David Gans's An Introduction to Non-Euclidean Geometry PDF

By David Gans

ISBN-10: 0122748506

ISBN-13: 9780122748509

Ebook by way of Gans, David

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Additional info for An Introduction to Non-Euclidean Geometry

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Proof. Consider the Saccheri quadrilateral ABCD (Fig. Ill, 2). Triangles ABC, BAD are congruent by side-angle-side (Theo. 4). Hence AC = BD. * Since Euclid proved Theorems 4 and 8 by superposition, we are regarding the facts they state as assumptions (see §2, Property 13). 40 III. PARALLELS WITH A COMMON PERPENDICULAR D C b d A B Fig. Ill, 2 Triangles ADC, BCD are congruent by side-side-side (Theo. 8). Therefore %ADC= <&BCD. Theorem 30. The line joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to each.

THE DEFECT OF A TRIANGLE A D 61 B Fig. Ill, 20 verified that in each case the defect relation is the same as in transversal subdivisions. As might be imagined, there are many other nontransversal subdivisions of triangle ABC, but it can be proved that this same defect relation holds in all cases. The key step of the proof* consists in showing that, by the use of additional line segments, any nontransversal subdivision S of triangle ABC can be converted into a transversal subdivision St of triangle ABC such that each triangle resulting from S either is left unchanged by the added segments or is divided transversally by them.

The summit angles of a Saccheri quadri- It might seem, at first, that this assumption is less in accord with experience than the assumption that the summit angles are right angles, but a little reflection will show that this is not necessarily the case. Ordinarily, to be sure, we do regard the summit angles as being right angles, but this is mainly because we have been conditioned to think in a Euclidean manner. The angles appear to be right angles and so, influenced by our study of Euclidean geometry, we say they are right angles.

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An Introduction to Non-Euclidean Geometry by David Gans

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