Euclid - the creation of mathematics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1999
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Einheitssacht. des kommentierten Werkes: Elementa |
| Beschreibung: | XVI, 343 S. Ill., graph. Darst. |
| ISBN: | 0387984232 |
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Datensatz im Suchindex
| _version_ | 1806962967825088512 |
|---|---|
| adam_text |
Contents
Preface v
Notes to the Reader xv
1 General Historical Remarks 1
2 The Contents of the Elements 3
2.1 Book I: Foundations of Plane Geometry 3
2.2 Book II: The Geometry of Rectangles 4
2.3 Book III: The Geometry of the Circle 5
2.4 Book IV: Regular Polygons in Circles 5
2.5 Book V: The General Theory of Magnitudes
in Proportion 5
2.6 Book VI: The Plane Geometry of Similar Figures . . . 6
2.7 Book VII: Basic Arithmetic 7
2.8 Book VIII: Numbers in Continued Proportion 7
2.9 Book IX: Numbers in Continued Proportion; the
Theory of Even and Odd Numbers, Perfect Numbers 8
2.10 Book X: Incommensurable Line Segments 8
2.11 Book XI: Foundations of Solid Geometry 9
ix
2.12 Book XII: Areas and Volumes; Eudoxus's Method
of Exhaustion 9
2.13 Book XIII: The Platonic Solids 10
3 The Origin of Mathematics 1: The Testimony of
Eudemus 11
Time Table 16
4 Euclid Book I 17
4.1 The Overall Composition of Book I 17
4.2 Definitions and Axioms 18
4.3 Book I, Part A: Foundations 21
4.4 Book I, Part B: The Theory of Parallels 31
4.5 Book I, Part C: Parallelograms and Their Areas . 36
4.6 Book I, Part D: The Theorem of Pythagoras 42
5 The Origin of Mathematics 2: Parallels and Axioms 47
6 The Origin of Mathematics 3: Pythagoras of Samos 51
7 Euclid Book II 61
8 The Origin of Mathematics 4: Squaring the Circle 73
9 Euclid Book III 79
9.1 The Overall Composition of Book III 79
9.2 The Definitions of Book III 80
9.3 Book III, Part A: Chords in Circles, Circles
Intersecting or Touching Each Other 81
9.4 Book III, Part B: Tangents 82
9.5 Book III, Part C\\ Angles in Segments of Circles . 84
9.6 Book III, Part C2: Chords, Arcs, and Angles 86
9.7 Book III, Part C3: More About Angles in Circles . 86
9.8 Book III, Part D: Intersecting Chords, Secants,
and Tangents 88
10 The Origin of Mathematics 5: Problems and Theories 93
11 Euclid Book IV 97
11.1 The Contents of Book IV 97
11.2 The Regular Pentagon 99
11.3 Speculations About the Pentagon 102
12 The Origin of Mathematics 6: The Birth of Rigor 109
13 The Origin of Mathematics 7: Polygons After Euclid 113
13.1 What We Missed in Book IV 113
13.2 What Euclid Knew 114
13.3 What Archimedes Did 115
13.4 What Gauss Proved 116
13.5 How Gauss Did It 117
13.6 The Moral of the Story 119
13.7 What Plotinus Has to Say About All This 120
14 Euclid Book V 121
14.1 Proportions Outside of Mathematics 121
14.2 General Remarks About Book V 123
14.3 Proportions in a Modern Version 124
14.4 The Definitions of Book V 127
14.5 The Propositions of Book V 129
15 Euclid Book VI 135
15.1 The Overall Composition of Book VI 135
15.2 The Basis of Similarity Geometry 136
15.3 The Basic Theorems of Similarity Geometry . 138
15.4 Book VI, Part C: Proportions and Areas (Products) 141
15.5 Book VI, Part D: Similar Rectilinear Figures 144
15.6 Book VI, Part E: The Application of Areas 146
16 The Origin of Mathematics 8: Be Wise, Generalize 151
16.1 Extending a Result 152
16.2 Weakened Hypotheses 152
16.3 Abstraction 153
16.4 The Case of Pythagoras's Theorem 153
16.5 The Generalization of Ptolemy 155
16.6 Marcel Proust on Abstraction 159
17 Euclid Book VII 161
17.1 The Historical Background 161
17.2 The Overall Composition of Book VII 163
17.3 Definitions 163
17.4 Book VII, Part A: The Euclidean Algorithm 165
17.5 Book VII, Parts B and C: Proportion for Numbers . . 167
17.6 Book VII, Part D: Proportions and Products 171
17.7 Book VII, Part E: The Greatest Common Divisor and
Prime Divisors 176
