PALMER: THE HYPERBOLIC SPIRAL. 7) 
cure a desired value of the constant of the spiral and to have OL 
the initial line of the spiral, as it should be. It is always desirable 
to obtain a spiral with a certain definite value for the constant, 
since this constant fixes the size of the spiral. 
ar: 
This can be attained, for the case of 6 measured in degrees and 
rin inches, by choosing a suitable space along the horizontal line 
of the drawing, as OM, to represent 360 (degrees) and making the 
constant k of the hyperbola equal tothe desired c of the spiral. 
Then by dividing the space OM into as many equal parts as the 
360 degrees of Fig. 3 and drawing a vertical to the hyperbola 
at each division point, each radius vector of Fig. 3 will have a 
vertical corresponding to it on Fig. 2, from which its true length 
can at once be set off by the dividers. 
A serious practical difficulty is found, however, in trying to set 
off accurately the lengths of the long verticals approaching OY in 
Fig. 2, for a very slight error to the right or left in drawing one of 
these long ordinates to the hyperbola produces an error many times 
as great in the length, which results in a serious irregularity in the 
spiral. This method has given satisfactory plottings, using the 
greatest care, but necessitates extreme care and the most skillful 
use of the drawing instruments. 
BY CALCULATING THE RADII. 
Apparently the simplest. and most natural way of constructing 
this spiral would be to merely figure by simple arithmetic the 
