Equivalence of the methods of Archimedes and Apollonius
Both Archimedes and Apollonius had propositions for finding a cone that contained a given elliptical section. Apollonius' purpose was to show that any section obtained from an oblique cone could be found in a right cone though not necessarily with the same orientation. He provided methods for pre-processing sections characterized by their conjugate diameters and based his section finding method on the sections major axis. In a right cone the section major axis lies on a plane of the cone axis which is perpendicular to the cone base and does not contain any "landscape" oriented ellipse.
It was possible that Archimedes found a right cone that contained the given section but it would have been a coincidence. His method was geared toward finding a circular base for an elliptical cone segment that was known to contain the given section. Thus, though both started with right elliptical cone segments, Apollonius, constructed a right circular cone containing the section while Archimedes found a circular base to convert the segment into a cone.
This worksheet shows that the same method finds both solutions.