Simon Jacob – Fibonacci numbers as an "arithmetic labyrinth"

Extract from Simon Jacob’s arithmetic book (1594 edition)

In the denominator, a number of printing errors crept into this edition: what is meant is the number 124,591,936,076,998.

Fibonacci numbers unexpectedly appear in the eighth edition (1594) of Simon Jacob's "Rechenbuch auff den Linien und mit Ziffern" (external link) (Arithmetic book on lines and with numbers).

This occurs in the section on shortening fractions. For this purpose the highest common divisor of numerator and denominator must be calculated. This happens in accordance with the Euclidean algorithm.

Jacob explains the principle using a gigantic fraction whose numerator and denominator, if divided by 19, turn out to be elements of the Fibonacci sequence. He then continues with some "wonderful" properties of this "arithmetic labyrinth": the square of a number in this sequence differs by 1 from the product of the two adjacent numbers, and the ratios of consecutive numbers tend to the golden ratio.

Listed in the margin are the first 27 elements of the Fibonacci sequence, with printing errors having crept in regarding two of them. How Jacob came to these results is unclear. From the context, it is unlikely that he relied on Fibonacci. The number sequence later known as the Fibonacci sequence appears at that time to have already been circulating in the mathematical literature.

Simon Jacob is assumed to have been born in Coburg in 1510 and died in Frankfurt am Main in 1564. Nothing is known about his education, but he was strongly influenced by Michael Stifel’s works. Jacob was one of the best-known reckoning masters of his day and wrote two relevant publications in German.