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

301 Pages·2017·5.1 MB·English

The Metaphysics of the
Pythagorean Theorem
SUNY series in Ancient Greek Philosophy
——————
Anthony Preus, editor
The Metaphysics of the
Pythagorean Theorem
Thales, Pythagoras, Engineering, Diagrams, and the
Construction of the Cosmos out of Right Triangles
Robert Hahn
Published by State University of New York Press, Albany
© 2017 State University of New York
All rights reserved
Printed in the United States of America
No part of this book may be used or reproduced in any manner whatsoever
without written permission. No part of this book may be stored in a retrieval system or
transmitted in any form or by any means including electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording, or otherwise without the prior per-
mission in writing of the publisher.
For information, contact State University of New York Press, Albany, NY
www.sunypress.edu
Production, Ryan Morris
Marketing, Anne M. Valentine
Library of Congress Cataloging-in-Publication Data
Names: Hahn, Robert, 1952– author.
Title: The metaphysics of the Pythagorean theorem : Thales, Pythagoras, engineering,
diagrams, and the construction of the cosmos out of right triangles / by Robert Hahn.
Description: Albany, NY : State University of New York, 2017. | Series: SUNY series
in ancient Greek philosophy | Includes bibliographical references and index.
Identifiers: LCCN 2016031414 | ISBN 9781438464893 (hardcover : alk. paper) | ISBN
9781438464916 (ebook)
Subjects: LCSH: Philosophy, Ancient. | Pythagoras. | Thales, approximately 634 B.C.–
approximately 546 B.C. | Euclid. | Mathematics, Greek. | Pythagorean theorem.
Classification: LCC B188 .H185 2017 | DDC 182—dc23
LC record available at https://lccn.loc.gov/2016031414
10 9 8 7 6 5 4 3 2 1
(cid:148)(cid:232)(cid:469) (cid:143)(cid:128)(cid:562) (cid:160)(cid:152)(cid:148)(cid:153)(cid:91) (cid:545)(cid:871)(cid:152) (cid:302)(cid:546)(cid:130)(cid:509)(cid:469) (cid:133)(cid:154)(cid:142)(cid:128)(cid:152)(cid:155)(cid:559)(cid:223)(cid:153)(cid:554) (cid:147)(cid:152)(cid:556)
At the Pythagoras statue in Samos.

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
G. 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 (μέση ἀνάλογον) 70
D. Arithmetic and Geometric Means 72
E. Overview and Summary: The Metaphysics of the Pythagorean Theorem 81
viii Contents
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 μέση ἀνάλογος
(Mean Proportional) 168
H. The “Other” Proof of the Mean Proportional: The Pythagoreans 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 Timaeus 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
Preface
In this Preface I have three objectives. I want to (A) explain how I came to write this book, (B)
describe the audiences for whom it has been written, and (C) provide a broad overview of the
organization of the book.
A
Almost a decade ago, I stumbled into this project. My research for a long time has focused on the
origins of philosophy, trying to understand why, when, and how it began in eastern Greece. My
approach had been to explore ancient architecture and building technologies as underappreciated
sources that contributed to that story, specifically, to Anaximander’s cosmology. Anaximander
identified the shape of the earth as a column drum, and reckoned the distances to the heavenly
bodies in earthly/column-drum proportions, just as the architects set out the dimensions of their
building in column-drum proportions. I argued that Anaximander, who with Thales was the first
philosopher, came to see the cosmos in architectural terms because he imagined it as cosmic ar-
chitecture; just as the great stone temples were built in stages, the cosmos too was built in stages.
My last book, Archaeology and the Origins of Philosophy, tried to show that since we learned about
ancient architecture from archaeological artifacts and reports, research in ancient archaeology
could illuminate the abstract ideas of philosophy. Philosophers rarely appeal to archaeological
evidence and artifacts to illuminate the ideas of the ancient thinkers; this last project showed how
archaeology is relevant and how it may be used in future research. Since my work had focused on
eastern Greece—Samos, Didyma, and Ephesus—I traveled to southern Italy to explore how my
work might continue there, in Crotona and Tarento where Pythagoras emigrated after leaving
Samos. As I sat at the temple to Athena in Metapontum, a place where, according to one legend,
Pythagoras fled as he was pursued by his enemies and where he took refuge until he died in a
siege, I recalled a book on Pythagoras I read with great admiration as a graduate student, Walter
Burkert’s Lore and Wisdom in Ancient Pythagoreanism. Burkert had argued that the evidence con-
necting Pythagoras with the famous theorem that bears his name was too late and unsecure, and
that Pythagoras probably had nothing to do with mathematics at all. That book consolidated an
avalanche of consensus among scholars discrediting “Pythagoras the mathematician.” I wondered
if that judgment merited a review, since my research on ancient architecture and building tech-
nologies supplied so many contemporaneous examples of applied geometry. Might Burkert have
gotten it wrong? And as I started to look into this matter, I tried to recall what the Pythagorean
ix