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