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

## 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
2.3 Anthyphairetic algorithms
(a) The Parmenides proposition
(b) An algorithm for calculating anthyphairese
(c) An algorithm for calculating convergents
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
2.5 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
3.6 The third attempt: Generalised sides and diagonals
(a) The method
(b) Historical observations
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
(a) Introduction
(b) The slaveboy meets Eudoxus
(c) Egyptian and early Greek astronomy
(a) Introduction
(b) Archytas meets the slaveboy
(c) Compounding ratios
(d) The Sectio Canonis
(e) Further problems
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

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
5.2 Elements X: A classification of some incommensurable lines
(a) Introduction
(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
5.3 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
(a) Division tables
(c) Tables of squares
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
7.4 Conclusions and some consequences
(a) Synthesis
(b) The slaveboy meets an accountant
7.5 Appendix: A catalogue of published tables
(a) Division tables
(c) Tables of squares

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

9 CONTINUED FRACTIONS
9.1 The basic theory
(a) Continued fractions, convergents, and approximation
(b) The Parmenides proposition and algorithm
(d) Analytic properties
(e) Lagrange and the solution of equations
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
9.3 Two recent developments
(a) Continued fraction arithmetic
(b) Higher dimensional algorithms

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
10.2 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
10.4 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?
10.5 ... 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