10 
Proceedings of Royal Society of Edinburgh. [sess. 
Trisectrix of Maclaurin { 2x(x 2 + y 2 ) = a(y 2 - 3x 2 ) ; see Basset’s 
Cubic and Quartic Curves , § 133}, and the Quartic Trisectrix 
((x 2 -\-y 2 ) 2 - 3a 2 (x 2 + y 2 ) - 2a 3 £C = 0), which is a particular case of 
Pascal’s Limagon. We shall see presently that Mr Miller’s instru- 
ment is closely related to the last of these curves. 
The interest excited by the invention of Peaucellier’s inverting 
linkage in 1864 led to the discovery that the division of any given 
angle into any given number of parts could be effected by a linkage. 
Two distinct forms of trisecting compass, one due to Sylvester and 
the other to Kempe, are described and figured in a little work 
published by the latter in the Nature Series, under the title How 
to Draw a Straight Line (Macmillan, 1877). 
The Sextic Trisectrix. 
Miller’s Trisector consists of two bars jointed at A, so that 
OA = AB = 2 a, say. 
To the middle point C of O A is fixed a perpendicular bar C P. 
To trisect an angle X O K, O is placed at O and B slid along 
O X until the intersection of B A P and C P, viz., P, falls on OK; 
then A 0 B is one-third of K 0 X. 
To find the locus of P, when B moves along 0 X, we have 
x = a cos 6 - a tan 2 0 sin 0 = a cos 39 sec 2 0 ; ) 
y — a sin 0 + a tan 2 0 cos 0 = a sin 30 sec 2 0 . ) 
