348 On the Curves of Trisection. 
tre, it is double GDB, an angle at the circumference. But 
s equal to DGC, they being angles formed by the 
radii cD and CG, and the chord DG ; and therefore, as 
woe has been proved to be equal to DGC , ACG is equal 
o GDB. Consequently ACG is equal to one half of GCB. 
Bisect, then, the arc GB in H, and draw CH, and the an- 
gles ACG, GCH, and HCB are equal. The angle ACB 
is therefore trisected. 
Tn like manner may any other angle, of which CB is one 
side, and the other side extending from C to any eens of 
the circumference between B and D, be trisected by dra 
ing from D a straight line through the intersection of the 
curve and of the side of the angle. By drawing this straight 
line through the point of intersection, it gives the point on 
the circumference, which is distant from A one third of the 
given arc. So that with this curve, all that is wanting in 
order to trisection, is to draw a straight line through a given 
point, and to set offa given distance. 
From the description of this curve it is evident, that an 
angle may be trisected by the rule and compasses, in the 
following manner. ACB being the proposed angle, and the 
semi-circles being drawn with the radius ieok one semi-circle 
double that of the other ; apply a straight line to D and ex- 
tend it across AC till by the compasses the distance from 
a point in AC to the circumference, as measured on the 
rule, be equal to the distance from the same point tothe cen- 
tre C. When the’ distance 0G is thus found equal to o€, 
G is the point marking one third of the arc AGB, and a 
straight line from the centre € to G will therefore cut off one 
third of the proposed angle. 
If the points o and K be connected by a straight Fads 0K 
this Tin is a tangent to the are IK, of which Co is the Se- 
cant ; that is, Co is a secant of the are measuring one third 
_of the given angle, and OK is isa tangent of the same are. “Al! 
- this is very obvious from an inspection of the instrument for 
saelis may be t risected without the ~ of the ex- 
io searbon ae fy Allthat i is necessary” is to find the poiné 
