50 



INTRODUCTION. 



sides be increased without limit, the polygons will approach infi- 

 nitely near to curves, and each portion of the curve described will 

 still be perpendicular to the line passing through the point of con- 

 tact. 



207. Definition. A circle being supposed to roll on 

 a straight line, tlie curve described by a point in the cir- 

 cumference is called a cycloid. 



208. Theorem. The evolute of a cycloid is an equal 

 cycloid, and the length of its arc is double that of the por- 

 tion of the tangent cut off by the vertical tangent. 



Let two equal circles AB, BC, 

 rolling on the parallel bases DA and 

 EB, at the distance of a diameter 

 of the circles, describe with tlie 

 points F and G the equal cycloids 

 EF and EG. Draw the diameter 

 FH ; then H will be the point that 

 coincided with D, and HArzDAzz 

 EBn arc BG, and the remainders AF and GC are equal, therefore 

 2lABF=:CBG (133), and FBG is a right line (96). But FG is per- 

 pendicular to AF (134), therefore it touches EF (206), and it is 

 always perpendicular to EG (206) ; therefore EG will coincide with 

 the involute of EF, for they set out together from E, and are always 

 perpendicular to the same line FG (193), which tluey could not be if 

 they ever separated. Consequently the curve EF is always equal 

 to FG (192), or 2FB, twice the portion of the tangent cut off by EB. 



209. Theorem. The fluxion of the cycloidal arc is to 

 that of the basis, as the evolved radius to the diameter of 

 the generating circle. 



For the increment GI=:2BK, and BK : BL'.'.BG 

 : BC, and 2BK : BL: : FG ; BC, which is therefore 

 the ratio of the fluxions. 



Scholium. If the fluxion of the base be constant' 

 that of the curve will vary as the distance of the 

 describing point from the point of contact. 



