The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles (SUNY series in Ancient Greek Philosophy)

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. Contents 8 Preface 10 A 10 B 13 C 13 Acknowledgments 16 Introduction Metaphysics, Geometry, and the Problems with Diagrams 18 A. The Missed Connection between the Origins of Philosophy-Science and Geometry: Metaphysics and Geometrical Diagrams 18 B. The Problems Concerning Geometrical Diagrams 21 C. Diagrams and Geometric Algebra: Babylonian Mathematics 24 D. Diagrams and Ancient Egyptian Mathematics: What Geometrical Knowledge Could Thales have Learned in Egypt? 29 E. Thales’s Advance in Diagrams Beyond Egyptian Geometry 42 F. The Earliest Geometrical Diagrams Were Practical: The Archaic Evidence for Greek Geometrical Diagrams and Lettered Diagrams 49 G. Summary 58 Chapter 1 The Pythagorean Theorem: Euclid I.47 and VI.31 62 A. Euclid: The Pythagorean Theorem I.47 63 (i) The Pythagorean theorem of Euclid I.47 (following Heath): 63 (ii) Reflections on the strategies of Euclid I.47: 65 (iii) The geometrical intuitions: the sequence of ideas that are connected in the proof: 71 B. The “Enlargement” of the Pythagorean Theorem: Euclid VI.31 83 (i) The Pythagorean theorem of Euclid VI.31 (following Heath): 83 (ii) Reflections on the strategies of Euclid VI.31: 86 (iii) The geometrical Intuitions—the sequence of ideas that are connected in the proof: 86 C. Ratio, Proportion, and the Mean Proportional (μέση ἀνάλογον) 87 D. Arithmetic and Geometric Means 89 E. Overview and Summary: The Metaphysics of the Pythagorean Theorem 98 Chapter 2 Thales and Geometry: Egypt, Miletus, and Beyond 108 A. Thales: Geometry in the Big Picture 109 B. What Geometry Could Thales Have Learned in Egypt? 114 B.1 Thales’s Measurement of the Height of a Pyramid 114 Technique 1: When the Shadow Length Was Equal to Its Height 114 B.2 Thales’s Measurement of the Height of a Pyramid 124 Technique 2: When the Shadow Length Was NOT Equal to Its Height 124 C. Thales’ Lines of Thought to the Hypotenuse Theorem 133 Chapter 3 Pythagoras and the Famous Theorems 152 A. The Problems of Connecting Pythagoras with the Famous Theorem 152 B. Hippocrates and the Squaring of the Lunes 154 C. Hippasus and the Proof of Incommensurability 158 D. Lines, Shapes, and Numbers: Figurate Numbers 165 E. Line Lengths, Numbers, Musical Intervals, Microcosmic-Macrocosmic Arguments, and the Harmony of the Circles 170 F. Pythagoras and the Theorem: Geometry and the Tunnel of Eupalinos on Samos 174 G. Pythagoras, the Hypotenuse Theorem, and the μέση ἀνάλογος (Mean Proportional) 185 H. The “Other” Proof of the Mean Proportional: The Pythagoreans and Euclid Book II 199 I. Pythagoras’s Other Theorem: The Application of Areas 206 I.1 The Application of Areas Theorems at Euclid I.42, 44, and 45 206 I.2 The Application of Areas Theorems in Euclid VI.25, 28, 29 by Ratios and Proportions 210 J. Pythagoras’s Other Theorem in the Bigger Metaphysical Picture: Plato’s Timaeus 53Cff 212 K. Pythagoras and the Regular Solids: Building the Elements and the Cosmos Out of Right Triangles 215 K.1 The Role of the Cosmic Figures in the Big Picture: Proclus’s Insight into the Metaphysical Purpose of Euclid 215 K.2 Did Pythagoras Discover the Cosmic Figures? 218 Proposition 21 221 Remark 223 K.3 Pythagoras’s Regular Solids and Plato’s Timaeous: The Reduction of the Elements to Right Triangles, the Construction of the Cosmos out of Right Triangles 227 Chapter 4 Epilogue: From the Pythagorean Theorem to the Construction of the Cosmos Out of Right Triangles 230 Notes 258 Bibliography 280 Image Credits 288 Index 290 Greek Terms 299 "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."--Publisher's description Metaphysics, Geometry, And The Problems With Diagrams -- The Pythagorean Theorem: Euclid I.47 And Vi. 31 -- Thales And Geometry: Egypt, Miletus, And Beyond -- Pythagoras And The Famous Theorems -- From The Pythagorean Theorem To The Construction Of The Cosmos Out Of Right Triangles. Robert Hahn. Includes Bibliographical References And Index.