The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek GeometrySpringer Science & Business Media, 1975 - 374 pages 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. |
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 |
353 | |
Ancient Authors | 355 |
357 | |
360 | |
366 | |
369 | |
Autres éditions - Tout afficher
The Evolution of the Euclidean Elements: A Study of the Theory of ... W.R. Knorr Aucun aperçu disponible - 1980 |
Expressions et termes fréquents
algebraic algorithm anthyphairetic apotome application of areas Archytas argument Aristotle Aristotle's arithmetic binomial Boethius Chapter commensurable construction context definition Diels discovery of incommensurability discussion divisible divisor Elements equal establish Euclid Euclidean Euclidem Eudemus Eudoxus extreme and mean fact Figure follows form 4N+3 formal geometric given gnomon Greek Hence heteromecic Hippasus Hippocrates hypotenuse Iamblichus incommen incommensurability incommensurable lines incommensurable magnitudes instance integers irrational lines J. L. Heiberg lemma mathematical mean proportional Metaphysics method metrical multiple Nicomachus number theory number triples odd number Pappus passage Philolaus Plato pre-Euclidean Proclus proof proportion theory proved Pythagorean triple reconstruction rectangle relation relatively prime right triangle Sectio Canonis Section segments side and diameter square integers square number study of incommensurability surable T. L. Heath Theaetetus Theodorus Theorem theory of incommensurability theory of irrationals theory of proportion tion triangular numbers unit W. D. Ross Waerden Zeuthen καὶ
Références à ce livre
Amazing Traces of a Babylonian Origin in Greek Mathematics Jöran Friberg Aucun aperçu disponible - 2007 |
Amazing Traces of a Babylonian Origin in Greek Mathematics Jöran Friberg Aucun aperçu disponible - 2007 |