Bolyai, Riemann and others – so I read – devised a new branch of geometry in the 19th Century by discarding Euclid’s axiom that parallel lines never meet. How was this different from the geometry that cartographers doing map projections had done for centuries, or indeed, that ancient Greeks not long after Euclid had used to discover the size and shape of the Earth?
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There are 5 Euclid’s axioms. There are several formulations of these axioms so I’ll choose the one I’m familiar with
* For any two distinct points, there exists a unique line segment between these two points
* Given a straight line segment, one can extend it indefinitely into a straight line
* Given a line segment, there exists a circle whose radius it that segment centered on one of the endpoint
* All right angles are congruent.
* [The parallel postulate]: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
(Note that the last axiom is NOT that “parallel lines never meet” ! Because “not meeting” is actually the definition of parallelism. The last axiom can be reformulated as “given a line and a point not on that line, there is a unique line that does not meet the first line and that is passing through the point”)
The geometry that cartographers used is spherical geometry. In spherical geometry, the parallel postulate fails for sure, because all lines necessarily meet. But some previous axioms also fail. The first one for example is not true anymore as there exists infinitely many line segment between the North pole and the South pole on the sphere. And you cannot extend a line segment between the North Pole and the South pole.
So even if spherical geometry could be considered as “non-Euclidean” because it doesn’t satisfy Euclid’s axioms, it’s very different from the one that we usually decribe as “non-euclidean”.
The non-euclidean geometry discovered by mathematicians in the 19th century, the so-called hyperbolic geometry, respects the first four axioms of Euclidean Geometry, and only the last one fails. And it fails in a completely different way than in spherical geometry. In hyperbolic geometry, given a line and a point, there exists infinitely many other lines not meeting the first line and passing through the point.
(To sum up : in Euclidean geometry, a unique parallel. In spherical geometry, no parallel. In hyperbolic geometry, infinitely many parallel).
That was a surprise, because for a long time, mathematicians were questioning if the fifth axiom of Euclidean geometry was really necessary or not. They didn’t expect to find a perfectly consistent model of geometry where not only that axioms fails, but that turned out to be an extremely rich world where a lot of new results could be proven and used in other domains.
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