Ruler and the Round: Classic Problems in Geometric Constructions

Rating:
Summary: RULER-AND-COMPASS IMPOSSIBILITY PROOFS DEMYSTIFIED
Review: This thoroughly readable and enjoyable book for a general audience was written by an accomplished and respected mathematician, former Professor and Chair at two of the world's best Departments of Mathematics, the University of Michigan and the University of Buffalo. Unlike many other writers of popular mathematics books, Kazarinoff does not betray his beloved field by over-simplification, pontification, or dogmatism. On the contrary, Kazarinoff has deep faith in the intelligence and critical abilities of his readers and he makes ever effort to help them to become genuine participants in a small, but richly fascinating and beautiful corner of mathematics. His aim is to help the readers to gain personal knowledge of several mathematically, philosophically, historically and culturally important mathematical facts. Although these facts can be stated in short simple sentences, they were discovered to be facts only after centuries of intense mathematical research by some of the world's greatest minds. The simplest example is the fact that it is impossible to trisect the angle, i.e. there exists no general construction method or algorithm using only straight-edge and compass for trisecting an arbitrarily given angle. "RULER" in the title refers to the straight-edge; "ROUND" refers to the compass.
The Kazarinoff book, again contrary to the vast majority of popular mathematics books, carefully explains the nature of the mathematical facts to be proved: the relevant fundamentals of geometry, what a construction is, exactly what can and can not be done with the straight-edge, exactly what can and can not be done with the compass. It has an intriguing and pedagogically effective discussion of the differences between what I call the collapsing compass and the non-collapsing compass. With the collapsing compass one draws a circle given the center and a point on the circumference, but one can not carry the length of the radius to other points to make copies of the circle, the compass "collapses". But with the non-collapsing compass, once one circle has been drawn it can be copied over and over wherever a center is given. An appendix presents the reader with enough practice using the straight-edge and collapsing compass that there will be no confusion or uncertainty about which facts are being proved. The practice is not just exercise: it is used to give a cogent and accessible proof of the fact, justifiably called astounding by Kazarinoff, that any figure constructible from given points by means of the straight-edge and non-collapsing compass can be done with the straight-edge supplemented only by a collapsing compass. Moreover, and here, as far as I know, RULER AND THE ROUND is absolutely unique: it provides a brief but informative discussion of exactly what a mathematical proof is.
In each case, Kazarinoff wants the readers to know exactly which fact is being argued for and exactly what a proof of it would be like -- so that the readers can make their own judgments of whether Kazarinoff has actually proved it. On page 5, at the end of the section called "PROOF" he says to the reader concerning the arguments to be presented: "I hope they convince you too". In what other popular mathematics book have you seen such respect for the reader, such openness, such modesty? In what other popular mathematics book have you seen concern for the reader's opinion? This book is an implicit insult to the elitist high-priests of popularization with their breezy enthusiasm, their hocus-pocus "proofs", their mumbo-jumbo, their scientistic dogmatism.
Ironically, it is Kazarinoff's openness the leads him to temper his realism with what to my mind seems to be an unacceptable level of cultural relativism and to temper his egalitarianism with a sometimes hard-edged elitism. Nevertheless, his frankness and independence are truly refreshing and his sincere effort to share with non-experts his profound mastery of the material can only evoke gratitude. Of course, there is room for disagreement about the details and about how well he fulfilled his goals. Judgments on these issues are to some extent subjective and will depend on the background of the person making the judgment.
I first read a library copy of this book in 1970 when it first appeared. A few months later, when I decided to reread it, the library copy was on-loan with a long waiting list. I tried to buy a copy but by then it was out of print. Recently, I went to Amazon.com to try to get a used copy and was thrilled to learn that it is back in print with a 2003 date.
Who should read this book? Mathematics majors should look at this short 130-page book as early as possible because it might reveal to them what subject they have chosen:, or at the very least it will reveal to them what a serious, accomplished mathematician thinks the subject is. Mathematics teachers, especially those who complain that their students do not know what a proof is, might pick up a pointer or two from reading it. Logicians might learn something from it, especially from the section on pages 5 and 6 about what proofs are. Historians and philosophers of mathematics will find many original and thought-provoking perspectives in this book. Kazarinoff does not belong to any of the identifiable "schools" of philosophy of mathematics-he gives no signs in this book of having paid any of them the slightest attention. He is not selling anything and he is not spinning anything. I can not think of a better book for people curious about mathematics. - Frango Nabrasa, Manatee FL