than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the magnitude C. For C if multiplied The Thirteen Books of Euclid's Elements - Page 14by Euclid, Johan Ludvig Heiberg - 1908Full view - About this book
| Archimedes - Geometry - 1897 - 326 pages
...greater than the half, if from the remainder [a part] greater than the half be subtracted, and so on **continually, there will be left some magnitude which will be less than the** lesser given magnitude." This last lemma is frequently assumed by Archimedes, and the application of... | |
| T. L. HEATH - 1897
...greater than the half, if from the remainder [a part] greater than the half be subtracted, and so on **continually, there will be left some magnitude which will be less than the** lesser given magnitude." This last lemma is frequently assumed by Archimedes, and the application of... | |
| Sir Thomas Little Heath - Mathematics - 1981 - 446 pages
...half (or the half itself), from the remainder more than its half (or the half), and if this be done **continually, there will be left some magnitude which will be less than the** lesser of the given magnitudes. This last lemma is frequently used by Archimedes himself (notably in... | |
| C.H.Jr. Edwards - Mathematics - 1994 - 368 pages
...applied. Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater **than its half, and from that which is left a magnitude...be left some magnitude which will be less than the** lesser magnitude set out.. This result, which we will call “Eudoxus' principle,” may be phrased... | |
| Morris Kline - Mathematics - 1990 - 390 pages
...1. Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater **than its half, and from that which is left a magnitude...half, and if this process be repeated continually,** then there will be left some magnitude which will be less than the lesser magnitude set out. At the... | |
| Douglas M. Jesseph - Mathematics - 1993 - 322 pages
...positive integers m and n such that ma > /3 and n/3 > a. from that which is left a magnitude greater **than its half, and from that which is left a magnitude...be left some magnitude which will be less than the** lesser magnitude set out. (Elements X, 1) The general procedure for an exhaustion proof is to begin... | |
| Paolo Mancosu - Mathematics - 1996 - 288 pages
...1. Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater **than its half, and from that which is left a magnitude...be left some magnitude which will be less than the** lesser magnitude set out. As an example of proof by exhaustion we will consider proposition 10 of book... | |
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...1. Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater **than its half, and from that which is left a magnitude...be left some magnitude which will be less than the** lesser magnitude set out. As an example of proof by exhaustion we will consider proposition 10 of book... | |
| G. E. R. Lloyd - Philosophy - 1996 - 250 pages
...1. Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater **than its half, and from that which is left a magnitude...be left some magnitude which will be less than the** lesser magnitude set out."'' Used, as for example in Elements xt t 2, in the investigation of the ratios... | |
| Fan Dainian - History - 1996 - 471 pages
...Elements: Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater **than its half, and from that which is left a magnitude...half, and if this process be repeated continually,** then there will be left some magnitude which will be less than any known small magnitude. Because Archimedes... | |
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