. Memoirs and proceedings of the Manchester Literary & Philosophical Society. ince any value may be assigned to X. Some of the simpler cases arc as follows : i8 Gee and A damson, Trisecting an Angle. (a) Let X = i ; the conic is then the parabola 2x2 + $crx + by = 0. (b) Let \= — I ; the equation then becomes : x~ + ax - yr - by = conic is thus a rectangular hyperbola, and thesolution of the trisection problem is the same as that ofChasles (Fig. J). (c) Let A =3/2 ; then gx2 + 14ax +y- + 6by = is the equation of an ellipse of a form convenient forthe purpose of construction. V. Us
. Memoirs and proceedings of the Manchester Literary & Philosophical Society. ince any value may be assigned to X. Some of the simpler cases arc as follows : i8 Gee and A damson, Trisecting an Angle. (a) Let X = i ; the conic is then the parabola 2x2 + $crx + by = 0. (b) Let \= — I ; the equation then becomes : x~ + ax - yr - by = conic is thus a rectangular hyperbola, and thesolution of the trisection problem is the same as that ofChasles (Fig. J). (c) Let A =3/2 ; then gx2 + 14ax +y- + 6by = is the equation of an ellipse of a form convenient forthe purpose of construction. V. Use of the Quadratrix of Hippias, the Sine Curve andthe Spiral of A rdiimedes. In order to construct these curves, the circumferenceof a circle must be divided into a large number of equalparts. Consequently, these curves may be used to divideany angle at the centre of the circle into any number ofequal parts. Hippias of Elias (about 420 B C.) used the Quadratrixin connection with the quadrature of a circle. The curvemay be defined as the locus of a point P (Fig. 12) which. ON MB Fig. 12. Quadratrix, Sine Curve Spiral of Archimedes. MaricJiesier Memoirs, Vol. lix. (1915), No. 13. 19 is the intersection of a radius vector OR and an ordinateNP, each of which lines moves at a uniform rate suchthat N describes the diameter BC in the same time as Rdescribes the semi-circumference BC. In other words,BN is the same fraction of the radius OB as the arc BRis of the quadrant BD. If RS be drawn parallel to OB to meet the ordinateNP at S, the locus of 5 is a sine (or cosine) curve. Thiscurve was used for trisection purposes by Dinostratus, adisciple of Plato (see Pappus), and by Tschirnhausen(1651-1708), to whom Saxony owed its porcelainmanufactory. The Spiral of Archimedes is also shown in Fig. is the locus of the point X on the radius vector ORsuch that OX increases uniformly as the angle BORincreases uniformly. In the particular curve shown, OXis equal to BN. The use of these
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectscience, bookyear1888
