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 0S 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 partof 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. t 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 until it ar- 
tives at 
2. This curve may be described by points, as follows. 
Take three quarters of the exterior semi-circle from B, and 
divide this are into any number of equal parts, and to each 
int of division draw a straight line from the centre. 
hen divide the whole interior semi-circle into the same 
number of equal parts, and from E draw a straight li 
through each point of the division, and raise a perpendicu- 
lar to DC at om 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. 
his curve may also be described, as follows. Extend a 
Straight rule from E towards the arc BIAG, and with a pair 
Vor. I No. 2. 19 
seces oO. 
