20 
Proceedings of Royal Society of Edinburgh. [sess. 
Use of the Quartic Trisectrix as a Trisecting Template. 
If we construct a template of the form OXYO (fig. 4), we may 
trisect any angle U 0 X with it as follows : — Place the template 
with 0 at the vertex of the angle and O X along one arm, and let 
its curved edge meet the other arm in B. Describe circles with 
O and B as centres and 0 F as radius. If A be that intersection 
of the two circles which falls within U 0 X, then U 0 A is one-third 
of the angle U O X. This property of the curve is well known. 
See Basset, Elementary Treatise on Cubic and Quartic Curves 
(Cambridge, 1901), § 297. 
From the present point of view the characteristic property of 
the quartic trisectrix is that it is generated by the intersection of 
two radii, through the fixed points 0 and O, moving so that twice 
the angle B O X is equal to thrice the angle BOX. 
It is obvious that another trisectrix would be obtained by 
causing B to move so that the angle B 0 X is thrice the angle BOX. 
The locus of B is then the cubic trisectrix of Maclaurin,* to which 
we have already referred. 
* See Maclfturin’s Fluxions (Edinburgh, 1742, vol, i. p. 262). 
{Issued separately February 17, 1902.) 
