Braun and Karl Schimper were the first ones to notice that the number of spirals in many plants matches adjacent pairs of numbers in the sequence: 1, 1, 2, 3, 5, 8, 13, 21 ... where each number is the sum of the previous two.  (They did not call it by its current name, Fibonacci, since at the time few people had heard of that medieval mathematician.)

But science works in mysterious ways.  At the very same moment in history, the Fibonacci sequence in plants was independently discovered by two young French brothers, Auguste and Louis Bravais.  They produced a beautiful 1837 paper detailing, with perhaps greater rigor, the mathematics of spiral lattices.  

The brothers were crushed to learn, just as they were about to submit their paper, that they had been scooped by Braun and Schimper! The French brothers saw things a bit more irrationally, however. They argued that instead of the Fibonacci angles 3/8, 5/13, 8/21, 13/34...  (in fractions of turn) posited by the German scientists, the angle between successive leaves may simply be the limit of these fractions. This angle, which they calculated to be 360 (3 - sqrt(5))/2  (about 137.51 degrees)  is what we now call the golden angle, as it is related to the irrational number   (1 + sqrt(5))/2, called the golden ratio.

It turns out the truth is somewhere in between: in real plants, the angle between successive leaves tends to vary in a range around the golden angle.