474 Geometrical Construction for rectifying any Arc. 

 Whence also, incidentally, BAT = ia, 



EAT = J;*, 



HAT = Ja, 



Now, ABC = 90°~f) 



- ^therefore ABE = 90°. 

 and CBE = | ) 



Similarly 



AEH=AEa+GEH=EAO+GEH 



= 90 p <-? + ^=90°. 

 4 4 



(2) AB = 2 sin ^, seen by dropping same perpendicular 

 from on AB. Therefore 



AE = 2 sin ~ -*- cos *- = 2 2 sin ^ 2 = AK 

 2 4 2 2 



(AH = 2 2 sin ^-r- cos -^ = 2 3 sin — not needed here 



though; 

 AI=AE-^cos ^2 =2 2 tan ^- 2 . 



It is interesting to note that with A as origin and AT the 

 initial line, the points B, E, H, . . . all lie on the curve whose 



, .. a. sin 6 -^ , . , 



polar equation is r= — -~ — . ±or, taking any one oi the 



radii vectores, say AH, when 0=^ then 2 3 = ^' whence 



AH = g sin 0.] 



