232 On Space of Four Dimensions. [April, 
We ought to say, 4 prater nos but for 4 prceter,’ we 
substitute the preposition 4 extra f which is something 
quite different, i.e,, we imagine these things in the 
space outside ourselves. This evidently is not per- 
ception, but it seems to be something firmly inter- 
woven with the nature of our sensual perceptive 
powers ; it is the form under which that conception 
of the 4 prceter nos 9 is given to us — the form of the 
sensual.” 
The want of these conceptions would necessarily be felt 
by us, if in some individuals, and these only occasionally : 
the will should be capable of producing physical movements, 
for whose geometro-mathematical definition a four-dimen- 
sional system of co-ordinates is necessary. 
To my knowledge Gauss was the first to direct from the 
point of view of the 44 Geometria Situs,” his attention to 
the theory of the twistings of flexible cords. In his manu- 
scripts left behind (Gauss’s Werke, vol. v., p. 605), we find 
the following remarks : — 
44 Of the 4 Geometria Situs which Leibnitz foresaw, and on 
which to throw a feeble glance was allowed only to 
a few mathematicians (Euler and Vandermonde), we, 
after a lapse of 150 years, know and possess hardly 
more than nothing. One of the principal problems 
on the boundary of the Geometria Situs and the Geo - 
metria Magnitudinis will be to calculate the number 
of the twistings of two closed and endless cords.” 
In my first treatise, 44 On Action at a Distance,” I have 
discussed in detail the truth, first discovered by Kant, later 
by Gauss and the representatives of the anti-Euclidian 
geometry, viz., that our present conception of space, fami- 
liar to us by habit, has been derived from experience, i.e. f 
from empirical faCts by means of the causal principle exist- 
ing a priori in our intellect. This in particular is to be said 
of the three dimensions of our present conception of space. 
If from our childhood phenomena had been of daily occur- 
rence, requiring a space of four or more dimensions for an 
explanation which should be free from contradiction, i.e. f 
conformable to reason, we should be able to form a concep- 
tion of space of four or more dimensions. It follows that 
the real existence of a four-dimensional space can only be 
decided by experience , i.e., by observation of facts. 
A great step has been made by acknowledging that 
the possibility of a four-dimensional development of space 
can be understood by our intellect, although, on account 
of reasons previously given, no corresponding image of 
it can be conceived by the mind. (Dass die moeglichkeit 
