302 
way for its reception. It may, where the 
immediate force of the evidence is not felt, 
serve as verification, a posteriori, of the prac- 
tical validity of the principle in question. 
But this does not affect the position af- 
firmed, viz., that the general principle must 
be seen in the particular instance—seen to 
be general in application as well as true in 
the special example. The employment of 
the uninterpretable symbol “ —1 in the 
intermediate processes of trigonometry fur- 
nishes an illustration of what has been said. 
I apprehend that there is no mode of ex- 
plaining that application which does not 
covertly assume the very principle in ques- 
tion. But that principle, though not, as I 
conceive, warranted by formal reasoning 
based upon other grounds, seems to deserve 
a place among those axiomatic truths which 
constitute in some sense the foundation of 
general knowledge, and which may properly 
be regarded as expressions of the mind’s 
own laws and constitution ’’ (p. 68). 
We are all familiar with the fact that al- 
gebraic reasoning may be conducted through 
intermediate equations without requiring a 
sustained reference to the meaning of these 
equations ; but it is paradoxical to say that 
these equations can, in any case, have no 
meaning, no sense, no interpretation. It 
may not be necessary to consider their 
meaning ; it may even be difficult to find 
their meaning, but that they have a mean- 
ing is a dictate of common sense. It is en- 
tirely paradoxical to say that, as a general 
process we can start from equations having 
a meaning and arrive at equations having a 
meaning, by passing through equations 
which have no meaning. The particular 
instance in which Boole sees the truth of 
the paradoxical principle is the successful 
employment of the uninterpretable symbol 
/ — 1inthe intermediate processes of trigo- 
nometry. As soon, then, as the Y — 1 oc- 
curring in these processes is demonstrated, 
the evidence for the principle fails. As a 
SCIENCE. 
[N.S. Von. X. No. 246. 
matter of fact, the doctrine of algebraists 
about “ — 1 has long been a dark corner 
in exact science ; and as a consequence it 
has been made the foundation for all sorts 
of crank theories. Recently I noticed that 
an ingenious individual had applied the 
v —1 and its successive powers to con- 
struct a mathematical theory of sensation. 
Before the introduction by Descartes of the 
geometrical idea of the opposite the use of 
—in algebra might have been made the 
foundation for a similar transcendental 
theory of reasoning. Algebra, as the anal- 
ysis of quantity in space, has a clear mean- 
ing for “ —1as the operation of turning 
through a right angle round a definite or an 
indefinite axis ; in the former case it is vec- 
tor in nature, because the axis must be 
specified ; in the latter it is scalar in na- 
ture, because the axis may be any suitable 
one. It follows that — denotes turning 
through two right angles, and this includes 
‘opposite’ as a particular case. Thus an 
instance is still wanting on which to build 
the transcendental theory of reasoning 
enunciated by Boole. 
The object of Boole’s work, ‘The Laws 
of Thought,”’ is to investigate the funda- 
mental laws of thought, to give expression 
to them in the symbolical language of a cal- 
culus, and upon that foundation to establish 
the science of logic. In the concluding chap- 
ter he considers the ight which the inquiry 
throws on the nature and constitution of 
the human mind. Now, as a matter of fact, 
the subject analyzed is quality, and its con- 
nection with the nature and constitution of 
the human mind is nowise more inanimate 
than is the connection of algebra the science 
of quantity. 
It is interesting to compare Boole’s inven- 
tory of the symbols and laws for a calculus 
of reasoning (analysis of quality) with the 
inventory made by De Morgan for the sym- 
bols and laws of algebra (the analysis of 
quantity). The symbols are the same, ex- 
