A mathematical "point," which has only position and neither length nor breadth nor thickness, can similarly be called space of no dimensions or "Zero-dimensional space." Also I wish to take the opportunity of defining one or two words which I may have occasion to use and have the merit of brevity. (1) Lines which are drawn through a point for the sake of determining direction are called in Geometrical parlance, "Axes." Thus in Fig. 1 AOA´ and BOB´ are axes. The former would be known as "the axis of A," the latter as "the axis of B." Similarly in Fig. 3 COC´ is "the axis of C." (2) The point in which two or more axes meet, is called the "Origin" and is commonly denoted by the letter O. (3) When convenient, I shall use the terms, "Two space," "Three space," "Four space," etc., instead of writing "Two-dimensional space," "Three-dimensional space," "Four-dimensional space," etc. in full each time. THE ANALOGY OF A TWO-DIMENSIONAL WORLD. The consideration of the analogy of a two dimensional world is necessary because, as Mr. C.H. Hinton says in his book, "The Fourth Dimension," p. 6. "The change in our conceptions, which we make in passing from the shapes and motions in two dimensions to those in three, affords a pattern by which we can pass on still further to the conception of an existence in four-dimensional space." Let us start then by imagining a very large, flat and perfectly smooth surface; such for instance as the top of a highly polished table or the surface of a sheet of still liquid. We have seen that such a surface constitutes space of two dimensions, because through any point in it we can only draw two lines at right angles to one another. In order to draw a third such line we must get out of the surface altogether and draw the line perpendicular to it. Next we must try to imagine that this surface is populated by a race of beings of an extraordinary thinness. In order to grasp the analogy properly we must imagine them to be so constituted that they are incapable of realising any direction in space which does not lie in the aforementioned flat surface on which they live. We can imagine this by supposing that their thickness, _i.e._:--their extension in the third dimension perpendicular to their surface,--is so small as to be invisible to them and also that their "nerve endings" all lie on their periphery. This last is equivalent to saying that they have no "sense organs" facing the third dimension and that therefore they cannot receive impressions, or respond to any stimuli that come to them from that direction. It follows, therefore, that unless they develope special sense organs which face the third dimension they will be acquainted only with such objects and events as lie, or take place, in their surface.