Springer | 1978 | 363 páginas | pdf | 16 Mb
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Table of contents :
Title......Page 1
Contents......Page 2
Chronological Table......Page 4
Introduction......Page 5
1.1 Current views of the theory's development......Page 14
1.2 The concept of 'Dynamis'......Page 16
1.3 The mathematic part of the 'Theaetetus'......Page 18
1.4 The usage and chronology of 'Dynamis'......Page 20
1.5 'Tetragonismos'......Page 21
1.6 The mean proportional......Page 22
1.7 The mathematics lecture delivered by Theodorus......Page 25
1.8. The mathematical discoveries of Plato's Theaethetus......Page 28
1.9 The 'independence' of Theaetetus......Page 31
1.10 A glance at some rival theories......Page 33
1.11 The so called 'Theaetetus problem'......Page 35
1.12 The discovery of incommensurability......Page 40
1.13 The problem of doubling the square......Page 43
1.14 Doubling the square and the mean proportional......Page 46
2.1 Introduction......Page 47
2.2 A survey of the most important terms......Page 49
2.3 Consonances and intervals......Page 52
B. Diastema as interval......Page 53
2.4 The diastema between two numbers......Page 55
2.5 A digression on the theory of music......Page 57
2.6 End points and intervals pictured as straight lines......Page 60
2.7 'Diplasion', 'Hemiolion', 'Epitriton'......Page 62
2.8 The euclidean algorithm......Page 65
2.9 The canon......Page 66
2.10 Arithmetical operations on the canon......Page 68
2.12 'Aναλογια as 'geometric proportion'......Page 70
2.13 'Aνάλογον......Page 72
2.14 The preposition άνά......Page 73
2.15 The elliptic expression άνά λόγον......Page 75
2.16 The subsequent history of ανάλογον as a mathematical term......Page 76
2.17 Out of the canon and musical means......Page 78
2.18 The creation of the mathematical concept of λόγος......Page 81
2.19 A digression on the history of the word λόγος......Page 82
2.20 The appication of the theory of proportions to arithmetic and geometry..Page 83
2.21 The mean proportional in music, arithmetic and geometry......Page 85
2.22 The construction of the mean proportional......Page 86
2.23 Conclusion......Page 88
3.1 'Proof' in Greek mathematics......Page 90
3.2 The proof of incommensurability......Page 97
3.3 The origin of anti-empiricism and indirect proof......Page 106
3.4 Euclid's foundations......Page 108
3.5 Aristotle and foundations of mathematics......Page 111
3.6 'Hypotheseis'......Page 114
3.7 The 'assumptions' in dialectic......Page 116
3.8 How 'hypotheseis' were used......Page 117
3.9 'Hypotheseis' and the method of indirect proof......Page 119
3.10 A question of priority......Page 120
3.11 Zeno, the inventor of dialectic......Page 122
3.12 Plato and the Eleatics......Page 123
3.13 'Hypotheseis' and the foundations of mathematics......Page 124
3.14 The definition of 'Unit'......Page 126
3.15 Arithmetic and the teaching of the Eleatics......Page 128
3.16 The divisibility of numbers......Page 130
3.17 The problem of the 'aithemata'......Page 132
3.18 Euclid's postulates......Page 133
3.19 The construction of oenopides......Page 134
3.20 The first three postulates in the 'Elements'......Page 136
3.21 The 'Koinai Ennoiai'......Page 138
3.22 The word άξιωμα......Page 139
3.23 Plato's όμολογήματα and Euclid's άξιώματα......Page 141
3.24 "The whole is greater than the part"......Page 143
3.25 A complex of axioms......Page 147
3.26 The difference between postulates and axioms......Page 149
3.27 Arithmetic and geometry......Page 150
3.28 The science of space......Page 151
3.29 The foundations of geometry......Page 154
3.30 A reconsideration of some problems relating to early greek mathematics......Page 156
Postscript......Page 163
1. The prevailing view......Page 164
3. Elements of a pythagorean theory about the areas of parallelograms......Page 165
4. How to find a square with the same area as a given rectangle......Page 171
5. Conclusion......Page 174
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