The fourth dimension

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 57

to AB, that is, not meet aB however far produced, and let

¢ lines beyond cp also not meet c 4B; let there be a certain region between cD and CE, in which no line drawn meets AB. CE and cp produced backwards through c will give a similar region on the other side of ©. Nothing so triumphantly, one may almost say so

A B fe) Fig. 32.

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insolently, ignoring of sense had ever been written before. |

Men had struggled against the limitations of the body, fought them, despised them, conquered them. But no one had ever thought simply as if the body, the bodily eyes, the organs of vision, all this vast experience of space, had never existed. The age-long contest of the soul with the body, the struggle for mastery, had come to a culmination. Bolyai and Lobatchewsky simply thought as if the body was not. ‘The struggle for dominion, the strife and combat of the soul were over; they had mastered, and the Hungarian drew his line.

Can we point out any connection, as in the case of Parmenides, between these speculations and higher space? Can we suppose it was any inner perception by the soul of a motion not known to the senses, which resulted in this theory so free from the bonds of sense? No such supposition appears to be possible.

Practically, however, metageometry had a great influence in bringing the higher space to the front as a working hypothesis. This can be traced to the tendency the mind has to move in the direction of least resistance. The results of the new geometry could not be neglected, the problem of parallels had occupied a place too prominent in the development of mathematical thought for its final solution to be neglected. But this utter independence of all mechanical considerations, this perfect cutting loose