PALMER: THE HYPERBOLIC SPIRAL. 161 
Preferably this curve should be made from transparent celluloid, 
about ,3,” thick. The transparency is very helpful in permitting 
accurate setting of the instrument. 
RECIPROCALS. 
From the character of the spiral outline of the instrument, we 
have the fact that any radius of the spiral, r, Fig. 5, is equal to 100 
times the reciprocal of the angle in degrees which the radius, r, 
makes with the initial line OL. So if we wish the reciprocal of 
any number, as, say, 119, it is readily found by setting off 119 by 
means of an ordinary protractor, upward from a horizontal line, as 
shown in Fig. 6, and applying the spiral instrument. 
Fig. 6: 
Fig. 7 shows the angle laid out by the protractor. Upon this 
angle, with the pole at the vertex of the angle and the initial line 
co-incident with OL, apply the instrument as shown in the figure. 
R 
Big. 7. 
Mark with the lead pencil along the outline of the spiral where it 
crosses OR. Then measure accurately, with decimal scale, the 
distance from O to this intersecting point, A. Then this measure- 
