PALMER: THE HYPERBOLIC SPIRAL. I7t 
In Fig. 13, let BPA, etc., be an indefinite arc of the given radius. 
Lay off OM equal the given length of the arc to be found, and mark 
the point a. Now join any point of the arc drawn, as P to M. 
Bigs te: 
Draw aC from a, parallel to MP, thus determining C. With ra- 
dius OC determine W, and produce OW to A. Then AB is the 
required arc. 
Be Ab OP. OM 
WD? OCs" Oa 
instrument. 
13) But arc WD=Oa, by the property of the 
Arc AB=OM- . 
MECHANICAL PROPERTIES OF THE SPIRAL INSTRUMENT. 
Although the mathematical properties noticed are interesting, 
and some of them make possible a number of drawing board oper- 
ations by use of the instrument described, their bearing from the 
standpoint of actual utility is not to be compared to that of the 
curve’s physical characteristics. The great value of the Hyperbolic 
Spiral lies in the fact that it possesses just the proper rate of varia- 
‘tion and range of curvature to make it the ideal outline for the 
drawing instruments commonly known as ‘‘irregular curves.” 
Irregular Curves. 
Irregular curves, so-called, are used by draftsmen for drawing 
smooth curves through ranges of points on a drawing determined 
by plotting or otherwise, where these points locate some curve of 
definite character not circular. These instruments are made by 
cutting out a variety of curves, usually of no definite mathematical 
character, from uniform thin sheets of pearwood, hard rubber, 
transparent celluloid or metal, as may be desired. The outiine of 
one of these instruments usually consists of many kinds of curves 
of different degrees of curvatures, chosen either entirely at a ven- 
ture, or perhaps from some experience with curves of approximately 
the same form. Often the ornamental appearance has more influ- 
