The “famous problems of geometry” referenced in the title of this slim little book are indeed famous, and all concern that portion of Euclidean geometry concerned with the possibility or impossibility of certain constructions with compass and straightedge. Specifically, the author considers the three classical Greek impossibility results (squaring the circle, doubling the cube and trisecting an angle) and the question of determining for which n the regular n-gon can be constructed. The Dover edition that is the subject of this review was originally published by Van Nostrand Reinhold in 1969 under the title Famous Problems of Mathematics: A History of Constructions with Straight Edge and Compasses; this Dover edition is, according to a statement on the copyright page, a “slightly corrected” version of the initial text.
The solutions to these four problems all involve a certain amount of abstract algebra as well as geometry, and for that reason most undergraduates rarely get to see more than the statements of the results, if indeed they even see those. There are some textbooks that do bravely plunge into the proofs (see, e.g., Axiomatic Geometry by Lee or Borceux’s An Axiomatic Approach to Geometry) but professors of geometry courses rarely have the time to develop all the necessary algebra in detail, and, likewise, professors of abstract algebra courses may often find it too big a detour to do the geometry, particularly the construction of regular n-gons. This is unfortunate, because the solutions to these classical problems are not only beautiful but also offer an interesting look at how different branches of mathematics can reinforce each other.
The book under review charts a middle course through these results. Not everything is proved in detail, but enough detail is provided to give a good sense of the ideas behind the proofs, accessible to students with no formal background in abstract algebra. This accessibility is achieved by sometimes just skipping a proof when necessary, and sometimes by using ad hoc arguments based on specific calculation, rather than appealing to general theory.
The book opens with some preliminary material on compass and straightedge constructions, geared towards showing that the set of all constructible numbers is a subset of the real numbers closed under the four basic arithmetic operations and taking of square roots. This is all done via a series of exercises, solutions to all of which appear at the end of the book. The author then introduces the term “field” (not as an abstract algebraic system, but as a subfield of the real numbers) and explains why the field of constructible numbers that was just discussed turns out to be precisely the set of real numbers that can be obtained by starting with the rational numbers and repeatedly doing the five operations just specified. Then, as an example of the ad hoc approach I referred to in the previous paragraph, Bold proves directly, without using concepts such as the degree of a field extension, that any real root of an irreducible cubic polynomial with rational coefficients cannot be constructible.
It is this result that turns out to be instrumental in solving two of the three ancient impossibility problems mentioned above (doubling the cube and trisecting an angle). Before getting to this, however, the author inserts a background chapter on complex numbers. After this, in chapter IV, Bold makes quick work of the cube-doubling problem and in the next chapter addresses, using some trigonometry, the matter of trisecting an angle.
Chapter V involves the question of squaring the circle, and the solution here requires some understanding of transcendental numbers. Bold’s chapter (VI) on this problem is more history than mathematical analysis; nothing is proved here, but the history of the determination that \(\pi\) is transcendental is traced.
Chapter VII, a lengthy one, addresses the history and resolution of the question of determining precisely which regular n-gons are constructible. Using earlier material on complex numbers, the author discusses in some detail the details of the construction in the cases n = 5, 7 and 17 (this last example is not often found in the elementary textbook literature) and then states without proof the general theorem the general criterion, which is that a regular n-gon is constructible with compass and straightedge if and only if n = 2km, where m is odd and is equal either to 1 or a product of distinct Fermat primes.
There follows one remaining chapter of text, titled “Concluding Remarks”, which comments on several related issues. The book shows its age here by stating that Fermat’s last theorem is unsolved, but this sort of thing seems inevitable every time an older book is reprinted without alteration.
One particularly nice feature of this book is the attention paid to history, and specifically to an aspect of history that I suspect is insufficiently well known, even among mathematically trained people: while most people have heard of compass and straightedge constructions, the fact is that the ancient Greeks were not wedded to these devices and in fact explored constructions with other tools. Archimedes knew, for example, that an arbitrary angle could be trisected with a compass and straightedge, provided the straightedge had one segment marked off on it. His argument is reproduced in this book. Likewise, Hippias (more than a hundred years before Euclid) used a curve called the quadratrix to both trisect an angle and construct the number \(\pi\). This fact is mentioned in the text, but the details are not provided. Little tidbits of interesting history like this appear throughout the text, as do several remarks that do not often find their way into standard texts (e.g., while it is impossible to trisect an arbitrary angle \(\theta\), it is possible by repeated bisection to construct an angle arbitrarily close to \(\theta/3\); the proof is a simple exercise in summing an infinite geometric series).
By contrast, one not-so-nice feature of the book is the total absence of an index. There should be a law against this sort of thing, even in slim mathematics books like this one.
To summarize and conclude: this is a pleasant little book, with some interesting things in it, and, particularly given its cheap price (less than eight dollars on amazon.com, as I write this), one that is particularly well-suited as a source of supplemental reading for a geometry course in which these topics are covered. Another likely audience for this book would be students who want to learn something about these ideas without having to wade through the technical details found in more advanced texts. The book is sufficiently clear that any college undergraduate with a reasonable grounding in high school mathematics should be able to read it without faculty assistance.
Mark Hunacek (email@example.com) teaches mathematics at Iowa State University.
|I||Achievement of the Ancient Greeks|
|II||An Analytic Criterion for Constructibility|
|IV||The Delian Problem|
|V||The Problem of Trisecting an Angle|
|VI||The Problem of Squaring the Circle|
|VII||The Problem of Constructing Regular Polygons|
|Suggestions for Further Reading|
|Solutions to the Problems|