Which came first: the Fibonacci Numbers or the Golden Ratio? I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?
 A: As previous answers have pointed out, both the golden ratio and the Fibonacci numbers go back thousands of years. However, I believe the connection
between the two was discovered around 1730. At that time, Daniel Bernoulli and
Abraham de Moivre independently came up with the generating function
for the Fibonacci numbers, and the resulting formula for the
$n$th Fibonacci number in terms of the golden ratio.
A: Golden ratio came first.  Wikipedia has a rather thorough article on it.  It's not nearly as pervasive in nature or architecture as people like to say it is.  It will show up in anything with regular pentagons, though.
A: The book A Mathematical History of the Golden Number by Roger Herz-Fischler is an exhaustive study of nearly all references to the golden ratio, from the earliest times, and is available as a free e-book. As has been pointed out by others, the golden ratio is older than the Fibonacci numbers. On page 53, Herz-Fischler notes that a pentagram appears as "a pot mark on a jar" dating from 3100 BC in Egypt.
A: The golden ratio in mathematics dates back to the Pythagoreans, circa 500 BC, it's true. But the Fibonacci numbers also have a long heritage going back to Pingala in India circa 200 BC.
However, the mystical claims about the golden ratio and Fibonacci numbers going back hundreds of millions of years in biology and showing up in every piece of ancient art and architecture seem to date back only to Pacioli in the 16th century AD.
A: What is the significance?  Most of the nice properties of the golden mean can be attributed to the fact that its continued fraction coefficients are uniformly bounded, as will be true in particular for any periodic continued fraction, which is to say any quadratic irrational, such as arises as the spectral radius of an indecomposable two-term linear recurrence relation.  Among such continued fractions, the unique one with the minimum possible upper bound of 1 naturally exhibits these effects most prominantly, and it arises from (arguably) the simplest such recurrence.
A: The golden ratio was used extensively in ancient art, but the man named Fibonacci (Leonardo of Pisa) lived around 1200 AD. It's possible that the Fibonacci series was known before Fibonacci but I'm not aware of this.  So I think it's safe to assume the golden ratio is older.
A: The answer for either of these is "hundreds of millions of years" due to their emergence/use in biological development programs, the self-assembly of symmetrical viral capsids (the adenovirus for example), and maybe even protein structure.  Because of their close relationship I'd be hard pressed to say which 'came first'.  
If you google for it, you'll find plenty of books and papers.  However, be extremely careful about examples without a well-explained functional role... there are an arbitrarily large number of coincidences out there if you're looking for them, and humans excel at numerology.  
A: Golden ration came first in nature long before humans evolved to think about Fibonacci numbers.
