FIGURE 8a
Projection of Equal Hyperbolic Areas
The points along the hyperbola that correspond to equal divisions of area are projected onto the axis, by drawing perpendicular lines from the axis to those points. This produces lengths, Ob,Oc,Od. Oa=1.
FIGURE 8b
Measuring the Lengths Along the Axis
When the perpendicular lines from the axis are extended to intersect the asymptote, they mark off the lengths b'=21+1/21 , c'= 22 +1/22 ,d'=23+1/23. When the points b, c, and d are projected back down to the asymptote they mark off lengths, b"=(21+1/21)/2; c"=(22+1/22)/2; d"=(23+1/23)/2;
Figure 8c
When the points a, b, c, and d are projected back down to the asymptote they mark off lengths, a"=1, b"=(21+1/21)/2; c"=(22+1/22)/2; d"=(23+1/23)/2;
FIGURE 8d
The Relationship Between Hyperbola and Catenary
When lengths Oa", Ob", Oc", Od", from the hyperbola, are set along a line at equal intervals, their endpoints form the catenary.