On the Curves of Trisection. 349 
K, where the tangent oK touches the circle. And this is easi- 
ly found by taking the secant Co in the compasses, on C as 
a centre, and drawing the arc oS, and then erecting on I the 
perpendicular In. The intersection of this perpendicular 
and the arc oS gives the point n, to which is to be drawn 
from the centre the line Cn. This line cuts the arc IF in 
the point K, where the tangent touches the circle, giving the 
arc IK one third of the arc IF, and measuring one third of 
the angle to be trisected. Bisect then KF, or set off IK 
from K towards F and you obtain the point L, to which 
draw CL, and the angle ICF is trisected without the aid of 
the exterior semi-circle. 
It is obvious that a similar curve may be formed on the 
other side of the diameter DB, and the two curves together 
would complete the curve of secants, forming a kind of oval, 
with a point at D, as in figure 7. 
Il. The Trisecting Curve of Sines. 
Two semi-circles being drawn, with radii as one to two, 
Bmo in figure 3, is a part of the Trisecting Curve of Sines, 
the property of which is, that HC is equal to Hm. 
--1. ‘This curve may be conceived to be generated by the 
motion of a point, as follows. Let EB be a radius moving 
on E-asa'centre to I, and G, and further, and let a point 
move with this radius, setting out at B, distant CF or the 
radius from ‘the interior semi-circle, which point keeps al- 
ways atthe same distance, as measured on the moving radi- 
us, from the circumference of the interior circle anti! it ar- 
rives at O. 
2. This curve may be described by points, as follows. 
Take three quarters of the exterior semi-circle from B, and 
divide this arc into any number of equal parts, and to each 
point of division draw a straight line from the centre. 
Then divide the whole interior semi-circle into the same 
number of equal parts, and from E draw a straight line 
through each point of the division, and raise a perpendicu- 
lar to DC at E. The intersection of these lines, and of the 
lines from the centre will give points of the curve of sines, 
through which with a steady hand the curve may be drawn. 
This curve may also be described, as follows. Extend a 
straight rule from E towards the are BIAG, and with a pair 
Vou. 1Y.....No..2. 19 
