Page:Popular Science Monthly Volume 83.djvu/389

Rh measured perpendicular to the $$X$$-axis are positive if measured above and negative if below the $$X$$-axis, such a distance being called the ordinate of the given point.

Thus it is evident that a point is fully determined in the plane if its abscissa and ordinate are given. Every school boy has met with this principle in the location of a place on the earth's surface by latitude and longitude, where the axes of reference are great circles of the globe.

Now our duodim has a far more extended space than the unodim, and can do many things that the unodim is totally ignorant of. His space may not necessarily be one of zero curvature,—for it is perfectly consistent with our definition of $$2$$-space for it to be the surface of a sphere, of an ellipsoid, of an egg-shaped figure, or what not. It is to be noticed that if the space has constant curvature (including no curvature), a body may be moved from any place to any other place on the surface without changing its shape.

If there are (Fig. 5) two triangles like $$E$$ and $$F$$ in which the sides

are equal each to each, but are arranged in reversed order, it is impossible in $$2$$-space for $$F$$ to be made to take the position of $$E$$. Here $$E$$ is the reflection of $$F$$ in a mirror. An inhabitant of $$3$$-space has no difficulty, however, in taking up $$F$$, turning it over and putting it on the position $$E$$.

A three-dimensional body as such is of course invisible to a $$2$$-space being. If a $$3$$-space body, say a cube, crosses a $$2$$-space, the $$2$$-space being is conscious only of its section with his world.

Fig. 6 represents the effect of a $$3$$-space body, a cube, passing through the $$2$$-space pictured by the plane "$$mn.$$" The section $$ABCD$$ of the plane $$mn$$ with the cube $$G$$ is all that a duodim would be conscious of. As $$G$$ passes through $$mn$$, this section, certainly if $$G$$ is a homogeneous body, will appear the same until suddenly it vanishes as $$G$$ passes beyond $$mn$$.