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ABSTRACT ONE of the most remarkable speculations of the present century is the speculation that the axioms of geometry may be only approximately true, and that the actual properties of space
may be somewhat different from those which we are in the habit of ascribing to it. It was Lobatchewsky who first worked out the conception of a space in which some of the ordinary laws of
geometry should no longer hold good. Among the axioms which lie at the foundation of the Euclidian scheme, he assumed all to be true except the one which relates to parallel straight lines.
An equivalent form of this axiom, and the one now generally employed in works on geometry, is the statement that it is impossible to draw more than one straight line parallel to a given
straight line through a given point outside it. In other words, if we take a fixed straight line, A B, prolonged infinitely in both directions, and a fixed point, P, outside it; then, if a
second straight line, also infinitely prolonged in both directions, be made to rotate about P, there is _only one_ position in which it will not intersect A B. Now Lobatchewsky made the
supposition that this axiom should be untrue, and that there should be a finite angle through which the rotating line might be turned, without ever intersecting the fixed straight line, A B.
And in following out the consequences of this assumption he was never brought into collision with any of the other axioms, but was able to construct a perfectly self-consistent scheme of
propositions, ail of them valid as analytical conceptions, but all of them perfectly incapable of being realised in thought. Access through your institution Buy or subscribe This is a
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checkout ADDITIONAL ACCESS OPTIONS: * Log in * Learn about institutional subscriptions * Read our FAQs * Contact customer support Authors * F. W. FRANKLAND View author publications You can
also search for this author inPubMed Google Scholar RIGHTS AND PERMISSIONS Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE FRANKLAND, F. _On the Simplest Continuous
ManiFoldness of Two Dimensions and of Finite Extent_ 1 . _Nature_ 15, 515–517 (1877). https://doi.org/10.1038/015515a0 Download citation * Issue Date: 12 April 1877 * DOI:
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