160 KANSAS UNIVERSITY QUARTERLY. 
value of the angle along YH, and the corresponding radius imme- 
diately maked off on AD, or read off directly in inches and decimals 
as desired. 
PROPERTIES OF THE HYPERBOLIC SPIRAL. 
MATHEMATICAL PROPERTIES. 
This spiral is found to possess many interesting properties. If 
a spiral of any chosen size be cut from a sheet of uniform thin 
metal, pearwood, hard rubber, celluloid or any other suitable mate- 
rial, so that by its use this particular size of spiral can be easily 
and quickly traced upon a drawing in any desired position, a num- 
ber of interesting mathematical operations may be performed 
graphically. Among these the most notable are the trisection or 
multisection of any angle, the finding of reciprocals of numbers, 
the laying out of the regular polygons and the rectification of arcs. 
Craphical Operations. 
THE SPIRAL INSTRUMENT. 
For convenience, choose a spiral with the constant term made 
equal to 100. Fig. 5 shows the outlines of an instrument con- 
: see 100 
structed as suggested, with a spiral whose equation is r=—— for 
6 
the working curve. The cut is just half the actual size of the in- 
strument. 
Figs 5: 
The curve EBCD, Fig. 5, is the spiral, O is the pole and OL its 
initial line. The curve FHK and the V-shape with vertex exactly 
at the pole are to permit the pole being placed exactly at a point 
on the drawing, and the construction is such that at the same 
time the initial line, OL, can be brought, readily, into co-incidence 
with any straight line through O. The space bounded by the 
curves TRSG, is cut out simply to lighten the instrument. And 
the curves DNG and QWL are merely to give a ‘‘finish” to it. 
