The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry

Couverture
Springer Science & Business Media, 1975 - 374 pages
0 Avis
The present work has three principal objectives: (1) to fix the chronology of the development of the pre-Euclidean theory of incommensurable magnitudes beginning from the first discoveries by fifth-century Pythago reans, advancing through the achievements of Theodorus of Cyrene, Theaetetus, Archytas and Eudoxus, and culminating in the formal theory of Elements X; (2) to correlate the stages of this developing theory with the evolution of the Elements as a whole; and (3) to establish that the high standards of rigor characteristic of this evolution were intrinsic to the mathematicians' work. In this third point, we wish to counterbalance a prevalent thesis that the impulse toward mathematical rigor was purely a response to the dialecticians' critique of foundations; on the contrary, we shall see that not until Eudoxus does there appear work which may be described as purely foundational in its intent. Through the examination of these problems, the present work will either alter or set in a new light virtually every standard thesis about the fourth-century Greek geometry. I. THE PRE-EUCLIDEAN THEORY OF INCOMMENSURABLE MAGNITUDES The Euclidean theory of incommensurable magnitudes, as preserved in Book X of the Elements, is a synthetic masterwork. Yet there are detect able seams in its structure, seams revealed both through terminology and through the historical clues provided by the neo-Platonist commentator Proclus.
 

Avis des internautes - Rédiger un commentaire

Aucun commentaire n'a été trouvé aux emplacements habituels.

Table des matières

I INTRODUCTION
1
II General Methodlogical Observations
5
III Indispensable Definitions
14
II THE SIDE AND THE DIAMETER OF THE SQUARE
21
I The Received Proof of the Incommensurability of the Side and Diameter of the Square
22
II Anthyphairesis and the Side and Diameter
29
III Impact of the Discovery of Incommensurability
36
IV Summary of the Early Studies
49
II Right Triangles and the Discovery of Incommensurability
174
III The Lesson of Theodorus
181
IV Theodorus and Elements II
193
THEAETETUS AND ARCHYTAS
211
I The Theorem of Archytas on Epimoric Ratios
212
II The Theorems of Theaetetus
225
III The Arithmetic Proofs of the Theorems of Theaetetus
227
IV The Arithmetic Basis of Theaetetus Theory
233

III PLATOS ACCOUNT OF THE WORK OF THEODORUS
62
𝛿ú𝜐𝜶𝜇𝜀𝜄ç
65
𝛾𝜚á𝜙𝜀𝜄𝜐
69
𝛿𝜋𝜎𝜙𝛼𝜄𝜀𝜐
75
IV Why Seperate Cases?
79
V Why Stop at Seventeen?
81
VI The Theorems of Theaetetus
83
VII Theodorus Style of Geometry
87
VIII Summary of Interpretive Criteria
96
IV A CRITICAL REVIEW OF RECONSTRUCTIONS OF THEODORUS PROOFS
109
II Algebraic Reconstruction
111
III Anthyphairetic Reconstruction
118
V THE PYTHAGOREAN ARITHMETIC OF THE FIFTH CENTURY
131
I Pythagorean Studies of the Odd and the Even
134
II The PebbleRepresentation of Numbers
135
III The PebbleMethods Applied to the Study of the Odd and the Even
137
IV The Theory of Figured Numbers
142
V Properties of Pythagorean Number Triples
154
THEODORUS
170
I Numbers Represented as Magnitudes
171
V Observations on PreEuclidean Arithmetic
238
THEAETETUS AND EUDOXUS
252
II Anthyphairesis and the Theory of Proportions
255
III The Theory of Proportions in Elements X
261
IV Theaetetus and Eudoxus
273
V Summary of the Development of the Theory of Irrationals
286
IX CONCLUSIONS AND SYNTHESES
298
II The Editing of the Elements
303
III The PreEuclidean Theory of Incommensurable Magnitudes
306
On the Extension of Theodorus Method
314
On the Anthyphairetic Proportion Theory
332
A LIST OF THE THEOREMS IN CHAPTERS VVIII AND THE APPENDICES
345
REFERENCING CONVENTIONS AND BIBLIOGRAPHY
353
Ancient Authors
355
Books
357
Articles
360
INDEX OF NAMES
366
INDEX OF PASSAGES CITED FROM ANCIENT WORKS
369
Droits d'auteur

Autres éditions - Tout afficher

Expressions et termes fréquents

Références à ce livre

Informations bibliographiques