The Conics Generated by the Method of Application of Areas
  The method of application of areas as presented in Euclid's Elements, is employed to generate the three conics as the loci of points with Cartesian coordinates satisfying quadratic equations with coefficients defined by the initial settings of the geometric constructions produced by the applications method. This conceptual reconstruction supports the view that the conics were most probably discovered as plane curves by fusing the method of the application of areas with the concept of locus, long before Apollonius studied them as conic sections. 
Dimitris Sardelis and Theodoros Valahas
math.GM