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 - Page 372by Euclid, Sir Thomas Little Heath, 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... | |
| C.H.Jr. Edwards - Mathematics - 1994 - 368 pages
...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 - Law - 1988 - 87 pages
...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 - Mathematics - 1990 - 390 pages
...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 - Mathematics - 1992 - 417 pages
...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 - Mathematics - 1993 - 322 pages
...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... | |
| Paolo Mancosu - Mathematics - 1996 - 288 pages
...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.** As an example of proof by exhaustion we will consider proposition 10 of book XII of the Elements. PROPOSITION... | |
| Frank Burk - Mathematics - 1998 - 292 pages
...that which is left a magnitude greater than its half, and if this process be repeated continuously, **there will be left some magnitude which will be less than the lesser magnitude set out.** In modem terminology, let A/ and e > 0 be given with 0 < e < M. Then form: M , M - rM = (l -r)A/,(l... | |
| Plato, Reginald E. Allen - Philosophy - 1998 - 351 pages
...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."** This theorem, which later came to be called the Postulate of Archimedes, is the foundation of much... | |
| Peter Machamer - Philosophy - 1998 - 462 pages
...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."** V, 3: "a ratio is a sort of relation in respect of size between two magnitudes of the same kind" (My... | |
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