To cut a given finite straight line in extreme and mean ratio.

Let *AB* be the given finite straight line.

It is required to cut *AB* in extreme and mean ratio.

Describe the square *BC* on *AB.* Apply the parallelogram *CD* to *AC* equal to the sum of *BC* and the figure *AD* similar to *BC.*

Now *BC* is a square, therefore *AD* is also a square.

And, since *BC* equals *CD,* subtract *CE* from each, therefore the remainder *BF* equals the remainder *AD.*

But *FE* equals *AB,* and *ED* equals *AE.*

Therefore *AB* is to *AE* as *AE* is to *EB.*

And *AB* is greater than *AE,* therefore *AE* is also greater than *EB.*

Therefore the straight line *AB* has been cut in extreme and mean ratio at *E,* and the greater segment of it is *AE.*

Q.E.F.