1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 143 
The negative pedals are given by 
The first negative pedal is the parabola 
The second negative pedal is given by 
whose arcs can be expressed by straight lines. Only these may increase 
beyond all limit, as the curve goes to infinity with the parabola. 
We thus form two sets of curves : in one set the arcs can be expressed by 
parabolic arcs and straight lines, and in the other set by straight lines only. 
§ 57. Prop. XIX. 
The pedals of the equilateral hyperbola 
x 2 - y 2 = a 2 , 
or 
p/r = a 2 /r 2 . 
The first positive pedal is the lemniseate 
( x 2 + y 2 ) 2 = a 2 (x 2 - y 2 ), 
or 
jp/r = ( r/a ) 2 . 
Two series of curves are obtained, in one of which arcs are expressible 
by hyperbolic arcs and straight lines, and in the other by arcs of lemni- 
scates and straight lines. 
§ 58. Prop. XXI. 
The radius of curvature of the curve 
Maclaurin proves the more general formula p — r dr /dp, from which the 
formula is easily deduced. 
[§ 59. Remarks on Pedals. 
B p 
0 
Fig. 44. 
