Ruler & Compasses

Ruler & Compasses

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 edition. Excerpt: in fig. 25. Then P LMP' = A P BCP' = D PBCP' = PM' L'P' = P' L'M'P, so that PP' belong to the involution determined by the pairs LL', MM'. This is a projective property, and is therefore true of the range cut out on a transversal by the sides of a quadrangle and any conic through the four vertices; hence all such conics cut the transversal in pairs of conjugate points of the same involution determined by the pairs of opposite sides. But these pairs of sides are themselves particular, degenerate cases of conics through the four points, so the theorem can be stated in the simple and general form: Conies through four fixed points cut any transversal in involution. Hence if five points ABODE on a conic are given, the other point of intersection X of any straight line EX through one of the five points is determined as the conjugate of E in the involution cut out on EX by the sides of the quadrangle A BCD. It can therefore be constructed with ruler only. Thus we have a ruler construction for any number of points on the conic through five given points, for we can find its intersection with any number of straight lines through any of the given points. Thus with ruler only, and without calculation, we can plot the curve with as much accuracy as time, patience and other limitations permit, and this is the nearest approach to actually drawing the curve that we can ever get with Euclidean instruments. But if we wish to find the two intersections of a conic with an arbitrary straight line not passing through any of the five points, the problem cannot be solved with ruler only. Let it be required to find the intersections XY of a given straight line with the conic through five given points ABCDE. Take any point P on XY; join AP and determine Fig. 26. more

Product details

  • Paperback | 38 pages
  • 189 x 246 x 2mm | 86g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236513231
  • 9781236513236