The truth of a picture lies in its total effect. It is vain to seek information about the landscape from an examination of the pigments. And in any method of thought it is the complexity of the whole that brings us to a knowledge of nature. Dimensions are artificial enough, but in the multiplicity of them we catch some breath of nature. We must therefore, and this seems to me the practical conclusion of the whole matter, proceed to form means of intellectual apprehension of a greater and greater degree of complexity, both dimensionally and in extent in any dimension. Such means of representation must always be artificial, but in the multiplicity of the elements with which we deal, however incipiently arbitrary, lies our chance of apprehending nature. And as a concluding chapter to this part of the book, I will extend the figures, which have been used to represent Kant’s theory, two steps, so that the reader may have the opportunity of looking at a four-dimensional figure which can be delineated without any of the special apparatus, to the consideration of which I shall subsequently pass on. CHAPTER X A FOUR-DIMENSIONAL FIGURE The method used in the preceding chapter to illustrate the problem of Kant’s critique, gives a singularly easy and direct mode of constructing a series of important figures in any number of dimensions. We have seen that to represent our space a plane being must give up one of his axes, and similarly to represent the higher shapes we must give up one amongst our three axes. But there is another kind of giving up which reduces the construction of higher shapes to a matter of the utmost simplicity. Ordinarily we have on a straight line any number of positions. The wealth of space in position is illimitable, while there are only three dimensions. I propose to give up this wealth of positions, and to consider the figures obtained by taking just as many positions as dimensions. In this way I consider dimensions and positions as two “kinds,” and applying the simple rule of selecting every one of one kind with every other of every other kind, get a series of figures which are noteworthy because they exactly fill space of any number of dimensions (as the hexagon fills a plane) by equal repetitions of themselves. The rule will be made more evident by a simple application. Let us consider one dimension and one position. I will call the axis _i_, and the position _o_. ———————————————-_i_ _o_ Here the figure is the position _o_ on the line _i_. Take now two dimensions and two positions on each. [Illustration: Fig. 63.]