The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry (Synthese Historical Library, 15)

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. Front Matter....Pages I-XI Introduction....Pages 1-20 The Side and the Diameter....Pages 21-61 Plato’s Account of the Work of Theodorus....Pages 62-108 A Critical Review of Reconstructions of Theodorus’ Proofs....Pages 109-130 The Pythagorean Arithmetic of the Fifth Century....Pages 131-169 The Early Study of Incommensurability: Theodorus....Pages 170-210 The Arithmetic of Incommensurability: Theaetetus and Archytas....Pages 211-251 The Geometry of Incommensurability: Theaetetus and Eudoxus....Pages 252-297 Conclusions and Syntheses....Pages 298-313 Back Matter....Pages 314-379 Wilbur Richard Knorr. Portions Of This Work Originally Included In The Author's Thesis, Harvard, 1973. Includes Indexes. Bibliography: P. 355-365.