PALMER: THE HYPERBOLIC SPIRAL. ‘ 165 
II. The general relation for this case has already been noticed. 
For Trisection: Vf after finding W, Fig. 8, by setting to B, we 
make BH—OW, and strike a spiral arc on AB, determming N, 
then ON is again a trisector of ~ BOA. For OH=3 of the pri- 
mary radius, OA. Hence “2 HOS=2 BOA, so when S moves 
to H, through 7 2 BOA, A moves through 3 BOA, to N, making 
mIBoON—2'- 7 BOA. it S).be joimed to O the othen\trisector 1s 
drawn, making four ways in which an angle may be trisected by 
means of this instrument. 
: 2 : a n-—a 
And so, in general terms, since either the () or the ( jen 
nN nN 
part gives the desired (= )en part of the angle on the drawing, 
n 
: n n : : : 
either (>) or ( ) times the primary radius may be used along 
n—a 
OL to determine it. 
III, With the curve in the position YKM, at the point of tri- 
section, K, the radius OY was equal 3 OA, or YA=1 OA; for, then, 
total angle B'OA=B’ OK+KOA=-0+1 0=4 0. .. The corres- 
ponding radius vector OY=3 the radius for 0, =? OA: 
For bisection YA becomes } OA, in the same way. Continuing, 
if we make YA—! OA, the 1 of the angle is determined, and so on. 
So that we have in terms of the primary radius and @ the following 
table of relations: 
(a) (b) (c) (d) 
Fraction of OA, Fraction of angle  Interceptonother — Ratio of intercepts, 
measured from A, measured downward side of angle, : 
YA. from A. 2 AOK. OM. Oa): 
1 
t I I 2 
1 1 
3 z 2 6 
1 1 
4 3 3 12 
1 1 
5 4 4 20 
1 1 
6 5 5 30 
etc. etc. etc. etc 
2 2 3 16 
5 3 2 4 
2 2 5 S65 
i 5 P 4 
3 3 4 28 
7 4 3 
etc etc. ere: ELC 
3 2 2 10 
5 2 3 9 
4 1 5 
5 4 4 16 
4 4 3 aL 
7 3 4 16 
5 5 2 14 
fi 2 5 25 
6 1 16 
i 6 6 36 
