Geometry with Trigonometry

High school students in the state of New York have, for well over a hundred years, endured a rite of passage known as the Regents Exams. These are statewide exams that are intended to measure academic performance. Archive copies are available online here; particularly in the subject of geometry, it is an instructive exercise to compare the current crop of exams with the ones that were given, say, back when I learned the subject, in the mid 1960s. The current exams do not focus on proof, and in fact several recent ones don’t ask the student to prove even a single theorem; the exams from the 1960s all ask for several proofs each. The New York experience seems to parallel the one in Iowa: students of mine who aspire to teach high school mathematics tell me that high school geometry courses here often spend little or even no time on proofs.

The teaching of geometry at the college level has also declined over this time period. When I was an undergraduate, the school I attended offered courses in advanced Euclidean geometry, foundations of geometry, projective geometry, geometric transformations, and advanced analytic geometry. Nowadays, most universities rarely offer more than one upper-level course in geometry. Patrick Barry, the author of the book under review, laments this decline, “in terms of both quantity and quality” of geometry teaching, which, he points out, has resulted in students having to rely “mainly on the earlier intuitive treatment” in secondary school.

This book is his response. Intended for college students who have already taken a less rigorous high school course in geometry, this book approaches the subject from several different points of view. It includes an axiomatic development, but also discusses analytic geometry (i.e., the use of coordinates), the use of geometric transformations, vector methods, trigonometry and complex numbers.

This is a second edition of a text first published more than 15 years ago. The new edition, according to the author, differs mostly in the chapter on vector and complex-number methods, “in which there is considerable alteration of detail.”

Before the text even begins, there is an interesting and valuable feature: the author gives a two-page account of the Greek and Latin origins of many common mathematical terms. Then, after an introductory chapter discussing history, the rise of logical reasoning and The Elements, there are four chapters (2–5) presenting an axiomatic development of familiar topics in the geometry of triangles and parallelograms. The material covered in these chapters does not, it seems, extend beyond what a high school student is taught, but it is taught rigorously and honestly. The author does not hide behind vague axioms; he proves things that need to be proved, notwithstanding the possible pedagogical difficulties that necessarily attend proving things that many students feel don’t warrant proof because they are “obvious”. His axiom system (like Birkhoff’s, circa 1932) assumes the existence of the real numbers and includes both a “ruler” and “protractor” axiom, allowing the measurement of distance and angle measure.

Coordinates are introduced in chapter 6, and then used as a tool to study circles in chapter 7. Again, most of the concepts covered in this latter chapter would probably be familiar to high school students (except perhaps for topics like the power of a point and the radical axis), but the approach taken here is much more formal and rigorous than would be taught in secondary schools.

Chapter 8 begins the study of isometries, concentrating in this chapter on translations and reflections, although Barry calls the latter “axial symmetries”. One nice feature here is the author’s proof that an isometry is necessarily bijective; many authors insert bijectivity into the definition of an isometry, although that is redundant. Of course, the fact that a distance-preserving function is one-to-one is obvious, but surjectivity requires a bit more work. This chapter doesn’t proceed very far, however; in fact, rotations won’t even be defined until chapter 10.

Before this, however, there is chapter 9, in which trigonometric functions are defined. This is not as easy as it sounds. Because the notion of “angle” is so elusive and hard to define, a considerable amount of technical fussing is required to do things precisely. After defining the trigonometric functions, the author puts them to good use by giving a trigonometric proof of the famous Steiner-Lehmus theorem (if, in a triangle, the angle bisectors of two of the interior angles have the same length, then these angles are equal and the triangle is isosceles). This theorem has an interesting history: it has the appearance of an easy exercise, especially because the corresponding results for altitudes and medians are pretty easy, but the result for angle bisectors turns out to be quite difficult indeed. (In fact, in an article appearing in The Best Writing on Mathematics 2015, John Conway and Alex Ryba discuss whether it is even possible to give a direct geometric proof of this theorem.)

Chapter 10 begins with a discussion of complex numbers and then discusses complex coordinates. From here we get to other topics like rotations and sensed angles. The subtle concept of orientation is also introduced here.

After this, there is a chapter discussing the use of complex numbers and vectors in geometry. These methods are used to prove some theorems in Euclidean geometry that are usually not discussed at the secondary level, including those of Ceva, Menelaus, Pappus and Miquel. Other more advanced topics covered in this chapter include triangle centers and the nine-point circle.

The book ends with a chapter introducing calculus methods into the study of trigonometry. Applications include a derivation of the formula for the area of a circle.

There is a bibliography, but it is a fairly short one (13 entries). Also, none of the entries are more recent than 1989, which made me wonder whether the author updated the bibliography for the new edition.

The book is carefully written, and proceeds at a fairly high level of sophistication — perhaps too high a level for most undergraduate geometry courses. The author’s attention to detail and formality is admirable, but it does carry with it a certain pedagogical risk. Notation becomes fairly cumbersome, for example, particularly when discussing slippery notions like “angle”. To illustrate: the author defines the angle-support of two rays with common endpoint (we can’t say angle, because these two rays define two angles) and denotes it using the fairly clunky notation \(\mid BAC\). He then gives the following, pretty horrible-looking, set-union definition of the exterior region of a non-straight angle-support: \(\{\Pi\, \backslash\, \mathcal{IR}|\underline{BAC}\}\,\cup\,|A,B\, \cup\, |A,C\).

This sort of thing (which, at the very least, may serve to discourage casual or “drop-in” readers of the book) may be inevitable in any really careful, rigorous treatment of these ideas, but it should be noted that a really careful, rigorous treatment may not be essential, or even desirable, at this level. There is a happy medium between completely intuitive accounts of the subject on the one hand, and completely rigorous accounts on the other. It is possible to fashion a course that is proof-based, but which doesn’t proceed from a fully axiomatic point of view; a text along these lines, for example, is Isaacs’ Geometry for College Students, which gives the student ample opportunity to read and fashion proofs but which retains the idea that geometry is still an essentially visual subject.

It’s all, of course, a matter of taste. Instructors who want to present a completely rigorous development of the subject and who are therefore willing to accept the pedagogical risks involved, will certainly want to look at this text. To my mind, though, this book probably makes a better reference than a text. It’s always nice having a source for this sort of thing, but I can’t help but feel that the formalism will, for many students anyway, overwhelm them to the point where they lose sight of some of the more beautiful aspects of geometry.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.