The daily morning journal and courier., 1902-01-13
WHAT THE FOURTH DIMENSION
 REALLY IS.

WHAT THE FOUhlI II DlMc4N oiCAS REALLY IS.
 Sunnose the world consisting of boundless flat plane to be inhabited by reasoning beings who can move about at pleasure on the plane, but are not ab1e to turn their heads up or down, or even to see or think of such terms as above them and below them, and things sround them can be pushed or pulled about in any direction, but cannot pc lifted from the plane. People ana things can pass around each other, put cannot step over anything. These dwell CTS in ''Aatland" could construct a, plane geometery which would be exactly ike ours in being based on the axioms Of Euclid Two parallel straight lines l wou1d never meet, though continued in- definitely.
 But suppose that the surface on which these things live, instead of be- ing an infinitely extended plane, is real ly the surface of an immense globe, like the earth on which we live. It needs no knowledge of geometry, but only an ex- animation of any elobular obJect-an apple, for example-to show that II we draw a line as traigbt as possible on sphere, and parallel to it draw small piece of a second line, and continue thls 1 in as straight a line as We can, the two lines will meet when we proceed m either direction one quarter of the way around the sphere For our "1atland' people these lines would both be per- fectly straight, because the only curva- ture would be in the direction down. wards. which they could never either perceive or discover.
 To explain hypergeometry proper we must first set forth what fourth di- menson of space means, and show how natural the way by which 1t may be ap- preached We continue our analogy from "flatland.' In this supposed lana l let us make a cross-two straight lines intersecting at right angles The ln- 1 habitants of this land understand the cross perfectly and conceive of it just as We do But let us ask them to draw a third line, intersecting in the same point and perpendicular to both the other lines. They would at once pro- nounce this absurd and impossible, It is equally absurd and impossible to us | if we require the third line to be drawn 1 on the paper. But we should reply: if you allow us to leave the paper or flat surface. then we can solve the problem by simple drawing the thild line through the paper perpendicular to its l surface"
 l Now, to pursue the analogy, suppose that, after we have drawn three mu- tually perpendicular lines. some being
 from another sphere proposes to us the
 drawing of a fourth line through the same point, perpendicular to all three Of the lines already there. We should an
 swer him in the same way that the in-
 habitants of "hatland' answered us:
 "The problem is impossible. You can
 not draw any such line in space as we
 understand it." If our visitor conceiv- ed of the fourth dimension, he would reply to us as we replied to the flat
 land" people: "The problem is absurd and impossible if you confine your line
 to space as you understand it. But for me there is a fourth dimension in
 space. Draw your line through that di-
 mensjon and the problem will be solv-
 ed. This is perfectly simple to me; it is
 impossible to you solely because your
 conceptions do not admit of more than
 three dimensions
 Supposing the inhabitants of fiat land" to be intellectual beings as we are. it would be intersting to them to
 be told what dwellers of space in three dimensions could do. Let us pursue the analogy by showing what dwellers in four dimensions might do. Place a
 dweller of "Flatland' inside a circle drawn on his plane, and ask him to step outside of 1t, without breaking through it. He would go all around and finding every inch of it closed, he would say 1t
 was impossible from the very nature of the conditions. But" we would reply,
 that is because of your limited concep- tions. We can step over it.
 "Step over it!" he would exclaim. "I | do not know what that means I can
 pass around anything if there is a way open, but I cannot imagine what you
 mean by stepping over it.
 But we should simply step over the
 line and reappear on the other side So if we confine being able to move in a fourth dimension in the walls of a dun
 geon of which the sides, the floor, and ' the ceiling were all impenetrable he Uould step outside of it without touch ing any part of the building, just as
 easily as we could step over circle
 drawn on the plane without touching it. lie would simply disappear from our view like spirit, and perhaps re- 1 appear the next moment outside the prison. To do this he would only have to make a little excursion in the fourth dimension.-Prof simon Newcomb in