182 
MR. GRAVES ON A RECTIFICATION OF THE 
§ 7- On the orders and ranks of logarithms. 
In [22] let y — R + — 1 S and a — A + >J — 1 B ; 
then, by [28], will 
f - 1 
y 
2 in + cos 
- l 
R 
f — 1 a 
ox x — 
\/li 2 + S- 
- y' — 1 1 V R 2 + S a 
2 i 7 r 4 cos 
- l 
[ 32 ] 
V A 2 + B a 
- y' - 1 1 */ A a + B s 
When I have thus separated respectively the real and imaginary parts of the 
numerator and denominator of [22], upon assigning particular values, 1 and 1, 
to i and i in [32] , I would indicate the order of a logarithm by the 1 in the 
denominator, and the rank it bears in that order by the 1 in the numerator; e. g. 
I would say of the resulting x that, in the base a, it was the 1th logarithm of y 
of the 1th order. 
By [20], all the values of (A + B)* are comprised in the formula 
or, (see [28] ) 
f f 1 (A + V — 1 B)j> 
f ( 2 i 7 r + cos 1 
*/- ll V A? + B 
’)} 
a/ A 2 + B 2 
When, in this formula, i assumes the particular value i, I would denominate 
A 
f (2 l IT + cos 1 
V A? + B a 
- 11 V A 2 + B 
the 1th value of (A + — 1 B) 1 
When, with respect to the base a , x is any logarithm of y of the «th order, the 
1th value of a x will = y. 
Employing the mode of expression above explained, I conceive that the chief 
novelty of my system consists, not in showing that any assigned quantity, 
relatively to a given base, has an infinite number of logarithms (which was 
known before), but in showing that it has an infinite number of orders of 
logarithms, and an infinite number of logarithms in each order. 
Thus, all the Neperian logarithms of 1 have been hitherto supposed to be 
comprised in the formula 
V — 12 in 
