which were all mathematically consistent and yet lacked all relation to
external existence. A considerable number of space-systems have thus
become established among which there is the system that served Einstein
to derive his space-time concept. Some of them have been more or less
fully worked out, while in certain instances all that has been done is
to show that they are mathematically conceivable. Among these there is
one which in all its characteristics is polarically opposite to the
Euclidean system, and which is destined for this reason to become the
space-system of levity. It is symptomatic of the remoteness from
reality of mathematical thinking in the onlooker-age that precisely
this system has so far received no special attention.1

For the purpose of this book it is not necessary to expound in detail
why modern mathematical thinking has been led to look for thought-forms
other than those of classical geometry. It is enough to remark that for
quite a long time there had been an awareness of the fact that the
consistency of Euclid's definitions and proofs fails as soon as one has
no longer to do with finite geometrical entities, but with figures
which extend into infinity, as for instance when the properties of
parallel straight lines come into question. For the concept of infinity
was foreign to classical geometrical thinking. Problems of the kind
which had defeated Euclidean thinking became soluble directly human
thinking was able to handle the concept of infinity.

We shall now indicate some of the lines of geometrical thought which
follow from this.

*

Let us consider a straight line extending without limits in either