To mathematicians, the ratio simply represented an elemental definition laid down by  Euclid of what he called the "extreme and mean ratio." But Golden Ratio (probably coined by some classical PR agent) is the term that has come down to us through the ages.

Euclid^[6] expressed it geometrically: a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

Algebraically, the Golden Ratio (represented by the symbol φ or Phi) then translates to:

Rewrite it as:                                    1+ 1/φ = φ 

Which transforms to:                   φ² - φ - 1 = 0

Solve the quadratic and bingo, you have the positive solution: 

Forget the square root of 5 for now. The proof is fundamental, though a bit tedious. But what Fr. Wroblewski led me to with his second challenge was a garden of unending delight: the world of continued fractions .

Take                                                     φ = 1+ 1/φ

Substituting for  φ  on the right hand side, you get:

                                   φ = 1+ 1/(1+ 1/φ)

which you can keep on doing each time you encounter φ, ad infinitum:

resulting in φ = 1+ 1/(1+ 1/(1+ 1/(1+ 1/(1+ 1/φ)))).......

and presented elegantly in the canonical form for infinite continued fractions:

Or take:                                                       φ² - φ - 1 = 0 

which can be rewritten as:                        φ² = 1 + φ

Taking the square root  of each side and keeping on substituting for  φ repeatedly (math jargon: recursively) likewise produces the infinite continued square root form for the Golden Ratio:

                                            

     O R G A S M
                                                                         

I didn't touch it, I swear, Father, nocturnal emissions don't count as mortal sin (actually Fr. Wrob was way cool about such things).

So there I was, not yet sixteen, I had had my first orgasm and was about to go off to college to be a mathematician.  Still to come, so to say, in Part 3.

WOOF

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Notes: 

[1] Vitruvius The Ten Books on Architecture (trans. Morris Hicky Morgan 1914) Dover Publications,New York, 1960.

Vitruvius wrote his treatise on Greek architecture of the high classical period (5th century BCE) some four hundred after the fact, dedicating his work to the Emperor Augustus  around 30 BCE. The works of Euclid (c. 300 BCE) should have been known to him, but no reference to Euclid or the Golden Ratio is found in the book.

Near-contemporary sources exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions. Vitruvius himself uses only rational numbers i.e. ratios like 3 : 2 for his proportions in his famous Chapter III: On Symmetry: In Temples and in the Human Body.  So it is probably mythic to believe that the Golden Ratio had a directive role in classical architecture.

[2]  Convergence describes the property of a series of numbers to approach a definite value.  In the case of the higher numbers of the Fibonacci Series, for example, dividing a number by its preceding one results as follows:

(55/34) = 1.61764...     (89/55) = 1.61818...    (144/89) = 1.61797....

illustrating how the quotient series approaches  φ = 1.61803.....

[3] Leonardo da Vinci incorporated many of the ratios of Vitruvius in his quest for the perfect representation of the human body. But even here, he was clearly interested in commensurate (i.e. measurable) proportions and no references are to be found to the Golden Ratio. Thus, his famous drawing is correctly called Vitruvian Man, even though many references try to force fit the golden ratio or rectangle onto this work.

[4] Nautilus Shell : photograph from Wikimedia Commons:

[5] Argand diagram. The spiral is essentially the same as the more conventional representation at the top of Part 1, using a different technique.

[6] Euclid defined the "extreme and mean ratio" in his Elements Bk.VI, Def. 3, as mentioned earlier.  But there is no recorded attempt to calculate its value in decimal terms till much later. Euclid knew about a close counterpart to continued fractions which could have been used for its computation for those so interested.

[7] Irrational numbers are not easy to define positively.  Rational numbers are defined as any number that can be expressed as the ratio of two integers.  Irrational numbers are those that cannot be so defined! They have no exact decimal equivalent, they do not terminate (thus the inevitable ... at the end), they do not show periodicity etc.

A surd is an irrational number which can be expressed as the n^th root of a number. The Square Root of 5 is indeed a surd, thus my joke was a pretty good one!