A mathematical history of the golden number:
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| Previous Title: | A mathematical history of the golden number |
|---|---|
| Main Author: | |
| Format: | Book |
| Language: | English |
| Published: |
Mineola, New York
Dover Publications, Inc.
[1998]
|
| Edition: | Unabridged republication |
| Series: | Dover books in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 195 Seiten Diagramme |
| ISBN: | 0486400077 |
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| 240 | 1 | 0 | |a A mathematical history of division in extreme and mean ratio |
| 245 | 1 | 0 | |a A mathematical history of the golden number |c Roger Herz-Fischler |
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Record in the Search Index
| _version_ | 1804142005114109952 |
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| adam_text | PREFACE TO THE DOVER EDITION XV
FOREWORD XVII
A GUIDE FOR READERS XIX
A. INTERNAL ORGANIZATION XIX
B. BIBLIOGRAPHICAL DETAILS XIX
C. ABBREVIATIONS XX
D. SYMBOLS XX
E. DATES XXI
F. QUOTATIONS FROM PRIMARY SOURCES XXI
INTRODUCTION 1
CHAPTER I. THE EUCLIDEAN TEXT 6
SECTION 1. THE TEXT . 6
SECTION 2. AN EXAMINATION OF THE EUCLIDEAN TEXT 25
A. PRELIMINARY OBSERVATIONS 26
B. A PROPOSAL CONCERNING THE ORIGIN OF DEMR 27
C. THEOREM XIII,8 31
D. THEOREMS XIII,1-5 33
E. STAGES IN THE DEVELOPMENT OF DEMR IN BOOK XIII 33
CHAPTER II. MATHEMATICAL TOPICS 35
SECTION 3. COMPLEMENTS AND THE GNOMON 35
SECTION 4. TRANSFORMATION OF AREAS 36
SECTION 5. GEOMETRICAL ALGEBRA, APPLICATION OF AREAS, AND SOLUTIONS OF
EQUATIONS 37
A. GEOMETRICAL ALGEBRA*LEVEL 1 37
B. GEOMETRICAL ALGEBRA*LEVEL 2 38
C. APPLICATION OF AREAS*LEVEL 3 40
D. HISTORICAL REFERENCES 41
E. SETTING OUT THE DEBATE 42
F. OTHER INTERPRETATIONS IN TERMS OF EQUATIONS 42
G. PROBLEMS IN INTERPRETATION 43
H. DIVISION OF FIGURES 43
/. THEOREMS VI,28,29 VS 11,5,6 44
J. EUCLID S DATA 44
IX
X A MATHEMATICAL HISTORY OF THE GOLDEN NUMBER
K. THEOREM 11,11 46
L. 11,11*APPLICATION OF AREAS, VARIOUS VIEWS 47
I. SZABO 47
II. JUNGE 47
HI. VALABREGA-GIBELLATO 48
SECTION 6. SIDE AND DIAGONAL NUMBERS 48
SECTION 7. INCOMMENSURABILITY 49
SECTION 8. THE EUCLIDEAN ALGORITHM, ANTHYPHAIRESIS, AND CONTINUED
FRACTIONS 50
CHAPTER III. EXAMPLES OF THE PENTAGON, PENTAGRAM,
AND DODECAHEDRON BEFORE -400 52
SECTION 9. EXAMPLES BEFORE PYTHAGORAS (BEFORE C. -550) 53
A. PREHISTORIC EGYPT 53
B. PREHISTORIC MESOPOTAMIA 53
C. SUMERIAN AND AKKADIAN CUNEIFORM IDEOGRAMS 55
I. FUYE S THEORY 56
D. A BABYLONIAN APPROXIMATION FOR THE AREA OF THE PENTAGON 56
I. STAPLETON S THEORY 57
E. PALESTINE 57
SECTION 10. FROM PYTHAGORAS UNTIL -400 58
A. VASES FROM GREECE AND ITS ITALIAN COLONIES, ETRURIA (ITALY) 58
B. SHIELD DEVICES ON VASES 59
C. COINS 60
