The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry (Synthese Historical Library) [Hardcover]

W.R. Knorr (Author)

Product Details

• Hardcover: 385 pages
• Publisher: Springer; 1 edition (February 28, 1974)
• Language: English
• ISBN-10: 9027705097
• ISBN-13: 978-9027705099
• Product Dimensions: 9.3 x 5.8 x 1.3 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,782,584 in Books (See Top 100 in Books)

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6 of 6 people found the following review helpful:

5.0 out of 5 stars The development of the crown-jewel theory of incommensurability, from which Euclid took dictation, April 2, 2006

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry (Synthese Historical Library) (Hardcover)

Around 430-410 B.C. the Greeks discovered that a side of a square is incommensurable with the diameter. Over one hundred years later Euclid wrote down a sophisticated theory of incommensurability in the Elements. What happened in between? First we look the discovery of incommensurability (chapter 2). Here we know very little for certain, but it seems that mathematicians did not perceive incommensurability as the big foundational problem that it's cracked up to be, so we feel that the theory of incommensurability was developed as a mathematically interesting subject in its own right. Plato has it that Theodorus proved the irrationality of the square roots of 3, 5, ... up to 17 (chapter 3). This work is lost. Knorr is not happy with the reconstructions that have been suggested (chapter 4). He offers his own reconstruction (chapter 6), which rests on the Pythagorean arithmetic tradition (chapter 5). To be able to treat incommensurability Theodorus had to geometrise Pythagorean mathematics, creating the so-called "geometric algebra" of Book II of the Elements, and in this way we see that "Book II could be transformed from the aimless miscellany it appears to be into a systematically developed theory" (p. 197) and that "Book II was not so much a 'geometrization of algebra' as it was a 're-geometrization of an algebraized geometry'" (p. 199). Continuing in this arithmetic tradition, Archytas's and Theaetetus's work on incommensurability developed the number theory of Book VII (chapter 7); again geometrisation is necessary for the theory to be applicable to incommensurable magnitudes. Eudoxus was then able to make further progress by initiating the recasting this work in the geometric form we know from Book X (chapter 8). In conclusion, "excepting the topological materials contained in Books I, III, VI and XI, virtually the whole of the Elements can be understood as a product of Euclid's effort to present the entire formal theory of irrationals within a self-sufficient compilation of treatises" (p. 288).