360 
amounts to asking a question, and this 
question may or may not have an answer 
according to circumstances. If we write 
down the symbols for the answer to the 
question in any of those cases where there 
is no answer, and then speak of them as if 
they meant something, we shall talk non- 
sense. But this nonsense is not to be 
thrown away as uselessrubbish. We have 
learned by very long and varied experience 
that nothing is more valuable than the non- 
sense which we get in this way; only it is 
to be recognized as nonsense, and by means 
of that recognition made into sense. We 
turn the nonsense into sense by giving a 
new meaning to the words or symbols which 
shall enable the question to have an answer 
that previously had no answer.”’ 
This is the true phenomenon in algebra; it 
is more logical thanits framer. How can it 
be possible, unless the algebraist founds his 
analysis upon realrelations? It is the logic 
of real relations which may outrun the im- 
perfect definitions and principles of the 
analyst and make it necessary for him to 
return to revise them. 
To get over the impossible subtraction 
he introduces instead of the discrete unit 
supposed by number, the idea of a step, 
making plus mean ‘ forwards’ and minus: 
‘backward.’ The summing of steps is in- 
dependent of the order in which they are 
taken, and a minus step is just as inde- 
pendent as a plus step. When these sym- 
bols occur in multipliers he gives them, not 
the meaning of ‘ forwards’ and ‘backwards,’ 
but that of ‘keep’ and ‘reverse.’ He gives 
them these meanings in addition to their 
former meanings, and leaves it to the con- 
text to show which is the right meaning in 
any particular case. It may be remarked 
that it is doubtful whether in any case two 
distinct meanings can be given to a symbol 
at one and the same time without produc- 
ing confusion. It seems to me, as already 
stated, that the most general meanings of 
SCIENCE. 
[N.S. Von. X. No. 246. 
+ and — are the angular ideas of an even 
and an odd number of semi-circumferences, 
but this reduces in certain cases to the linear 
ideas of direct and opposite. 
From the idea of step he passes to the 
idea of operation, on the theory that a prod- 
uct may be composed either of a step and 
an operation or of two operations. Asa 
matter of fact, an operation is merely a 
relationship which may subsist between 
two quantities; and we may have two dis- 
tinct products, one expressing a related 
quantity, the other a compound relation- 
ship. The analysis of operations is a special 
part of the more general analysis of rela- 
tionships. According to Clifford’s view, 
because a sum of operations of the kind 
consideréd is independent of the order of 
the operations, it follows that 
at+b=b+a4 ab = ba 
a(b +c) = ab + ac ~ (a + b)e =ac + be. 
As regards the advance from numbers to 
quantity he says (‘ Philosophy of the Pure 
Sciences,’ p. 240): ‘‘ For reasons too long 
to give here, I do not believe that the pro- 
visional use of unmeaning arithmetical 
symbols can ever lead to the science of 
quantity; and I feel sure that the attempt 
to found it on such abstractions obscures its 
true physical nature. The science of num- 
ber is founded on the hypothesis of the dis- 
tinctness of things; the science of quantity 
is founded on the totally different hypothe- 
sis of continuity. Nevertheless, the rela- 
tions between the two sciences are. very 
close and extensive. The scale of numbers 
is used, as we shall see, in forming the men- 
tal apparatus of the scale of quantities, and 
the fundamental conception of equality of 
ratios is so defined that it can be reasoned 
about in terms of arithmetic. The opera- 
tions of addition and subtraction of quanti- 
ties are closely analogous to the operations 
of the same name performed on numbers, 
and follow the same laws. The composi- 
