In fig. 17b, where we encountered an incommensurable expression of the geometric mean, what we really came upon is an unique ration known as the Golden Section.60 The unique feature to this ratio is that the remaining line segment will always be in the same ratio to the prior line segment it is compared to. Thus DC is in the same ratio to AB, as AB is to AC (fig. 20).
The construction in fig. 17b is therefore used to extend a given line, where a second segment is added to the first in order to provide the complete ratio. Figs. 22a & 22b are now provided to show how a given line can be cut into the same ratio.
The construction requires building a double square on the line and using the diagonal of the resulting rectangle to find the cut. This too is presented in the Elements as Proposition 30, Book VI. The construction used is provided by fig. 23 which is similar to fig. 17 and fig. 18. The Proposition states: "To cut a given finite straight line in extreme and mean ratio." Thus the term 'mean and extreme ratio' is also ascribed to the Golden Section. There are two other geometrical shapes that share the same nomenclature: the Golden Rectangle and the Golden Triangle. The rectangle is again fig. 17b. The triangle is generated with the diagonals of the Golden Rectangle. (figs. 23a &23b)
The Golden section is associated with the Pythagoreans on two counts. The first association is literary through the adjective 'Golden'. The second is again geometrical and through the Pentagram.
There is a poem that comes down to us known as the Golden Verses. It provides the initiate with the exoteric tradition of the Pythagorean Community. The poem blends popular wisdom with the esoteric insights that are promised to the student who follows its Golden Path. Its transcription is dated about 350 to 300 BC, but it may have been part of an oral tradition that predates this time. The following excerpt illustrates the point:
before you have reviewed each of the day's deeds three times:
"Where have I transgressed? What have I accomplished? What
duty have I neglected?"
Beginning from the first one go through them in detail, and then,
if you have brought about worthless things, reprimand yourself,
but if you have achieved good things, be glad.
Work had at this, meditate on this, you should passionately
this will put you in the footsteps of divine Virtue,
yes, by him who imparted the tetraktys,
font of ever-flowing nature. But to work!
and pray to the gods to grant fulfillment.
When you have mastered these things,
you will come to know the essence of immortal gods and mortal men, etc.....
The Pythagoreans were also associated with the pentagram. A pentagon's diagonals form a star-pentagon, or pentagram, which served as a symbol of health, as well as an icon for the sect. Lucian, writing in the 2nd century AD, explains:
"The divine Pythagoras chose not to leave anything of his own, but if we may judge by Ocellus the Leucaman and Archytas and his other disciples, he did not prefix "Joy to you" or "Do well", but told them to begin with "Health to you". At any rate all his school in serious letters to each other began straightway with "Health to you", as a greeting most suitable for both body and soul, encompassing all human goods. Indeed the pentagram, the triple intersecting triangle which they used as a symbol of their sect [literally, those of the same teaching], they called "Health."
Now, the Golden Section, as we will see, is essential to the construction of the pentagon. There are two basic constructions for the pentagon. The first is non-rigorous. The second rigorous and found in Euclid as Proposition 11, Book IV, and requires the construction of a Golden Triangle which is given in Proposition 10.
Let us turn to the non-rigorous construction provided in figs. 24-28.