; describing the curve, as in figure 2. 
348 On the Curves of Triseetion. 
tre, it is double GDB, an angle at the. circumference. But 
GDB. is equal to DGC, they being angles: formed by the 
radii CD. and CG, and. the chord DG ;, and) therefore, as 
ACG has been proved to be equal to DGG; ACG. is. equal 
to GDB, | Consequently ACG is equal to one half of GCB. 
Bisect, then, the are GBin H, and draw CH, and the, an- 
gles ACG, GCH, and HCB are equal..,,Fhe angle ACB 
is therefore bidideterdy ' 
In like manner may any oth serangle, of which CB is one 
side, and the other side extending from C to.any point.of 
the circumference between Band D, be.trisected hy draw- 
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 civcumaference, which is distant from A one third, of the 
given arc. So that with this carve, all: that is wanting. in 
order to trisection, is to draw a straight line through a given 
point, and to set offa given diatanver ry 
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-cireles being drawn with the radius of, one semi-circle 
_ double that of the other; apply, a straight line to Drand 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 to the cen- 
tre C. When the distance oG is thus found equal to,eC, 
G is the point marking one third of the. are AGB, and a 
straight line from the centre C to.G will therefore cut, off one 
third of the proposed angle. 
If the points o and K he connected by a, straightdine, oK, 
_ this line is. a tangent to the arc IK, of which Co. is the Se- 
cant 3 that i is, Covis a secant of the are measuring one third 
of the given angle, and oK isa tangent of the same. are. All 
this is very obyious from an inspection of the instramentifol 
Or) CK, being ) by, the 
aint 
construction, in figure 1, equal to KG, and, Go. equal to.0G, 
_/and CK being the radius of the interior semi-circle; itis. evi- 
__dent, that oK is the, tangent and Co. the secant of the’arc KI. 
The curye of secants being drawn, the angle ICF, and 
' any other angle, may be trisected without the aid of the ex- 
terior semi-circle. “All that is necessary is to find the point 
