1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 139 
x 2 — if 2 — a 2 
p/r = a 2 lr 2 (n= — 2 ), 
the pole being at the centre. 
(V.) The Cardioid (first positive pedal of the circle), 
p/r - (r/2a)i 
(i.e. p 2 = r 3 /2a). 
(VI.) The Lemniscate (first positive pedal of the rectangular hyperbola 
of IV), 
or 
p/r = r 2 / a 2 
(x 2 + y 2 ) 2 = a\x 2 -fi) 
in Cartesian co-ordinates. 
(VII.) Maclaurin also gives later the logarithmic spiral* whose p — r 
equation is 
p/r = C ; n = 0 , 
(vide Section IV), but this is not an algebraic curve. 
§ 53. Prop. XIV. 
Property of the curve p/r = (r/a) n . 
Let B be the point p — r = a\ this point is a vertex on the curve and 
its pedals. 
The following relation holds (vide fig. 40) : — 
L PiOQi = (n+\) L POQ. 
Dem. 
Let the polar co-ordinates of P be (r, 0) and of P 1 (p, (p). Since 
pjr = ( r/a) n , therefore 
dp / , x dr 
— = (n+ 1 )—. 
V r 
But by the pedal transformation 
dO deb 
and therefore 
d<p = (n + \)d0 ; etc. 
Cor. 1. In particular, if 0 and <p are measured from the initial position 
OB, then 
cf> = (n + 1)0. 
§ 54. Prop. XV. 
Maclaurin’ s theorem regarding the rectification of such curves. 
If P traces out the curve p]r = ( r/a) n , starting from the vertex B, while 
* Since p/r=pjr 1 = etc., a logarithmic spiral can be described to pass through the 
points P, P l5 P 2 , etc. 
