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
| 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... | |
| Plato, Reginald E. Allen - Philosophy - 1998 - 351 pages
...mathematics. Euclid, writing about ¿oo BC, is resolutely finitist. Elements X, Definition i states that **“two unequal magnitudes being set out, if from the...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... | |
| Frank Burk - Mathematics - 1998 - 292 pages
...their diameters, an obvious result for regular polygons. “Truth” was to be based on Eudoxus' Axiom: **Two unequal magnitudes being set out, if from the...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.... | |
| Peter Machamer - Philosophy - 1998 - 462 pages
...act out, if from the greater chere be subacracced a magnitude greater than its half, and from chat **which is left a magnitude greater than its half, and...magnitude which will be less than the lesser magnitude** sec out.” V, ¿: “a ratio is a sort of relation in respect of size between two magnitudes of the... | |
| Kangshen Shen, John N. Crossley, Anthony Wah-Cheung Lun, Hui Liu - Mathematics - 1999 - 596 pages
...such an n comes from the method of exhaustion, which appears in Proposition 10.1 of Euclid's Elements. **“Two unequal magnitudes being set out, if from the...will be less than the lesser magnitude set out.”** At the end of the Proposition Euclid added: “And the theorem can be similarly proved even if the... | |
| Douglas M. Jesseph - Mathematics - 2000 - 419 pages
...as proposition 1 of book 10 of the Elements in the form of the claim: OISPUTEO FOUNOATI ONS / 161 BD **Two unequal magnitudes being set out, if from the...which will be less than the lesser magnitude set out.** (Elements 10, prop. 1) The angle of contact appears to violate this condition, since no amount of division... | |
| Douglas M. Jesseph - Mathematics - 2000 - 419 pages
...as proposition 1 of book 10 of the Elements in the form of the claim: DISPUTED FOUNDATIONS 1161 BD **Two unequal magnitudes being set out, if from the...which will be less than the lesser magnitude set out.** (Elements 10, prop. 1) The angle of contact appears to violate this condition, since no amount of division... | |
| Reinhard Laubenbacher, David Pengelley - Mathematics - 1999 - 275 pages
...Elements [51, Vol. 3, p. 14], as Proposition 1 in Book X: Two unequal magnitudes being set out, iffrom **the greater there be subtracted a magnitude greater...which will be less than the lesser magnitude set out.** Clearly, this is exactly what Archimedes uses to obtain his Corollary to Proposiiswhat makes it possible... | |
| James Roy Newman - Mathematics - 2000 - 768 pages
...it was proved in the first theorem of the tenth book that, if two unequal magnitudes be set out, and **if from the greater there be subtracted a magnitude greater than its half, and from** the remainder a magnitude greater than its half, and so on continually, there will be left some magnitude... | |
| Audun Holme - Mathematics - 2002 - 378 pages
...of Exhaustion is established. We find it already in Proposition 1, quoted here from [16]: Euclid X.1 **Two unequal magnitudes being set out, if from the...some magnitude which will be less than the lesser** of the magnitudes. This proposition is used to prove Proposition X1I.2, namely that the areas of two... | |
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