I am afraid that the title of my address is not quite explicit. It lacks that precision of statement which ought to characterise the utterances of a mathematician. I have undertaken to speak to you about the End of the World, but I have not told you which end. The world—or space-time—is a four-dimensional continuum, and consequently offers a choice of a great many directions in which we might start off to look for an end; and it is by no means easy to describe “from the standpoint of mathematical physics” the direction in which I intend to lead you. I have therefore to examine at some length the preliminary question, Which end? But that leads us so deeply into the whole problem that by the time I have finished with it my address will be over.

In selecting a subject for submission to your attention, I have avoided topics belonging to the customary ranges of the school room, the lecture theatre, and the examination arena. My wish is to discuss the notion of dimensions in abstract geometry. In recent years, the “fourth dimension” has been obtruded upon an unsophisticated world in phrases which occasionally defy comprehension. Little more than a year ago, our Sabbath peace was perturbed by a newspaper announcement that a fifth dimension had been discovered, though the explanation which introduced the discovery was little more than incoherent verbiage. Novelists have ventured upon the intangible theme of multiple space : did not one of them, in an attenuated remembrance of Cambridge training, malign a great mathematician by assigning to his genius a fatuous dogma that ghosts are possible by means of the fourth dimension? Theologians of a sort, on both sides of the Atlantic, have not disdained to make a new departure by placing heaven in the fourth dimension : one of them, indeed, has postulated an unlimited number of dimensions as constituting the heaven of heavens, the dwelling-place of The Most High. Nay, coming nearer to our mathematical science, some popular expositions of the successive theories of relativity have stated, almost pontifically as an article in a creed, that Time is the fourth dimension. Even more bewildering was the rhetoric launched at us some six months ago when we were told, in a metaphorical summary, that the four-dimensional space-time continuum had