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 26by Euclid, Johan Ludvig Heiberg - 1908Full view - About this book
| 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... | |
| John N. Crossley - Mathematics - 1987 - 222 pages
...ones are trivial to classify, of course, because as Euclid notes in Proposition X.5, Coesnensurable **magnitudes have to one another the ratio which a number has to a number.** Given the separation between lines and numbers, there is no clear way of introducing numbers corresponding... | |
| John N. Crossley - Mathematics - 1987 - 222 pages
...ones are trivial to classify, of course, because as Euclid notes in Proposition X3, Coimnensurable **magnitudes have to one another the ratio which a number has to a number.** of introducing numbers corresponding to ratios between lengths except by specifying their relations... | |
| Plato, Thomas L. Pangle - Philosophy - 1988 - 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... | |
| Morris Kline - Mathematics - 1990 - 390 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... | |
| 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... | |
| 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... | |
| 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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