350 
combination, giving a symbolic algebra 
which may hereafter become the grammar 
of a hundred distinct significant algebras. 
If any one were to assert that + and — 
might mean reward and punishment and 
A, B, C, etc., might stand for virtues and 
vices, the reader might believe him, or con- 
tradict him, as he pleases, but not out 
of this chapter. The one exception above 
noted, which has some share of meaning, 
is the sign = placed between two symbols, 
asin A= B. It indicates that the two sym- 
bols have the same resulting meaning, by 
whatever steps attained. That A and B,if 
quantities, are the same amount of quantity, 
that if operations, they are of the same ef- 
fect, ete.” Let us apply to the theory 
quoted the logical maxim that the excep- 
tion proves the rule, prove being used in 
the old sense of test. Well then, I say, 
because one symbol at least is found to be 
refractory to the theory, it follows that the 
theory is fallacious. 
De Morgan proceeds to give an inventory 
of the fundamental symbols and laws of al- 
gebra, that for the symbols being 0,1, +, 
—, X,+, ( ) and letters. With respect 
to it the following questions may be asked : 
Why should)" be included, while the in- 
verse idea, denoted by log is left out? What 
of the functional symbols sin and cos? Can 
they be derived from the above? As — de- 
notes opposite and ~ reciprocal, what are 
the signs for sum and product? Can they 
be derived from the above? 
His inventory of the fundamental laws is 
expressed under fourteen heads, but some 
of them.are merely definitions. The laws 
proper may be reduced to the following, 
which he admits are not all independent of 
one another : 
I. Law of signs: ++=-+, +-—or 
= = = 
XxX =X, XSor] xX ==, += == xX. 
II. Commutative law: a+b=6-+4, 
ab = ba. 
SCIENCE. 
[N. 8. Von. X. No. 246. 
III. Distributive law: a(6 +c) = ab + ace. 
IV. Index laws: a’ x a=a't*, 
(a) =a", (ab)°=a'd’. 
V.a—a=0, 
These last may be called the rules of re- 
duction. What Gregory gave was a classi- 
fication of the more important operations 
occurring in algebra; De Morgan professes 
to give a complete inventory of the laws 
which the symbols of algebra must obey, 
for he says ‘‘Any system of symbols which 
obeys these rules and no others, except they 
be formed by combination of these rules, 
and which uses the preceding symbols and 
no others, except they be new symbols in- 
vented in abbreviation of combinations of 
these symbols, is symbolic algebra.” 
Compare this inventory with Gregory’s 
classification. De Morgan brings x and + 
under the same rule as + and —; he ap- 
plies the commutative law to a sum as well 
as toa product; he introduces the third in- 
dex law, which makes the index distribu- 
tive over the factors of the base; he leaves 
out the logarithmic and trigonometrical 
principles and introduces what may be 
called the rules of reduction. From his 
point of view, none of them are rules ; 
they are laws, that is, arbitrarily chosen re- 
lations to which the algebraic symbols must 
be subject. He does not mention the law 
pointed out by Gregory, afterwards called 
the law of association. It is an unfortu- 
nate thing for the formalist that a? is not 
equal to b% , for then his commutative law 
would have full scope; as it is, the index 
operations prove exceedingly refractory, so 
that in some of the beautifully formal sys- 
tems they are left out of account altogether. 
Here already we have sufficient indication 
that to give an inventory of the laws which 
the symbols of algebra must obey, is as am- 
biguous a task as to give an inventory of 
the a priori furniture of the mind. 
Like De Morgan, George Boole was a 
mathematician who investigated and wrote 
a+a=l1. 
