. An elementary treatise on the differential calculus founded on the method of rates or fluxions. adratrix. 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 H
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. An elementary treatise on the differential calculus founded on the method of rates or fluxions. adratrix. 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, and a tangent at the opposite extremi-ty of the diameter AB; if the or-dinate DR be produced, so thatPR = BC, the locus of P will be thewitch. Taking the origin at A, anddenoting the radius of the circleby a, we have DRt = x{2a — x). also, by the construction of thelocus, Fig. 40. 2a DR x therefore xj? = 4a* (2a — x) is the rectangular equation of tie curve. This curve was given under the name of versiera in a treatiseon Analytical Geometry by Donna Maria Agnesi, an Italianmathematician of the eighteenth century. § XXXI.] EXAMPLES. Examples. 1. Show, by means of the equation of the witch, that the axis of;is an asymptote, and that the curve has two imaginary asymptotesparallel to the axis of x. 2. Find the points of inflexion in the witch, and the inclination ofthe curve at these points. CI* ± !*.):?£) = * J*. The Folium of Descartes, 264. This name has been given to a cubic defined by theequation x + y — ^axy — o. The form of this curve is represented inFig. 41; the axes are tangent to the curve atthe origin. By the method employed in Art.228, the equation of the asymptote is foundto be x + y + a = o.