# Book:David Fowler/The Mathematics of Plato's Academy/Second Edition

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## David Fowler:

## David Fowler: *The Mathematics of Plato's Academy: A New Reconstruction (2nd Edition)*

Published $\text {1999}$, **Clarendon Press Oxford**

- ISBN 0 19 850258 3

### Contents

- PREFACE TO THE SECOND EDITION
- PREFACE TO THE FIRST EDITION

- ACKNOWLEDGEMENTS

- List of plates

- Note on the transcriptions of papyri

- PART ONE: INTERPRETATIONS

- 1 THE PROPOSAL

- 1.1 Socrates meets Meno’s slaveboy

- 1.2 The characteristics of early Greek mathematics

- (a) Arithmetised mathematics
- (b) Non-arithmetised geometry
- (c) Numbers and parts: the arithmoi and more
- (d) Ratio (logos) and proportion (analogon)
- (e) The language of Greek mathematics

- 1.3 Socrates meets the slaveboy again

- 1 4 Notes and references

- 2 ANTHYPHAIRETIC RATIO THEORY

- 2.1 Introduction

- 2.2 Some anthyphairetic calculations
- (a) The diagonal and side
- (b) The circumference and diameter
- (c) The surface and section

- 2.2 Some anthyphairetic calculations

- 2.3 Anthyphairetic algorithms
- (a) The Parmenides proposition
- (b) An algorithm for calculating anthyphairese
- (c) An algorithm for calculating convergents

- 2.3 Anthyphairetic algorithms

- 2.4 Further anthyphairetic calculations
- (a) Eratosthenes’ ratio for the obliquity of the ecliptic
- (b) The Metonic cycle
- (c) Aristarchus’ reduction of ratios
- (d) Archimedes’ calculation of circumference to diameter
- (e) Pell’s equation
- (f) The alternative interpretation of Archimedes’ Cattle Problem

- 2.4 Further anthyphairetic calculations

- 2.5 Notes and references

- 3 ELEMENTS II: THE DIMENSION OF SQUARES

- 3.1 Introduction

- 3.2 Book II of the Elements

- 3.3 The hypotheses

- 3.4 The first attempt: The method of gnomons

- 3.5 The second attempt: Synthesising ratios
- (a) Introduction
- (b) The extreme and mean ratio
- (c) The nth order extreme and mean ratio
- (d) Elements XIII, 1-5
- (e) Further generalisations

- 3.5 The second attempt: Synthesising ratios

- 3.6 The third attempt: Generalised sides and diagonals
- (a) The method
- (b) Historical observations

- 3.6 The third attempt: Generalised sides and diagonals

- 3.7 Summary

- 3.8 Notes and references

- 4 PLATO’S MATHEMATICS CURRICULUM IN REPUBLIC VII

- 4.1 Plato as mathematician

- 4.2 Arithmetike te kai logistike

- 4.3 Plane and solid geometry

- 4.4 Academic astronomy
- (a) Introduction
- (b) The slaveboy meets Eudoxus
- (c) Egyptian and early Greek astronomy

- 4.4 Academic astronomy

- 4.5 Academic music theory
- (a) Introduction
- (b) Archytas meets the slaveboy
- (c) Compounding ratios
- (d) The Sectio Canonis
- (e) Further problems

- 4.5 Academic music theory

- 4.6 Appendix: The words logistike and logismos in Plato, Archytas, Aristotle, and the pre-Socratic philosophers
- (a) Plato
- (b) Archytas
- (c) Aristotle
- (d) Pre-Socratic philosophers

- 4.6 Appendix: The words logistike and logismos in Plato, Archytas, Aristotle, and the pre-Socratic philosophers

- 5 ELEMENTS IV, X, AND XIII: THE CIRCUMDIAMETER AND SIDE

- 5.1 The circumdiameter and side, and other examples
- (a) The problem
- (b) The pentagon
- (c) The extreme and mean ratio
- (d) Surd quantities
- (e) Anthyphairetic considerations

- 5.1 The circumdiameter and side, and other examples

- 5.2 Elements X: A classification of some incommensurable lines
- (a) Introduction
- (b) Preliminary comments
- (c) Commensurable, incommensurable, and expressible lines and areas
- (d) Interlude: surd numbers and alogoi magnitudes
- (e) The classification of Book X, and its use in Book XIII
- (f) Euclid’s presentation of the classification

