Euclidean and Hyperbolic Geometry: An Introduction
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1[1
Introduction
segment PQ:
In Euclidean geometry the perpendicular distance between the rays remains equal to the distance from P to Q as we move to the right.
However, in the early nineteenth century two alternative geometries were proposed. In hyperbolic geometry (from the Greek hyperballein,
"to exceed") the distance between the rays increases. In elliptic geometry (from the Greek elleipein, "to fall short") the distance decreases and the rays eventually meet. These non-Euclidean geometries were later incorporated in a much more general geometry developed by C. F. Gauss and G. F. B. Riemann (it is this more general geometry that is used in Einstein's general theory of relativity).1
We will concentrate on Euclidean and hyperbolic geometries in this book. Hyperbolic geometry requires a change in only one of Euclid's axioms, and can be as easily grasped as high school geometry. Elliptic geometry, on the other hand, involves the new topological notion of
"nonorientability," since all the points of the elliptic plane not on a given line lie on the same side of that line. This geometry cannot easily be approached in the spirit of Euclid. I have therefore made only brief comments about elliptic geometry in the body of the text, with further indications in Appendix A. (Do not be misled by this, however; elliptic geometry is no less important than hyperbolic.) Riemannian geometry requires a thorough understanding of the differential and integral calculus, and is therefore beyond the scope of this book (it is discussed briefly in Appendix A).
Chapter 1 begins with a brief history of geometry in ancient times, and emphasizes the development of the axiomatic method by the
Greeks. It presents Euclid's five postulates and includes one of Legendre's attempted proofs of the fifth postulate. In order to detect the
1 Einstein's special theory of relativity, which is needed to study subatomic particles, is based on a simpler geometry of space-time due to H. Minkowski. The
1[1
Introduction
segment PQ:
In Euclidean geometry the perpendicular distance between the rays remains equal to the distance from P to Q as we move to the right.
However, in the early nineteenth century two alternative geometries were proposed. In hyperbolic geometry (from the Greek hyperballein,
"to exceed") the distance between the rays increases. In elliptic geometry (from the Greek elleipein, "to fall short") the distance decreases and the rays eventually meet. These non-Euclidean geometries were later incorporated in a much more general geometry developed by C. F. Gauss and G. F. B. Riemann (it is this more general geometry that is used in Einstein's general theory of relativity).1
We will concentrate on Euclidean and hyperbolic geometries in this book. Hyperbolic geometry requires a change in only one of Euclid's axioms, and can be as easily grasped as high school geometry. Elliptic geometry, on the other hand, involves the new topological notion of
"nonorientability," since all the points of the elliptic plane not on a given line lie on the same side of that line. This geometry cannot easily be approached in the spirit of Euclid. I have therefore made only brief comments about elliptic geometry in the body of the text, with further indications in Appendix A. (Do not be misled by this, however; elliptic geometry is no less important than hyperbolic.) Riemannian geometry requires a thorough understanding of the differential and integral calculus, and is therefore beyond the scope of this book (it is discussed briefly in Appendix A).
Chapter 1 begins with a brief history of geometry in ancient times, and emphasizes the development of the axiomatic method by the
Greeks. It presents Euclid's five postulates and includes one of Legendre's attempted proofs of the fifth postulate. In order to detect the
1 Einstein's special theory of relativity, which is needed to study subatomic particles, is based on a simpler geometry of space-time due to H. Minkowski. The