AUGUST 27, 1897. ] 
the outlines are given briefly; they consist of the fol- 
lowing : 
Positive and negative numbers can only find an ap- 
plication when the thing counted has an opposite 
which when conceived of as united with it has the 
effect of destroying it. Accurately speaking, this 
supposition can only be made where the things 
enumerated are not substances (objects thinkable in 
themselves), but relations between any two objects. 
It is postulated that these objects are arranged after 
a definite fashion in a series, e. g., A, B, C, D, * * * 
and that the relation of A to B can be regarded as 
equal to that of B to C, etc. The notion of opposi- 
tion involves nothing further than the interchange of 
the terms of the relation so that if the relation of (or 
transition from) A to B is considered as +-1 the rela- 
tion of B to A must be represented by —1. So far 
then as such a series is unlimited on both sides, every 
real integer represents the relation of a term arbi- 
trarily taken as origin to a definite term of the series. 
If, however, the objects are of such a kind that 
they cannot be arranged in one series, even though 
unlimited, but only in series of series, or, what 
amounts to the same thing, they form a manifoldness 
of two dimens‘ons; if there is the same connection 
between the relations of one series to another, or the 
transitions from one to another, as in the case of the 
transition from one term of a series to another term 
of the same series, we shall evidently need for the 
measurement of the transition from one term of the 
system to another, besides the previous units -+-1 and 
—1, two others opposite in character +7 and —. 
Obviously we must also postulate that the unit 7 
shall always mark the transition from a given term 
of the one series to a definite term of the immediately 
adjacent series. In this way the system can be ar- 
ranged in a two-fold manner in series of series. 
The mathematician leaves entirely out of considera- 
tion the nature of the objects and the content of their 
relations. He has simply to do with the enumera- 
tion and comparison of the relations. So far as he 
has assigned sameness of nature to the relations des- 
ignated by +-1 and —1, considered in themselves, he 
is warranted in extending such sameness to all four 
elements +1, —1, ++i, —i. 
These relations can be made intuitive only by a rep- 
resentation in space and the simplest case, where there 
is no reason for arranging the objects in any other than 
quadratic fashion, is that in which an unlimited plane 
is divided into squares by two systems of parallel lines 
intersecting at right angles, and the points of inter- 
section are selected as the symbols. Every such 
point has four adjacent points, and if we designate the 
relation A to a neighboring point by +1, the relation 
to be denoted by —1 is determined of itself, while we 
SCIENCE. 
305 
can select which of the two others we please for +7, 
or can take the point to be denoted by +7 at pleas- 
ure on the right or left. This distinction between 
right or left so soon as we have fixed (at pleasure) 
upon forwards and backwards in the plane, and above 
and below with respect to the two sides of the plane 
is completely determined in itself, although we can 
convey our own intuition of this difference to others 
only by reference to actually existent material things. 
But when we have decided upon the latter we see 
that it is still a matter of choice as to which of the 
two series intersecting at one point we shall regard as 
the principal series and which direction in it shall be 
considered as haying to do with positive numbers. 
We see further that if we wish to take +-1 for the re- 
lation previously expressed by -+7, we must neces- 
sarily take +-i for the relation previously expressed 
by —1. In the language of mathematicians this 
means that -+-7 is a mean proportional between +1 
and —1, or corresponds to the symbol  —1. We 
say purposely not the mean proportional because —i 
has just as good a right to that designation. Here 
then the demonstrability of an intuitive signification 
of j/—1 has been fully justified and nothing more is 
necessary to bring this quantity into the domain of 
objects of arithmetic. 
We have thought to render. the friends of mathe- 
matics a service by this brief exposition of the princi- 
pal elements of a new theory of the so-called imagi- 
nary quantities. If people have considered this sub- 
ject from a false point of view and thereby found a 
mysterious obscurity, this is largely due to an unsuit_ 
able nomenclature. If +1, —1, ~=il had not been 
called positive, negative, imaginary (or impossible) 
unity, but perhaps direct, inverse, lateral unity, such 
obscurity could hardly have been suggested. The 
subject which, properly enough, in the present trea. 
tise has been touched upon only incidentally the au_ 
thor has reserved for a more elaborate treatment in 
the future where also the question will be answered 
as to why the relations between things which present 
a manifoldness of more than two dimensions cannot 
furnish still other classes of magnitudes admissible 
in general arithmetic. 
Such was Gauss’s masterly presentation 
of the underlying principles of the treat- 
ment of the imaginary. In Germany the 
impulse given by his commanding influence 
is felt even to the present day. 
Buée’s memoir Sur les Quantités Imaginaires, 
read before the Royal Society of London in 
1805 and covering 65 pages of the Philo- 
sophical Transactions of 1806, is somewhat 
