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Summary

Explores Thales’s speculative philosophy through a study of geometrical diagrams.

Bringing together geometry and philosophy, this book undertakes a strikingly original study of the origins and significance of the Pythagorean theorem. Thales, whom Aristotle called the first philosopher and who was an older contemporary of Pythagoras, posited the principle of a unity from which all things come, and back into which they return upon dissolution. He held that all appearances are only alterations of this basic unity and there can be no change in the cosmos. Such an account requires some fundamental geometric figure out of which appearances are structured. Robert Hahn argues that Thales came to the conclusion that it was the right triangle: by recombination and repackaging, all alterations can be explained from that figure. This idea is central to what the discovery of the Pythagorean theorem could have meant to Thales and Pythagoras in the sixth century BCE. With more than two hundred illustrations and figures, Hahn provides a series of geometric proofs for this lost narrative, tracing it from Thales to Pythagoras and the Pythagoreans who followed, and then finally to Plato’s Timaeus. Uncovering the philosophical motivation behind the discovery of the theorem, Hahn’s book will enrich the study of ancient philosophy and mathematics alike.

“This book would be an excellent resource for anyone studying Euclidean geometry, or it could serve as a text for a geometry seminar … Highly recommended.” — CHOICE

At Southern Illinois University Carbondale, Robert Hahn is Professor of Philosophy and Director of the Ancient Legacies Program, through which he leads traveling seminars to Greece, Turkey, and Egypt. He is the author of Archaeology and the Origins of Philosophy; Anaximander in Context: New Studies in the Origins of Greek Philosophy (with Dirk L. Couprie and Gerard Naddaf); and Anaximander and the Architects: The Contributions of Egyptian and Greek Architectural Technologies to the Origins of Greek Philosophy, all published by SUNY Press.

Table of Contents

Preface
Acknowledgments

Introduction: Metaphysics, Geometry, and the Problems with Diagrams

A. The Missed Connection between the Origins of Philosophy-Science and Geometry: Metaphysics and Geometrical Diagrams

B. The Problems Concerning Geometrical Diagrams

C. Diagrams and Geometric Algebra: Babylonian Mathematics

D. Diagrams and Ancient Egyptian Mathematics: What Geometrical Knowledge Could Thales have Learned in Egypt?

E. Thales’s Advance in Diagrams Beyond Egyptian Geometry

F. The Earliest Geometrical Diagrams Were Practical: The Archaic Evidence for Greek Geometrical Diagrams and Lettered Diagrams

G. Summary

1. The Pythagorean Theorem: Euclid I.47 and VI.31

A. Euclid: The Pythagorean Theorem I.47

B. The “Enlargement” of the Pythagorean Theorem: Euclid VI.31

C. Ratio, Proportion, and the Mean Proportional (μέση ἀνάλογον)

D. Arithmetic and Geometric Means

E. Overview and Summary: The Metaphysics of the Pythagorean Theorem

2. Thales and Geometry: Egypt, Miletus, and Beyond

A. Thales: Geometry in the Big Picture

B. What Geometry Could Thales Have Learned in Egypt?

B.1 Thales’s Measurement of the Height of a Pyramid

B.2 Thales’s Measurement of the Height of a Pyramid

C. Thales’ Lines of Thought to the Hypotenuse Theorem

3. Pythagoras and the Famous Theorems

A. The Problems of Connecting Pythagoras with the Famous Theorem

B. Hippocrates and the Squaring of the Lunes

C. Hippasus and the Proof of Incommensurability

D. Lines, Shapes, and Numbers: Figurate Numbers

E. Line Lengths, Numbers, Musical Intervals, Microcosmic-Macrocosmic Arguments, and the Harmony of the Circles

F. Pythagoras and the Theorem: Geometry and the Tunnel of Eupalinos on Samos

G. Pythagoras, the Hypotenuse Theorem, and the μέση ἀνάλογος (Mean Proportional)

H. The “Other” Proof of the Mean Proportional: The Pythagoreans and Euclid Book II

I. Pythagoras’s Other Theorem: The Application of Areas

J. Pythagoras’s Other Theorem in the Bigger Metaphysical Picture: Plato’s Timaeus 53Cff

K. Pythagoras and the Regular Solids: Building the Elements and the Cosmos Out of Right Triangles

4. Epilogue: From the Pythagorean Theorem to the Construction of the Cosmos Out of Right Triangles