17.8 Book VII, Part F: The Least Common Multiple . 182
18 The Origin of Mathematics 9: Nicomachus and
Diophantus 183
18.1 Nicomachus: Introduction to Arithmetic 183
18.2 Diophantus: The Arithmetica 188
19 Euclid Book VIII 193
19.1 The Overall Composition of Books VIII and IX . 193
19.2 Book VIII, Part A: Numbers in
Continued Proportion 194
19.3 Book VIII, Part B: The Geometry of Numbers . . 196
20 The Origin of Mathematics 10: Tools and Theorems 203
21 Euclid Book IX 207
21.1 Book IX, Parts B, C, D, and E: More About Numbers
in Continued Proportion 207
21.2 Book IX, Part F: The Number of Primes 209
21.3 Book IX, Part G: Odd and Even Numbers, and
Perfect Numbers 210
22 The Origin of Mathematics 11: Math Is Beautiful 213
23 Euclid Book X 223
23.1 Commensurability and Its Relation to
Other Notions 227
24 The Origin of Mathematics 12: Incommensurability
and Irrationality 229
25 Euclid Book XI 255
25.1 The Overall Composition of Book XI 255
25.2 The Definitions of Book XI 256
25.3 Foundations of Solid Geometry 257
25.4 The Affinities of Books I and XI 258
25.5 The Duplication of the Cube 263
26 The Origin of Mathematics 13: The Role of
Definitions 267
27 Euclid Book XII 271
27.1 The Overall Composition of Book XII 271
27.2 The Circle 271
27.3 The Pyramid 274
27.4 Cylinders, Cones, and Spheres 278
28 The Origin of Mathematics 14: The Taming of the
Infinite 279
29 Euclid Book XIII 283
29.1 The Overall Composition of Book XIII 283
29.2 Division in Extreme and Mean Ratio and
the Pentagon 284
29.3 About the Decagon 286
29.4 The Regular Solids 292
29.5 The Classification of the Regular Solids 296
30 The Origin of Mathematics 15: Symmetry Through
the Ages 299
30.1 Nature 300
30.2 Art 300
30.3 Philosophy 305
30.4 Mathematics 308
31 The Origin of Mathematics 16: The Origin of the
Elements 313
Notes 317
Bibliography 329
Index 341 |
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| indexdate | 2024-08-10T01:42:52Z |
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| isbn | 0387984232 |
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| spellingShingle | Artmann, Benno 1933-2010 Euclid - the creation of mathematics Euclide <ca 3e siècle av. J.-C> / Éléments Euclid Euclid Elements Euclides ca. v3. Jh. (DE-588)118638955 gnd Euclides ca. v3. Jh. Elementa (DE-588)4266253-9 gnd Euclide, Éléments d' ram Griekse oudheid gtt Mathématiques grecques Mathématiques grecques ram Wiskunde gtt Mathematik Mathematics, Greek Mathematik (DE-588)4037944-9 gnd |
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| title | Euclid - the creation of mathematics |
| title_auth | Euclid - the creation of mathematics |
| title_exact_search | Euclid - the creation of mathematics |
| title_full | Euclid - the creation of mathematics Benno Artmann |
| title_fullStr | Euclid - the creation of mathematics Benno Artmann |
| title_full_unstemmed | Euclid - the creation of mathematics Benno Artmann |
| title_short | Euclid - the creation of mathematics |
| title_sort | euclid the creation of mathematics |
| topic | Euclide <ca 3e siècle av. J.-C> / Éléments Euclid Euclid Elements Euclides ca. v3. Jh. (DE-588)118638955 gnd Euclides ca. v3. Jh. Elementa (DE-588)4266253-9 gnd Euclide, Éléments d' ram Griekse oudheid gtt Mathématiques grecques Mathématiques grecques ram Wiskunde gtt Mathematik Mathematics, Greek Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Euclide <ca 3e siècle av. J.-C> / Éléments Euclid Euclid Elements Euclides ca. v3. Jh. Euclides ca. v3. Jh. Elementa Euclide, Éléments d' Griekse oudheid Mathématiques grecques Wiskunde Mathematik Mathematics, Greek Griechenland (Altertum) |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008558000&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT artmannbenno euclidthecreationofmathematics AT euclides euclidthecreationofmathematics |