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Updated: May 10, 2025
"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!
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 idea which Lobatchewsky worked out was that through a point more than one parallel to a given line could be drawn; that is to say, if through the point P we have already supposed another line were drawn making ever so small an angle with CD, this line also would never meet the line AB. It might approach the latter at first, but would eventually diverge.
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