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Books Books 141 - 150 of 162 on AB be the given straight line ; it is required to divide it into two parts, so that....  
" AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part. "
Introduction and books 1,2 - Page 402
by Euclid, Sir Thomas Little Heath, Johan Ludvig Heiberg - 1908
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Mathematical Expeditions: Chronicles by the Explorers

Reinhard Laubenbacher, David Pengelley - Mathematics - 1999 - 275 pages
...Hint: See Figure 5.6. Exercise 5.9: Proposition 1 1 in Book II states: To cut a given straight line so that the rectangle contained by the whole and one...segments is equal to the square on the remaining segment. Which type of quadratic equation can be solved with the help of this proposition and how? Exercise...
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Proportion: Science, Philosophy, Architecture

Richard Padovan - Architecture - 1999 - 388 pages
...1 Spiral generared by the 0 tr iangle. rectangle. He defines it thus: 'To cut a given straight lme so that the rectangle contained by the whole and one...segments is equal to the square on the remaining segment. M: Referring to Figure 8.7, this means that if AB is the given straight line, then AB x BR = AR2. Euclid...
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Apollonius of Perga's Conica: Text, Context, Subtext

Michael N. Fried - History - 2001 - 499 pages
...from the following example.48 Proposition II. 1 1 of the Elements reads: To cut a given straight line so that the rectangle contained by the whole and one...the segments is equal to the square on the remaining segment.49 Euclid's purely geometrical, constructive proof amounts to the following: Let AB be the...
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Greek Science of the Hellenistic Era: A Sourcebook

Georgia Lynette Irby-Massie, Paul Turquand Keyser - History - 2002 - 392 pages
...proportion used in the Parthenon, which we call (sqrt(5)+1)/2 = 1.618...] To cut a given straight line so that the rectangle contained by the whole and one...segments is equal to the square on the remaining segment. Let AB be the given straight line (Figure 2.2); thus it is required to cut AB so that the rectangle...
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Greek Science of the Hellenistic Era: A Sourcebook

Georgia Lynette Irby-Massie, Paul Turquand Keyser - History - 2002 - 392 pages
...proportion used in the Parthenon, which we call (sqrt(5)+1)/2 = 1.618...] To cut a given straight line so that the rectangle contained by the whole and one...segments is equal to the square on the remaining segment. Let AB be the given straight line (Figure 2.2); thus it is required to cut AB so that the rectangle...
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Mathematical Journeys

Peter D. Schumer - History - 2004 - 199 pages
...particular. Chapter II. Proposition ll of Euclid's Elements directs us "to cut a given straight line so thai the rectangle contained by the whole and one of the...is equal to the square on the remaining segment." In modern parlance, we are asked to find a point -v along a unit line segment such that the ratio l...
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Mathematics and the Divine: A Historical Study

Teun Koetsier, Luc Bergmans - Mathematics - 2004 - 716 pages
...that this proportion occurs: in Book II, prop. 11, Euclid instructs us to "cut a given straight line so that the rectangle contained by the whole and one...is equal to the square on the remaining segment". 10 This implies that the division of AB in H must be made according to extreme and mean ratio (AE ...
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A History of Abstract Algebra

Israel Kleiner - Mathematics - 2007 - 184 pages
...algebraically as (a + b)2 a2 + 2ab + b2. Proposition II. 1 1 states: "To cut a given straight line so that the rectangle contained by the whole and one...is equal to the square on the remaining segment." It asks, in algebraic language, to solve the equation a(a x) x2. See [7, p. 70]. Note that...
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The Thirteen Books of Euclib's Elements.

Euclid - Geometry - 1925
...us independent of any recollection of results. PROPOSITION 3. If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment. For let the straight line...
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Greek Science of the Hellenistic Era: A Sourcebook

Georgia Lynette Irby-Massie, Paul Turquand Keyser - Foreign Language Study - 2002 - 392 pages
We all want to understand the world around us, and the ancient Greeks were the first to try and do so in a way we can properly call scientific. Their thought and writings laid ...
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