Engineering Calculus Notes 146

Engineering Calculus Notes 146 - 134 CHAPTER 2. CURVES (d)...

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134 CHAPTER 2. CURVES (d) x 2 + 4 x 16 y 2 + 32 y + 4 = 0 Theory problems: 3. Show that Equation ( 2.2 ) (the statement of Prop. 13, Book VI of the Elements ) is equivalent to the standard equation for a circle. History notes: Spiral of Archimedes: Archimedes in his work On Spirals [ 3 ], studied the curve with polar equation r = ( a a positive constant) (see p. 150 ). 4. Quadrature of the Circle: According to Heath [ 24 , vol. 1, p. 230] and Eves [ 13 , p. 84], Archimedes is said to have used the spiral to construct a square whose area equals that of a given circle. This was one of the three classical problems (along with trisecting the angle and duplicating the cube) which the Greeks realized could not be solved by ruler-and-compass constructions [ 24 , vol 1, pp. 218F], although a proof of this impossibility was not given until the nineteenth century. However, a number of constructions using other curves ( not constructible by compass and straightedge) were given. Our exposition of Archimedes’ approach follows [
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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