The fourth dimension

CHAPTER XIV.*

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.

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

* 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 formule of orthogonal

transformation,’ Noy, 29th, 1903. 208