3

.

as

justified by the fact that they find an adequate substratum in in-
numerable objects. As regards them, it is true, an understanding has for
some time been arrived at. But the opposite of real, that is imaginary
number, formerly, and even now sometimes ,improperly termed impossi
ble--is still rather tolerated than naturalised, is rather regarded
a mere symbolic artifice, to which is utterly denied any objective sub-
stratum, though there is no wish to despise the precious contribution
yielded by this symbolizm to the treasure-house of real quantitative
relations

The author has for many years been regarding this highly important
part of mathematics from a different point of view.in accordance with
which a substratum can be supplied for imaginary, just as well as for #
negative quantity.

Positive and negative number can only then find an application,
when what is numbered has an opposite which, conceived as combined with
it, reduces it to nothing. Exactly taken, this assumption is only then
an actuality when a not substances objects conceivable per se, ut relat
etween pairs of objects
i
what is numbered. It is assumed that
these objects are arr nged in a
ĭ ow in some particular way, s A, B, C, D,
and that the relation of A

i on s

to B can be regarded s equal to that of

* G

from their constitution and contents; he has to do only with the eas-
urement and comparison of the relations with each other.hus, u t as
he assign similarity of nature to the relations denoted by +1 and
he is entitled to extend that characteristic to the four elements +1
1,, andį.

1

These relations can be presented to perception only by a repre
sentation in space, and the simplest case Is where no reason exist
for arranging the symbols of the objects otherwise than in a quadr
tic form, that is, in n an unlimited plane, divided into squares by two
systems of parallel lines intersecting at right angies, we select the
points of intersection as the symbol. Every such point A mas four
neighbors, and, if we de note the relation of A to a neighboring point,
by 1, then that denoted by -1 is determined of itself, while we
select for 2 whichever we ike of the other two, or can take the
point that refers to +1 either to the right or left. This distinction
between right and left isik self quite determined as soon as we
have settled the forward, the up and down in relation to the sides of
the plane, although we we can establish our perception of this diet!
t no
tion only by reference to material objects actually present, but, when
we have sectied this, we
that it none the les d-p nder оп our

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