The Mathematics of Plato's Academy: A New Reconstruction

Contents......Page all_25445_to_00478.cpc0018.djvu
Note on the transcriptions of papyri......Page all_25445_to_00478.cpc0024.djvu
Part One: Interpretations......Page all_25445_to_00478.cpc0026.djvu
1.1 Socrates meets Meno\'s slaveboy......Page all_25445_to_00478.cpc0028.djvu
a) Arithmetised mathematics......Page all_25445_to_00478.cpc0045.djvu
b) Non-arithmetised geometry......Page all_25445_to_00478.cpc0047.djvu
c) Numbers and parts: the arithmoi and mere......Page all_25445_to_00478.cpc0050.djvu
d) Ratio (logos) and proportion (analogon)......Page all_25445_to_00478.cpc0052.djvu
e) The language of Greek mathematics......Page all_25445_to_00478.cpc0057.djvu
1.3 Socrates meets the slaveboy again......Page all_25445_to_00478.cpc0060.djvu
1.4 Notes and references......Page all_25445_to_00478.cpc0064.djvu
2.1 Introduction......Page all_25445_to_00478.cpc0067.djvu
a) The diagonal and side......Page all_25445_to_00478.cpc0069.djvu
b) The circumference and diameter......Page all_25445_to_00478.cpc0072.djvu
a) The Parmenides proposition......Page all_25445_to_00478.cpc0078.djvu
b) An algorithm for calculating anthyphaireses......Page all_25445_to_00478.cpc0080.djvu
c) An algorithm for calculating convergents......Page all_25445_to_00478.cpc0084.djvu
a) Eratosthenes\' ratio for the obliquity of the ecliptic......Page all_25445_to_00478.cpc0087.djvu
b) The Metonic cycle......Page all_25445_to_00478.cpc0088.djvu
c) Aristarchus\' reduction of ratios......Page all_25445_to_00478.cpc0089.djvu
d) Archimedes\' calculation of circumference to diameter......Page all_25445_to_00478.cpc0090.djvu
e) Pell\'s equation......Page all_25445_to_00478.cpc0092.djvu
f) The alternative interpretation of Archimedes\' Cattle Problem......Page all_25445_to_00478.cpc0095.djvu
2.5 Notes and references......Page all_25445_to_00478.cpc0097.djvu
3.1 Introduction......Page all_25445_to_00478.cpc0102.djvu
3.2 Book II of the Elements......Page all_25445_to_00478.cpc0103.djvu
3.3 The hypotheses......Page all_25445_to_00478.cpc0108.djvu
3.4 The first attempt: The method of gnomons......Page all_25445_to_00478.cpc0111.djvu
b) The extreme and mean ratio......Page all_25445_to_00478.cpc0121.djvu
c) The nth order extreme and mean ratio......Page all_25445_to_00478.cpc0122.djvu
d) Elements XIII, 1-5......Page all_25445_to_00478.cpc0124.djvu
e) Further generalisations......Page all_25445_to_00478.cpc0126.djvu
a) The method......Page all_25445_to_00478.cpc0130.djvu
b) Historical observations......Page all_25445_to_00478.cpc0134.djvu
3.7 Summary......Page all_25445_to_00478.cpc0138.djvu
3.8 Notes and references......Page all_25445_to_00478.cpc0139.djvu
4.1 Plato as mathematician......Page all_25445_to_00478.cpc0140.djvu
4.2 Arithmetike te kai logistike......Page all_25445_to_00478.cpc0142.djvu
4.3 Plane and solid geometry......Page all_25445_to_00478.cpc0150.djvu
a) Introduction......Page all_25445_to_00478.cpc0154.djvu
b) The slaveboy meets Eudoxus......Page all_25445_to_00478.cpc0155.djvu
c) Egyptian and early Greek astronomy......Page all_25445_to_00478.cpc0158.djvu
a) Introduction......Page all_25445_to_00478.cpc0162.djvu
b) Archytas meets the slaveboy......Page all_25445_to_00478.cpc0163.djvu
c) Compounding ratios......Page all_25445_to_00478.cpc0170.djvu
d) The Sectio Canonis......Page all_25445_to_00478.cpc0175.djvu
e) Further problems......Page all_25445_to_00478.cpc0183.djvu
a) Plato......Page all_25445_to_00478.cpc0185.djvu
b) Archytas......Page all_25445_to_00478.cpc0186.djvu
d) Pre-Socratic philosophers......Page all_25445_to_00478.cpc0187.djvu
b) The pentagon......Page all_25445_to_00478.cpc0189.djvu
d) Surd quantities......Page all_25445_to_00478.cpc0193.djvu
e) Anthyphairetic considerations......Page all_25445_to_00478.cpc0194.djvu
a) Introduction......Page all_25445_to_00478.cpc0198.djvu
b) Preliminary comments......Page all_25445_to_00478.cpc0201.djvu
c) Commensurable, incommensurable, and expressible lines and areas......Page all_25445_to_00478.cpc0202.djvu
d) Interlude: surd numbers and alogoi magnitudes......Page all_25445_to_00478.cpc0208.djvu
e) The classification of Book X, and its use in Book XIII......Page all_25445_to_00478.cpc0210.djvu
f) Euclid\'s presentation of the classification......Page all_25445_to_00478.cpc0224.djvu
5.3 The scope and motivation of Book X......Page all_25445_to_00478.cpc0226.djvu
5.4 Appendix: The words alogos and (ar)rhetos in Plato, Aristotle, and the pre-Socratic philosophers......Page all_25445_to_00478.cpc0228.djvu
Notes......Page all_25445_to_00478.cpc0230.djvu
Part Two: Evidence......Page all_25445_to_00478.cpc0234.djvu
