542 DOC.
523
APRIL
1918
It
seems
to
me
that
no argument can
be
raised
against your objection
to
the
conception
of
the
world
as
quasielliptic
with
respect
to
the
spatial
relations.[2]
Your
objection
seems
correct to
me,
upon elementary examination,
just
for
even
(spatial)
dimensions.
I
imagine
an
nth
dimensional
spherical
manifold
Rn (in an
n+1th
dimensional
Euclidean
space
Rn+1)
x12+...+x2n+l
=
R21.
An
elementary
orthogonal
"nbein”
is
situated
within
an
element
of
Rn.
It
can
be moved
along
Rn.
I
call
the
point
0
= x1
=
x2
=
...
xn,
xn+1
=
±1
the
“south
pole”
or
“north
pole”
of
Rn, respectively.
The
nbein
lies
initially
at
the south
pole
in such
a
way
that
its
legs
b1
...
bn
are
parallel
to
the
positive
axis
orientations
x1 ...
xn,
respectively.
Now I
move
the nbein
on
the
sphere
Rn
from
the
south
pole
to
the north
pole
in
the
following way:
1)
The motion
is
restricted
to
motion
along
the
meridian
x2
=
x3
•••
=
xn
=
0.
2)
The
legs
b2,
b3
...
bn
constantly
remain
perpendicular to
this
meridian while
in
motion
(determined
in
space
Rn+1).
Upon
arrival
of
the nbein
at
the north
pole,
observed from within
Rn+1,
the
orientation
is
hence
given by
the
scheme
b1 b2
...
bn

+
...
+,
i.e.,
the orientation
is
the
same
as
initially, except
that the
first
leg
has reversed
its
direction
Now
if
Rn represents
an
elliptic space,
then
centrically symmetrical points are
identical.
Our
nbein
situated
on
the north
pole
is
then
identical
to
one
to be
found
at
the south
pole
with
legs
pointed
in
opposite
directions
(seen
from
Rn+1).
This
corresponds
to
the
scheme
b1 b1
...
bn
+
It
now comes
down to
whether
during
this round
trip
in
the
elliptic
world
the
nbein
changes
into
a congruent or
symmetrical
one
(viewed
from
Rn).
It
is
easily