A mathematical history of division in extreme and mean ratio:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Waterloo, Ont.
Wilfrid Laurier Univ. Pr.
1987
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 191 S. |
| ISBN: | 0889201528 |
Internformat
MARC
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| 245 | 1 | 0 | |a A mathematical history of division in extreme and mean ratio |
| 264 | 1 | |a Waterloo, Ont. |b Wilfrid Laurier Univ. Pr. |c 1987 | |
| 300 | |a XVI, 191 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 648 | 7 | |a Geschichte Anfänge-1800 |2 gnd |9 rswk-swf | |
| 650 | 4 | |a Géometrie algébrique - Histoire | |
| 650 | 4 | |a Geschichte | |
| 650 | 4 | |a Ratio and proportion |x History | |
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Datensatz im Suchindex
| _version_ | 1824619408475029504 |
|---|---|
| adam_text |
TABLE OF CONTENTS
PREFACE xi
A GUIDE FOR READERS xiii
A. Internal Organization xiii
B. Bibliographical Details xiii
C. Abbreviations xiv
D. Symbols xiv
E. Dates xv
F. Quotations from Primary Sources xv
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D. Historical References 41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
V
Vi A MATHEMATICAL HISTORY OF DEMR
K. Theorem 11,11 46
L. 11,11—Application of Areas, Various Views 47
i. Szabo Al
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
/. 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
/. Pythagoras 64
ii. Hippasus 64
ill. 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
/'. 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 vii
Section 13. Miscellaneous Theories 74
/. Michel 74
(V. Fowler: An Anthyphairesis Development of DEMR 14
Hi. Knorr: Anthyphairesis and DEMR 75
(V. hard: Theorem IX,15 75
Section 14. Theorems XIII.1 5 76
/. Bretschneider 76
ii. Allman 76
Hi. Michel 76
iv. Dijksterhuis and Van der Waerden 76
v. Lasserre 76
vi. Fritz 16
vii. 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 11
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 3IB 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
i. Tannery 88
ii. Allman 88
Hi. 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
iVi. 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
Viii A MATHEMATICAL HISTORY OF DEMR
Hi. 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
/. Equations and Application of Areas 95
ii. Incommensurability 95
Hi. 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
/'. The Area of the Pentagram 108
ii. The Area of the Decagon 109
hi. 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. Ptolemy 113
A. The Chords of 36° and 72° in Almagest 113
B. Chord(I08°)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 ix
B. AbuKamil 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
iii. 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
ill. 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
/. Pre 1600 Numerical Approximations to DEMR 157
i. Unknown Annotator to Paccioli's Euclid 157
ii. Holtzmann 158
iii. 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
X A MATHEMATICAL HISTORY OF DEMR
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
BIBLIOGRAPHY 176 |
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| illustrated | Not Illustrated |
| indexdate | 2025-02-20T23:04:26Z |
| institution | BVB |
| isbn | 0889201528 |
| language | English |
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| physical | XVI, 191 S. |
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| spelling | Herz-Fischler, Roger Verfasser aut A mathematical history of division in extreme and mean ratio Waterloo, Ont. Wilfrid Laurier Univ. Pr. 1987 XVI, 191 S. txt rdacontent n rdamedia nc rdacarrier Geschichte Anfänge-1800 gnd rswk-swf Géometrie algébrique - Histoire 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 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000498902&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Herz-Fischler, Roger A mathematical history of division in extreme and mean ratio Géometrie algébrique - Histoire Geschichte Ratio and proportion History Goldener Schnitt (DE-588)4021529-5 gnd |
| subject_GND | (DE-588)4021529-5 |
| title | A mathematical history of division in extreme and mean ratio |
| title_auth | A mathematical history of division in extreme and mean ratio |
| title_exact_search | A mathematical history of division in extreme and mean ratio |
| title_full | A mathematical history of division in extreme and mean ratio |
| title_fullStr | A mathematical history of division in extreme and mean ratio |
| title_full_unstemmed | A mathematical history of division in extreme and mean ratio |
| title_short | A mathematical history of division in extreme and mean ratio |
| title_sort | a mathematical history of division in extreme and mean ratio |
| topic | Géometrie algébrique - Histoire Geschichte Ratio and proportion History Goldener Schnitt (DE-588)4021529-5 gnd |
| topic_facet | Géometrie algébrique - Histoire Geschichte Ratio and proportion History Goldener Schnitt |
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