SEPTEMBER 15, 1899.] 
stituted. He quotes with approval a saying 
of Stout’s that a word is an instrument for 
thinking about the meaning which it ex- 
presses, whereas a substitutive sign is a 
means of not thinking about the meaning 
which it symbolizes ; and he adds that the 
use of substitutive signs in reasoning is to 
economize thought. 
It seems to me that a sign economizes 
thought in precisely the same way that a 
word economizes thought, but to greater 
degree. A word is introduced to dispense 
with a long phrase or description, and in 
using the word one no more thinks of its 
meaning than in using an algebraic symbol 
does one think of the particular meaning it 
is made to stand for, for the time being. 
There seems to be a lurking fallacy that 
thought is economized by dispensing with 
it altogether. I prefer the saying of Clif- 
ford, with reference to (a + 6)? =a’? + 2ab 
-+ 6? and its expression in English: ‘‘ Two 
things may be observed on this comparison 
—first, how very much the shorthand ex- 
pression gains in clearness from its brevity; 
secondly, that it is only shorthand for 
something which is just straightforward 
common sense and nothing else. We may 
always depend upon it that algebra which 
cannot be translated into good English and 
sound common sense is bad algebra.” 
In his statement of the fundamental 
principles of algebra Whitehead follows 
Grassmann toa large extent. He divides 
them into two classes, the general and the 
special; the former apply to the whole of 
ordinary and universal algebra; the latter 
apply to special branches only. The gen- 
eral principles are as follows: Addition 
follows the commutative and associative 
laws; multiplication follows the distribu- 
tive law, but does not necessarily follow the 
commutative and associative laws. The 
theory looks beautiful and plausible, but it 
does not stand the test of comparison with 
actual analysis, for quaternions is one of 
SCIENCE. 
363 
the principal branches of universal algebra, 
and in it the addition of indices is in gen- 
eral non-commutative, and the power of a 
binomial of indices is not formed after the 
distributive law. 
But in addition to this formal bond we 
find in the book.another bond uniting the 
several parts into one whole. In the pref- 
ace Mr. Whitehead says: ‘The idea of a 
generalized conception of space has been 
made prominent in the belief that the prop- 
erties and operations involved in it can be 
made to form a uniform method of inter- 
pretation of the various algebras. Thus it 
is hoped in this work to exhibit the alge- 
bras, both as systems of symbolism and also 
as engines for the investigation of the pos- 
sibilities of thought and reasoning con- 
nected with the abstract general idea of 
space.’’ The chance for any arbitrary sys- 
tem of symbolism applying to anything 
real is very small, as the author admits; 
for he says that the entities created by con- 
ventional definitions must have properties 
which bear some affinity to the properties 
of existing things. Unless the affinity or 
correspondence is perfect, how can. the one 
apply to the other? How-can this perfect 
correspondence be secured, except by the 
conventions being real definitions, the equa- 
tions true propositions and the rules expres- 
sions of universal properties? The placing. 
of the algebra of logic on a space basis has 
been criticised, but in reply it may be 
pointed out that logicians have been accus- 
tomed ever since the time of Euler to 
prove their principal theorems by means of 
diagrams. 
Our conclusion about the fundamental 
rules of algebra is: If the elements of a 
sum or of a product are independent of 
order, then the written order of the terms 
is indifferent, and the product of two such 
sums is the sum of the partial products ;. 
but when the elements of a sum or of a 
product have a real order, then the written 
