1915-16.] The “ Geometria Organ i ca ” of Colin Maclaurin. 141 
dr = dcr — — ds = do-— -ds . 
p r 
But 
ds. = dp sec if/ = (ft + 1 ) - ds . 
r 
where 
p/r = (v/a) n . 
Hence 
dsq = 0 4- l)(do- — dr), 
so that 
arc BP 3 = (n + l)(arc BB T 1 - PN^ 
= (h + 1 )(arc BNj^ + N-^P). 
( 1 ) 
( 2 ) 
The signs of BP 1} BN 1? and N X P must be attended to. Thus when n>l 
the angle BONp which is equal to (l—n)6, is of opposite sign to that of 
BOP and of BOPj ; the arcs BP X and BN X are therefore of opposite sign, and 
N 1 P has the same sign as arc BP r Numerically, the arc BP X is less than 
(n + 1 ) times the length of the line PN 1 by (?^ + l) times the arc BN r ] 
This interesting theorem Maclaurin proceeds to apply to deduce 
various conclusions regarding the rectification of the series of curves 
formed by a given curve, p/r = (r/a) n , along with its positive and negative 
pedals. 
Cor. 1. If two consecutive curves of the series admit of rectification, 
so do all. 
Cor. 2. If one of the curves admits of rectification, but not the next 
in the series, then half only of the curves of the series admit of rectification. 
Cor. 5. When the first negative pedal passes through the pole, the 
theorem for the total lengths from B may be written 
= ( n + ljoq. 
For in such a case PN X vanishes. 
§ 55. Prop. XVII. 
When the given curve is a circle of radius a/2, and the radial equation 
is p/r — r/a, the first negative pedal reduces to a p>oint B. 
Fig. 43. 
The first positive pedal is the cardioid BP V and for it 
arc BP 1 = 2 chord BP .... 
Thus half the complete cardioid = 2a, and the whole length = 4 a. 
• ( 1 ) 
