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The Circle: A Mathematical Exploration Beyond the Line

Alfred S. Posamentier and Robert Geretschläger
Prometheus Books
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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It’s a reasonably safe bet that anybody with an interest in mathematics education, or mathematics textbooks (particularly those that involve geometry) will have heard of Alfred Posamentier. For many years, he has been a well-known figure in the education community, and has also been a very prolific author and co-author, with (according to the book’s bibliographic blurb) over sixty books published in the areas of mathematics and mathematics education.

These books vary in tone. Some have been more or less traditional textbooks at the high school or college level; others have been what might be called “vignette books”, consisting of a collection of bite-sized problems, paradoxes and other brief discussions. I’ve reviewed two of them (Magnificent Mistakes in Mathematics and Geometry: Its Elements and Structure), and, in connection with my teaching, read portions of others (Challenging Problems in Geometry, Advanced Euclidean Geometry, and The Secrets of Triangles: A Mathematical Journey) and, to be honest, have (perhaps not surprisingly, given the sheer number of books that he has produced) found them to be of variable quality. The book now under review is, I think, one of his better efforts. Unlike, say, Magnificent Mistakes, this one has a strong central theme, the entries are generally quite interesting, some of them were new to me, and they did not feel forced or “shoe-horned” in.

As the title implies, this is a book about circles. They are explored from many different perspectives — mathematical, historical, artistic, architectural, etc. If you have a question about circles, then (with a few exceptions, noted later) chances are that you can find the answer (though perhaps not the proof) somewhere within these pages.

The first chapter begins by reviewing the basics of circle geometry, listing (without proof) theorems that are typically proved in a high school course: for example, formulas for the measure of an angle formed by a tangent to, and a secant of, a given circle. It then discusses something that may be new to readers: the Reuleaux triangle, a figure (the sides of which are circular arcs) that, like the circle, has constant width; i.e., if placed between two parallel tangent lines, can be continuously rotated between the lines.

The second and third chapters continue to explore geometric theorems about circles, including those that are associated with other geometric figures, such as the circumcircle of a triangle, the nine-point circle (which should, I think, have merited a chapter of its own), and circles in which quadrilaterals are inscribed. Not only are results about circles proved, but it is shown that circles play a role in many theorems about other geometric objects. For example, there is a formula (Heron’s formula) for the area of a triangle in terms of the three side lengths. There is no general Heron-like formula for quadrilaterals, but it turns out that for a cyclic quadrilateral (one inscribed in a circle) there is an analog.

Chapter four discusses “circle packing problems”, and also includes a digression to spheres and a discussion of Hales’ solution to the Kepler conjecture, about the optimal way to pack spheres. This is then followed by a chapter on “equicircles”, a term that I had never heard of prior to seeing this book. Any triangle has both an inscribed circle and three exscribed ones, each one tangent to a side of the triangle and the extensions of the other two sides. These circles, collectively, are the equicircles; their properties are discussed at some length in this chapter.

The next chapter addresses a famous class of compass-and-straightedge construction problems collectively known as the problem of Apollonius. The idea here is this: given three geometric objects (each one a point, line or circle; repetition of types is allowed), construct a circle that is tangent to all three. (“Tangent to a point” is just a shorthand way of saying “containing the point”.) So, for example, the simplest example is when all three objects are points; in that case a suitable circle exists if and only if the points are noncollinear, and in that case the construction is a routine exercise. There are ten cases in all to consider, the remaining nine being less routine; one rarely finds all ten worked out in any one text, but they are here. This chapter strikes me as a very valuable reference for teachers of geometry.

Inversion in a circle is the theme of chapter 7. The inversion map is defined and the basic properties of inversion established. Applications to the Apollonius problem, Steiner chains and other geometric problems are given. Unfortunately, no mention is made in this chapter of the relationship between circle inversion and non-Euclidean geometry.

The next chapter is another useful reference on a classical problem involving compass-straightedge constructions. The question arises: what constructions are possible given only a compass? (We view a line as having been constructed if we can construct two points on it.) The surprising answer turns out to be that anything that can be constructed with a compass and straightedge can be constructed with just a compass. This is the Mohr-Mascheroni theorem, another result whose proof is not easily found in the undergraduate textbook literature. The authors give a proof in this chapter. (A proof using inversions can be found in An Axiomatic Approach to Geometry, the first of the Borceux trilogy of geometry texts.)

As a change of pace, the next chapter, containing many pictures, discusses the use of circles in art and architecture. After this, it’s back to mathematics with two more chapters, one (written by Christian Spreitzer) on hypocycloids and epicycloids (curves that are obtained by rolling a circle) and the other on spherical geometry, where great circles appear as the “lines” on a sphere.

The chapters discussed above constitute the main body of the text, but there is also some “back-matter”. First, there is an Afterword, written by Erwin Rauscher, on circles in culture, and this is followed by eight Appendices, filling in proof details that the authors decided to defer. The book closes with five pages of supplementary notes (the authors apparently prefer to put notes in the back of the book rather than use footnotes).

There were a few topics that were omitted from this book that might have been added. The isoperimetric property of the circle is not discussed, and I would also have thought that the number \(\pi\) would play more of a role in this text (though the authors do give a reference to another book co-authored by Posamentier, unseen by me, titled Pi: A Biography of the World’s Most Mysterious Number). And, as previously noted, there is no discussion here of non-Euclidean geometry.

Readers of a certain age may be reminded of the much older (first published in 1957, revised slightly by adding an introductory background chapter in 1997) Circles: A Mathematical View, written by Dan Pedoe. The two books have nonempty intersection, but neither is a subset of the other. Pedoe’s book, for example, discusses Möbius transformations and non-Euclidean geometry, and also discusses the isoperimetric property of the circle. It does not, however, discuss all ten cases of the Apollonius problem (although it does consider the “three circles” case, which, as Posamentier and Geretschlager tell us, is frequently referred to by itself as the Problem of Apollonius), and does not contain the kind of coverage of circles in art, architecture and culture that this text does. I think it reasonable to say that this book does for a modern audience what Pedoe’s book did for its audience 60 years ago.

The book under review is sufficiently clearly written that a competent undergraduate should find it accessible, but because the subject matter of this book is fairly specialized, I wouldn’t suggest it as a text for a traditional course in upper-level undergraduate geometry. I suspect it’s not really intended as one, because, for example, there are no exercises at all. However, the instructor of such a course might want to keep a copy close at hand for reference and lecture-topic suggestions. I also recommend its acquisition by college libraries.

Mark Hunacek ( teaches mathematics at Iowa State University. 

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