for computing Logarithms. 309 
the logarithms of fmall numbers from thole of great 
ones, may be confidered as another proof of the miftake 
before mentioned. I have only to add here, that thefe 
theorems are new to me; and if they are fo to the pub- 
lic, I humbly prefume they will be acceptable. 
THEOREM I. 
The log. of Lti = 2 x log. of Ittll + log. of IHJ ] — . 
& P ° B-P + l & Zp+qV-qq 
DEMONSTRATION. 
Zp + qf ^ Zp+2qf _ 1 x Zp+Zq]' 1 _ 4-pp + 8/ff -f q_qq 
2 frq^—qq 2 ? +fP 2 /* + ~ 11 1 4pp+4-pq 
PJL - til : confequently, log. i£. + + a 1 . 2? 
pXp + q P 2j> + jl *-qq * 2p + f 
-log.ttl. q. e. D. 
COROLLARY. 
If q - 1 , and we write 72 for />, the theorem becomes 
l°g- — = 2 log. + log. -> which expreffion 
perhaps is of more frequent ufe than that above. 
THEOREM II. 
Log. 
p+q 
* log. 2^- log. ^ 
2 i>+?r 
qq 
T t 2 
D E- 
