332 On the Curves of Trisection. 
But Gm is the sine of the angle GCA; therefore ml, 
which is perpendicular also to AC from the same point m, 
is oo sine of the angle ACI, equal to the angle GCA. That 
ml is the Sine of one third of the given angle ACB. 
Be letting fall therefore a perpendicular from the point of 
intersection of the curve of sines and of the side of the given 
angle, the intersection of this perpendicular and of the arc 
measuring the given angle cuts off one third of that arc, or 
gives the point of that arc, to which point a line drawn from 
the centre will cut off one third of the given angle. 
The consideration of the nature of this curve suggests a 
method of trisecting an angle by the rule and compasses 
alone. t the angle to be trisected be ACB. Produce 
BC and draw the two semi-circles. Extend a rule from E 
to the side AC, and taking the radius of the interior circle in 
the compasses move the rule, cutting the circumference HF 
and the side AC, until the distance between them by the 
edge of the rule be found, by monarel the compasses, to be 
equal in.the ad ; that is, until Hm be equal to HC. 
mgle GCA is fund peat g the stpoint u, through it draw CG, and the 
as before to 
tt ae em e ACB. 
el oe a slight change in this cit the third part of any 
angle, not larger than 135°, may be obtained by it mechan- 
ically gure 5 the instrument has the addition of two 
rules, namely, the rule CL, (parallel with EG) revolving on 
C, where it is fastened by a pin to the rule DB, and the rule 
KN, moveable about a pin at K, where it is connected 
with the rule CL, and moveable also about a pin at m, 
where it is connected with the rule Hm; and the dis- 
tances CH, Hm, mK, and CK being each equal to the 
ees 
other, and one face of the rule CL to being in the 
straight line j Joining C and K continued. 
‘Let ACB be iven angle to be trisected. Apply the 
face of the rule D to the side of the angle CB, and the 
centre C at the angular point. Then move the sliding rule 
Hm, and of course the other moveable rules, till the point: i 
tules EG and KN are of the same width) by the side AC 
ines through the angle formed by the rules EG and KN. 
rule CL gives the line for one third of ACB, or 
cuts off one third of the angle to be trisected. For as it has 
