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Geometry: Euclid and Beyond

Robin Hartshorne
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Morteza Seddighin
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This is a well written book on geometry and its history. The author starts with Euclid's postulates and gradually presents geometry from a modern standpoint and links the subject to advanced theories such as Field Theory, Galois Theory and Inversion Theory.

The book is divided into eight chapters.  In chapter 1 the author discusses Euclid's system of axioms. This chapter also contains some newer results, such as the Euler line and the nine-point circle. In chapter 2 the author discusses Hilbert's axioms and how they complete Euclid's axioms, and defines Hilbert's plane as an abstract set of objects (points) together with an abstract set of subsets (lines) which satisfy the axioms.

In chapter 3 the author introduces geometry over fields and proves several theorems of Euclidean geometry using coordinate techniques. In chapter 4 the reader is introduced to segment arithmetic, in which one can define addition and multiplication of line segments in a Hilbert plane that satisfies the parallel axiom. Using the equivalence classes of line segments the author presents rigorous proofs for some theorems regarding similarity of triangles.

In chapter 5 the author very clearly explains the difference between the notion of Area as it was conceived by Euclid and the modern conception of Area as defined by a "measure of area" function on the Hilbert plane. To understand chapter 5 of this book one has to be familiar with some topics from modern abstract algebra such as ordered abelian groups and tensors. Here one finds an elegant modern solution to Hilbert's Third Problem (first solved by Max Dehn in 1900). The problem is to find two solid figures of equal volume that are not equivalent by dissection even after possibly adding on other figures that are equivalent by dissection.

In chapter 6 the author links the construction problems to field extensions and uses Galois theory to prove that the duplication of a cube and the trisection of an angle are impossible by ruler and compass. We also get an account of Gauss's Theorem that a regular 17-sided polygon is constructible by ruler and compass.  The proof presented here is very close to the proof given by Carl Friedrich Gauss himself in 1796 and it does not use modern algebra such or Galois theory. At the end of chapter 6 the author shows that the use of compass and marked ruler is equivalent to finding successive real roots of quadratic and cubic equations.

Chapter 7 is on non-Euclidean geometry. Here the author uses Inversion Theory to verify the axioms of hyperbolic geometry for the Poincaré model. He presents a very condensed summary of inversion theory and related topics, such as the theory of transformations on the complex plane. He also describes some classical theorems of Euclidean geometry, such as the solution of the Apollonius problem, by utilizing inversion theory. The author proves several advanced theorems of hyperbolic geometry in this chapter. In chapter 8 the theory of polyhedra is presented. This chapter includes Euler's and Cauchy's Theorems and discusses symmetry groups of polyhedra.

This is a very accurate account of geometry which presents the evolution of geometry from Euclid to modern days. The author looks at geometry from a historical perspective. As a result the book includes several problems of historical interest. For example a two dimensional version of the problem of Alhazen (whose actual name was Abu Ali al-Hasan ibn al-Haytham) is among the exercises. The original problem, which asks us to find, given a light source and a spherical mirror, the point on the mirror where the light will be reflected to the eye of an observer, has a long and interesting history.

Since several of the topics discussed in this book have their roots in the work of Arabic/Islamic mathematicians, the reviewer expected to see more samples of the work of Arabic/Islamic mathematicians. For example, the work of Omar Khayyam includes a complete classification of cubic equations with geometric solutions found by means of conic sections and commentaries on the difficult postulates of Euclid's book. In the latter, he anticipated some of the theorems of non-Euclidean geometry. Both of these would have made for valuable additions to the book. Perhaps the author will consider adding them in a future edition.

Morteza Seddighin ( is associate professor of mathematics at Indiana University East. His research interests are functional analysis and operator theory.

* Euclid's Geometry * Hilbert's Axioms * Geometry over Fields * Segment Arithmetic * Area * Construction Problems and Field Extensions * Non-Euclidean Geometry * Polyhedra