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The Ancient Tradition of Geometric Problems

Wilbur R. Knorr
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

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, a math help site that fosters inquiry learning.


  1 Sifting History from Legend
  2 Beginnings and Early Efforts
    (I) The Duplication of the Cube
    (II) The Quadrature of the Circle
    (III) Problems and Methods
  3 The Geometers in Plato's Academy
    (I) Solutions of the Cube-Duplication
    (II) Geometric Methods in the Analysis of Problems
    (III) Efforts toward the Quadrature of the Circle
    (IV) Geometry and Philosophy in the 4th Century
  4 The Generation of Euclid
    (I) A Locus-Problem in the Aristotelian Corpus
    (II) Euclid's Analytic Works
    (III) The Analysis of Conic Problems: Some Reconstructions
    (IV) "An Angle-Trisection via "Surface-Locus"
    (V) Euclid's Contribution to the Study of Problems
  5 Archimedes?The Perfect Eudoxean Geometer
    (I) Circle-Quadrature and Spirals
    (II) Problem-Solving via Conic Sections
    (III) Problem-Solving via Neuses
    (IV) An Anonymous Cube-Duplication
    (V) The Impact of Archimedes' Work
  6 The Successors of Archimedes in the 3rd Century
    (I) Eratosthenes
    (II) Nicomedes
    (III) Diocles
    (IV) "On the Curve called "Cissoid"
    (V) "Dionysodorus, Perseus and Zenodorus"
    (VI) In the Shadow of Archimedes
  7 Apollonius?Culmination of the Tradition
    (I) "Apollonius, Archimedes and Heraclides"
    (II) Apollonius and Nicomedes
    (III) Apollonius and Euclid
    (IV) Apollonius and Aristaeus
    (V) Origins and Motives of the Apollonian Geometry
  8 Appraisal of the Analytic Field in Antiquity
    (I) The Ancient Classifications of Problems
    (II) "Problems, Theorems and the Method of Analysis"
    (III) ". . . And many and the greatest sought, but did not find."
    (IV) Epilogue