230 On Space of Four Dimensions. [April, 
I now proceed to apply the higher conception of space to 
the theory of twisting a perfectly flexible cord. Let us con- 
sider such a cord to be represented by a b, showing us, when 
stretched, a development of space in one dimension, — 
( a - — b). 
If the cord is bent so that during this action its parts always 
remain in the same plane, a development of space in two 
dimensions will be required for this operation. The follow- 
ing figure may be given to the cord, — 
and all its parts, if conceived of infinite thinness, may be 
considered as lying in the same plane, i.e., in a development 
of space in two dimensions. If the flexible cord, without 
being broken, has to be brought back into the former figure 
of a straight line, in such a manner that during this opera- 
tion all its parts remain in the same plane, this can only be 
effected by describing with one end of the cord a circle 
of 360°. 
For beings with only fr^o-dimensional perceptions these 
operations with the cord would correspond to what we, with 
our ^r^-dimensional perception, call a knot in the cord. 
Now if a being, limited on account of its bodily organisation 
to the conception of only two dimensions of space, possessed, 
nevertheless, the ability of executing by his will operations 
with this cord which are only possible in the space of three 
dimensions, such a being would be able to undo this two- 
dimensional knot in a much simpler way. Merely the 
turning over of part of the cord would be required, so that 
after the operation, when all parts again lie in the same 
plane, the cord would have passed through the following 
positions : — 
_______ <n _ST“ . ~ . 
By the same operations, but in an inverted sense, such a 
being would be able again to form the knot without needing 
that circumstantial process, during which all parts of the 
thread have to remain in the /m>-dimensional space of per- 
ception. 
If this consideration, by way of analogy, is transferred to 
a knot in space of three dimensions, it will easily be seen 
that the tying as well as the untying of such a knot can 
only be effected by operations, during which the parts of the 
