318 On the Multisection of an Angle 



blem can be solved by pure Geometry, that is, by circles and 

 straight lines. 



There are several other curves, however, by means of which 

 an angle may be trisected, but these curves are difficult to be 

 described, as the hyperbola, trisectrix, quadratrix, &c. The 

 tri section of an angle, therefore, by means of these curves, 

 possesses little practical interest; 



The object of the writer is to point out that the properties 

 of the cycloid afford a ready means of trisecting or multi- 

 secting any angle. The cycloid differs from the other curves 

 above-mentioned in the comparative facility with which a 

 perfectly accurate figure may be obtained. Indeed there is 

 no reason why, by a simple mechanism, the cycloid may not 

 be figured with as much mathematical precision as the circle 

 itself. 



The cycloid, also, is a curve which is nearly allied to the 

 circle, and its properties, which are derived from this rela- 

 tion, are readily understood. There is another property of 

 the cycloid which the writer has not seen noticed before, 

 viz., that there is a point in the curve where its cord be- 

 comes a tangent of the generating circle, and is exactly equal 

 to the arc of the circle contained between this point and the 

 base of the cycloid, which becomes the opposite, tangent. At 

 this point, therefore, two tangents of the generating circle are 

 each equal to the arc of the circle, which they enclose, that is, 

 each tangent is equal to twice its own arc. 



This property of the cycloid appeared to the writer to offer 

 some clue to a geometrical quadrature of the circle. In this 

 expectation he was disappointed. It Avas, however, in direct- 

 ing his attention to the possible solution of this problem by 

 means of the cycloid, that he discovered that property of this 

 curve by which any angle may be readily divided into any 

 number of equal parts. 



In order to obtain a perfectly accurate figure of the cycloid, 

 the writer has designed a simple instrument, Avhich is figured 

 in the accompanying plate. It consists of two wheels, of dif- 

 ferent sizes, attached together by the same pivot, and confined 

 between two parallel bars, on one of Avhich each wheel rolls. 

 A lead point, to describe the curve, is fixed in the flange 

 of the larger wheel, exactly opposite its rolling edge. 



The tAvo wheels are in contact with each other, and, when 

 in motion, rotate in opposite directions. The upper bar is in 

 a different plane from the lower, to receive the smaller wheel. 



The side pieces of the instniment are at right angles to the 



