On the Curves of Trisection. 355 
these, is therefore a right angle, and FA. is perpendicular to 
AC or HC. AL 
Or the same thing may be demonstrated thus. By the 
construction GC, GA, and GF are equal to each other. 
Liet Gthen be the centre of a circle drawn through the 
points CAF. It is evident that FAC is an angle in a semi- 
circle, which is a right angle, FA is therefore perpendicular 
to HC. 
A perpendicular on the opposite side of AC, drawn from 
the same point A, will necessarily be equal to ‘AF, and cut 
off an arc equal to the arc HF’; that is, will cut off an are, 
HR, measuring one third of the given angle HCB. 
‘The whole curve therefore, BoAE, though formed by a 
complex operation, may well be called the Tr isecting 
Curve of Sines. 
By making in the same manner a corresponding curve on 
the other side of the diameter, the curve of sines will be 
completed, and the whole figure will resemble in form, 
Woveh not in properties, the Cardioide of Carre. 
In figure 8 the two trisecting curves, completed on each 
side of the diameter, are placed together. DoF np is the 
Trisecting Curve of Secants, and EmBsp is the Trisecting 
Curve of Sines. Any angle may be trisected with the great- 
est ease by either of them; as the angle ACB on the one 
side of the diameter, merely by drawing through the point 
o, (where the side AC intersects the curve of secants,) the 
straight line DG, which gives the are AG, one third of the 
arc AGB measuring the proposed angle,—or merely by: 
erecting at the point m (where the side AC intersects the 
curve of sines) the perpendicular mG, which also gives the 
are AG, one third of the are AGB.—In like manner may 
the angle HCB, on the opposite side of the diameter, be 
trisected by drawing through the point x of the curve of se- 
cants the straight line DI, or by erecting at the point S of 
the curve of sines the perpendicular sf, for by both meth- 
ods the are HI is obtained, one third of the given arc HIB. 
And by the same methods may any angle whatever, (on: 
either side of the diameter), formed with oP by AED revel 
ving on C, be trisected. 
) By inverting the position off one of eae, clirses, , (as the: 
curve of sines, so that its point B shall be at D), it/is-obvi- 
ous, that while the angles ACB, HCB may be trisected by 
