P. HISTORY OF ' AATHEMATICAL - School of Mathematics

114 A HISTORY OF MATHEMATICAL NOTATIONS
Sehellbach' of Berlin adopted for the operation of logarithmirung,
a
in place of the incomplete formula, log a = c, the equation X =c; he
b
expresses the theorem relating to change of modulus thus,
"a b a"
X X X.
b k k
As the notation here proposed is not always convenient, he suggests
an alternative notation. Just as one has aXb and a.b, ~ and a:b, so
a
one may choose X and a: b. He writes (a+b) : c for log (a+b) to the
b
modulus c, also (a: b) : c for loge (log~) .
480. Complex numbers.-Martin Ohm, in his treatment of logarithms
of complex numbers, lets "log" represent the infinitely many
logarithms of a complex number, "L" the tabular logarithm of the
modulus. He states that when the concept of the general power ax
is given the concept of the general logarithm b? a, if by it is meant
every expression x such that one has a x =b or ex log a = b."2 If z has
an infinite number of values for each value of log a, one sees the reason
for the appearance in a ? b of two independent arbitrary constants, as
is seen also in the investigations due to J. P. W. Stein, John Graves,
and W. R. Hamilton." Ohm says that, since b and a are taken completely
general, and aX has an infinity of values, b? a is wholly undetermined,
unless it is stated for what value of log a the power aX is to
be taken. He adopts the special notation b?(alla), which means the
logarithm oN to the base a, when log a= a. He shows that b?(alla) =
10gb is a complete equation. Ifa=e=2.718 .... , then Ohm's logaa
rithms reduce to those previously developed by Euler.
De Morgan! in developing the general theory of logarithms used
"log" for the numerical or tabular value of the logarithm, and let Ax
1 K. H. Schellbach, "Ueber die Zeichen der Mathematik," Crelle's Journal,
Vol. XII (1834), p. 70-72.
2 Martin Ohm, System der Mathematik, Vol. II (2d ed., 1829), p, 438,415.
3 See F. Cajori, American Mathematical Monthly, Vol. XX (1913), p. 173-78.
• A. de Morgan, Tram. Cambridge Philoll. Soc., Vol. VII (1842), p. 186.