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?
[Math] Which came first: the Fibonacci Numbers or the Golden Ratio
enumerative-combinatoricsho.history-overview
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Found it! (Sorry, Doug, ha ha.)
Augustus de Morgan added several appendices to his Elements of Arithmetic in the fifth edition, 1846 (available on Google Books). Appendix 10, pages 201-210, is "on combinations." The relevant paragraph is on 202-203.
Required the number of ways in which a number can be compounded of odd numbers, different orders counting as different ways. If $a$ be the number of ways in which $n$ can be so made, and $b$ the number of ways in which $n+1$ can be made, then $a+b$ must be the number of ways in which $n+2$ can be made; for every way of making $12$ out of odd numbers is either a way of making $10$ with the last number increased by $2$, or a way of making $11$ with a $1$ annexed. Thus, $1+5+3+3$ gives $12$, formed from $1+5+3+1$ giving $10$. But $1+9+1+1$ is formed from $1+9+1$ giving $11$. Consequently, the number of ways of forming $12$ is the sum of the number of ways of forming $10$ and of forming $11$. Now, $1$ can only be formed in $1$ way, and $2$ can only be formed in $1$ way; hence $3$ can only be formed in $1+1$ or $2$ ways, $4$ in only $1+2$ or $3$ ways. If we take the series $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, &c. in which each number is the sum of the two preceding, then the $n$th number of this set is the number of ways (orders counting) in which $n$ can be formed of odd numbers. Thus, $10$ can be formed in $55$ ways, $11$ in $89$ ways, &c.
He established "increasing" and "annexing" in deriving the formula for the number of what we now call compositions. He does not treat either of the other two restrictions mentioned above.
I suggest the article A Circular Argument (Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162.) It may be relevant to your questions. It suggests that (a variant of) the limit $\lim_{x\to 0}\frac{\sin{x}}{x}=1$ is important to the area result of Archimedes which you mention and that the reasoning may be ... circular. Here is: a freely available version.
revised version I think that it is a bit subtle. The right question might be: Who first treated the question as one which could make sense. The answer to that is probably Archimedes. Once you have that (in an acceptably defined way) the result may not be that hard.
Consider first questions simply of inequalities. If a circle is inscribed in a square the Euclid would agree that the area of the circle is less than that of the square because the whole is greater than the part. But Euclid never says that the perimeter is greater than the circumference because they are different kinds of things. Mark Saphir notes that in Book VI Proposition 33, Euclid proves that in circles of equal radii the lengths of two arcs are in equal proportion to the (central) angles cutting them off. Just sticking to one circle for now with center $O$ we understand what it would mean to say that $\angle AOB < \angle COD$ or that $\stackrel{\frown}{AB} < \stackrel{\frown}{CD}$ and also what it would mean to say that one is twice the other. And hence we have that proposition: $\frac{\angle AOB}{\angle COD}=\frac{\stackrel{\frown}{AB}}{\stackrel{\frown}{CD}}$ (But $\frac{\angle AOB}{\stackrel{\frown}{AB}}=\frac{\angle COD}{\stackrel{\frown}{CD}}$ would not make sense.) Again, Euclid could describe the situation that the radius of one circle is twice that of another. And would even agree that the area of the second is four times that of the first. However he would not say that the circumference of the second was larger than that of the first (let alone twice as much.)
Archimedes introduces the concept of concavity and the postulate:
If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D.
This is intuitive (as befits a postulate) but is not obvious. With this in hand he can say that for a circle of diameter d, the circumference C is something such that p<C<P where p and P are the perimeters of polygons (of some number of sides, he used 96) inscribed and circumscribed about a fixed circle. If this is granted then p/d < C/d < P/d and, because we know the bounds are independent of d (thanks to similarity of polygons), we have that his bounds are independent. Implicitly, letting the number of sides increase, we have that C/d must be similarly independent.
Here we see the idea of arc length (for convex curves) as the limit of the length of inscribed polygonal paths (or perhaps the common limit, if it can be demonstrated, of inscribed and tangential paths.)
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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.