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By David Gans

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Taurinus (1794-1874) was the first to publish his results, which were substantial, but while accepting the new geometry as logically tenable he rejected it on physical grounds, believing Euclidean geometry to be the only possible system applicable to space. J. Bolyai (1802-1867) and N. I. Lobachevsky (1793-1856) are usually accorded top honors with Gauss as the discoverers of non-Euclidean geometry, for they not only published extensive developments of the subject, believing it to be logically sound, but were convinced that it was as applicable to the physical world as Euclidean geometry.

The point A is on neither ray, but is between each point of one ray and each point of the other. A ray with endpoint A which contains a point B is called the ray AB. The figure consisting of two rays AB, AC is called an angle and denoted by ^BAC or <$:CAB; the rays are called the sides of the angle and A is called its vertex. An angle with collinear rays is called a straight angle. 4. An angle (together with its vertex) separates the rest of the plane into two regions. If an angle is not a straight angle, these regions are called the interior and the exterior of the angle.

First, from Legendre's work we see that Postulate 5 can be proved if we use statement ( j) and the basis E. Conversely, we must show that statement ( j) can be proved if we use Postulate 5 and the basis E. Consider any angle with vertex A and sides g, h, and any point D within this angle (Fig. II, 10). Draw A B E Fig. II, 10 line DG parallel to h (Prop. 31). Since h meets g, so does line DG, say in B. Take any point E on g to the right of B. Draw line DE. Since a + b equals two right angles (Prop.

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