70 
[726 
726. 
A FORMULA BY GAUSS FOR THE CALCULATION OF LOG 2 
AND CERTAIN OTHER LOGARITHMS. 
[From the Messenger of Mathematics, vol. viii. (1879), pp. 125, 126.] 
Gauss has given, Werke, t. il, p. 501, a formula which is in effect as follows: 
a,.. _m.il02Sy /1048576y /6560y /15624y ,9801y 
\1024/ \1048575/ 1,6.5617 v 15625/ \980()/ ’ 
= 2» 5»/S!i«Y / 2» Y /6.2».4iy /2».8». 7.81V ( 3611» y 
V 2“ j 1.5». 8.11.31.417 V 3» )\ 5‘ ) \2".5».7»7 ’ 
where on the right-hand side the several prime factors have the indices following, viz. 
2, index is (59 + 160 + 15 + 24-50-12) = 196, 
3 „ 
(16 + 
16- 
8-24 
) = 0, 
5 
(59 + 
10 + 
CO 
1 
1—1 
Gi 
1 
00 
1 
8) = 0, 
7 
( 8- 
8 
) = o, 
11 „ 
( 8- 
8 
) = o, 
31 „ 
( 8- 
8 
) = o, 
41 „ 
( 5 + 
3- 
8 
) = o, 
or the right-hand side is = 2 196 as it should be. The value of log 2 calculated from 
2M6 — 1059 log 2 = = *301020, viz. there is an error of a unit in fifth place of 
decimals. The actual value of 2 1 »« has been given me by Mr Glaisher: 
2i»« = 10043 36277 66186 89222 13726 30771 
32266 26576 37687 11142 45522 06336* 
Supposing log 2 calculated by the form, we then have 
41 = (tM?) 212 * 102 > giving- log 41, 
and 
3 8 = 10 . . 2 4 .41, giving log 3 ; 
and formulse may be obtained proper for the calculation of the logarithms of if, 11.31, 
and 7.31. 
* The value was deduced by Mr Glaisher from Mr Shanks’s value of 2 193 in his Rectification of the Circle, 
(1853), p. 90.