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Updated: May 9, 2025
If this axiom of parallels does not follow from the other axioms, then from these latter we may construct a system of geometry in which the axiom of parallels shall not be true. This was done by Lobatchewsky and Bolyai, the one a Russian the other a Hungarian geometer, about 1830.
"The audacious speculations of Bolyai, the amazing theories of Gauss that through a point more than one line could be drawn parallel to a given line; the possibility that the angles of a triangle are together greater than two right angles, if drawn upon immense curvatures the breathless intuitions of Beltrami and Lobatchewsky all these I hurried through, and emerged, panting but unsatisfied, upon the verge of my my new world, my Higher Space possibilities in a word, my disease!
Apparently straight parallel lines drawn upon it would then diverge, as supposed by Bolyai. We cannot, however, imagine such a surface extended indefinitely without losing its properties. The analogy is not so clearly marked as in the other case.
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