"Early mathematicians had no difficulty accepting the first four of Euclid’s axioms. But the fifth was thought not to be so obvious as the other four, and mathematicians tried into the 19th century to derive the fifth postulate from the first four. All such attempts were doomed to failure, but in making the effort, investigators developed a great deal of modern mathematics. It was finally discovered (in the 19th century by Bolyai, Gauss, Lobachevsky, Beltrami, and others) that the fifth postulate is in fact independent of the first four and that one gets a perfectly legitimate geometry by discarding Euclid’s parallel axiom and replacing it with a different one: 'The hyperbolic parallel axiom: Given a line l and a point P not on l, there are infinitely many lines through P that are parallel to l.'" (Bruce Crauder et al, "Functions and Change: A Modeling Approach to College Algebra and Trigonometry", 2008)
"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
"Beltrami started this train of thought in 1865, by asking which surfaces can be mapped onto the plane in such a way that their geodesics go to straight lines. He found that the answer is precisely the surfaces of constant Gaussian curvature. For example, great circles on the sphere can be mapped to lines on the plane, and the map that does the trick (to be precise, for the hemisphere) is central projection [...]. Rays from the center O to any great circle form a plane, which of course meets any other plane in a line. Thus projection from O to a plane sends great circles to lines, though only half of the sphere is mapped." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful [...]. What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone. This was noticed by the third great contributor to non-Euclidean geometry after Beltrami and Klein - the French mathematician Henri Poincaré [...]" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
"[...] quaternions give a nice approach to symmetric objects in three-dimensional space: the regular polyhedra. But this leads in turn to the regular polytopes, a family of four-dimensional symmetrical objects as remarkable as the regular polyhedra. One then becomes convinced that four-dimensional space is not just a set of quadruples; it is a world of genuine geometry." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
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