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
| Reinhard Laubenbacher, David Pengelley - Mathematics - 1999 - 275 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.** is what makes it possible to eventually undershoot it, since it does not recede as the halving process... | |
| Victor J. Katz - Mathematics - 2000 - 261 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.** About proposition II, in proving it we observe that it depends on the result of proposition III. As... | |
| Serafina Cuomo - Mathematics - 2000 - 234 pages
...the greater there be subtracted more than its half, from the remainder more than its half and so on, **there will be left some magnitude which will be less than the lesser** of the first two"). and the "Eudoxus-Archimedes" one, based on ratios ("Of unequal magnitudes the greater... | |
| K. Neal - Education - 2002 - 174 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 proposition utilises Book V definition four, and it therefore links ratios to commensurability.... | |
| Audun Holme - Mathematics - 2002 - 378 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** of the magnitudes. This proposition is used to prove Proposition XII.2, namely that the areas of two... | |
| Chris Pritchard, Mathematical Association of America - Mathematics - 2003 - 541 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.** Prop. 9: The squares on straight lines commensurable in length have to one another the ratio which... | |
| Hans Niels Jahnke - Mathematics - 422 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); or whether one considers this theorem to be another, a so-called dieisive form of... | |
| Avicenna Study Group. Conference - Philosophy - 2004 - 262 pages
...half, and from that which is oo 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.** 12 10 Euclid, Elements (1956), 2:114. 11 Usul al-handasa, 153. Ibn Siha's version of this statement... | |
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