Fibonacci and digit-by-digit computation; An example of reverse engineering in computational mathematics

TL;DR

This paper investigates Fibonacci's historical method for approximating a root of a cubic polynomial, reconstructing possible techniques and identifying the most likely approach used by Fibonacci.

Contribution

It reconstructs Fibonacci's method for approximating a cubic root, demonstrating that three different methods yield the same result and identifying the most probable one.

Findings

All three methods produce the same approximation.

The most likely method used by Fibonacci is identified.

The study sheds light on historical computational techniques.

Abstract

The Fibonacci numbers are familiar to all of us. They appear unexpectedly often in mathematics, so much there is an entire journal and a sequence of conferences dedicated to their study. However, there is also another sequence of numbers associated with Fibonacci. In The On-Line Encyclopedia of Integer Sequences, a sequence of numbers which is an approximation to the real root of the cubic polynomial. Fibonacci gave the first few numbers in the sequence in the manuscript Flos from around 1215. Fibonacci stated an error in the last number and based on this error we try, in this paper to reconstruct the method used by Fibonacci. Fibonacci gave no indication on how he determined the numbers and the problem of identifying possible methods was raised already the year after the first transcribed version of the manuscript was published in 1854. There are three possible methods available to Fibonacci to solve the cubic equation; two of the methods have been shown to give Fibonacci's result. In this paper we show that also the third method gives the same result, and we argue that this is the most likely method.