08-30-2026, 09:20 PM
Non-Euclidean Geometry
Book name: Non-Euclidean Geometry: Sixth Edition
Author: H. S. M. Coxeter
Publication date: 1998
Publisher: Mathematical Association of America (now MAA Press, an imprint of the American Mathematical Society)
Series: Spectrum, Volume 23
Length: 336 pages
Field: Geometry — especially projective, elliptic, and hyperbolic geometry
Coxeter's Non-Euclidean Geometry is a classic systematic treatment of geometries in which Euclid's parallel postulate is no longer assumed. The book begins with the historical development of the subject, explaining how the work of Gauss, Lobachevsky, Bolyai, Riemann and others led mathematicians to recognize that logically consistent alternatives to Euclidean geometry exist. Coxeter then takes a distinctive route: rather than beginning immediately with distances and angles, he develops real projective geometry from basic concepts such as points, lines, planes, incidence, order and continuity. Projective geometry becomes the common framework from which Euclidean, elliptic and hyperbolic geometries can be understood.
After establishing the foundations of projective geometry, Coxeter introduces polarities, conics, quadrics and homogeneous coordinates. Transformations preserving incidence—collineations—play a central role, and suitable projective polarities are used to generate the metric structures of elliptic and hyperbolic geometry. The book develops elliptic geometry first in one, two and three dimensions, then turns to Euclidean and hyperbolic geometry. This approach reveals that these apparently different geometries are closely related rather than isolated theories. Algebraic methods are gradually introduced alongside the synthetic arguments, allowing Coxeter to derive formulas for elliptic and hyperbolic trigonometry through general linear transformations.
The later chapters investigate hyperbolic planes, circles, triangles, area and geometric models. A particularly important theme is the relationship between curvature and the angle sum of a triangle: unlike Euclidean geometry, where the angles total $\pi$, hyperbolic triangles have an angle defect while elliptic triangles have an angle excess, with triangle area closely connected to that difference. Coxeter also examines Euclidean representations of non-Euclidean spaces, helping the reader visualize otherwise unfamiliar geometric structures. The sixth edition adds a section on Coxeter's concept of inversive distance, extending the discussion of circles and inversive geometry. An appendix treats angles and arcs in the hyperbolic plane.
Main topics
Key takeaways
BOOK
Book name: Non-Euclidean Geometry: Sixth Edition
Author: H. S. M. Coxeter
Publication date: 1998
Publisher: Mathematical Association of America (now MAA Press, an imprint of the American Mathematical Society)
Series: Spectrum, Volume 23
Length: 336 pages
Field: Geometry — especially projective, elliptic, and hyperbolic geometry
Coxeter's Non-Euclidean Geometry is a classic systematic treatment of geometries in which Euclid's parallel postulate is no longer assumed. The book begins with the historical development of the subject, explaining how the work of Gauss, Lobachevsky, Bolyai, Riemann and others led mathematicians to recognize that logically consistent alternatives to Euclidean geometry exist. Coxeter then takes a distinctive route: rather than beginning immediately with distances and angles, he develops real projective geometry from basic concepts such as points, lines, planes, incidence, order and continuity. Projective geometry becomes the common framework from which Euclidean, elliptic and hyperbolic geometries can be understood.
After establishing the foundations of projective geometry, Coxeter introduces polarities, conics, quadrics and homogeneous coordinates. Transformations preserving incidence—collineations—play a central role, and suitable projective polarities are used to generate the metric structures of elliptic and hyperbolic geometry. The book develops elliptic geometry first in one, two and three dimensions, then turns to Euclidean and hyperbolic geometry. This approach reveals that these apparently different geometries are closely related rather than isolated theories. Algebraic methods are gradually introduced alongside the synthetic arguments, allowing Coxeter to derive formulas for elliptic and hyperbolic trigonometry through general linear transformations.
The later chapters investigate hyperbolic planes, circles, triangles, area and geometric models. A particularly important theme is the relationship between curvature and the angle sum of a triangle: unlike Euclidean geometry, where the angles total $\pi$, hyperbolic triangles have an angle defect while elliptic triangles have an angle excess, with triangle area closely connected to that difference. Coxeter also examines Euclidean representations of non-Euclidean spaces, helping the reader visualize otherwise unfamiliar geometric structures. The sixth edition adds a section on Coxeter's concept of inversive distance, extending the discussion of circles and inversive geometry. An appendix treats angles and arcs in the hyperbolic plane.
Main topics
- Historical origins of non-Euclidean geometry
- Foundations of real projective geometry
- Projective transformations and collineations
- Conics, quadrics and polarities
- Homogeneous coordinates
- Elliptic geometry in dimensions 1, 2 and 3
- Euclidean and hyperbolic geometry
- Hyperbolic circles and triangles
- Elliptic and hyperbolic trigonometry
- Triangle area and angular excess/defect
- Euclidean models of non-Euclidean geometries
- Inversive geometry and inversive distance
Key takeaways
- Projective geometry provides a unifying foundation for Euclidean, elliptic and hyperbolic geometries rather than treating them as unrelated subjects.
- Metric ideas can emerge from non-metric projective concepts. Coxeter first studies incidence, order and polarity, and only later introduces distance and angle.
- The book emphasizes the deep connection between transformations and geometry: understanding the transformations preserving a geometry is one of the best ways to understand the geometry itself.
- It is mathematically serious but does not require advanced analysis. Coxeter states that a reader familiar with algebra through the elementary ideas of group theory should be able to follow the treatment.
- This is especially valuable for readers interested in classical geometry, projective geometry, hyperbolic geometry, geometric transformations, or the historical foundations of geometry. The MAA's Basic Library List recommends it for undergraduate mathematics libraries.
BOOK
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