Abstract
When I had the honour of being asked by the Council of the Royal Society to give the following address, I chose the subject partly because it had been brought under the notice of the fellows by my predecessor, Professor Kelland. His memoir was written comparatively early in the history of the subject; and he seems to have been but little acquainted with what others had done even up to the time at which he wrote. Accordingly, although the subject is treated very ably in his paper, it is treated from only one point of view; and, indeed, one side of it is left out of sight altogether. The relation of the whole theory to the question of the origin and mutual independence of the axioms of geometry has been made much clearer of late, and I believed that some account of the more modern views might be of interest.
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