168 KANSAS UNIVERSITY QUARTERLY. 
(4) Set at M and draw OV. Get Z HOV. 
Z HOV=(6'--6") ate xis nal 
i r 
I I 
And ratio to 8, ay = ar) 
These relations, and others, may be derived readily from proper- 
ties of the curve expressed by its equation r— but none of them 
a 
Sar 2 
appear to be of a sufficiently simple form to be of any further 
interest. 
CONSTRUCTION OF REGULAR POLYGONS. 
The spiral iustrument affords an interesting means for construct- 
ing the regular polygons. There are two cases: (a) The circum- 
scribing circle given to construct a regular polygon of any required 
number of sides. (b) Having given the length of one side, to 
construct upon it a regular polygon of a required number of sides. 
(a) For this case we have evidently again merely the division of 
an angle into a desired number of equal parts. Bearing in mind 
that the angle to be divided is here 360, and hence that the two 
sides coincide in OL, Fig. 10, any desired inscribed polygon can 
Rig, 10. 
be had at once, as follows: Draw the given circumscribing circle, 
apply the instrument and mark A. Then lay off OA from O, as 
many times as the polygon is to have sides, say seven, here, ob- 
taining point M. Then draw the arc MP, and PO determines one 
side of the polygon, BC, which can now be set off the remaining 
number of times around the circle. 
