PAPPUS REBORN PAPPUS OF ALEXANDRIA AND THE CHANGING FACE OF ANALYSIS AND ...

189 Pages·2009·1.13 MB·English

PAPPUS REBORN
PAPPUS OF ALEXANDRIA AND THE CHANGING FACE OF ANALYSIS AND
SYNTHESIS IN LATE ANTIQUITY
A thesis submitted in partial fulfilment of the requirements for the
Degree
of Master of Arts in History and Philosophy of Science
in the University of Canterbury
by Bronwyn Rideout
University of Canterbury
2008
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Figure 1: Map of Alexandria from El-Abbadi, Mostafa. The Life and Fate of
the Ancient Library of Alexandria. Mayenne, France: Imprimerie Floch, 1990:
http://ils.unc.edu/dpr/path/alexandria/geography.htm
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Contents
I Preface 10
0.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
0.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
II Contexts of Analysis 13
0.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1 Setting the Scene 18
1.1 Societal Influences in Literacy . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 Egyptians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.2 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1.3 Romans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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1.1.4 Jewish Inhabitants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 Women and Literacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3 Affluence and Literacy vs. Mathematics . . . . . . . . . . . . . . . . . . . . 28
1.4 Mathematicians in the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Mathematical Professions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Historiographical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6.1 Deuteronomic Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.6.2 The Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 The Domain of Analysis 45
2.1 The Domain of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 The Domain and Book 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Provenance and Originality of Book 7 . . . . . . . . . . . . . . . . . . . . . . 51
3 Everyone Love Archimedes: A Brief Prelude to Pappian Analysis and Syn-
thesis 53
3.1 Sphere and Cylinder, Book 2, Proposition 3 . . . . . . . . . . . . . . . . . . 54
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3.2 Thoughts on Archimedean Analysis . . . . . . . . . . . . . . . . . . . . . . . 58
4 Hermodorus’ New Power: Analysis in Book 7 60
4.1 Pappus in Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.2 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.3 Loose Threads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.4 Division of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 A Modern Opinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Pappus in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Proposition 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Proposition 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Analysis and Synthesis: An Evaluation Thus Far . . . . . . . . . . . . . . . . 87
5 Dear Pandrosion: Mathematical Etiquette and Analysis 90
5.1 Rules for being a Mathematician . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Pappus’ Practice in Book 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
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5.3 The Student’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Book 3: A Renewed Look At Analysis and Synthesis . . . . . . . . . . . . . 107
6 Sing the Wrath of Demeter: Analysis and the Underworld of Arithmetic 109
6.1 The Style of Book 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Concerns about Heath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Arithmetic and Geometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4 Analysis and Synthesis and Book 2 . . . . . . . . . . . . . . . . . . . . . . . 122
7 Epilogue 125
III Translation and Commentary 127
8 Book 2 Text, Translation, and Commentary 129
8.1 Greek Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Greek Text, with translation and commentary of Book 2 from Pappus of
Alexandria’s Mathematical Collection . . . . . . . . . . . . . . . . . . . . . . 130
8.3 General Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
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8.4 Mathematical Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
IV Bibliography 181
9 Bibliography 182
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List of Figures
1 MapofAlexandriafromEl-Abbadi, Mostafa. TheLifeandFateoftheAncient
LibraryofAlexandria. Mayenne,France: ImprimerieFloch,1990: http://ils.unc.edu/dpr/path/alexandria/geography.htm 2
3.1 Proposition 3, Book 2, Sphere and Cylinder from Heiberg . . . . . . . . . 54
4.1 Proposition 21, Book 7 of the Collection from F. Hultsch. . . . . . . . . . . 76
4.2 Proposition 19, Book 7 of the Collection from F. Hultsch . . . . . . . . . . 78
4.3 Proposition 72, Book 7 of the Collection from F. Hultsch . . . . . . . . . . 82
4.4 Proposition 71, Book 7 of the (Collection) from F. Hultsch . . . . . . . . . . 84
5.1 A Planar way to solve the Delian Problem, Book 3 of the Collection from F.
Hultsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Eratosthenes Ruler, Handbook of Greek Mathematics from Heath . . . 99
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5.3 ArepresentationofwhatisgoingoninPappus’criticismoftheDelianproblem.
Ataratioof4 : 2, PK isfoundbysubtractionofΘP fromKΘ. Theremainder
issplitevenlybetweenPΣandΣκ. Fromhere, Pappuscancalculatethevalues
of his first set of equivalent ratios. . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1 Proposition 14, Book 2 of the Collection from A. P. Treweek . . . . . . . . 121
6.2 Proposition 17, Book 2 of the Collection from A. P. Treweek . . . . . . . . 121
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Part I
Preface
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