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HsHHbIÉmhméhìicì 
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THE DUPLICATION OF THE CUBE 
Accordingly, to find the point P, we have to construct 
(1) a parabola with 0 as vertex, OM as axis, and latus rectum 
equal to OB, 
(2) a hyperbola with asymptotes OM, ON and such that 
the rectangle contained by straight lines PM, PN drawn 
from any point P on the curve parallel to one asymptote and 
meeting the other is equal to the rectangle AO . OB. 
The intersection of the parabola and hyperbola gives the 
point P which solves the problem, for 
AO : PN = PN: PM = PM: OB. 
Second solution. 
Supposing the problem solved, as in the first case, we have, 
since AO : OM = OM: ON = ON: OB, 
(1) the relation OB. OM = ON 2 = PM 2 ,