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The subject which I wish to outline in the briefest manner is the old problem of introducing into Algebra and Geometry such concepts as, for example, irrational numbers, imaginary numbers, the infinite, non-Euclidean geometry, four-dimensional geometry. This is a matter which must have been, at one time or another, of interest to each of you, and I owe you an apology, or at least an explanation, for talking to you about something with which you are presumably familiar. My reason for choosing this topic is to show how the introduction of all of these concepts is guided, in a general sense, by a few simple and plausible principles. It seemed to me that it might be of some interest to compress into the frame of one short talk the consideration of questions which have presented themselves to you at various times and divers occasions.