08-30-2026, 09:34 PM
Book:Foundations of Euclidean and Non-Euclidean Geometry
Author: Richard L. Faber
Publication date: 1983
Publisher: Marcel Dekker, New York
Series:Monographs and Textbooks in Pure and Applied Mathematics, Vol. 73
ISBN: 0-8247-1748-1 / 978-0-8247-1748-3
Length: xi + 329 pages in the original Dekker edition. A later Taylor & Francis catalog record lists 352 pages.
Summary
Faber's book is a rigorous introduction to the foundations of geometry, built around the question of what happens when Euclid's axioms—particularly the famous fifth or parallel postulate—are examined rather than simply accepted. It develops the portion of geometry that does not depend on the parallel postulate, often called absolute or neutral geometry, and then shows how different assumptions concerning parallel lines lead to different geometric systems. In Euclidean geometry, through a point outside a line there is exactly one parallel; in hyperbolic geometry there are more than one. This axiomatic viewpoint makes the book as much about the logical structure of mathematics as about geometric constructions themselves. Faber discusses such important devices as Saccheri quadrilaterals, which historically played a central role in attempts to prove the parallel postulate and instead helped reveal non-Euclidean geometry.
A substantial historical component traces the centuries-long struggle with Euclid's fifth postulate, through mathematicians such as Saccheri, Lambert, Gauss, Lobachevsky, and Bolyai. The book explains how failed attempts to derive the parallel postulate eventually led to the realization that a logically coherent geometry could exist in which the postulate is false. Faber's treatment of Gauss, for example, documents his private investigations of non-Euclidean geometry decades before its public acceptance. The mathematical development then explores the consequences of these alternative axioms: triangle angle sums, parallelism, perpendiculars, quadrilaterals and the structure of the hyperbolic plane. Thus the book connects axiomatic geometry, mathematical logic, history, and classical geometric reasoning rather than treating non-Euclidean geometry merely as a collection of unusual formulas.
The book is most appropriate for university mathematics students, teachers, and readers interested in the logical foundations of geometry. Its emphasis is more theoretical and proof-oriented than computational. A reader familiar only with school Euclidean geometry will encounter a deeper question: which familiar geometric statements actually follow from the basic incidence and congruence axioms, and which secretly depend on Euclid's parallel postulate? That makes Faber's book particularly useful for understanding why the discovery of non-Euclidean geometry was such an important event—it demonstrated that apparently self-evident properties of physical space are not inevitable mathematical truths but consequences of chosen axioms. The book is classified specifically under geometry and non-Euclidean geometry and includes a bibliography and index.
Key takeaways
Overall: This is a serious classical text for someone who wants to understand why Euclidean geometry works, exactly which assumptions it requires, and how entirely different but logically consistent geometries arise from changing those assumptions.
BOOK
Author: Richard L. Faber
Publication date: 1983
Publisher: Marcel Dekker, New York
Series:Monographs and Textbooks in Pure and Applied Mathematics, Vol. 73
ISBN: 0-8247-1748-1 / 978-0-8247-1748-3
Length: xi + 329 pages in the original Dekker edition. A later Taylor & Francis catalog record lists 352 pages.
Summary
Faber's book is a rigorous introduction to the foundations of geometry, built around the question of what happens when Euclid's axioms—particularly the famous fifth or parallel postulate—are examined rather than simply accepted. It develops the portion of geometry that does not depend on the parallel postulate, often called absolute or neutral geometry, and then shows how different assumptions concerning parallel lines lead to different geometric systems. In Euclidean geometry, through a point outside a line there is exactly one parallel; in hyperbolic geometry there are more than one. This axiomatic viewpoint makes the book as much about the logical structure of mathematics as about geometric constructions themselves. Faber discusses such important devices as Saccheri quadrilaterals, which historically played a central role in attempts to prove the parallel postulate and instead helped reveal non-Euclidean geometry.
A substantial historical component traces the centuries-long struggle with Euclid's fifth postulate, through mathematicians such as Saccheri, Lambert, Gauss, Lobachevsky, and Bolyai. The book explains how failed attempts to derive the parallel postulate eventually led to the realization that a logically coherent geometry could exist in which the postulate is false. Faber's treatment of Gauss, for example, documents his private investigations of non-Euclidean geometry decades before its public acceptance. The mathematical development then explores the consequences of these alternative axioms: triangle angle sums, parallelism, perpendiculars, quadrilaterals and the structure of the hyperbolic plane. Thus the book connects axiomatic geometry, mathematical logic, history, and classical geometric reasoning rather than treating non-Euclidean geometry merely as a collection of unusual formulas.
The book is most appropriate for university mathematics students, teachers, and readers interested in the logical foundations of geometry. Its emphasis is more theoretical and proof-oriented than computational. A reader familiar only with school Euclidean geometry will encounter a deeper question: which familiar geometric statements actually follow from the basic incidence and congruence axioms, and which secretly depend on Euclid's parallel postulate? That makes Faber's book particularly useful for understanding why the discovery of non-Euclidean geometry was such an important event—it demonstrated that apparently self-evident properties of physical space are not inevitable mathematical truths but consequences of chosen axioms. The book is classified specifically under geometry and non-Euclidean geometry and includes a bibliography and index.
Key takeaways
- Main area: Foundations of geometry, Euclidean geometry, and non-Euclidean/hyperbolic geometry.
- Central theme: Euclid's parallel postulate and what changes when it is removed or replaced.
- Distinctive strength: Combines rigorous proofs with the historical development of non-Euclidean geometry.
- Level: Best suited to undergraduate mathematics and above, especially readers interested in geometry from an axiomatic rather than purely computational viewpoint.
Overall: This is a serious classical text for someone who wants to understand why Euclidean geometry works, exactly which assumptions it requires, and how entirely different but logically consistent geometries arise from changing those assumptions.
BOOK
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