The San Francisco call. [volume] (San Francisco [Calif.]) 1895-1913, 1895-04-28
he Fourth Dimension.
~ ENRY RrEL1 PROF MATHEMATICS NEVADA STATE UNIVERSITY


Much has been said during the last dec {
ede concerning the assumption by mathe- |
poaticians of fourth direction at right |
ANGLES TO the three known directions OF
space.
It will be the object of this article to pre-
sent to the readers of the CALL some OF the
interesting consequences that follow the
assumption. as well as the arguments based
upon analogy that lead to these conse-
euences Let it be distinctly understood
that this is a fiction and an assumption,
and that no scholar desires to argue the
likelihood of the real existence Of such
direction.
First, consider a point. 1t has neither
length, breadth nor thickness. tits space
OF no dimension. Call it Pointland. Sup
pose an individual lived in this point
Snch a creature could have no dimension
itself and could have nonotion of direction
or dimension. 1t could have no thought
OF ANYTHING outside of its little universe,
of which il would be the sole monarch.
Having performed this rather painful
effort of the imagination, change the scene
and consider space of one dimension.
This would be line, which might be called
Eineiand. Suppose an individual or
number of individuals to exist IN
this line Each of these creatures
could have length but no other
dimension. Imagine one of these creatures
provided with an eye at each end. 1t
would be able to see its fellow, but all 1t
could see would be a point, and the only
way one individual could differ from an
other would be in length 1t might have
motion to and fro in the line, but could
have no motion to one side or the other OF
the same. If an obstacle lmpeded It's way
IL would have no alternative but to stop or
run back. The creature could have no no
ton OF space on one side or the other OF its
line OF motion. 1t could form no concep-
ton of any direction except the two back
ward and forward. 1t would be greatly
superior to the individual of Pointland
and still a very inferior creature. 1t
might be conscious of its length, but
would be unable to see itself. Its two sur
faces would be its two ends.
Suppose now an individual to exist in
what we will call falkland. Space of this
kind contains two directions and would al
low an individual of this order consider
able freedom of motion. Suppose 1t to be
q square or a triangle and to have an eve at
each vertex. The borders of the figure
would be its external surfaces, those that
would be observable by its fellow-creatures.
The interior of the square would be hidden
creatures as the length OF the line would be
hidden from the inhabitants of Lineland.
Such an individual could observe the crea-
ture of Lineland crawling back and forth
in its own narrow universe and In compass
sion might speak to the Lineman OF di-
rection to this side and that Its voice
would come from an entirely new direction
to the Lineman and he would not be able
TO understand anything about the new
direction spoken of. Should the square
push himself across the line of motion of
the Lineman, that creature would see a
point. nothing more, and that point would
appear to come from nowhere and
vanish into nowhere. 1t would appear
like a creature of Lineland, but would
seem to be possessed of a power that the
Linemen had not-that of vanishing and
Teqppearing at its own pleasure. The man
from Flatland would have the power of
looking over his fellow-creatures and COM
pairing their sizes and shapes with his
own. although to do this he would have to
travel round therl and observe them from
all sides He would only with a single eye
be able to see one line al a time. He would
have no notion OF a direction up or down, I
and could form no conception OF creature :
whose shape extended into space of three /
dimensions Infinitely superior in order |
10 the Lineman, in that he is able to move |
in two directions instead Of one, he is yet a
creature of very inferior order compared to
the one from what we will call Spaceland
Let the individual from Spaceland be
represented by a cube, an object having
length. breadth and thickness Its surfaces
are planes or spaces of two dimensions.
Suppose it is to be provided with eyes at
all of its veTtices 1t will be able to look
down with ease upon the creatures of Flat
land to talk to them from the realm of
space and tell them of a direction up and
down as the Flatlander did to the Line
lander. Like the Lineman, they will not
understand the words OF the man from
Spaceland, and should the cube appear
among them they will only see square,
figure similar to their own, and when the
cube rises it to them will appear to vanish
mysteriously into realm OF which they
know nothing. Suppose the cube took one
Of the Flatlanders up with him and showed
him the homes of the Flatlanders below
him and explained to him the true state of
affairs and then took him back to his home
and left him. The Flatlander, conscious
then dimlv of all that lay above and about
him. might attempt to teach his fellows Of
the wonderful land he had visited, the
realm Of space. His companions would
probably laugh him to scorn and perhaps
Shut him up in a madhouse for what they
would consider his lunacy.
Now do the analogleS necessarily stop
here? MAY there not be still fourth
direction which three dimensional crea-
f,IreS cannot conceive or any better than
he Linelander can conceive Of Flatland or
tho Flatlander of space? If so, it undoubt-
cfIy lies all around and about this space,
and NAV be peopIed by creatures as much
above humanity as it is above the fictitious
creatures of Flatland. As man can look
down upon a fat surface and see the
inside of the squares and triangles, pos
siblv the fourth dimensional being can
look in and through him, perhaps, discern
his thoughts and desires As a cube ap-
nearing among the Flatlanders could only
appear as a square, one like themselves, so
q fourth dimensional creature could only
appear among men as three-dimensional
creature. like themselves, but able to go
and come, appear and disappear, at will.
Let Ms try to form an idea of fourth d.
mensional solid We begin with a point,
snace of no dimension. If we assume the
point to move it will generate line, a
space of one dimension, consisting of two
ends. external surfaces that are points and
one line.
Now assume the line, a space OF one
dimension, to move in new direction. 1t
will generate square, space OF two oil
mansions Each point generates line
and there will be also the first position OF
a. I;ro apd tho second in the new figure,