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## Über dieses Buch

The discovery of hyperbolic geometry, and the subsequent proof that this geometry is just as logical as Euclid's, had a profound in­ fluence on man's understanding of mathematics and the relation of mathematical geometry to the physical world. It is now possible, due in large part to axioms devised by George Birkhoff, to give an accurate, elementary development of hyperbolic plane geometry. Also, using the Poincare model and inversive geometry, the equiconsistency of hyperbolic plane geometry and euclidean plane geometry can be proved without the use of any advanced mathematics. These two facts provided both the motivation and the two central themes of the present work. Basic hyperbolic plane geometry, and the proof of its equal footing with euclidean plane geometry, is presented here in terms acces­ sible to anyone with a good background in high school mathematics. The development, however, is especially directed to college students who may become secondary teachers. For that reason, the treatment is de­ signed to emphasize those aspects of hyperbolic plane geometry which contribute to the skills, knowledge, and insights needed to teach eucli­ dean geometry with some mastery.

## Inhaltsverzeichnis

### Chapter I. Some Historical Background

Abstract
Some understanding of basic arithmetic and geometric concepts, some knowledge of mathematical relations, were in all probability a part of human culture long before there was any recording of knowledge. The earliest records that do exist tend to confirm this view. Unfortunately, such records are not so much a story of culture at the time as they are clues to such a story. The interpretation of these clues by research scholars is constantly adding to our understanding of early civilizations, but the history of mathematical knowledge is far from complete.
Paul Kelly, Gordon Matthews

### Chapter II. Absolute Plane Geometry

Abstract
In this chapter we will review the basic properties of Absolute Plane geometry, based on the Birkhoff axioms. All the theorems to be considered are also theorems of Euclidean geometry and hence, for the most part, will be familiar to the reader. For this reason, proofs will not be given except for a few theorems not ordinarily encountered in a beginning course in Euclidean geometry. Our intent is to sketch a natural progression for the theorems and to introduce notations and conventions that we will need in the later study of Hyperbolic geometry.
Paul Kelly, Gordon Matthews

### Chapter III. Hyperbolic Plane Geometry

Abstract
The properties in Chapter II belong to both absolute geometry and to Euclidean geometry, but the axioms there are sufficient to imply only a part of Euclidean geometry. For example, they do not imply that the angle sum of a triangle is 180°. To establish this, and many other facts of Euclidean geometry, some assumption equivalent to Euclid’s “parallell postulate” is necessary. In traditional beginning geometry courses, the extra axiom most commonly adopted it the Playfair axiom:
“If P not on line t, then there exists exactly one line through P that does not intersect t.”
Paul Kelly, Gordon Matthews

### Chapter IV. A Euclidean Model of the Hyperbolic Plane

Abstract
At the end of Chapter I, we mentioned the discovery that a non-euclidean geometry could have a euclidean representation. In this chapter, we want to look at one such representation, due to H. Poincare (1854–1912), which is called “the Poincaré model of hyperbolic geometry”. Not only is this model attractively ingenious, but, as we shall explain in detail, it implies that if there is a logical inconsistency in hyperbolic geometry then there is a logical inconsistency in euclidean geometry. Thus, however non-intuitive hyperbolic geometry may appear, it cannot be refuted on logical grounds unless there is a similar refutation of the highly intuitive relations of euclidean geometry.
Paul Kelly, Gordon Matthews

### Backmatter

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