The purpose of this problem is to prepare for the construction of the regular decagon and pentagon. The division of a line in extreme and mean ratio is called "the golden section," and is probably "the section" mentioned by Proclus when he says that Eudoxus "greatly added to the number of the theorems which Plato originated regarding the section." The expression "golden section" is not old, however, and its origin is uncertain.
If a line AB is divided in golden section at P, we have
AB × PB = (AP)2.
Therefore, if AB = a, and AP = x, we have
a(a - x) = x2,
or x2 + ax - a2 = 0;
whence x = - a/2 ± a/2√5
= a(1.118 - 0.5)
= 0.618a,
the other root representing the external point.
That is, x = about 0.6a, and a - x = about 0.4a, and a is therefore divided in about the ratio of 2 : 3.
There has been a great deal written upon the æsthetic features of the golden section. It is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio. A painting has the strong feature in the center, or more often at a point about 0.4 of the distance from one side, that is, at the golden section of the width of the picture. It is said that in nature this same harmony is found, as in the division of the veins of such leaves as the ivy and fern.