156 KANSAS UNIVERSITY QUARTERLY. 
PLOTTING THE SPIRAL. 
BY TRANSFORMING THE COMMON HYPERBOLA. 
The first method which suggests itself from a consideration of 
the characteristics of the spiral, is that of replotting a common 
hyperbola to polar co-ordinates. If we take an accurate plotting 
of one branch of an hyperbola, which can readily be made by the 
well known rule, and draw a number of verticals, equally spaced, 
as in Fig. 2, and then lay out the series of equally spaced radial 
lines shown in Fig. 3, a Hyperbolic Spiral may be plotted upon 
the latter, using Fig. 2 as an auxiliary diagram. 
Commencing, say, with the longest vertical of Fig. 2, lay it off 
with the dividers upon the radial line No. 3, and the next on No. 
4, and soon. The points thus marked off are points on the line of 
the hyperbolic spiral and may now be joined by a smooth curve. 
It is plain from the characteristics of this hyperbola, as well as 
from an inspection of the equation of the spiral, that the curve may 
be continued indefinitely in either direction, never reaching either 
the pole, O, or the initial line. 
This gives a correct construction for the spiral, but it is neces- 
sary to plan the diagram of Fig. 2 in a correct way in order to se- 
