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The Metaphysics of the Pythagorean Theorem

Robert Hahn
SUNY Press
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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

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