Let us now imagine the tesseract and the slanting space both together to pass transverse to our space, a distance of one unit, we have in 1_h_ a section of the tesseract, whose axes are parallels to the previous axes. The slanting space cuts them at a distance of five units along each. Drawing the plane through these points in 1_h_ it will be found to cut the cubical section of the tesseract in the hexagonal figure drawn. In 2_h_ (fig. 72) the slanting space cuts the parallels to the axes at a distance of four along each, and the hexagonal figure is the section of this section of the tesseract by it. Finally when 3_h_ comes in the slanting space cuts the axes at a distance of three along each, and the section is a triangle, of which the hexagon drawn is a truncated portion. After this the tesseract, which extends only three units in each of the four dimensions, has completely passed transverse of our space, and there is no more of it to be cut. Hence, putting the plane sections together in the right relations, we have the section determined by the particular slanting space: namely an octahedron. CHAPTER XIV.[6] A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT There are two directions of inquiry in which the research for the physical reality of a fourth dimension can be prosecuted. One is the investigation of the infinitely great, the other is the investigation of the infinitely small. [6] The contents of this chapter are taken from a paper read before the Philosophical Society of Washington. The mathematical portion of the paper has appeared in part in the Proceedings of the Royal Irish Academy under the title, “Cayley’s formulæ of orthogonal transformation,” Nov. 29th, 1903. By the measurement of the angles of vast triangles, whose sides are the distances between the stars, astronomers have sought to determine if there is any deviation from the values given by geometrical deduction. If the angles of a celestial triangle do not together equal two right angles, there would be an evidence for the physical reality of a fourth dimension. This conclusion deserves a word of explanation. If space is really four-dimensional, certain conclusions follow which must be brought clearly into evidence if we are to frame the questions definitely which we put to Nature. To account for our limitation let us assume a solid material sheet against which we move. This sheet must stretch alongside every object in every direction in which it visibly moves. Every material body must slip or slide along this sheet, not deviating from contact with it in any motion which we can observe.