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Books Books 1 - 6 of 6 on PROPOSITION 6. If two magnitudes have to one another the ratio which a number has....  
" PROPOSITION 6. If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable. For let the two magnitudes A, B have to one another the ratio which the number D has to the number E "
The Thirteen Books of Euclid's Elements - Page 26
by Euclid, Johan Ludvig Heiberg - 1908
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Coloured Quadrangles: A Guide to the Tenth Book of Euclid's Elements

Christian Marinus Taisbak - Religion - 1982 - 78 pages
...number has to a number. 22 CM Taisbak And the converse: X 6 If two magnitudes have to one another a ratio which a number has to a number, the magnitudes will be commensurable. The transmitted proof of X 5 suffers from various defects. To our view, such a statement cannot be...
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The Laws of Plato

Plato - Philosophy - 1980 - 562 pages
...Philology 53 (1958): 97-98. This passage is also discussed in Athenaeus XV 670 ff. 37. "Commensurable magnitudes have to one another the ratio which a number has to a number" (Euclid X, proposition 5). On the subject of commensurability and incommensurability (or irrational...
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Mathematical Thought from Ancient to Modern Times:

Morris Kline - Mathematics - 1990 - 428 pages
...have a few theorems that relate them. For example, Proposition 5 of Book X states that commensurable magnitudes have to one another the ratio which a number has to a number. In the three books under discussion, as in other books, Euclid assumes facts that he does not state...
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The Mathematics of Measurement: A Critical History

John J. Roche - Mathematics - 1998 - 330 pages
...apply rigorously to incommensurable ratios. According to Euclid, ‘incommensurable magnitudes have not to one another the ratio which a number has to a number” 3 . Another objection to the reduction of a ratio to a number, also deriving from Euclid, was that...
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A History of Analysis

Hans Niels Jahnke - Mathematics - 422 pages
...already mentioned that from the 5th century BC the Greeks knew that some geometrical “magnitudes do not have to one another the ratio which a number has to a number” (Elements X, 7). In the history of mathematics this discovery of incommensurable magnitudes is often...
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Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King

Carl Huffman - Philosophy - 2005
...required to recognize incommensurability, since incommensurability arises when two magnitudes “have not to one another the ratio which a number has to a number” (Euclid x. 7). “The analysis of certain classes of problems in geometry, eg the construction of irrational...
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