HE who elects to write on a mathematical topic is confronted with a choice between two evils. He may decide to handle his subject mathematically, using the conventional mathematical symbols, and whatever facts, formulas and equations the subject may demand—save himself who can! Or he may choose to abandon all mathematical symbols, formulas and equations, and attempt to translate into the vernacular this language which the mathematician speaks so fluently. In the one case there results a finished article which only the elect understand, in the other, only a rather crude and clumsy approximation to the truth. A similar condition exists in all highly specialized branches of learning, but it can safely be said that in no other science must one fare so far, and accumulate so much knowledge on the way, in order to investigate or even understand new problems. And so it is with some trepidation that the attempt is made to discuss in the following pages one of the newest and most important branches of mathematical activity. For the writer has chosen the second evil, and, deprived of his formulas, to borrow a figure of Poincaré's, finds himself a cripple without his crutches.After this mutually encouraging prologue let us introduce the subject with a definition. What is relativity? By relativity, the theory of relativity, the principle of relativity, the doctrine of relativity, is meant a new conception of the fundamental ideas of mechanics. By the relativity mechanics, or as we may sometimes say, the new mechanics, is meant that body of doctrine which is based on these new conceptions. Now this is a very simple definition and one which would be perfectly comprehensible to everybody, provided the four following points were made clear: first, what are the fundamental concepts of mechanics, second, what are the classical notions about them, third, how are these modified by the new relativity principles, and fourth, how did it come about that we have been forced to change our notions of these fundamental concepts which have not been questioned since the time of Newton? These four questions will now be discussed, though perhaps not in this order. The results reached are, to say the least, amazing, but perhaps our astonishment will not be greater than it was when first we learned, or heard rather, that the earth is round, and that there are persons directly opposite us who do not fall off, and stranger yet, do not realize that they are in any immediate danger of doing so.