by Keith Devlin
I had always thought that Fibonacci was some Italian dude who discovered the sequence now named after him. You know, the one that shows up in sunflowers and all sorts of other natural phenomena: 1, 2, 3, 5, 8, 13, 21, 34… It is obtained by a simple formula: the next number in the series is equal to the sum of the previous two numbers. The Finonacci sequence has all sorts of interesting mathematical properties; it therefore excites an almost spiritual reverence among some mathematicians. I’ll bet you could start a religion deifying the Fibonacci sequence and find a goodly number of adherents to bilk.
But that’s all wrong. In the first place, his name wasn’t Fibonacci; that name was bestowed upon him by Guilliaume Libri, a mathematician who ‘rediscovered’ him in 1838. The original fellow had a number of names. Leonardo of Pisa was the one he most commonly used, but he also occasionally used ‘Leonardo Bigollo’. The nickname we now use for him (Fibonacci) arose from the fact that he was descended from the Bonacci family and in some documents his name appears as ‘Leonardo filius Bonacci’, meaning ‘Leonardo of the Bonacci family’. In some early documents this Latin official name was shortened to ‘Leonardo fi Bonacci’, which is doubtless what induced Mr. Libri to misname him. But if you’re cool, you don’t call him ‘Fibonacci’, you call him ‘Leonardo of Pisa’. Be cool, man.
Here’s another kick in the butt: the Fibonacci sequence is not his most important contribution to civilization, not by a long shot. Leonardo’s role in history is far greater than most people realize. He’s the guy who dragged Western mathematics out of the Dark Ages and into the modern world.
He didn’t invent much of it; most of what he knew came from the Arabs. His father was a merchant who lived in the town of Bugia, on the coast of Algeria. The town is now known as Bejaia. In those days, traders in a foreign city lived in their own assigned quarter of the city, for reasons both of convenience and protection. The city fathers appreciated their presence as essential to the prosperity of the city, but the townspeople sometimes took umbrage at foreign ways and frictions would lead to brawls if the two alien parties were allowed to mix freely.
Leonardo grew up in Bugia, learning the trade from his father and from other merchants. The greatest impact came from the Muslim merchants, who used what we now call the ‘Hindu-Arabic number system’ — 0, 1, 2, 3… The Italians all used the clumsy old Roman numerals. Those numerals were just fine for simple calculations entailing only addition and subtraction, but trading systems were growing too complicated to be handled with Roman numerals. The essence of the problem lay in multiplication and division, which were nasty calculations with Roman numerals and much, much easier with Hindu-Arabic numerals. All sorts of innovations were requiring multiplication and division: partnerships, more sophisticated tax systems, group investments, and, above all, currency exchange. Nobody used decimal currency systems back then; usually there were four to six denominations for each currency. For example, the ancient Greek system used obols for pennies; 6 obols made a drachma; 100 drachmae made a mina; and 60 minae made a talent. Converting between such systems is a headache even in modern times; it was a major problem for the Europeans using Roman numerals.
Young Leonardo took a shine to these new Hindu-Arabic numerals and quickly mastered them. He didn’t stop there; he learned Arabic and studied everything he could of Islamic mathematics, which at the time was far in advance of anything in Europe. He learned about the new field of mathematics we now call algebra, and he accumulated a vast store of computational techniques from his Islamic tutors.
Leonardo was more than a quick study; he had a brilliant and creative mind, and he developed his own mathematical ideas. By the time he returned to Italy, he was easily the best mathematician in Christendom. He set about writing a book presenting all his ideas. That book, the Liber Abbaci, or Book of Calculations, was a sensation. Italian merchants instantly recognized the enormous advantages conveyed by a knowledge of Hindu-Arabic numerals. There was, of course, resistance; Roman numerals remained the only officially recognized system for several centuries in some of the more hidebound European towns. But the Hindu-Arabic numerals quickly won over the Italian merchants.
Liber Abbaci had two serious flaws. First, it was written in Latin, the univeral language of scholarship, but not what most merchants used. Second, it not only explained how to calculate, it explained the mathematics behind those calculations. The core of the book lay in the hundreds of solved problems addressing every possible situation a merchant might encounter. Many of those solved problems were taken directly from Islamic books. But Leonardo couldn’t resist the temptation to explain the wonderful mathematical principles at work. This abstruse material was all well and good, but most merchants were in too much of a hurry to devote time to that.
Leonardo responded to the complaints by producing another book: Libro di minor guisa. This might be thought of as the “Reader’s Digest” version of the Liber Abbaci; it was shorter, presented simpler problems, and included no theory. This was what merchants wanted, and this book was also very popular.
The value of the shorter book is demonstrated by the number of imitations it attracted. Remember, this was before the printing press; every book had to be copied by hand. In effect, every scribe was his own author. There was no tradition of respect for proper credit assignment. The only thing that was important was the material in the book. Moreover, Leonardo’s book used Pisan currency for its examples. Every city in Italy had its own currency, so it was necessary to produce books using other currencies. This in turn required reworking the problem sets, so a certain amount of authorship went into every copy of the book. These authors had no problem presenting these books as their own works. Thus, a huge family of books, all called in some fashion ‘Book of Calculations’, sprang up all over Italy. We still have hundreds of copies of these books, but not a single copy of Leonardo’s book Libro di minor guisa remains, and we have only a handful of remaining copies of the Liber Abbaci. But the books of calculation continued on; today’s arithmetic textbooks can trace a direct line of descent back to Leonardo of Pisa.
Leonardo became immensely famous and was the toast of Italy. The city of Pisa granted him a generous annual pension. He wrote many other books, none as influential as his books on calculation. But those books demonstrated that Leonardo was a brilliant mathematician.
The Fickle Finger of Fate moved on, and within a few centuries Leonardo of Pisa was forgotten, until he was rediscovered in 1838 by Guilliame Libri, at which point his fame was restored.
But that fame is misplaced. The Fibonacci sequence is not Leonardo’s main contribution to civilization. No, his main contribution was bringing the Hindu-Arabic numerals to Europe, and, as far as Christendom was concerned, inventing algebra.