"The discovery of incommensurability meant that there existed geometric magnitudes that could not be measured by numbers! ...It followed that geometric magnitudes could not be manipulated without hesitation in algebraic computations just as though they were numbers. ...The Greek answer ...is presented in the geometric algebra of Books II and VI... The product of two lengths a and b is not a third length, but rather the area of a rectangle with sides a and b. ... The principle Greek technique for the geometric solution of algebraic equations was based on the "application of areas." For example, given a segment... of length a, the construction in Proposition I.44 (Proposition 44 of Book I) ...a rectangle with base AB of length a and area equal to that of a given square of edge b... provides a solution of the equation ax = b^2. This corresponds to geometric division [x = \frac{b^2}{a}], and we say that the given area b^2 has been applied to the given segment AB. Proposition VI.28... shows how to apply a given area b^2 to a given segment of length a, but "deficient" by a square. ...This construction provides a geometric solution of the quadratic equation ax - x^2 = b^2."