6.1 Ageometretos Medeis Eisito......Page all_25445_to_00478.cpc0236.djvu
6.2 Early written evidence......Page all_25445_to_00478.cpc0241.djvu
6.3 The introduction of minuscule script......Page all_25445_to_00478.cpc0254.djvu
a) Numerals......Page all_25445_to_00478.cpc0259.djvu
b) Simple and compound parts......Page all_25445_to_00478.cpc0264.djvu
c) P. Hib. i 27, a parapegma......Page all_25445_to_00478.cpc0266.djvu
d) O. Bodl. ii 1847, a land survey ostracon......Page all_25445_to_00478.cpc0268.djvu
a) Division tables......Page all_25445_to_00478.cpc0271.djvu
b) Multiplication and addition tables......Page all_25445_to_00478.cpc0275.djvu
c) Tables of squares......Page all_25445_to_00478.cpc0276.djvu
a) Archimedes\' Measurement of a Circle......Page all_25445_to_00478.cpc0277.djvu
b) Aristarchus\' On the Sizes and Distances of the Sun and Moon......Page all_25445_to_00478.cpc0283.djvu
c) P. Lond. ii 265 (p. 257)......Page all_25445_to_00478.cpc0285.djvu
d) M.P.E.R., N.S. i 1......Page all_25445_to_00478.cpc0290.djvu
e) Demotic mathematical papyri......Page all_25445_to_00478.cpc0295.djvu
a) Synthesis......Page all_25445_to_00478.cpc0299.djvu
b) The slaveboy meets an accountant......Page all_25445_to_00478.cpc0303.djvu
7.5 Appendix: A catalogue of published tables......Page all_25445_to_00478.cpc0305.djvu
a) Division tables......Page all_25445_to_00478.cpc0306.djvu
b) Multiplication and addition tables......Page all_25445_to_00478.cpc0311.djvu
c) Tables of squares......Page all_25445_to_00478.cpc0313.djvu
Part Three: Later Developments......Page all_25445_to_00478.cpc0314.djvu
8.1 Egyptian land measurement as the origin of Greek geometry?......Page all_25445_to_00478.cpc0316.djvu
8.2 Neusis-constructions in Greek geometry......Page all_25445_to_00478.cpc0320.djvu
b) The evidence......Page all_25445_to_00478.cpc0326.djvu
c) Discussion of the evidence......Page all_25445_to_00478.cpc0334.djvu
a) Continued fractions, convergents, and approximation......Page all_25445_to_00478.cpc0341.djvu
b) The Parmenides proposition and algorithm......Page all_25445_to_00478.cpc0350.djvu
c) The quadratic theory......Page all_25445_to_00478.cpc0359.djvu
d) Analytic properties......Page all_25445_to_00478.cpc0366.djvu
e) Lagrange and the solution of equations......Page all_25445_to_00478.cpc0368.djvu
b) Continued fractions and the hypergeometric series......Page all_25445_to_00478.cpc0371.djvu
c) Continued fractions and probability theory......Page all_25445_to_00478.cpc0375.djvu
d) Gauss\'s number theory......Page all_25445_to_00478.cpc0377.djvu
e) Gauss\'s legacy in number theory......Page all_25445_to_00478.cpc0380.djvu
a) Continued fraction arithmetic......Page all_25445_to_00478.cpc0383.djvu
b) Higher dimensional algorithms......Page all_25445_to_00478.cpc0389.djvu
b) Some general remarks about our evidence......Page all_25445_to_00478.cpc0393.djvu
c) Our evidence concerning incommensurability and associated topics......Page all_25445_to_00478.cpc0394.djvu
d) The supposed effects of the discovery of incommensurability......Page all_25445_to_00478.cpc0399.djvu
e) Objections to some proposed interpretations......Page all_25445_to_00478.cpc0400.djvu
f) Our difficulties in defining ratio......Page all_25445_to_00478.cpc0402.djvu
g) Some examples of anthyphairetic geometry......Page all_25445_to_00478.cpc0404.djvu
10.2 Ratio as the equivalence class of proportionality......Page all_25445_to_00478.cpc0406.djvu
a) Introduction......Page all_25445_to_00478.cpc0408.djvu
b) From heuristic to deduction via algorithms......Page all_25445_to_00478.cpc0409.djvu
c) A compendium of examples......Page all_25445_to_00478.cpc0411.djvu
d) The geometry lesson......Page all_25445_to_00478.cpc0415.djvu
e) The overall structure of the Theaetetus......Page all_25445_to_00478.cpc0417.djvu
b) Lexica, dictionaries, and the scholarly literature......Page all_25445_to_00478.cpc0418.djvu
c) Proclus......Page all_25445_to_00478.cpc0419.djvu
e) The Euclidean proposition......Page all_25445_to_00478.cpc0422.djvu
f) Pre-Euclidean evidence on elements and mathematical style......Page all_25445_to_00478.cpc0423.djvu
g) When, where, and why was the Euclidean style introduced, and when were mathematics books first called Elements?......Page all_25445_to_00478.cpc0429.djvu
10.5 ... but why is there no evidence for these ratio theories?......Page all_25445_to_00478.cpc0431.djvu
11 Epilogue: A Brief Intellectual Autobiography......Page all_25445_to_00478.cpc0439.djvu
Bibliography......Page all_25445_to_00478.cpc0446.djvu
Index of Cited Passages......Page all_25445_to_00478.cpc0468.djvu
Index of Names......Page all_25445_to_00478.cpc0471.djvu
General Index......Page all_25445_to_00478.cpc0476.djvu