I need help understanding what apollnius did when he defined a parabola and what he proved. "First let the diameter PM of the section be parallel to one of the sides of the axial triangle as AC, and let QV be any ordinate to the diameter PM. Then if a straight line PL (supposed to be drawn perpendicular to PM in the plane of the section) be taken of such a length that PL:PA = BC² : BA.AC, it is to be proven that QV² = PL.PV

Let HK be drawn through V parallel to BC. Then, since QV is also parallel to DE, it follows that the plane through H, Q, K is parallel to the base of the cone and therefore produces a circular section whose diameter is HK. Also QV is at right angles to HK." From http://www.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html

What is ordinate? I look online everything says cooridnates, but I think ordinate means something else here. What are the ":" for and what do they mean? How about that dot as in PL.PV...exactly what is apollonius' main point here?


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