_{n+1}/F

_{n}of consecutive Fibonacci numbers [1]. One may wonder who noticed this first? Leonardo Pisano (c. 1170 – c. 1250, also known as Fibonacci)? Not even close!

Evidence that this fact was noticed as early as the beginning of the 16th century was discovered by Leonard Curchin and Roger Herz-Fischler (handwritten annotation in the 1509 Luca Pacioli's edition of Elements [2]). When it comes to published work which associates the EMR with the limit of the sequence of quotients of consecutive Fibonacci numbers, Johannes Kepler wrote [3] the following, in 1611, about "this proportion that the geometers of today call divine":

It is impossible to provide a perfect example in round numbers. However, the further we advance from the number one, the more perfect the example becomes. Let the smallest numbers be 1 and 1... Add them, and the sum will be 2; add to this the greater of the 1s, result 3; add 2 to this, and get 5; add 3, get 8; 5 to 8, 13; 8 to 13, 21. As 5 is to 8, so 8 is to 13 approximately, and as 8 to 13, so 13 is to 21, approximately.However, as communicated in 1995 by Peter Schreiber [4], in a rare book by the German reckoning master Simon Jacob (d. 1564) one can find a remark that the sequence of quotients of consecutive Fibonacci numbers approaches the EMR.

NOTES

[1] The term "extreme and mean ratio" goes back to Euclid's "Elements". Luca Pacioli introduced the "divine proportion" term in 1509, while the term "golden section" was introduced in 1835 by Martin Ohm. The exact value for the EMR is (1+sqrt(5))/2. The phrase "Fibonacci sequence" was coined by Edouard Lucas (1842-1891).

[2] Leonard Curchin and Roger Herz-Fischler, "De quand date le premier rapprochement entre la suite de Fibonacci et la division en extrÃªme et moyenne raison?" (French) ["When were the first parallels drawn between the Fibonacci sequence and the golden section?"], Centaurus 28 (1985), no. 2, 129-138.

[3] Kepler, Johannes. Vom sechseckigen Schnee. (German) [On hexagonal snowflakes]. Translated from the Latin and with an introduction and notes by Dorothea Goetz. Ostwald's Classics of the Exact Sciences, 273. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.

[4] Peter Schreiber, "A Supplement to J. Shallit’s Paper "Origins of the analysis of the Euclidean algorithm"", Historia Mathematica, Vol. 22, Issue 4, 422-424 (1995). http://poncelet.math.nthu.edu.tw/disk5/js/history/1033a.pdf