266 Proceedings of Royal Society of Edinburgh. [sess. 
For sine, 
For cosine, 
For tangent, 
For cotangent, 
Q 
-log. 
sin. a 
A 1 
-j(i 
CM 
<d 
1 
Q 
- log- 
cos. a 
A 1 
+i(l 
-a) A 2 
Q 
-log. 
tan. a 
A 1 
+1(1 
<M 
<1 
i 
. Q 
-log. 
cot. a 
A 1 
+«1 
-a?) A 2 
This requires first the division of the difference Q - A by A 1 to 
obtain an approximate value of x , and then a second division with 
the approximate value of J(1 -x) thus found, in order to obtain a 
closer approximation to x. And for arcs very near 0° or 90° when 
the functions vary rapidly and the use of third differences would 
be required, it is recommended to find the corresponding natural 
sine, tangent, etc., for which only first differences are necessary, 
and then to find the logarithm of this, in order to obtain the 
correct values. These processes are cumbersome. 
In Hutton’s Tables ( 10th ed., p. xxxvii.) Maskelyne’s rules, given 
in his introduction to Taylor’s Logarithms (1792), are stated as “often 
useful,” but no examples of their use are given. These rules are 
quite empirical, but for small arcs, under 2°, they are very con- 
venient and accurate, and have often been published, as in Gal- 
braith’s Mathematical Tables (1827), Shortrede’s Logarithmic 
Tables , etc. They are expressed by the formulae — 
log. sin. a!' = log. sin. 1" + log. a" - J(log. sec. a" - 1 0) 
log. tan. a = log. tan. 1" + log. a" + |(log. sec. a ' - 10). 
When a = 5°, or a" — 1800”, the error in the 7 th place of decimals 
is -5,611 or 0”-023; 
„ a = 10°, the error in the 6th and 7th places of decimals is 
- 90,428, or 0”*757 in sine and 0"*711 in the tangent ; 
,, 10° 34', the error in the 5th, 6th, and 7th places of deci- 
mals is - 112,861, or l”in sine and 0”'966 in tangent; 
,, a = 15°, the error in the 5th, 6th, and 7th places of decimals 
is - 463,445, or 5” - 9 in sine and 5”'5 in tangent, etc. ; 
the divergence rapidly increasing above 10°. 
