ich the sides, the floor, and the ceiling were
all impenetrable, he would step outside of it without touching any part
of the building, just as easily as we could step over a circle drawn on
the plane without touching it. He would simply disappear from our view
like a spirit, and perhaps reappear the next moment outside the prison.
To do this he would only have to make a little excursion in the fourth
dimension.

[Illustration with caption: FIG. 3]

Another curious application of the principle is more purely
geometrical. We have here two triangles, of which the sides and angles
of the one are all equal to corresponding sides and angles of the
other. Euclid takes it for granted that the one triangle can be laid
upon the other so that the two shall fit together. But this cannot be
done unless we lift one up and turn it over. In the geometry of
"flat-land" such a thing as lifting up is inconceivable; the two
triangles could never be fitted together.

[Illustration with caption: FIG 4]

Now let us suppose two pyramids similarly related. All the faces and
angles of the one correspond to the faces and angles of the other. Yet,
lift them about as we please, we could never fit them together. If we
fit the bases together the two will lie on opposite sides, one being
below the other. But the dweller in four dimensions of space will fit
them together without any trouble. By the mere turning over of one he
will convert it into the other without any change whatever in the
relative position of its parts. What he could do with the pyramids he
could also do with one of us if we allowed him to take hold of us and
turn a somersault with us in the fourth dimension. We should then come
back into our natural space, but changed as if we were seen in a
mirror. Everything on us would be changed from right to left, even the
seams in our clothes, and every hair on our head. All this would be
done without, during any of the motion, any change having occurred in
the positions of the parts of the body.

It is very curious that, in these transcendental speculations, the most
rigorous mathematical methods correspond to the most mystical ideas of
the Swedenborgian and other forms of religion. Right around us, but in
a direction which we cannot conceive any more than the inhabitants of
"flat-land" can conceive up and down, there may exist not merely
another universe, but any number of universes. All that physical
science can say against the suppo