350 On the Curves of Trisection. 
of compasses, opened to the distanee of CE, or radius, of 
the interior circle, set off this same distance from. the point 
of intersection of the rule, and interior circle towards, the 
exterior circle. If, for instance, the rule intersect the cir- 
cle in H, then set off the radius EC or HC, from H to m 
Then will m be one point of the curve. In this way. may 
a sufficient number of points be found to enable one with a 
steady hand to draw the curve. 
3. This curve may also be described mechanically, by. a 
continued motion, as follows. Let a straight rule CH in 
figure 4, be fastened by a pin at Cso asto be moveable 
about C. Let Hm be another rule of the same length, with 
a hole at m for a pencil to pass through. Let this rule be 
pinned to the rule CH at H so as to move about H, as CH 
moves about the centre C. Let EG be another straight 
rule, a little longer than three times CH, with a slit through 
the length of it, and moveable about E, where it is fastened 
with a pin, the distance. EC being equal to CH. In the 
slit of this rule let Hm be placed, so as to slide in it with 
ease and yet with accuracy. 
The. instrument being thus constructed, put a_ pencil 
through the. perforation at m, and. by pushing the rule with. 
it to Bon one side and to o on the other, the curve will be 
described. But when the point.m is_at 0,.the rule CH will 
be in the position CE, and the rules EG and Hm. in the po- 
sition Eo. 
As the largest angle, which can. be trisected by. means of 
this curve, is an angle of 135°, if the given angle is: larger 
than 135°, it must be bisected, and the parts trisected sepa- 
ratel 
List ACB, in figure 3, be the given angle, to be trisected. 
Describe the curve of sines, which intersects the side CA 
inm. From) the. point m, raise mI perpendicular to, AC. 
The perpendicular ml is the sine of one third. of the angle 
ACB. From the centre C draw Cl. Of course the. are 
Al is one third of the arc AIB, and the angle ACI is. one 
third of the angle ACB. Wherefore, by bisecting IB, or 
setting off the arc AI towards B, and joining the point thus 
found with the centre C, the angle ACB will be trisected. .. 
The demonstration of which is as follows. 
