Page:Popular Science Monthly Volume 83.djvu/388

384 any curvature. It is well to note that in an exactly similar way our space may be curved without our being conscious of it. So he might feel just as certain that his space is "straight-line" space as we, the high and mighty $$3$$-space beings, do that our space is Euclidean (or space of zero curvature).

Again, suppose the tube to be bent in an egg-shaped curve where the curvature is not constant. Here the unodim's world would still be one-dimensional, but as his body would be bent a little more in one part of his world than in another, it is possible that he may feel that there is some variety in his space. He may walk a little straighter at times and less straight at others. Whether his $$1$$-space is straight or curved, and, if curved, whatever may be the variety of its convolutions, the unodim can not know of the existence of a world of $$2$$-space or $$3$$-space.

If a $$2$$-space body, say a square, passes through his $$1$$-space world, he sees only the $$1$$-space section of the square.

In Fig. 3, $$xy$$, the $$1$$-space world, is represented as being in the same

plane with the square $$mn$$. The square may cut $$xy$$ at right angles or obliquely. In any case the unodim sees at any moment only the part of the square common to his world and is not conscious that there is any more to the square.

Next let us consider $$2$$-space.

Assume a $$2$$-space being, which we shall call a duodim, that is, a flat being (theoretically with no thickness) with length and breadth and confined to a surface having length and breadth but no thickness. Such a being could move to the right or left or forward or backward, we will say, but neither up nor down from the surface. In fact, he knows neither up nor down: the surface is his world.

His position in his world is easily located by the Cartesian system of coordinates, that is, with reference to its distance from, say, two straight lines at right angles to each other. For illustration, define his world as the geometrical plane formed by the two lines $$xx', yy'$$ intersecting each other at right angles. Employing the usual notation, we consider distances measured perpendicular to the $$Y$$-axis as positive if measured to the right, and negative if to the left of the $$Y$$-axis. Such a distance is called the abscissa of the given point. Similarly, distances