The Golden Section

The Golden Section is that part of the segment that is medium proportional between the whole segment and the remaining part.

In the third definition of the book VI of the Elements, Euclid says that a segment is divided in medium and extreme ratio, when the entire segment has to his longer part the same ratio as this longer part has to the shorter one. The actual determination of the dividing point is given by the proposition 30 of the same book. If a is the length of the above segment, and we indicate as x the length of the longer part, the problem is described by the following proposition:

a : x = x : (a - x),

i.e. the following 2^nd degree equation:

a(a - x) = x²;

From the above, it is possible to formulate the same problem in the following fashion: to divide the segment in two parts so that the rectangle having as sides the whole segment and one of the two parts is equivalent (i.e. has the same area) to the square built on the other part. Under this form the problem is solved in the Elements in the proposition 11 of book II. The length of the longer part of the segment a is the positive root of the 2^nd degree equation above:

;

This value is often indicated as the Golden Section of the segment a.

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