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.