D. DODECAHEDRA 61
E. ADDITIONAL MATERIAL 61
CONCLUSIONS 61
CHAPTER IV. THE PYTHAGOREANS 63
I. PYTHAGORAS 64
II. HIPPASUS 64
WJ. HIPPOCRATES OF CHIOS 64
IV. THEODORUS OF CYRENE 64
V. ARCHYTAS 64
SECTION 11. ANCIENT REFERENCES TO THE PYTHAGOREANS 65
A. THE PENTAGRAM AS A SYMBOL OF THE PYTHAGOREANS 65
B. THE PYTHAGOREANS AND THE CONSTRUCTION OF THE DODECAHEDRON 65
C. OTHER REFERENCES TO THE PYTHAGOREANS 67
SECTION 12. THEORIES LINKING DEMR WITH THE PYTHAGOREANS 68
I. THE PENTAGRAM 68
II. SCHOLIA ASSIGNING BOOK IV TO THE PYTHAGOREANS 68
HI. EQUATIONS AND APPLICATION OF AREAS 68
IV. THE DODECAHEDRON 69
V. A MARKED STRAIGHT-EDGE CONSTRUCTION OF THE PENTAGON 69
VI. A GNOMON THEORY 69
VII. ALLMAN S THEORY: THE DISCOVERY OF INCOMMENSURABILITY 70
VIII. FRITZ*JUNGE THEORY: THE DISCOVERY OF INCOMMENSURABILITY 70
IX. HELLER S THEORY: THE DISCOVERY OF DEMR 71
X. NEUENSCHWANDER S ANALYSIS 72
XI. STAPLETON 72
CHAPTER V. MISCELLANEOUS THEORIES 74
TABLE OF CONTENTS XI
SECTION 13. MISCELLANEOUS THEORIES 74
I. MICHEL J4
II. FOWLER: AN ANTHYPHAIRESIS DEVELOPMENT OF DEMR 74
HI. KNORR: ANTHYPHAIRESIS AND DEMR 75
IV. HARD: THEOREM IXJ5 75
SECTION 14. THEOREMS XIII.1-5 76
I. BRETSCHNEIDER 76
II. ALLMAN 76
ILL. MICHEL 76
IV. DIJKSTERHUIS AND VAN DER WAERDEN 76
V. LASSERRE 76
VI. FRITZ 76
VI7. KNORR 76
VIII. HEIBERG 76
IX. HERZ-FISCHLER 76
CHAPTER VI. THE CLASSICAL PERIOD: FROM THEODORUS TO EUCLID 77
SECTION 15. THEODORUS 77
I. KNORR 77
II. MUGLER 78
SECTION 16. PLATO 78
A. PLATO AS A MATHEMATICIAN 78
B. MATHEMATICAL INFLUENCE OF PLATO 79
C. PLATO AND DEMR 81
D. PASSAGES FROM PLATO 82
I. THE DODECAHEDRON IN PHAEDO HOB AND TIMAEUS 55C 82
II. THE DIVIDED LINE IN THE REPUBLIC 509D 84
HI. TIMAEUS 31B 84
IV. HIPPIAS MAJOR 303B 85
SECTION 17. LEODAMAS OF THASOS 86
SECTION 18. THEAETETUS 86
A. THE LIFE OF THEAETETUS 86
B. THE CONTRIBUTIONS OF THEAETETUS 87
/. TANNERY 88
II. ALLMAN 88
/II. SACHS 88
IV. VAN DER WAERDEN 88
V. BULMER-THOMAS 88
VI. WATERHOUSE 89
VII. NEUENSCHWANDER 89
SECTION 19. SPEUSIPPUS 89
SECTION 20. EUDOXUS 90
A. INTERPRETING SECTION 90
I. BRETSCHNEIDER 90
II. TANNERY 91
HI. TROPFKE 91
IV. MICHEL 92
V. GAISER 92
VI. BURKERT 92
VII. FOWLER 93
B. CONTRIBUTIONS OF EUDOXUS TO THE DEVELOPMENT OF DEMR 93
I. BRETSCHNEIDER 93
II. ALLMAN 93
XII . A MATHEMATICAL HISTORY OF THE GOLDEN NUMBER
I II. SACHS 93
IV. VAN DER WAERDEN 93
V. LASSERRE 93
VI. KNORR 93
C. COMMENTARY 93
SECTION 21. EUCLID 95
SECTION 22. SOME VIEWS ON THE HISTORICAL DEVELOPMENT OF DEMR 95
A. A SUMMARY OF VARIOUS THEORIES 95
I. EQUATIONS AND APPLICATION OF AREAS 95
II. INCOMMENSURABILITY 95
III. SIMILAR TRIANGLES DEVELOPMENT BASED ON XIII,8 95
IV. ANTHYPHAIRESIS 95
B. SUMMARY OF MY CONCLUSIONS 95
C. A CHRONOLOGICAL PROPOSAL 96
D. A PROPOSAL CONCERNING A NAME 99
