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Books Books 1 - 10 of 17 on Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude....  
" 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 14
by Euclid, Johan Ludvig Heiberg - 1908
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The Works of Archimedes

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...
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THE WORKS OF ARCHIMEDES

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...
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The Historical Development of the Calculus

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...
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Mathematical Thought from Ancient to Modern Times:

Morris Kline - Mathematics - 1990 - 428 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...
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Berkeley's Philosophy of Mathematics

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...
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Adversaries and Authorities: Investigations Into Ancient Greek and Chinese ...

Geoffrey Ernest Richard 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,...
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Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

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...
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Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

Paolo Mancosu Assistant Professor of Philosophy University of California at Berkeley - Mathematics - 1995 - 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...
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Chinese Studies in the History and Philosophy of Science and ..., Volume 179

Fan Dainian, Robert S. Cohen - 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...
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