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Geometry: Its Elements and Structure

Alfred S. Posamentier and Robert L. Bannister
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Back in the mid-to-late 1960s, I was a high school student in New York, and, like most other high school students there, I experienced the Regents Exams. These were statewide standardized exams in certain core areas (including, of course, mathematics) that were required of most, in not all, students. I remember them vividly. In particular, I remember my 10th grade geometry exam, which required the production of a number of proofs. This emphasis on proof was entirely consistent with the nature of the course I had just taken, which required us to do lots of them over the course of the academic year.

This no longer seems to be the case. Several of the students in the geometry courses I teach who are planning to go on to be secondary teachers have told me that proofs no longer occupy any kind of central position at all in these courses. Out of curiosity, I checked out the two most recent Regents exams in geometry (in June 2014 and January 2014; they can be found online at and was surprised to find how little proofs mattered in them. In both exams, most of the questions were multiple choice; in June, the only thing resembling a proof was the last question, where the students were given the coordinates of a quadrilateral and asked to prove certain things about it, and in January, though a bona-fide proof did appear, its steps were written out for the student, who only had to fill in the reasons for each step. So, in neither of these two exams did the student, even once, have to construct an actual proof of a general geometric proposition.

I do not, to put it mildly, view this as a positive trend. It will of course come as no surprise to college professors of my generation to hear that students seem much less prepared to handle the rigors of undergraduate proof-oriented mathematics than were the students of yesteryear. Geometry, to my mind, is all about proofs, and has been such ever since the ancient Greeks turned geometry from an inductive applied subject to a deductive one. Pretending otherwise, I believe, cheats the students and makes it more difficult for them to deal with actual mathematics later in their academic careers.

And that brings me to the book under review, a Dover reprint of a text that was first published in the 1970s and which, reflecting that date, commendably bucks the current trend by doing geometry at the high school level from a proof-oriented standpoint. (The original version, published by McGraw-Hill, lists a third author, J. Houston Banks, whose name does not appear on this current incarnation of the book.) The approach used is mostly synthetic (reasoning from postulates) but there is some discussion of coordinate geometry and vectors as well.

The back cover of the book indicates that this text is suitable for college courses, but it seems clear to me that the primary intended audience is high school students. Except for a brief final chapter introducing non-Euclidean geometry, the subject matter of the text is entirely Euclidean, both plane and solid, and the topics covered are (except for scattered sections that will be discussed later) those typically taught at the high school level: congruence criteria, parallel lines and parallelograms, the basic rules concerning circles (and the angles between chords, secants and tangents), similarity, area and straightedge-and-compass constructions. Some of the more sophisticated results of Euclidean geometry that one associates with a college-level course (e.g., the Euler line and nine-point circle, the theorems of Ceva and Menelaus, the Steiner-Lehmus theorem, geometric transformations) do not appear. And, as explained in more detail below, I don’t think the axiomatic development provided here meets the standards of a college-level course.

The book looks and reads like one intended for high school, rather than college, students: proofs are given in two-column format, definitions and postulates are boxed for emphasis, each chapter ends with a brief section entitled “How is Your Mathematical Vocabulary?” that summarizes key terms that were previously defined, and review exercises are provided as well. In addition, much more attention is paid to results in high school algebra (e.g., a/b = c/d if and only if ad = bc) than one would expect in a college mathematics text, and the exercises in the text are for the most part quite simple and probably not challenging enough for a college classroom. So I must respectfully disagree with the back-cover blurb: this book would not be a contender for the college Euclidean geometry course that I teach, for which, in the past, I have used Isaacs’ Geometry for College Students as a text but for which, next fall, I will try the newly published Classical Geometry by Leonard et al.

What about its suitability for high school, however? I confess to being somewhat handicapped in my analysis here, simply because I have never taught at that level. I think, however, that the book offers some positive, but also some negative, features as a text at that level.

