The Discovery of Non-Euclidean Geometry 11 proved theorems of non-Euclidean geometry, it was Lobachevsky and Bolyai (and pre- sumably also Gauss) who first recognized them as such. However, even after this ground- breaking work, there was still no proof that non-Euclidean geometry was consistent (i.e., that it could never lead to a contradiction). The coup de grace ˆ for attempts to prove the fifth postulate was provided in 1868 by another Italian mathematician, Eugenio Beltrami (1835–1900), who proved for the first time that non-Euclidean geometry was just as con- sistent as Euclidean geometry. Thus the ancient question of whether Euclid’s fifth postulate followed from the other four was finally settled. The versions of non-Euclidean geometry studied by Lobachevsky, Bolyai, Gauss, and Beltrami were all essentially equivalent to each other. This geometry is now called hyper- bolic geometry. Its most salient feature is that Playfair’s postulate is false: in hyperbolic geometry it is always possible for two or more distinct straight lines to be drawn through the same point, both parallel to a given straight line. As a consequence, many aspects of Euclid’s theory of parallel lines (such as the result in Proposition I.29 about the equality of corresponding angles made by a transversal to two parallel lines) are not valid in hy- perbolic geometry. In fact, as we will see in Chapter 19, the phenomenon of parallel lines approaching each other asymptotically but never meeting—which Saccheri declared to be “repugnant to the nature of the straight line”—does indeed occur in hyperbolic geometry. One might also wonder if the Euclidean theory of parallel lines could fail in the op- posite way: instead of having two or more parallels through the same point, might it be possible to construct a consistent theory in which there are no parallels to a given line through a given point? It is easy to imagine a type of geometry in which there are no par- allel lines: the geometry of a sphere. If you move as straight as possible on the surface of a sphere, you will follow a path known as a great circle—a circle whose center coincides with the center of the sphere. It can be visualized as the place where the sphere intersects a plane that passes through the center of the sphere. If we reinterpret the term “line” to mean a great circle on the sphere, then indeed there are no parallel “lines,” because any two great circles must intersect each other. But this does not seem to have much relevance for Euclid’s geometry, because line segments cannot be extended arbitrarily far—in spherical geometry, no line can be longer than the circumference of the sphere. This would seem to contradict Postulate 2, which had always been interpreted to mean that a line segment can be extended arbitrarily far in both directions. However, after the discovery of hyperbolic geometry, another German mathematician, Bernhard Riemann (1826–1866), realized that Euclid’s second postulate could be reinter- preted in such a way that it does hold on the surface of a sphere. Basically, he argued that Euclid’s second postulate only requires that any line segment can be extended to a longer one in both directions, but it does not specifically say that we can extend it to any length we wish. With this reinterpretation, spherical geometry can be seen to be a consistent ge- ometry in which no lines are parallel to each other. Of course, a number of Euclid’s proofs break down in this situation, because many of the implicit geometric assumptions he used in his proofs do not hold on the sphere see our discussion of Euclid’s Proposition I.16 be- low for an example. This alternative form of non-Euclidean geometry is sometimes called elliptic geometry. (Because of its association with Riemann, it is sometimes erroneously referred to as Riemannian geometry, but this term is now universally used to refer to an entirely different type of geometry, which is a branch of differential geometry.)

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