. An elementary treatise on the differential calculus founded on the method of rates or fluxions. sin^ __ sin e Fig. 38. whencetherefore bcos 6 sin cp — 2 sin 8 cos 6 = sin 26; e = ^cp = i CAB. Example. 1. Find, from the equation of the conchoid, the tangents at theorigin, and thence determine when the curve has a crunode, when anacnode, and when a cusp. The Quadratrzx of Dinostratus. 261. Let the radius of the circle in Fig. 39 revolve uni-formly, completing the semicircle AEB in the same time as § XXXI.] THE QUADRATRIX OF DINOSTRATUS. 287 that required by the ordinate RP to move uniformly ov
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. sin^ __ sin e Fig. 38. whencetherefore bcos 6 sin cp — 2 sin 8 cos 6 = sin 26; e = ^cp = i CAB. Example. 1. Find, from the equation of the conchoid, the tangents at theorigin, and thence determine when the curve has a crunode, when anacnode, and when a cusp. The Quadratrzx of Dinostratus. 261. Let the radius of the circle in Fig. 39 revolve uni-formly, completing the semicircle AEB in the same time as § XXXI.] THE QUADRATRIX OF DINOSTRATUS. 287 that required by the ordinate RP to move uniformly over thediameter; the intersection P of the ordinate and the radiuswill then describe the curve known as the Fig. 39. Denoting the radius by a, the angle at the centre by 6, andtaking the origin at A, we have, by the mode of construction, - = .£. 7t ~ 2a Eliminating 6, we obtain and y = (a — x) tan 0, j, = (a-x)tan—, the equation of the quadratrix. 262. It is evident that, if AR be divided into any numberof equal parts, and ordinates be erected at the points thusdetermined, the corresponding radii of the circle will divide theangle A CP into the same number of equal parts. Hence, bymeans of this curve, an angle rflay be divided into any numberof equal parts. The curve was employed for this purpose byDinostratus, a disciple of Plato; he also employed it in thequadrature of the circle. The latter application, from which 288 CERTAIN HIGHER PLANE CURVES. [Art. 262. was derived the name of the curve, depends upon the resultdeduced below. By evaluation, we obtain CD = (a — x) tan nx 2a 2a 7t hence we have ABCD = 7t. The Witch of A guest. 263. Given a point A on the circumference of a circle
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