142 Proceedings of the Koyal Society of Edinburgh. [Sess. 
The pedal of the cardioid is given by 
p/r - (r/a)$ 
and 
arc BP 2 = - (arc BP + P 1 P). 
Hence it cannot be measured by a straight line alone, nor by an arc 
of a circle alone save for the complete curve up to 0, when 
BP 2 0 = | BPO = Stra/i. 
The whole curve cannot be cut in any given ratio, for otherwise the 
quadrature of the circle would follow. 
The third positive pedal 
r \aj 
has an arc 
4 
BP, 
(BPj + PjP, 
= i(2 chord BP + line P,P 2 ) 
o 
and 
BP s O = ^BO = ^. 
3 3 
In general the nth pedal is given by 
p/r — (r/a) ll(n+1) . 
If S n denote the arc of the pedal from B to 0, 
s„=^Js n , 
n 
n + 1 n—\ 
when n is odd, 
,S„_4 
n n - 2 
_ («+!)(«-!) . ■ ■ 2 0p 
n(u - 2) ... 1 
_ (n + l)(w - 1) . . . 3 7r a 
n(n - 2) ... 2 Y 
when n is even. 
The areas of the pedals are also discussed. 
§ 56. Prop. XVIII. 
The 'pedals of the straight line 
r \a 
The first positive pedal reduces to a point. 