- 5.2 Elements X: A classification of some incommensurable lines

- 5.3 The scope and motivation of Book X

- 5.4 Appendix: The words alogos and (ar)rhetos in Plato, Aristotle, and the pre-Socratic philosophers

- Notes

- Part Two: Evidence

- 6 the Nature of our evidence

- 6.1 A FEQMETPH TOE MHAEIE EIEITQ

- 6.2 Early written evidence

- 6.3 The introduction of minuscule script

- 7 NUMBERS AND FRACTIONS

- 7. 1 Introduction
- (a) Numerals
- (b) Simple and compound parts
- (c) P. Hib. i 27, a parapegma
- (d) O. Bodl. ii 1847, a land survey ostracon

- 7. 1 Introduction

- 7.2 Tables and ready reckoners
- (a) Division tables
- (b) Multiplication and addition tables
- (c) Tables of squares

- 7.2 Tables and ready reckoners

- 7.3 A selection of texts
- (a) Archimedes’ Measurement of a Circle
- (b) Aristarchus’ On the Sizes and Distances of the Sun and Moon
- (c) P. Lond. ii 265 (p. 257)
- (d) M.P.E.R., N.S. i 1
- (e) Demotic mathematical papyri

- 7.3 A selection of texts

- 7.4 Conclusions and some consequences
- (a) Synthesis
- (b) The slaveboy meets an accountant

- 7.4 Conclusions and some consequences

- 7.5 Appendix: A catalogue of published tables
- (a) Division tables
- (b) Multiplication and addition tables
- (c) Tables of squares

- 7.5 Appendix: A catalogue of published tables

- PART THREE: LATER DEVELOPMENTS

- 8 LATER INTERPRETATIONS

- 8.1 Egyptian land measurement as the origin of Greek geometry?

- 8.2 Vewj/j-constructions in Greek geometry

- 8.3 The discovery and role of the phenomenon of incommensurability
- (a) The story
- (b) The evidence
- (c) Discussion of the evidence

- 8.3 The discovery and role of the phenomenon of incommensurability

- 9 CONTINUED FRACTIONS

- 9.1 The basic theory
- (a) Continued fractions, convergents, and approximation
- (b) The Parmenides proposition and algorithm
- (c) The quadratic theory
- (d) Analytic properties
- (e) Lagrange and the solution of equations

- 9.1 The basic theory

- 9.2 Gauss and continued fractions
- (a) Introduction
- (b) Continued fractions and the hypergeometric series
- (c) Continued fractions and probability theory
- (d) Gauss’s number theory
- (e) Gauss’s legacy in number theory

- 9.2 Gauss and continued fractions

- 9.3 Two recent developments
- (a) Continued fraction arithmetic
- (b) Higher dimensional algorithms

- 9.3 Two recent developments

- 10 APPENDIX: NEW MATERIAL ADDED TO THE SECOND EDITION

- 10.1 A new introduction: The story of the discovery of incommensurability
- (a) The standard story
- (b) Some general remarks about our evidence
- (c) Our evidence concerning incommensurability and associated topics
- (d) The supposed effects of the discovery of incommensurability
- (e) Objections to some proposed interpretations
- (f) Our difficulties in defining ratio
- (g) Some examples of anthyphairetic geometry

- 10.1 A new introduction: The story of the discovery of incommensurability

- 10.2 Ratio as the equivalence class of proportionality

- 10.3 Further reflections on the method of gnomons, the problem of the dimension of squares, and Theodorus’ lesson in Theaetetus, 147c-158b
- (a) Introduction
- (b) From heuristic to deduction via algorithms
- (c) A compendium of examples
- (d) The geometry lesson
- (e) The overall structure of the Theaetetus

- 10.3 Further reflections on the method of gnomons, the problem of the dimension of squares, and Theodorus’ lesson in Theaetetus, 147c-158b

- 10.4 Elements
- (a) Pre-Euclidean Elements
- (b) Lexica, dictionaries, and the scholarly literature
- (c) Proclus
- (d) The logical structure of Euclid’s Elements
- (e) The Euclidean proposition
- (f) Pre-Euclidean evidence on elements and mathematical style
- (g) When, where, and why was the Euclidean style introduced, and when were mathematics books first called Elements?

- 10.4 Elements

- 10.5 ... but why is there no evidence for these ratio theories?

- 11 EPILOGUE: A BRIEF INTELLECTUAL AUTOBIOGRAPHY

- Bibliography

- Index of Cited Passages

- Index of Names

- General Index