A detailed analysis of the role of work of each author in the context of the mathematics DEMR in the Elements and of the historical implica- of the time and on the transmission of results and con- tions is followed by a discussion of other mathematical cepts. [...] 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. [...] The last chapter of XV XVI A MATHEMATICAL HISTORY OF THE GOLDEN NUMBER that book was going to deal with the various theories of the useful for resolving certain enigmas in the field of art history, shape of the Great Pyramid, in particular those involving the may consult my article: golden number; see [Fischler, 1979b]. [...] The form of the bibliographical entry used depends A second bibliographical feature is the use of the upon the nature of the work. [...] A and B is the same as that of the ratio of A GUIDE FOR READERS XXI the magnitudes C and D. See my com- vth century The fifth century before the first century ments at the beginning of this list and also of the Julian system.
Authors
- Bibliography, etc. Note
- Bibliography: p. 176-191
- Control Number Identifier
- CaOOCEL
- Dewey Decimal Classification Number
- 512.7
- General Note
- Issued as part of the desLibris books collection
- ISBN
- 0889201528 9781459305458
- LCCN
- QA481
- LCCN Item number
- H47 1987eb
- Modifying agency
- CaBNVSL
- Original cataloging agency
- CaOTU
- Physical Description | Extent
- 1 electronic text (xvi, 191 p.)
- Published in
- Canada
- Publisher or Distributor Number
- CaOOCEL
- Rights
- Access restricted to authorized users and institutions
- System Control Number
- (CaBNVSL)slc00200951 (OCoLC)756538581 (CaOOCEL)406304
- System Details Note
- Mode of access: World Wide Web
- Title proper/short title
- Mathematical history of the golden number
- Transcribing agency
- CaOTU
Table of Contents
- TABLE OF CONTENTS 10
- PREFACE TO THE DOVER EDITION 16
- FOREWORD 18
- A GUIDE FOR READERS 20
- A. Internal Organization 20
- B. Bibliographical Details 20
- C. Abbreviations 21
- D. Symbols 21
- E. Dates 22
- F. Quotations from Primary Sources 22
- INTRODUCTION 24
- CHAPTER I. THE EUCLIDEAN TEXT 29
- Section 1. The Text 29
- Section 2. An Examination of the Euclidean Text 48
- A. Preliminary Observations 49
- B. A Proposal Concerning the Origin of DEMR 50
- C. Theorem XIII,8 54
- D. Theorems XIII,1-5 56
- E. Stages in the Development of DEMR in Book XIII 56
- CHAPTER II. MATHEMATICAL TOPICS 58
- Section 3. Complements and the Gnomon 58
- Section 4. Transformation of Areas 59
- Section 5. Geometrical Algebra, Application of Areas, and Solutions of Equations 60
- A. Geometrical Algebra—Level 1 60
- B. Geometrical Algebra—Level 2 61
- C. Application of Areas—Level 3 63
- D. Historical References 64
- E. Setting Out the Debate 65
- F. Other Interpretations in Terms of Equations 65
- G. Problems in Interpretation 66
- H. Division of Figures 66
- I. Theorems VI,28,29 vs II,5,6 67
- J. Euclid's Data 67
- K. Theorem II,11 69
- L. II,11—Application of Areas, Various Views 70
- Section 6. Side and Diagonal Numbers 71
- Section 7. Incommensurability 72
- Section 8. The Euclidean Algorithm, Anthyphairesis, and Continued Fractions 73
- CHAPTER III. EXAMPLES OF THE PENTAGON, PENTAGRAM, AND DODECAHEDRON BEFORE –400 75
- Section 9. Examples before Pythagoras (before c. –550) 76
- A. Prehistoric Egypt 76
- B. Prehistoric Mesopotamia 76
- C. Sumerian and Akkadian Cuneiform Ideograms 78
- D. A Babylonian Approximation for the Area of the Pentagon 79
- E. Palestine 80
- Section 10. From Pythagoras until –400 81
- A. Vases from Greece and its Italian Colonies, Etruria (Italy) 81
- B. Shield Devices on Vases 82
- C. Coins 83
- D. Dodecahedra 84
- E. Additional Material 84
- Conclusions 84
- CHAPTER IV. THE PYTHAGOREANS 86
- i. Pythagoras 87
- ii. Hippasus 87
- iii. Hippocrates of Chios 87
- iv. Theodorus of Cyrene 87
- v. Archytas 87
- Section 11. Ancient References to the Pythagoreans 88
