Monday, 3 October 2022

# 6 Fibonacci sequence.,… from old math term notes

 Fibonacci's Sequence This sequence is named for Leonardo of Pisa (1175-about 1250). Now he is also called Leonardo Fibonacci, which seems to have been a contraction of Filius Bonacci, son of Bonacci, although the name was never applied during his lifetime. He was one of the three men who are most responsible for introducing the Arabic numerals and methods of algebra to the Western World. In his classic book Liber abaci he poses, and solves the famous rabbit problem which produces the now famous sequence, {0, 1, 1, 2, 3, 5, 8, 13, 21...} in which each value is the sum of the two previous values.

Although it is almost certain that he knew, Fibonacci never wrote that each term was found by adding the two previous terms. The first record of such a statement occurred almost 400 years after Fibonacci by Kepler. Almost another hundred years would pass before R. Simson, for whom the Simson line is mis-named, recognized that each term was the convergent of the continued fraction

. Since all the partial quotients are one, this is the simplest, and therefore slowest to converge, of all the infinite continued fractions. Yet another 150 years would pass before the explicit formula for fn would be found by J. Binet. Letting the golden ratio be represented by  he was able to express the Nth term of the series as 

It is not clear when we began to call it the Fibonacci sequence. As late as the mid-1800's French Mathematician Gabrielle Lame' proved that the number of n-digit Fibonacci numbers is at least 4 and at most 5; but Lame' did not use the name "Fibonacci numbers". In fact, they were often called Lame's numbers because of his proof. Lame also contributed to the story of Fermat's Last Theorem in 1839 when he confimed that there could be no integer values for which x+ y= z7

According to Paul J. Nahin, author of AN IMAGINARY TALE, the name Fibonacci was not common until centuries after Leonardo's death, and during his lifetime he was called Bigollo, a slang term for a loafer drawn from the word bighellone. Julio Gonzalez Cabillon has written, "The name 'Fibonacci' most probably originated with the historian of mathematics Guillaume Libri (1803-1869)."

In September of 2001, Heinz Lueneburg posted a note that seemed to suggest that there may have been earlier uses than Julio suggested. He writes (with some editing by me):

I was in Rome and checked the Boncompagni paper
I quoted in my posting of August 28. The paper starts with the diskussion of what is known of persons of the Bonacci family other than Leonardo: One Matteo Bonacci is known because he is mentioned as a witness of the treaty Pisa and Genova signed on February 13, 1188.
Then he lists the names of authors who use the name "Leonardo Pisano".
Then he lists the names of authors who use the name "Fibonacci".
John Leslie 1820
Cossali 1797-99
Giovanni Gabriello Grimaldi 1790-1792
Libri 1838-1841
Chasles 1837
Nicollet 1811-1818
S. Ersch & I. G. Gruber 1818 and subsequent years
August de Morgan 1847. He also uses Bonacci.

Then he lists the names of authors who explain "Fibinacci = filio Bonacci"
Flaminio dal Borgo 1765
Tiraboschi 1822-1828
Ranieri Tempesti 1787
Giovanni Andres 1808-1817
Grimaldi 1790-1792
Libri 1838-1841

Then his arguments for Fibonacci = de filiis Bonacci follow.
Then he discusses the sobriquet Bigollone, Bigollo, Bigoloso.
Finally, in the major part of the paper, he discusses the various manuscripts of the various works of Fibonacci still in existence.
The question "who gets the credit?" is still open.
Regards, Heinz Lueneburg 

Those interested in learning more about the sequence may find a good reference at the Dr. Math Faq page. And there is a large body of information related to the Fibonacci sequence and the Golden Ratio at this page from England. The page is also the source of the photo at right of the statue to Fibonacci in the cemetary at the Duomo (Cathedral) in Pisa. 

A nice biorgraphy of Leonardo of Pisa can be found at a page by Clark Kimberling. The page shows an earlier picture of the same statue prior to its restoration when it stood on the street named for Fibonacci. My thanks to Gian Marco Rinaldi who corrected a previous mistake about the statue. He also introduced me to a web page where Alberto Rodríguez Santos maintains a site which has a history of the many moves of the statue around the city over the years. On my last day of my first visit to Pisa I missed a turn and came upon the Via Fibonacci quite by accident.

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