PALMER: THE HYPERBOLIC SPIRAL. 159 
Now if any value of 6 be measured off in degrees to the scale of 
OM, that is as the correct fractional part of OM (as OK), the cor- 
L : Cie. 
responding value of r in the equation r= a is at once/seen at DB; 
for, drawing KP, and PO cutting AD at B, we have by similar 
triangles we aes orm, 
DOV KO: 6 
To utilize this prepare the diagram of radial lines for the plot- 
ting, as in Fig. 3, using as many lines as possible. It is well to 
divide each quadrant into eighteen parts by trisecting the quadrant, 
then trisecting each part by the dividers and then bisecting each of 
these parts. Then each space will represent five degrees. For 
the sake of accuracy these divisions should be performed on as 
large a circle as can be drawn about the pole O on the sheet of 
paper used. 
Now divide YH, of the auxiliary diagram, Fig. 4, into the same 
number of parts (72), continuing spaces of the same size to the 
right of H. Then, beginning at 9—360, 1. e., at H, draw the line 
to O, determining the length of r, and with the dividers accurately 
set off this length on the corresponding radial line of the plotting, 
which is OL... Use the points to the right and-left of H as far as 
possible, points to the left of H giving constantly increasing values 
of r and being laid off on the plotting in the reverse order of the 
numbers, 1. e., OL, Oa, Ob, etc. Inthe case of values of YP less 
than YA, r will be found on BA produced upward. 
For accuracy’s sake, when drawing the radials to O, a small 
circle, say ;';” in diameter, should be drawn about the point O, 
and all lines stopped at the circle to prevent obscuring the point 
and thus to make it possible to draw each line exactly radial with 
respect toO. This suggestion should be observed in tayine out 
the radial lines of the plotting also. 
When all the values of r which can be secured from the diagram 
have been transferred to the plotting there will be a series of 
points which determine the desired spiral. Through these draw a 
smooth curve and the result will be the hyperbolic spiral of the 
predetermined dimensions. Some of the radii should now be 
checked by calculation, and should be correct to .or”. 
This construction can be much facilitated by using co-ordinate 
paper, ruled in inches and tenths, for the diagram of Fig. 4. OM 
can then be chosen so that an exact number of the smaller spaces 
represent a degree, when no drawing will be required beyond lo- 
cating YH. A straight edge can then be set at once to any desired 
