"The Pythagoreans started work on a class of problems known as application of areas. The simplest of these was to construct a polygon equal in area to a given polygon and similar to another given one. Another was to construct a specified figure with an area exceeding or falling short of another by a given area. The most important form... is: Given a line segment, construct on a part of it or on the line segment extended, a parallelogram equal to a given rectilinear figure in area and falling short (in the first case) or exceeding (in the second case) by a parallelogram similar to a given parallelogram. ... With propositions 28 and 29... [Kline describes the most important Pythagorean form in Euclid, Book VI. Prop. 27-29, with modern notation: ax \pm \frac{b}{c}x^2 = S for area S.] one can solve any quadratic equation [as lengths] when one or both roots are positive. In Proposition 28 the parallelogram constructed falls short... and in Proposition 29 the parallelogram exceeds... The respective parallelograms were called in Greek ellipsis and hyperbolè. A construction on the entire given line as base, as in Book 1, Proposition 44, was called parabolè. These terms were carried over to the conic sections for a reason which will be obvious when we study Apollonius' work."