1915-16.] The “ Geometria Organica” of Colin Maclaurin. 133 
Let O be a fixed point in the plane of a curve C, on which P and Q 
are two infinitely near points. PP X is the tangent at P to the curve, and 
OP x is drawn perpendicular to P^P. Then as P moves on its locus Pj 
generates a curve, the pedal of the given curve for the pole O. 
§ 48. Prop. IX. 
Draw PN X ± r OP, and QN X ± r OQ ; OQ x ± r QQ X ; OP 2 ± r P t Q r 
Then the following pairs of similar triangles arise : — 
AOPPj » AQjPjR,, 
A A PRjPj, 
AOR x P a A Qj.RjPj. 
Also P 1 Q 1 R 1 , PQR, P x OP 2 , POP ^ are similar ; and 
OP/OP! = OPJOPg. 
Denote OP by r, and OP x by p. If the curve C is given by the equation 
fix, y) — 0 (1) referred to axes through 0, 
op =V(* 2 +y 2 )> 
PP 2 has for equation 
Y-y = (X-x)y' (2) 
and 
p = {y-xy')IV( l + y' 2 ) ( 3 ) 
The elimination of x, y, and y' leads to a single relation 
(4) 
which is sufficient to characterise the curve, and it is this equation which 
Maclaurin calls the Radial Equation of the curve. 
Cor. 1. From the locus of P x may be similarly described its pedal, the 
locus of P 2 . We may thus derive an infinite series of curves (the positive 
pedals of C). 
From the radial equation of C can be easily deduced the radial equation 
of the locus of P 1 . 
Let p ± and rq correspond to the locus of P r 
Then 
pJ r i =p/ r i 
r 1 = p i 
••• P=“> \> rmr^/p, (5) 
Cor. 2. The series of curves may be continued in the opposite sense, 
viz. by drawing PN X and QN 1 perpendicular to OP and OQ, and finding 
the locus of Nj (the negative pedal) ; or N 2 may be found by drawing 0N 1? 
so that PONj is the complement of OPP r 
