162 KANSAS UNIVERSITY QUARTERLY. 
ment OP is 100 times the reciprocal of 119, and it only remains to 
move the decimal point two places to the left, when the result is 
the desired reciprocal of 119. Thus the reciprocal of any number 
within the physical limits of the particular instrument may be 
found. Mathematically there is no limit, as the spiral makes an 
infinite number of turns about the pole and extends an infinite dis- 
tance outward. 
MULTISECTION OF ANGLES. 
First Method.—1. To Bisect an Angle: Take any angle, LOD, 
Fig. 8. Place the instrument as before, initial line coinciding with 
OL, pole at vertex of angle. Strike a spiral arc at A, across OD. 
Draw arc AB and set off BC=-OB from point B. Draw the arc 
EC, connect E to O, and OE bisects angle BOA. For, by the 
properties of the spiral, / COE} / BOA, since the radius OE 
was made —2 OA, the angles being inversely as the radii. 
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Migs. 8. 
2. To Trisect an Angle: Similarly, if it is desired to trisect 
the angle, set off OB three times and MOG=i BOA. Orif any 
fractional part be desired, as the one-fifth or one-seventh, set off 
this ‘‘primary radius,” OA, a number of times equal the denomina- 
tor of the fraction, as five or seven. 
3. In General: For any fractional part of the angle, set off OA, 
or OB, a number of times equal to the fraction inverted. 
Second Method. —Better than this, the same result may be reached 
as follows: Take, for illustration, the case of trisection again. As 
before, draw the arc AB, Fig. 8, and set off the primary radius, 
OB, three times. Then turn the instrument till the spiral passes 
