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Elementary Theory of Equations

Elementary Theory of Equations


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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 edition. Excerpt: ...of a cube whose volume shall be double that of a given cube. Take the edge of the given cube as the unit of length and denote by x the length of an edge of the required cube. Then x3--2 = 0. Since no one of the divisors of 2 is a root of this cubic equation, the theorem stated in 8 implies the impossibility of the duplication of a cube by ruler and compasses. 10. f Cubic Equations with a Constructive Root. It remains to prove the theorem in 8 from which we have drawn such important conclusions. Suppose that (20) x3 + oa? + ffx + 7 = 0 (o, 3, y rational) is a cubic equation having a root Xi such that a line of length Xi or--Xi can be constructed by ruler and compasses. We shall prove that one of the roots of (20) is rational. The construction is in effect the determination of various points as the intersections of auxiliary straight lines and circles. Choose rectangular axes of coordinates. The coordinates of the intersection of two straight lines are rational functions of the coefficients of the equations of the two lines. To obtain the coordinates of the intersection of the straight line y = mx + b with the circle (x-p)2 + y-q)2 = r2, we eliminate y and obtain a quadratic equation for x. Thus x, and hencealso y, involves no irrationality (besides irrationalities in m, b, p, q, r) other than a square root. Finally, the intersections of two circles are given by the intersections of one of them with their common chord, so that this case reduces to the preceding. Hence the coordinates of the various points located by the construction, and therefore also the length Xi of the segment joining two of them, are found by a finite number of rational operations and extractions of real square roots, performed upon rational...


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Product Details
  • ISBN-13: 9781154987461
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 40
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1154987469
  • Publisher Date: 01 Oct 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 91 gr

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