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... Books 10-13 and appendix - Seite 372von Euclid - 1908Vollansicht - Über dieses Buch
| Archimedes - 1897 - 532 Seiten
...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... | |
| Archimedes - 1912 - 377 Seiten
...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... | |
| Ernest William Hobson - 1913 - 57 Seiten
...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 lesser magnitude set out."** This principle is deduced by Euclid from the axiom that, if there are two magnitudes of the same kind,... | |
| Sir Thomas Little Heath - 1921
...there be subtracted more than its half (or the half itself), from the remainder more than its half (or **the half), and if this be done continually, there...some magnitude which will be less than the lesser** of the given magnitudes. This last lemma is frequently used by Archimedes himself (notably in the second... | |
| Sir Thomas Little Heath - 1921
...there be subtracted more than its 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** niagmtude which will be less than the lesser of the given magnitudes. This last lemma is frequently... | |
| C.H.Jr. Edwards - 1994 - 368 Seiten
...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 lesser magnitude set out.** . This result, which we will call "Eudoxus' principle," may be phrased as follows. Let A/o and e be... | |
| Gray L. Dorsey - 1988 - 87 Seiten
...and from the greater a magnitude greater than its half is subtracted, and this process is repeated **continually, there will be left some magnitude which will be less than the lesser magnitude set out.** This means that any magnitude is infinitely divisible; that there is no smallest magnitude, because... | |
| Morris Kline - 1990 - 434 Seiten
...which is left a magnitude greater than its 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 conclusion of the proof Euclid says the theorem can be proven if the parts subtracted be halves.... | |
| Robert M. Young - 1992 - 436 Seiten
...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 lesser magnitude set out.** Application of this principle shows that, for any e > 0, we obtain after a finite number of steps an... | |
| Douglas M. Jesseph - 1993 - 335 Seiten
...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 lesser magnitude set out.** (Elements X, 1) The general procedure for an exhaustion proof is to begin with upper and lower bounds... | |
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