352 On the Curves of T'risection. 
But-Gm, is the, sine of the angle GCA ; therefore ml,.; 
which is perpendicular also to AC from the same. point m, 
is the sine of the angle ACI, equal to the angle GCA. That 
is, mI is the Stne of one third of the given angle ACB.. . 
By 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 are 
measuring the given angle cuts off one third of that are, 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. Let 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 cirele 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 meansof the compasses,, to be 
equal to the radius ; that is, until Hm be equal to H@, 
Having thus found the point H, through it draw CG, and the 
angle GCA is found, and may be demonstrated. as before. to 
equal to one third of the given angle ACB. / sili 
By aslight change in this instrument the third part of any 
angle, not larger than 135°, may be obtained, by it. mechan- 
ically. In figure 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 
other, and one face of the rule CL towards L being in the 
straight line joining © and K continued., 
Let ACB be the given angle to be trisected. Apply the. 
face of the rule DB 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 m 
is on the side AC, which can be easily determined, (if the . 
rules EG and KN,are, of the same width) by the side AC « 
assing through the angle formed by the rules EG and KN. 
hen the rale CL gives the line for one third of ACB, or 
cuts off one third of the angle to.be trisected.. For as it has « 