On the plus side, I like the emphasis on proofs, which are generally written in a clear and easy to understand way, and are accompanied by lots of diagrams. The strong emphasis on review, mentioned earlier, is also an attractive feature for a text designed for high school students. Another nice feature is the authors’ attention to history; interspersed throughout the book are sections (typically very brief, just a few paragraphs long) entitled “A Look at the Past” that provide supplemental information of an historical nature (e.g., a discussion of Euclid’s Elements, or the proof of Pons Asinorum — base angles of an isosceles triangle are equal — that appears there). The large number of exercises is also a plus.

On the other hand, I thought the book tried to cover too much, resulting in a very fat (600+ pages long) text that seemed crowded and dense. The inclusion of solid geometry may well be justified by the fact that most high schools cover it (although I personally believe the basic ideas are best introduced, at least initially, via a study of plane geometry), but there are other topics included here that I felt did not add materially to the text. There is a chapter on vectors, for example, but it spent a lot of time introducing and defining the basic ideas, and not much time actually proving anything about geometry using vectors; the whole chapter thus struck me as a lot of work not leading to any real payoff, and therefore easily omitted. Likewise, there is a chapter on coordinate geometry, but this also seemed unnecessary to me, since most high schools offer separate courses on this subject.

In addition, there are, scattered throughout the text, a number of sections called Mathematical Excursions, covering various “enrichment topics” such as finite geometries (that’s the term used in the book, but it is actually a misnomer, because the axiom set given by the authors does not require anything to be finite), equivalence classes, limits, spherical geometry, and the impossibility of various constructions. A good idea in principle, but I thought that most of these were too subtle or difficult for an intended audience of high school students. The material on finite geometry, for example, shows up on page 42 of the text, before the students have even become acclimated to the study of basic geometry; this seems awfully early to have the students thinking in these terms.

Another issue I had with the book concerned the level of rigor demonstrated in the axiomatic development of the subject. Let me be clear at the outset that I have no problem at all with not doing geometry at this level completely rigorously, and allowing the students to make reasonable inferences from diagrams and common sense. In fact, at the high school level, this seems to me to be the preferred approach: doing an extremely rigorous development of the subject entails all kinds of pedagogical pitfalls, including the need to state and prove facts which most students would think obvious. (For a college-level text that does a very nice job of this, see Axiomatic Geometry by Lee.)

But what I didn’t like in this book was the odd juxtaposition of such informal reasoning in some places and very precise, almost pedantic, analysis in others. As an example of the former, the authors define the center of a regular n-gon to be the “center of its circumscribed or inscribed circle”. This definition is not accompanied, however, by any discussion of how we know either of these circles exist, much less have the same center. Yet at the same time, the authors go to the trouble of postulating a principle of substitution (“If a = b, then either a or b may be replaced by the other in any statement without changing the truth or falsity of the statement”) that I think most high school students would simply take on faith without question, along with numerous principles of logic that are implicitly used throughout the text.

As another example, consider the text’s treatment of angle bisectors. First, early in the book, the bisector of an angle is defined as a ray emanating from the vertex of the angle. Then, about 75 pages later, it is proved that this ray bisector exists and is unique, followed immediately by the definition of an angle bisector of angle A in triangle ABC as the line segment, lying on the ray bisector, with one endpoint A and the other endpoint on the segment BC. But there is no discussion as to how we know the ray bisector actually intersects the segment BC. (Contrast Lee’s text, where this fact is explicitly proved.) The problem is not the assumption that this intersection exists, which of course is visually obvious; rather, I think the problem is the juxtaposition of this assumption with the authors’ earlier decision to prove the (equally obvious) fact that the ray bisector is unique. I can’t help but feel that the student is being sent mixed messages by this: when are assumptions allowed, and when are they not?

To summarize and conclude: while I don’t think this book would serve as an axiomatic development of geometry at the college level, it does seem, particularly if used by an instructor willing to put in some careful thought about omitting certain topics and perhaps modifying some of the axioms, like an interesting text for a proof-oriented high school course.

Mark Hunacek ( teaches mathematics at Iowa State University.

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