CHAPTER VII. THE POST-EUCLIDEAN GREEK PERIOD (C. -300 TO 350) 100
SECTION 23. ARCHIMEDES 100
A. APPROXIMATIONS TO THE CIRCUMFERENCE OF A CIRCLE 100
B. BROKEN CHORD THEOREM 102
C. TRIGONOMETRY 102
SECTION 24. THE SUPPLEMENT TO THE ELEMENTS 102
A. THE TEXT 102
B. QUESTIONS OF AUTHORSHIP 106
C. CHRONOLOGY 106
SECTION 25. HERO 108
A. APPROXIMATIONS FOR THE AREA OF THE PENTAGON AND DECAGON 108
I. THE AREA OF THE PENTAGRAM 108
II. THE AREA OF THE DECAGON 109
III. THE DIAMETER OF THE CIRCUMSCRIBED CIRCLE OF A PENTAGON 110
IV. COMMENTARIES 110
B. A VARIATION ON 11,11 111
C. THE VOLUMES OF THE ICOSAHEDRON AND DODECAHEDRON 111
I. THE TEXT 111
II. COMMENTARY 112
SECTION 26. ROLEMY 113
A. THE CHORDS OF 36 AND 72 IN ALMAGEST 113
B. CHORD(108)LDIAMETER IN GEOGRAPHY 114
C. TRIGONOMETRY BEFORE PTOLEMY 114
SECTION 27. PAPPUS 115
A. CONSTRUCTION OF THE ICOSAHEDRON AND DODECAHEDRON 115
B. COMPARISON OF VOLUMES 118
CHAPTER VIII. THE ARABIC WORLD, INDIA, AND CHINA 121
SECTION 28. THE ARABIC PERIOD 121
I. AUTHORS CONSULTED 121
II. EQUATIONS 122
A. AL-KHWARIZMI 122
I. ALGEBRA 122
II. PREDECESSORS OF AL-KHWARIZMI 123
TABLE OF CONTENTS XIIL
B. ABU KAMIL 124
I. ON THE PENTAGON AND DECAGON 124
II. ALGEBRA 127
C. ABU L-WAFA 128
D. IBN YUNUS 129
E. AL-BIRUNI 130
I. THE BOOK ON THE DETERMINATION OF CHORDS IN A CIRCLE 130
II. CANON MASUIDIUS 131
SECTION 29. INDIA 131
SECTION 30. CHINA 133
CHAPTER IX. EUROPE: FROM THE MIDDLE AGES THROUGH
THE EIGHTEENTH CENTURY 134
SECTION 31. EUROPE THROUGH THE 16TH CENTURY 134
A. AUTHORS CONSULTED 134
I. THE MIDDLE AGES 134
II. VERSIONS OF THE ELEMENTS AND SCHOLIA 136
HI. ITALY FROM FIBONACCI THROUGH THE RENAISSANCE 136
IV. 16TH CENTURY NON-ITALIAN AUTHORS 137
V. PRE-1600 NUMERICAL APPROXIMATIONS TO DEMR 137
VI. FIXED COMPASS AND STRAIGHT-EDGE CONSTRUCTIONS 137
VII. APPROXIMATE CONSTRUCTIONS OF THE PENTAGON 137
B. FIBONACCI 137
I. PLANAR CALCULATIONS 137
II. VOLUME COMPUTATIONS OF THE DODECAHEDRON AND ICOSAHEDRON 138
III. FIBONACCI AND ABU KAMIL 141
IV. EQUATIONS FROM ABU KAMIL S ALGEBRA 143
V. THE RABBIT PROBLEM, FIBONACCI NUMBERS 144
VI. SUMMARY 144
C. FRANCESCO 144
D. PACCIOLI 149
E. CARDANO 151
F. BOMBELLI 152
G. CANDALLA 155
H. RAMUS 156
/. STEVIN 157
J. PRE-1600 NUMERICAL APPROXIMATIONS TO DEMR 157
I. UNKNOWN ANNOTATOR TO PACCIOLI S EUCLID 157
II. HOLTZMANN 158
HI. MASTLIN 158
K. APPROXIMATE CONSTRUCTIONS OF THE PENTAGON 158
SECTION 32. THE 17TH AND 18TH CENTURIES 159
A. KEPLER 159
I. MAGIRUS*THE RIGHT TRIANGLE WITH PROPORTIONAL SIDES 159
II. FIBONACCI APPROXIMATIONS TO DEMR 160
B. THE FIBONACCI SEQUENCE 161
C. FIXED COMPASS AND COMPASS ONLY CONSTRUCTIONS 162
I. MOHR 162
II. MASCHERONI 162
BY WAY OF A CONCLUSION 163
XIV A MATHEMATICAL HISTORY OF THE GOLDEN NUMBER
