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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]

By W.R. Knorr (Author)
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Item description for The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry (Synthese Historical Library) by W.R. Knorr...

The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry (Synthese Historical Library) by W.R. Knorr

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Item Specifications...


Studio: Springer
Pages   385
Est. Packaging Dimensions:   Length: 9.3" Width: 5.8" Height: 1.3"
Weight:   1.65 lbs.
Binding  Hardcover
Release Date   Feb 28, 1974
Publisher   Springer
ISBN  9027705097  
ISBN13  9789027705099  


Availability  54 units.
Availability accurate as of May 27, 2017 04:17.
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1Books > Special Features > New & Used Textbooks > Humanities > Philosophy > History & Surveys
2Books > Special Features > New & Used Textbooks > Sciences > Mathematics > Geometry
3Books > Subjects > History > Europe > Greece > General
4Books > Subjects > History > Europe > Greece
5Books > Subjects > Nonfiction > Philosophy > General
6Books > Subjects > Nonfiction > Philosophy > Greek & Roman
7Books > Subjects > Nonfiction > Philosophy > History & Surveys
8Books > Subjects > Professional & Technical > Professional Science > Mathematics > Geometry & Topology > General Geometry
9Books > Subjects > Science > Mathematics > Geometry & Topology > General Geometry
10Books > Subjects > Science > Mathematics > Geometry & Topology



Reviews - What do customers think about The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry (Synthese Historical Library)?

The development of the crown-jewel theory of incommensurability, from which Euclid took dictation  Apr 2, 2006
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).
 

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