- A. The Pentagram as a Symbol of the Pythagoreans 88
- B. The Pythagoreans and the Construction of the Dodecahedron 88
- C. Other References to the Pythagoreans 90
- Section 12. Theories Linking DEMR with the Pythagoreans 91
- i. The Pentagram 91
- ii. Scholia Assigning Book IV to the Pythagoreans 91
- iii. Equations and Application of Areas 91
- iv. The Dodecahedron 92
- v. A Marked Straight-Edge Construction of the Pentagon 92
- vi. A Gnomon Theory 92
- vii. Allman's Theory: The Discovery of Incommensurability 93
- viii. Fritz–Junge Theory: The Discovery of Incommensurability 93
- ix. Heller's Theory: The Discovery of DEMR 94
- x. Neuenschwander's Analysis 95
- xi. Stapleton 95
- CHAPTER V. MISCELLANEOUS THEORIES 97
- Section 13. Miscellaneous Theories 97
- i. Michel 97
- ii. Fowler: An Anthyphairesis Development of DEMR 97
- iii. Knorr: Anthyphairesis and DEMR 98
- iv. Itard: Theorem IX,15 98
- Section 14. Theorems XIII,1-5 99
- i. Bretschneider 99
- ii. Allman 99
- iii. Michel 99
- iv. Dijksterhuis and Van der Waerden 99
- v. Lasserre 99
- vi. Fritz 99
- vii. Knorr 99
- viii. Heiberg 99
- ix. Herz-Fischler 99
- CHAPTER VI. THE CLASSICAL PERIOD: FROM THEODORUS TO EUCLID 100
- Section 15. Theodorus 100
- i. Knorr 100
- ii. Mugler 101
- Section 16. Plato 101
- A. Plato as a Mathematician 101
- B. Mathematical Influence of Plato 102
- C. Plato and DEMR 104
- D. Passages from Plato 105
- Section 17. Leodamas of Thasos 109
- Section 18. Theaetetus 109
- A. The Life of Theaetetus 109
- B. The Contributions of Theaetetus 110
- Section 19. Speusippus 112
- Section 20. Eudoxus 113
- A. Interpreting "Section" 113
- B. Contributions of Eudoxus to the Development of DEMR 116
- C. Commentary 116
- Section 21. Euclid 118
- Section 22. Some Views on the Historical Development of DEMR 118
- A. A Summary of Various Theories 118
- B. Summary of My Conclusions 118
- C. A Chronological Proposal 119
- D. A Proposal Concerning a Name 122
- CHAPTER VII. THE POST-EUCLIDEAN GREEK PERIOD (c. –300 to 350) 123
- Section 23. Archimedes 123
- A. Approximations to the Circumference of a Circle 123
- B. Broken Chord Theorem 125
- C. Trigonometry 125
- Section 24. The Supplement to the Elements 125
- A. The Text 125
- B. Questions of Authorship 129
- C. Chronology 129
- Section 25. Hero 131
- A. Approximations for the Area of the Pentagon and Decagon 131
- B. A Variation on II,11 134
- C. The Volumes of the Icosahedron and Dodecahedron 134
- Section 26. Ptolemy 136
- A. The Chords of 36° and 72° in Almagest 136
- B. Chord(108°)/Diameter in Geography 137
- C. Trigonometry before Ptolemy 137
- Section 27. Pappus 138
- A. Construction of the Icosahedron and Dodecahedron 138
- B. Comparison of Volumes 141
- CHAPTER VIII. THE ARABIC WORLD, INDIA, AND CHINA 144
- Section 28. The Arabic Period 144
- i. Authors Consulted 144
- ii. Equations 145
- A. Al-Khwarizmi 145
- B. Abu Kamil 147
- C. Abu'l-Wafa' 151
- D. Ibn Yunus 152
- E. Al-Biruni 153
- Section 29. India 154
- Section 30. China 156
- CHAPTER IX. EUROPE: FROM THE MIDDLE AGES THROUGH THE EIGHTEENTH CENTURY 157
- Section 31. Europe Through the 16th Century 157
- A. Authors Consulted 157
- B. Fibonacci 160
- C. Francesca 167
- D. Paccioli 172
- E. Cardano 174
- F. Bombelli 175
- G. Candalla 178
- H. Ramus 179
- I. Stevin 180
- J. Pre-1600 Numerical Approximations to DEMR 180
- K. Approximate Constructions of the Pentagon 181
- Section 32. The 17th and 18th Centuries 182
- A. Kepler 182
- B. The Fibonacci Sequence 184
- C. Fixed Compass and Compass Only Constructions 185
- By Way of a Conclusion 186
- APPENDIX I. "A PROPORTION BY ANY OTHER NAME": TERMINOLOGY FOR DIVISION IN EXTREME AND MEAN RATIO THROUGHOUT THE AGES 187
- A. "Extreme and Mean Ratio" 187
- B. "Middle and Two Ends" 189
- C. Names for DEMR 190
- APPENDIX II. "MIRABLIS... EST POTENTIA ...": THE GROWTH OF AN IDEA 194
- CORRECTIONS AND ADDITIONS 199
- BIBLIOGRAPHY 203
