- Author: Paris Pamfilos
- Publisher: Springer
- Publication Date: 02/11/2024
- Number of Pages: 595
- Format: Hardcover
- Price: $99.99
- ISBN: Volume 1: 978-3-031-48905-1 Volume 2: 978-3-031-48909-9
- Category: textbook
[Reviewed by Mark Hunacek , on 06/05/2026]
These two volumes comprise an attractive introduction to Euclidean geometry at the junior/senior undergraduate level. They definitely belong on the shelf of any faculty member who teaches, or is interested in, this subject.
Volume 1 (a fairly thick book, about 600 pages long) covers what might be called “Advanced Euclidean Plane Geometry”. There are five chapters, arranged in roughly ascending order of difficulty. The book starts out with the very basic axioms about points and lines, and establishes some results (e.g., that the sum of the angles of a triangle is 180 degrees) that the student has undoubtedly seen in high school, but quite possibly without proofs, and almost certainly not in the kind of detail discussed here: for example, the author takes the time to distinguish between results that are true in “absolute geometry” (sometimes called “neutral geometry” in other books, this refers to what can be proved without assuming a parallel postulate) and those that are true when the Euclidean parallel postulate is assumed. For example, in neutral geometry one can prove that the sum of the angles of a triangle is less than or equal to 180 degrees (this is the Saccheri-Legendre theorem); with the parallel postulate, that inequality becomes an equality. Chapter 2 discusses basic results about circles and polygons (including ruler and compass constructions, and a number of results, like the Steiner-Lehmus theorem, which would likely not have been covered in high school), and chapter 3 introduces the concept of area of a polygon and uses area-based proofs to establish, for example, the Pythagorean theorem. Again, the author gives more detail than a student reader will likely have seen before: He not only proves in the first three chapters that the angle sum theorem and the Pythagorean theorem follow from the parallel postulate, for example, but also proves that these theorems are in fact equivalent to that postulate: i.e., that either of these results, along with the other axioms, can be used to prove the parallel postulate). In chapters 4 and 5, the focus is on more advanced Euclidean theorems that seldom, if ever, get mentioned in high school. Chapter 4, “The Power of a Circle”, deals with topics like the golden ratio, Apollonian circles, and inversion in a circle. Chapter 5, “From the Classical Theorems”, is a veritable flurry of advanced results, including but not limited to Feuerbach’s theorem, Heron’s formula, the Steiner-Lehmus theorem, the theorems of Menelaus and Ceva, the existence of the nine-point circle (called here the Euler circle), and the theorems of Pappus and Desargues. (The book hints at projective geometry but does not explore the subject in any depth.)
Volume 2, which is about two-thirds the length of volume 1, discusses a potpourri of special topics, most involving solid geometry. While this volume presupposes a basic understanding of Euclidean geometry, actual physical possession of volume 1 is not necessary to read volume 2; there are occasional references to volume 1, but the results referred to are generally clear. The first chapter, titled “Circle Measurement”, covers classical topics like the definition of $\pi$ and the area of a circle; there is also a section on the less well-known isoperimetric inequality. The last section is titled “Anthyphairesis”, a word that was new to me; it turns out to refer to an ancient Greek algorithm for computing continued fractions. The author uses the process to prove that the diagonal of a square is incommensurate with a side—i.e., $\sqrt{2}$ is irrational. Chapter 2 discusses geometric transformations in the plane, starting with isometries and then moving to homotheties, similarities and inversions in circles. Interesting features of this chapter include a section on the Poincare disc model of the hyperbolic plane (a rare exception to the very Euclidean nature of these books, this is discussed here because inversions play a central role) and Archimedian tilings of the plane. Chapters 3-5 discuss topics in solid geometry: lines and planes in space, familiar solid figures (prisms, cylinders, spheres, cones) and area and volume. Chapter 6 is on conic sections, and chapter 7 is on geometric transformations in space.
$$In both volumes, the author’s approach is synthetic; vectors and analytic geometry are not used (although trigonometry sometimes is). Also, it should be noted that the proofs given here make free use of diagrams and geometric reasoning rather than following a rigid axiomatic development. This is, I think, entirely appropriate. On those few occasions when I attempted to show my own geometry classes what a real axiomatic development was like, I found it difficult to motivate the students to sit through proofs of results that seemed intuitively obvious. The important thing, of course, is that the author be honest, and tell the students up front that this is not a rigorous axiomatic treatment. Pamfilos does this. (For a textbook that does do a rigorous axiomatic treatment, see Lee’s Axiomatic Geometry (AMS 2013).)
Are these two volumes suitable as texts? Each one contains enough material to support a one-year course, and each book allows for judicious skipping to also accommodate a one-semester one (though a section dependence chart would have been useful). Given the arrangement of material, it seems likely that volume 1 is far more likely than volume 2 to be useful as a text at an American university; there just aren’t that many universities that offer courses on the topics of volume 2. To facilitate each volume’s use as a text, there are a huge number (more than 1400, according to the preface) of exercises, quite of few of which are accompanied by hints, some of them so detailed that they essentially amount to solutions. These exercises appear both interspersed through the text and in exercise sections at the end of each chapter. The book also benefits from many illustrations, quite of few of which are in color. Open either volume at random and it is much more likely than not that at least one of the two pages showing has an illustration on it.
On the other hand, there are aspects of these books that make their use as a text a little bit problematic. They are a translation from the Greek (the author refers in the preface-- which, by the way, is the same in both volumes -- to two previous Greek editions). This may account for occasionally idiosyncratic terminology (e.g., the “trace” of an altitude rather than the “foot”). The author’s writing style is, likewise, occasionally idiosyncratic: In section 3.5 of volume 1, for example, we have the sentences “Square is called the rectangle, which, in addition, has all its sides equal.” and “Similar are called two right triangles which, besides the right angle, have two more corresponding angles equal (therefore all their angles correspondingly equal).“ (It’s also unclear why the author feels the need to initially limit the definition of “similar triangles” to right triangles, especially since, in section 3.7, he discusses the concept of similar triangles for arbitrary triangles and just sort of assumes the definition originally given carries over.). The notation in these volumes can also get a little klunky. Students who are used to reading things like “triangle ABC” may raise an eyebrow at "triangle $\text{AB} \Gamma$" and even get starry-eyed when reading “Select a point $\Xi$ on the circle $\varepsilon$ and apply an inversion relative to the circle $\xi(\Xi$) ...". But these are minor issues, and certainly do not seriously impede the comprehensibility of the books.
$$The books struck me as quite carefully written. I only noticed one error, and that one was more of a typo than a substantive mistake: in corollary 2.1 on page 50 of volume 2, the author states that two isometries that agree on three points must agree everywhere; clearly, he meant “three noncollinear points”. And another reader has posted a comment on Mathematics Stack Exchange that exercise 1.43 in volume 1 is incorrect. But these are the only errors I am aware of. (There is, to my knowledge, no errata page for the books.)
Bottom line: Even if not used as a text, each volume, being a treasure trove of interesting geometric facts, makes a splendid reference and is also useful as a source of problems. I’ll conclude this review as I began it: if you are a faculty member who teaches, or even just enjoys, Euclidean geometry, you should definitely take a look at this set of books.
Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.