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An Axiomatic Approach to Geometry: Geometric Trilogy I

Francis Borceux
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Talk about perfect timing: this book arrived in mid-December, about a month before the start of a course I am scheduled to teach in the spring semester on the foundations of geometry. The course starts with a critical examination of Euclid’s Elements, and discusses the role of axiom systems and models in modern mathematical reasoning, the controversy surrounding Euclid’s fifth postulate, and the basic theorems and models of hyperbolic geometry. There is no question but that it will now be a better course for my having seen this excellent book.

This is the first in a trilogy, all three volumes of which are now available. The two other books are entitled, respectively, An Algebraic Approach to Geometry and A Differential Approach to Geometry. Algebraic Approach primarily covers affine, Euclidean and projective geometry from the perspective of linear algebra, and then culminates in a chapter introducing the basics of algebraic geometry. Differential Approach covers classical differential geometry of curves and surfaces, and shows how the power of calculus can be used to shed light on geometric ideas. Together with the book now under review, these books do an excellent job of conveying the broad sweep of geometry, and give considerable insight into the various ways in which one can approach the subject.

The approach in this volume is, as the title of the book makes clear, synthetic. (For the most part, anyway; some of the appendices use algebra.) The axiomatic approach to geometry accounts for much of its history and controversies, and this book beautifully discusses various aspects of this. It starts with a short chapter on the pre-Greek history of geometry, first looking briefly at the early pre-history (cave drawings, etc.) of geometry, then moving on to ancient Egypt and Mesopotamia. The discussion here, by design, does not give an exhaustive historical account, but instead just gives some flavor of these topics.

The next three chapters move to the development of geometry in ancient Greece, both before and after Euclid’s Elements. Chapter 2 looks at the works of some of the early, pre-Euclid, Greeks (both “big names” like Thales and Pythagoras, and less well known people such as Hippias, Anaxagoras, Archytas and Menaechmus, all of whom worked on and found solutions to some of the classical construction problems such as trisecting an angle, duplicating a cube or squaring a circle, all of which are described by the author in this chapter and then examined in more detail at the end of the book.

Chapter 3 is devoted to the Elements; each of the thirteen books of this treatise is given a section here in which the major results of that book are described in modern language, with some commentary by the author. (One criticism: the author uses his own indexing system, rather than Euclid’s; Euclid’s proposition 1 of book 1, for example (the construction of an equilateral triangle with given base) is not identified as such, but is instead denoted Proposition 3.1.5, to conform to the fact that it appears in section 1 of this book’s chapter 3.)

Chapter 4 then surveys the work of some of the important Greek mathematicians following Euclid, including Archimedes and Appollonius (the author devotes five sections of text to each of these two), and Heron, Menelaus, Ptolemy and Pappus, each of whom must make do with only one section each.

In the next chapter, some of the “modern” (very loosely speaking, of course!) theory of Euclidean geometry is presented: topics here include Viete’s formula for \(\pi\), the existence and basic properties of the centroid and orthocenter of a triangle, Ceva’s theorem, Morley’s theorem, inversions in the plane and three-space, and the Mohr-Mascheroni theorem (that any construction that can be done with straightedge and compass can be done with compass alone). The discussion here is not just historical and expository; the results in question are actually proved.

The focus then shifts to the development of foundational and non-Euclidean issues. First up, in chapter 6, is projective geometry. After the by-now-obligatory discussion of perspective drawing, the extended Euclidean plane is defined and its basic properties discussed, including proofs of the theorems of Desargues and Pappus. Then axioms are given for an arbitrary (abstract) projective plane, and the basic properties of these are established. Finally, the author discusses abstract projective planes which satisfy Desargues’s theorem (he calls them “Arguesian”, for some reason, rather than “Desarguesian”, the term I learned) and Pappus’s theorem, and spends the rest of the chapter discussing the beautiful connections between the satisfaction of these theorems in an abstract plane and the coordinatization of these planes by a division ring or field.

The next chapter discusses the development of non-Euclidean (i.e., hyperbolic, although the author doesn’t use that term for some time) geometry, following an historical approach of first considering a number of failed attempts to prove Euclid’s fifth postulate from the other axioms (or what I, and other textbooks such as Greenberg’s Euclidean and non-Euclidean Geometries, call “neutral geometry”) and then explaining how these gradually led to the realization that denial of the Fifth postulate actually led to the creation of a mathematically consistent geometry. The basic facts about hyperbolic geometry are then proved, and several models provided. (Elliptic geometry is not discussed, probably because it is inconsistent with the axioms of neutral geometry.) In a final chapter, the author gives a rigorous axiomatic development of Euclidean geometry, following Hilbert; as he points out, many people might have expected the book to open with this chapter, but he saved it for last to be faithful to the historical development of the subject.

This is not all, however. There follow three appendices, in which is provided a very detailed discussion of impossibility theorems in classical straightedge-and-compass constructions. The subject is taken far further here than in most geometry textbooks. For example, there is not just a statement, but also a proof, of the Gauss-Wantzel theorem, which gives a necessary and sufficient condition, in terms of Fermat primes, for the construction of a regular n-gon. Other impossible-construction results, for example the impossibility of trisecting an arbitrary angle, are also proved in this chapter; this of course requires the development of a fair amount of abstract algebra, which explains why all of this is relegated to a set of appendices.

I thoroughly enjoyed this book, and highly recommend it for instructors who are preparing courses in this material or who just want a great reference on their shelves. I am not sure, however, whether it would go over very well as an undergraduate text. (It probably is not intended as such; the webpage for the book lists the level as “Graduate”.) As the above description should make clear, the author covers a lot of material in 400 pages, and the exposition, though clear, is reasonably rapid and succinct, requiring some considerable degree of mathematical maturity; this is not, I think, a book for beginners. There are exercises and problems appearing at the end of each chapter, but few of them struck me as trivial.

While I doubt I will be using this as the assigned text for my foundations of geometry course, I will always be keeping it close at hand as I prepare lectures for it. Any decent college library should own this book (and the two others in the trilogy as well).

Mark Hunacek ( teaches mathematics at Iowa State University.