The investigation of the properties of numbers is much facilitated by the fact that relations between numbers are themselves able to be represented as numbers—_e.g._, 12, and 3 are both numbers, and the relation between them is 4, another number. The way is thus opened for a process of constructive theory, without there being any necessity for a recourse to another class of concepts besides that which is given in the phenomena to be studied. The discipline of number thus created is of great and varied applicability, but it is not solely as quantitative that we learn to understand the phenomena of nature. It is not possible to explain the properties of matter by number simply, but all the activities of matter are energies in space. They are numerically definite and also, we may say, directedly definite, _i.e._ definite in direction. Is there, then, a body of doctrine about space which, like that of number, is available in science? It is needless to answer: Yes; geometry. But there is a method lying alongside the ordinary methods of geometry, which tacitly used and presenting an analogy to the method of numerical thought deserves to be brought into greater prominence than it usually occupies. The relation of numbers is a number. Can we say in the same way that the relation of shapes is a shape? We can. To take an instance chosen on account of its ready availability. Let us take two right-angled triangles of a given hypothenuse, but having sides of different lengths (fig. 46). These triangles are shapes which have a certain relation to each other. Let us exhibit their relation as a figure. [Illustration: Fig. 46.] Draw two straight lines at right angles to each other, the one HL a horizontal level, the other VL a vertical level (fig. 47). By means of these two co-ordinating lines we can represent a double set of magnitudes; one set as distances to the right of the vertical level, the other as distances above the horizontal level, a suitable unit being chosen. [Illustration: Fig. 47.] Thus the line marked 7 will pick out the assemblage of points whose distance from the vertical level is 7, and the line marked 1 will pick out the points whose distance above the horizontal level is 1. The meeting point of these two lines, 7 and 1, will define a point which with regard to the one set of magnitudes is 7, with regard to the other is 1. Let us take the sides of our triangles as the two sets of magnitudes in question. Then the point 7, 1, will represent the triangle whose sides are 7 and 1. Similarly the point 5, 5—5, that is, to the right of the vertical level and 5 above the horizontal level—will represent the triangle whose sides are 5 and 5 (fig. 48). [Illustration: Fig. 48.]