The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles

by Robert Hahn

Overview

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.

Product Details

ISBN-13:	9781438464916
Publisher:	State University of New York Press
Publication date:	05/01/2017
Series:	SUNY series in Ancient Greek Philosophy
Sold by:	Barnes & Noble
Format:	eBook
Pages:	300
File size:	17 MB
Note:	This product may take a few minutes to download.

About the Author

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 ix

Acknowledgments xv

Introduction: Metaphysics, Geometry, and the Problems with Diagrams 1

A The Missed Connection between the Origins of Philosophy-Science and Geometry: Metaphysics and Geometrical Diagrams 1

B The Problems Concerning Geometrical Diagrams 4

C Diagrams and Geometric Algebra: Babylonian Mathematics 7

D Diagrams and Ancient Egyptian Mathematics: What Geometrical Knowledge Could Thales have Learned in Egypt? 12

E Thales's Advance in Diagrams Beyond Egyptian Geometry 25

F The Earliest Geometrical Diagrams Were Practical: The Archaic Evidence for Greek Geometrical Diagrams and Lettered Diagrams 32

D Summary 41

Chapter 1 The Pythagorean Theorem: Euclid I.47 and VI.31 45

A Euclid: The Pythagorean Theorem I.47 46

B The "Enlargement" of the Pythagorean Theorem: Euclid VI.31 66

C Ratio, Proportion, and the Mean Proportional (μεση αναλογον) 70

D Arithmetic and Geometric Means 72

E Overview and Summary: The Metaphysics of the Pythagorean Theorem 81

Chapter 2 Thales and Geometry: Egypt, Miletus, and Beyond 91

A Thales: Geometry in the Big Picture 92

B What Geometry Could Thales Have Learned in Egypt? 97

B.1 Thales's Measurement of the Height of a Pyramid 97

B.2 Thales's Measurement of the Height of a Pyramid 107

C Thales' Lines of Thought to the Hypotenuse Theorem 116

Chapter 3 Pythagoras and the Famous Theorems 135

A The Problems of Connecting Pythagoras with the Famous Theorem 135

B Hippocrates and the Squaring of the Lunes 137

C Hippasus and the Proof of Incommensurability 141

D Lines, Shapes, and Numbers: Figurate Numbers 148

E Line Lengths, Numbers, Musical Intervals, Microcosmic-Macrocosmic Arguments, and the Harmony of the Circles 153

F Pythagoras and the Theorem: Geometry and the Tunnel of Eupalinos on Samos 157

G Pythagoras, the Hypotenuse Theorem, and the μεση ανα λογος (Mean Proportional) 168

H The "Other" Proof of the Mean Proportional: The Psythagoreans and Euclid Book II 182

I Pythagoras's Other Theorem: The Application of Areas 189

J Pythagoras's Other Theorem in the Bigger Metaphysical Picture: Plato's Timqeus 53Cff 195

K Pythagoras and the Regular Solids: Building the Elements and the Cosmos Out of Right Triangles 198

Chapter 4 Epilogue: From the Pythagorean Theorem to the Construction of the Cosmos Out of Right Triangles 213

Notes 241

Bibliography 263

Image Credits 271

Index 273

Editorial Reviews

"Our understanding of the world in which Thales and Anaximander worked is sharpened by Hahn's discussion of contemporary design-techniques." — Aestimatio

"It is a mark of Hahn's generosity that he includes, in every chapter, clear analysis of the geometric problems together with a thorough history of both ancient sources and modern mathematical theory and interpretations before proposing his own extensions; this allows the non-specialist reader to perceive, if not participate in, the often quite divergent current debates about early Greek thought. The handsome production-values and clarity of organization of this book … are exemplary, as are its general index and index of Greek terms." — Bryn Mawr Classical Review

"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

From the Publisher