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 the supreme genius of Euclid to have expressed this particular axiom in such language as challenges at once the attention and the caution of the student. It may, however, be said that nearly all our difficulties in connection with the conceptions of space take their origin in the ambiguities which arise from the assumption which the twelfth axiom implies. Some modern mathematicians have gone so far as to deny the existence of this axiom altogether as a truth of nature, and it is most important to notice that when free from the embarrassment which the assumption of Euclid involves, a geometry emerges which removes our difficulties. It seems to show that space is finite rather than infinite, so far as we can assign definite meaning to the words, but it would lead me into matters somewhat inconvenient for these pages if I were to pursue the matter with any further detail. I may, however, say that it can be demonstrated that all known facts about space are reconcilable with the supposition that if we follow a straight line through space using for the word straight the definition which science has shown properly to belong to it then, after a journey which is not infinite in its length, we shall find ourselves back at the point from which we started. If anyone should think this a difficulty, I would recommend him to try to affix a legitimate definition to the word straight. He will find that the strictly definable attributes of straightness are quite compatible with the fact that a particle moving along a straight line will ultimately be restored to the point from which it departed.

It is quite true that this seems to be a paradox, but it will not be so considered by the geometer. The truth it implies is indeed quite a familiar doctrine in modern