# Ratio of Consecutive Fibonacci Numbers/Historical Note

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## Historical Note on Ratio of Consecutive Fibonacci Numbers

This result appears first to have been published by Simon Jacob in $1560$, but the work it appeared in was not well-known and subsequent mathematicians appear to have been unaware of it.

This result was observed by Johannes Kepler, but not established rigorously by him:

*It is so arranged that the two lesser terms of a progressive series added together constitute the third ... and so on to infinity, as the same proportion continues unbroken. It is impossible to provide a perfect example in round numbers. However ... Let the smalllest numbers be $1$ and $1$, which you must imagine as unequal. Add them, and the sum will be $2$; add this to $1$, result $3$; add $2$ to this, and get $5$; add $3$, get $8$ ... as $5$ is to $8$, so $8$ is to $13$, approximately, and as $8$ is to $13$, so $13$ is to $21$, approximately.*

However, it was Robert Simson in $1753$ who first explicitly stated that the Ratio of Consecutive Fibonacci Numbers tend to a limit.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1995: Peter Schreiber:
*A Supplement to J. Shallit's Paper “Origins of the Analysis of the Euclidean Algorithm”*(*Hist. Math.***Vol. 22**: pp. 422 – 424) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$