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The Ancient Tradition of Geometric Problems

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This book is in the MAA's basic library list.

MAA Review

This is a look at geometric constructions in ancient Greece from the point of view of “who knew what, and when did they know it.” The challenges here are that very little textual material has survived from the Greek era, much of it is in textbooks (such as Euclid’s Elements) that were not written by the original discoverers, and attributions were spotty.

The book focuses on the three classical problems of trisecting an angle, squaring the circle, and doubling the cube. These problems are still studied today because they turn out to be impossible if only straightedge and compass are used. This was not proven until the 1800s, but the Greeks had a number of other approaches to these problems using additional tools, and these successful constructions are discussed in the present work.

The book’s method is direct examination and textual analysis of the available texts, with occasional reference to later Arabic writers. The present edition is a 1993 reprint of the 1986 work published by Birkhäuser. The author published a supplemental volume, Textual Studies in Ancient and Medieval Geometry (Birkhäuser, 1989) that gives a more detailed textual criticism of many of the same Greek and Arabic texts. The present work is also based on textual criticism, but the exposition emphasizes the results rather than the methods and is aimed at an audience generally interested in history.

The endnotes are extensive and amount to about 20% of the book. They give not only sources for quotations, but references to related material and further discussions of some of the technical points.

The book is primarily a history book, with some elementary geometry and a large number of geometric drawings. It is very specialized and scholarly. Many popular math books discuss these three classic problems. A good exposition, with elementary proofs of their impossibility with straightedge and compass, and discussion of other approaches with additional drawing tools, is in chapter 3 of Courant & Robbins & Stewart, What Is Mathematics? An Elementary Approach to Ideas and Methods. The three problems (along with the construction of regular polygons) are the subject of a classic monograph by Felix Klein, Famous Problems of Elementary Geometry.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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