Engineering Calculus Notes 130

# Engineering Calculus Notes 130 - 118 CHAPTER 2 CURVES •...

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Unformatted text preview: 118 CHAPTER 2. CURVES • linear problems 3 are those involving other loci, such as spirals (see p. 149 and Exercises 4- 5 ), quadratrices (Exercise 6 ) and conchoids (Exercise 7 ). One classic problem is that of duplicating the cube : given a cube, we are to construct a second cube whose volume is twice that of the first (or some other specified multiple). Hippocrates of Chios (460-380 BC) reduced this [ 25 , p. 131], [ 31 , p. 41] to the problem of two mean proportionals : given line segments a and b , to construct two other segments, y and x , whose lengths satisfy | a | : | y | = | y | : | x | = | x | : | b | . Early solutions of this problem [ 25 , pp. 154-170] used “linear” loci, but two solutions by Menaechmus ( ca. 350 BC ), a follower of Plato, appear to be the first investigation and use of conic sections. The impossibility of duplicating the cube by compass and straightedge was first proved in the nineteenth century, using some deep algebraic and analytic results....
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