computations it frequently 
happens that we have to use the logarithmic sines or tangents of 
small arcs, or to find the small angles corresponding to their arti- 
ficial sines or tangents ; and in all trigonometrical tables directions 
are given to guide the learner in these operations. It might seem, 
then, superfluous to refer to such a matter. A variety of methods 
in the solution of a simple problem has, however, sometimes advan- 
tages, and the method I have been in the habit of using, though 
obvious enough, is not usually given. 
In Vega’s great Thesaurus Logarithmorum Completus (1794), 
based on A. Vlack’s Tables , the rules for the functions of small 
arcs make use of second differences ; * and the 7th and later editions 
of Hutton’s Tables (1830, 10th ed., 1846) follow the same method. 
Thus — . , 
Q being the required (or given) logarithmic function, A the next 
less tabular value, and A x , A 2 , the first and second differences,— 
| 
Then Q = A + xA 1 + —IT- — - A 2 + etc., or approximately, Q = A + 
A 
x A 1 - ^x(l - x) A 2 . 
That is, 
log. sin.(« + x) = log. sin. a + [x A 1 - \ x(l - x) A 2 ]. 
log. cos. (a +x) = log. cos. a - [x A 1 + |#(1 - x) A 2 ], 
log. tan. (a + x) = log. tan. a + \x A 1 + J x(l — x) A 2 ]. 
log. cot. (a + x) = log. cot. a-\x A 1 + |cr(l - x) A 2 ]. 
In the case of the tangent and cotangent the upper sign in the last 
term is used when a + x<ih°, and the lower when a + x> 45°. 
For the angle: — we have respectively for the excess of the angle 
above the tabular value for a, — 
* The rule given in Shortrede’s Logarithmic Tables (1858) applies only 
when the fraction of a second is 
VOL. XXII. 26/9/98. S 
