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In Euclid’s Elements, we find a task that later played an important role in the development of science. It was called Dividing a segment in the extreme and mean ratio. In the Elements, this task occurs in two forms. The first form is formulated in Proposition II.11 of Book II [32].

Proposition II.11 (see Fig. 1.4). The given segment AD is divided into two unequal segments AF and FD so that the area of the square AGHF, constructed on a larger segment AF, must equal to the area of the rectangle ABCD, constructed on a larger (AF) and smaller (FD) segments.

Let’s try understanding the essence of this task by using Fig. 1.4.

If we denote the length of the larger segment AF through b (it is equal to the side of the square AGHE), and the side of the smaller segment through a (it is equal to the vertical side of the rectangle ABCD), then the condition of the Proposition II.11 can be written in the form