3 io 
Mr. hellins’s theorems 
DEMONSTRATION. 
IttM x lt±l - lt±13 - 111 : therefore, log. p -±l ~ 
2 p + q 2 p‘ 9-P P P 
Ip + q 
2,p 
] 0 (t. 1111 + log. ^—tl . But it has been proved above, 
° 2 P+q Q 2 P 
that log.** =» log. '-£* + log. ^ 
■p+qx 
If now we 
take this equation from twice the laft there will remain 
p + q -• 2 p + 2 q , 2 p + q , 
— = 2 l0 S- U+q + 2 l0 g* -Tf - 2 l0 §' 
1 1 
a log. — - log, 
2 p_+2J 
2p + q 
2/> + ?)‘ 
- log 
2* + $V 
' 2p+iY-qi 
E • D# 
: that is, log. ^ log 
2 P + q 
2 p 
log, 
COROLLARY. 
Putting q- 1, and « -p, as above, we have 
»+ 1 
2 77 + t 
lQ g- — = 2 lo S- — ~ lo S 
2 72+ Ij 
272+ I 
I fhall now fet down fome examples of the ufe of 
thefe theorems beginning with theorem i . 
the JirJl example of the utility of this theorem may be in 
'computing the logarithm of the number 2. 
It is well known to mathematicians, that the compu- 
tation of this logarithm was formerly a very laborious 
1 ‘ talk : 
