1901 - 2 .] Application of Miller s Trisedor. 303 
•centre of a small cylindrical liole made at A through the piece 
AIFD. 
The method of using the Trisector so as to divide the angle, as 
proposed, into five equal angles, is to insert through that hole at A 
a sharp-pointed cylindrical pin, fitting it, but not too tightly, into 
the vertex of the angle, and then to move the point B of the piece 
B H G along the other side, B A, of the angle until the edges or 
borders I R and E G, which are straight, meet in a point which 
may be termed 0 in the perpendicular K Q. The straight line 
A 0 is then drawn from the vertex A, and may be produced to N ; 
and the angles C A 0 and 0 A L are bisected by the lines A M and 
AP. The angle BAL is, by this operation, divided into five 
equal angles, namely, BAG, CAM, MAO, 0 A P, and PAL. 
For 
As the sides A C and B C of the triangle BAG are, by the 
•construction of the Trisector, equal, so that triangle is isosceles, 
and its angles B A C and ABC are equal. 
But as the point C is, by construction, in the straight 
line B C E G, so the angle A C 0 is exterior to the isosceles 
triangle B A C, and is therefore equal to twice that angle 
BA C. 
Again, as by construction the sides C I and A I of the right- 
angled triangles C 0 I and A 0 I are equal, and as they have 
a common side I O, so their corresponding angles I C 0 or 
AGO and I A O or C A O are also equal. 
But the angle A C O is equal to twice the angle BAG. 
Therefore the angle C A O is also equal to twice that angle. 
Further, as the right-angled triangles 0 1 A and OKA 
have equal sides, A I and A K, and the same hypotenuse A 0, 
so their corresponding angles I A 0 or C A 0, and OAK or 
0 A L, are equal to each other. The angle 0 A L is therefore 
equal to twice the angle B A C. 
Moreover, as the angles CAM and MAO are halves of 
the angle C A 0, and the angles 0 A P and PAL are halves 
of the angle O A L, so each of them is exactly equal to 
the angle BAG. The angle B A L is therefore divided into 
five equal angles — namely, B A C, CAM, MAO, 0 A P, 
and PAL. 
