1915-16.] The “ Geometria Organica” of Colin Maclaurin. 107 
0 1 and 0 2 are two fixed points, P any point on a fixed line l. If 
OfiPN = a is constant and Q is taken on PN so that 0 2 QN is constant, = /3, 
then the locus of Q is a cubic . 
[We may note that, if O x P and 0 2 Q cut in R, the locus of R is a segment 
of a circle on 0 1 0 2 . Hence another method of generating the curve. 
Of course, Maclaurin is restricted to the use of linear loci only.] 
Maclaurin first shows that we may, without loss of generality, suppose 
a — /3 = ~, so that 0 1 P and 0 2 Q cut on the line at infinity, and the lemma 
is a particular case of Prop. VII. 
For draw 0 1 Bl r 0 1 0 2 as in fig. 20, and make 0 1 0 2 B = ~ — /3, so that B is a 
fixed point. 
Draw O x R parallel to 0 2 Q, meeting PQ in R, so that, by Lemma I, R 
generates a straight line. Draw RS parallel to OjB, and QS perpendicular 
to 0 2 Q ; also RT parallel to 0 1 0 2 , cutting 0 2 Q in T. Then SRTQ is cyclic, 
and RTS = RQS = ~~/3= 0,0^. 
But RT = 0 , 03 . Hence A RTS ~ A Oj 0 2 B and RS = 0,B is constant. 