APPENDIX I. A PROPORTION BY ANY OTHER NAME : TERMINOLOGY FOR DIVISION
IN EXTREME AND MEAN RATIO THROUGHOUT THE AGES 164
A. EXTREME AND MEAN RATIO 164
B. MIDDLE AND TWO ENDS 166
C. NAMES FOR DEMR 167
APPENDIX II. MIRABLIS... EST POTENTIA... : THE GROWTH OF AN IDEA 171
CORRECTIONS AND ADDITIONS 176
BIBLIOGRAPHY 180
|
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| id | DE-604.BV024975473 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:24:53Z |
| institution | BVB |
| isbn | 0486400077 |
| language | English |
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| owner_facet | DE-11 DE-255 |
| physical | XXII, 195 Seiten Diagramme |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Dover Publications, Inc. |
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| spelling | Herz-Fischler, Roger 1940- Verfasser (DE-588)172136024 aut A mathematical history of division in extreme and mean ratio A mathematical history of the golden number Roger Herz-Fischler Unabridged republication Mineola, New York Dover Publications, Inc. [1998] XXII, 195 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Dover books in mathematics Geschichte <Anfänge-1800> gnd rswk-swf Geschichte Anfänge-1800 gnd rswk-swf Geschichte Ratio and proportion History Goldener Schnitt (DE-588)4021529-5 gnd rswk-swf Goldener Schnitt (DE-588)4021529-5 s Geschichte Anfänge-1800 z DE-604 Geschichte <Anfänge-1800> z 1\p DE-604 Vorangegangen ist A mathematical history of the golden number (DE-604)BV000796243 Digitalisierung ZIKG München application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019642918&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Herz-Fischler, Roger 1940- A mathematical history of the golden number Geschichte Ratio and proportion History Goldener Schnitt (DE-588)4021529-5 gnd |
| subject_GND | (DE-588)4021529-5 |
| title | A mathematical history of the golden number |
| title_alt | A mathematical history of division in extreme and mean ratio |
| title_auth | A mathematical history of the golden number |
| title_exact_search | A mathematical history of the golden number |
| title_full | A mathematical history of the golden number Roger Herz-Fischler |
| title_fullStr | A mathematical history of the golden number Roger Herz-Fischler |
| title_full_unstemmed | A mathematical history of the golden number Roger Herz-Fischler |
| title_old | A mathematical history of the golden number |
| title_short | A mathematical history of the golden number |
| title_sort | a mathematical history of the golden number |
| topic | Geschichte Ratio and proportion History Goldener Schnitt (DE-588)4021529-5 gnd |
| topic_facet | Geschichte Ratio and proportion History Goldener Schnitt |
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