If now he conceives the square A to move in the, to him, unknown dimension it will trace out a cube, and the bounding squares will form cubes. Will these completely surround the cube generated by A? No; there will be two faces of the cube made by A left uncovered; the first, that face which coincides with the square A in its first position; the next, that which coincides with the square A in its final position. Against these two faces cubes must be placed in order to completely enclose the cube A. These may be called the cubes left and right or A_l_ and A_r_. Thus each of the enclosing squares of the square A becomes a cube and two more cubes are wanted to enclose the cube formed by the movement of A in the third dimension. [Illustration: Fig. 97.] The plane being could not see the square A with the squares A_n_, A_f_, etc., placed about it, because they completely hide it from view; and so we, in the analogous case in our three-dimensional world, cannot see a cube A surrounded by six other cubes. These cubes we will call A near A_n_, A far A_f_, A above A_a_, A below A_b_, A left A_l_, A right A_r_, shown in fig. 97. If now the cube A moves in the fourth dimension right out of space, it traces out a higher cube—a tesseract, as it may be called. Each of the six surrounding cubes carried on in the same motion will make a tesseract also, and these will be grouped around the tesseract formed by A. But will they enclose it completely? All the cubes A_n_, A_f_, etc., lie in our space. But there is nothing between the cube A and that solid sheet in contact with which every particle of matter is. When the cube A moves in the fourth direction it starts from its position, say A_k_, and ends in a final position A_n_ (using the words “ana” and “kata” for up and down in the fourth dimension). Now the movement in this fourth dimension is not bounded by any of the cubes A_n_, A_f_, nor by what they form when thus moved. The tesseract which A becomes is bounded in the positive and negative ways in this new direction by the first position of A and the last position of A. Or, if we ask how many tesseracts lie around the tesseract which A forms, there are eight, of which one meets it by the cube A, and another meets it by a cube like A at the end of its motion. We come here to a very curious thing. The whole solid cube A is to be looked on merely as a boundary of the tesseract. Yet this is exactly analogous to what the plane being would come to in his study of the solid world. The square A (fig. 96), which the plane being looks on as a solid existence in his plane world, is merely the boundary of the cube which he supposes generated by its motion.