"The discovery of incommensurable quantities threw an awful wrench in the machinery of geometry... The difficulty was finally overcome by Eudoxus' theory of proportion. But there was an indirect scare... In Euclid the theory of proportion and similar figures is postponed until the last possible moment, quite contrary to our present practice. Meanwhile, theorems which we prove by proportion were handled by the method of Application of Areas... The credit for discovering this seems to belong to the Pythagoreans:'According to the familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient, and belongs to the Pythagorean use.'What Proclus means...is... An area is applied to a given line segment, if we construct thereon a parallelogram of the given area and having a given angle. If the side of the parallelogram include not only the segment, but a prolongation, that part which is built on the extension is called the excess. On the other hand, if we use but a part of the segment, the parallelogram of the same height built on the unused part is called the defect. Let us see how the Greeks actually used the method. The deux ex machina was a simple figure called the 'gnomon'... Let us say that two plane figures are equivalent if they can be divide into the same number of figures... congruent, in pairs. They shall be called equivalent by completion if, by adding equivalent figures to them, the results are equivalent. ...we come to the actual use of the gnomon... in Euclid, II. 5... This is the identity\alpha \beta + (\frac{\alpha - \beta}{2})^2 = (\frac{\alpha + \beta}{2})^2Suppose... we wish to find the fourth proportional to... \alpha, \beta, \gamma. Euclid would reword this... apply to the line \alpha an area equal to that included by the lines \beta and \gamma. ...We construct a rectangle with non-parallel sides \beta and \gamma, extend the \beta side by the length \alpha, and complete the gnomon."