by means of the Cycloid. ^19 



parallel bars, and project beyond tliera. They are intended 

 to rest on the paper, or plain surface, on Avhicli the curve is to 

 be figured. The bars arc tongucd to receive the wheels, which 

 are grooved to retain them in their true position. . 



The larger wheel rolls on the lower bar, and exactly corre- 

 sponds with it, and, when the instrument is'used, there is just 

 sufficient space between the large wheel and the paper to allow 

 free motion to the marking point. The parallel bars are 

 so placed in contact with the wheels as to prevent any sliding 

 motion. The instrument, as figured in the plate, is reduced 

 one half. The curve itself was figured by the instrument, and 

 has not been reduced. 



PROBLEM. 



To trisect an angle by means of the cycloid. 



Let AB be the axis, BC the base, and AC the curve of a 

 semi-cycloid ; let D be the centre of the generating circle, 

 and let ADE be any angle which it is proposed to trisect. 



Through D draw Dj\I parallel to BC and intersecting the 

 circumference of the circle at F. Through E, di'aAv EG 

 parallel to BC and intersecting the curve of the cycloid at G. 

 Take EH, equal to two-thirds of EG (VI. 9. Euc.), and take 

 DI = EH, join IH. 



Then, because EH is parallel and equal to DI, III is 

 parallel and equal to DE (I. 33. Euc). 



Upon I as a centre, through H describe an arc of a circle 

 HK, intersecting the cycloid curve at K, join IK and draw 

 DL parallel to IK, and intersecting the circle AFB, at L join 

 LK. Then since DL is equal and parallel to IK, LK is equal 

 to DI, is equal to EH, is equal to two-thirds of EG. But 

 EG is equal to the arc AE, and LK is equal to the arc AL. 

 Therefore the arc AL equals two-tliirds of the arc AE, which 

 is therefore trisected at L. 



In the same way any angle may be divided into any other 

 number of equal parts. 



Note. — It is perhaps worthy of notice that the cycloid 

 curve affords a very near a})proximation to a true quadrature 

 of the circle. When the generating circle advances from the 

 point C at the extremity of the base through one-fourth of a 

 revolution, the generating point comes to M and the angle 

 CMO = 29° 43', which is so near to 30° that the triangle 

 CMN is very nearly an equilateral triangle. On this supposi- 



