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

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


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


 * 4.5 Academic music theory
 * (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
 * (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


 * 5.3 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


 * 7.2 Tables and ready reckoners
 * (a) Division tables
 * (b) Multiplication and addition 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
 * (b) Multiplication and addition 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
 * (c) The quadratic theory
 * (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