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 greater than its half, and if this process be repeated continually, there will be left some magnitude... 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 it to... | |
| 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 it to... | |
| C.H.Jr. Edwards - Mathematics - 1994 - 368 pages
...purpose the following consequence (Euclid Xl) of the ArchimedesEudoxus axiom is repeatedly applied. **Two unequal magnitudes being set out, if from the...which will be less than the lesser magnitude set out..** This result, which we will call “Eudoxus' principle,” may be phrased as follows. Let M 0 and c... | |
| Morris Kline - Mathematics - 1990 - 390 pages
...proposition in Book X is important for developments in later books of the Elements. Proposition 1. **Two unequal magnitudes being set out, if from the...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
...states a criterion by which a sequence of approximations can be brought arbitrarily close to a limit: **Two unequal magnitudes being set out, if from the...subtracted a magnitude greater than its half, and** 2. In effect, definition 4 states the "Archimedean axiom" for the real numbers, which says that for... | |
| G. E. R. Lloyd - Philosophy - 1996 - 250 pages
...developed by Eudoxus in the -4th century, depends on the principle expressed in Euclid Elements x 1. **Two unequal magnitudes being set out, if from the...will be less than the lesser magnitude set out."''** Used, as for example in Elements xt t 2, in the investigation of the ratios of circles to their diameters,... | |
| Paolo Mancosu - Drama - 1999 - 275 pages
...that each of the 2" parts is less than the smaller given magnitude. In Euclid's words: PROPOSITION 1. **Two unequal magnitudes being set out, if from the...which will be less than the lesser magnitude set out.** As an example of proof by exhaustion we will consider proposition 10 of book XII of the Elements. PROPOSITION... | |
| Paolo Mancosu - Mathematics - 1996 - 288 pages
...that each of the 2" parts is less than the smaller given magnitude. In Euclid's words: PROPOSITION 1. **Two unequal magnitudes being set out, if from the...which will be less than the lesser magnitude set out.** As an example of proof by exhaustion we will consider proposition 10 of book XII of the Elements. PROPOSITION... | |
| Fan Dainian - History - 1996 - 471 pages
...Archimedes used 'the method of exhaustion.' Its core is Proposition 1 in chapter 10 of the Elements: **Two unequal magnitudes being set out, if from the...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... | |
| Peter M. Engelfriet - Mathematics - 1998 - 488 pages
...The wording of the postulate here seems to be a combination of V.Def.4 with theorem I of Book X:¿ **Two unequal magnitudes being set out, if from the...which will be less than the lesser magnitude set out.** Unfortunately, Clavius does not give a motivation for his choice. Although he devoted an extensive... | |
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