09-04-2026, 12:50 AM
A Course in Modern Geometries
Author: Judith N. Cederberg
Publication date: 1989 (1st edition; Springer eBook release: 9 March 2013)
Publisher: Springer New York
Summary
A Course in Modern Geometries is an undergraduate-level introduction to several major geometries beyond the standard Euclidean treatment. Cederberg begins with axiomatic systems and finite geometries, using small mathematical models to show how geometries can be constructed from explicitly stated axioms. The second chapter develops Euclidean and non-Euclidean geometry, emphasizing the role of the parallel postulate and showing how changing an axiom leads to fundamentally different geometric worlds.
The book then moves from synthetic geometry to a more algebraic viewpoint. It studies transformations of the Euclidean plane—including isometries and other transformation groups—and represents many of these transformations with matrices, providing a strong connection with linear algebra. The final major section introduces projective geometry, treating it both synthetically and analytically. This progression makes the book particularly useful as a bridge between classical geometry, linear algebra, and eventually abstract algebra.
The text was designed especially for junior- and senior-level mathematics students, including future secondary-school mathematics teachers. Its four main chapters are Axiomatic Systems and Finite Geometries, Non-Euclidean Geometry, Geometric Transformations of the Euclidean Plane, and Projective Geometry.
Key takeaways
Springer book page
Author: Judith N. Cederberg
Publication date: 1989 (1st edition; Springer eBook release: 9 March 2013)
Publisher: Springer New York
Summary
A Course in Modern Geometries is an undergraduate-level introduction to several major geometries beyond the standard Euclidean treatment. Cederberg begins with axiomatic systems and finite geometries, using small mathematical models to show how geometries can be constructed from explicitly stated axioms. The second chapter develops Euclidean and non-Euclidean geometry, emphasizing the role of the parallel postulate and showing how changing an axiom leads to fundamentally different geometric worlds.
The book then moves from synthetic geometry to a more algebraic viewpoint. It studies transformations of the Euclidean plane—including isometries and other transformation groups—and represents many of these transformations with matrices, providing a strong connection with linear algebra. The final major section introduces projective geometry, treating it both synthetically and analytically. This progression makes the book particularly useful as a bridge between classical geometry, linear algebra, and eventually abstract algebra.
The text was designed especially for junior- and senior-level mathematics students, including future secondary-school mathematics teachers. Its four main chapters are Axiomatic Systems and Finite Geometries, Non-Euclidean Geometry, Geometric Transformations of the Euclidean Plane, and Projective Geometry.
Key takeaways
- Geometry is not a single system: different choices of axioms produce different legitimate geometries.
- Non-Euclidean geometry demonstrates the profound consequences of altering Euclid's parallel postulate.
- Transformation geometry connects geometry with matrices, groups, and linear algebra.
- Projective geometry extends Euclidean ideas by incorporating points at infinity and studying properties invariant under projection.
- The book is especially valuable for someone interested in the transition from classical geometry to modern algebraic thinking.
Springer book page
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