Leonardo Fibonacci

Chapter 1 · The Boy at Bugia

Sent abroad, and shown a better way to count

c. 1170 – c. 1200 · Pisa · Bugia · Egypt · Syria · Sicily

Leonardo was born at Pisa around 1170, into a republic whose ships reached every port of the Mediterranean. His father Guglielmo held a post as a customs official at Bugia, on the coast of what is now Algeria, and had his son brought out to join him there.

What happened next decided the rest of his life. His father arranged for him to be taught calculation by a Muslim master, and the method he was taught used the nine figures and a sign for nothing — the system that had come from India through the Arabic-speaking world. Leonardo was, by his own account, captivated by it.

He then travelled: Egypt, Syria, Byzantium, Sicily, Provence. Everywhere he went he studied the local ways of reckoning and compared them. His conclusion, stated plainly in the book he later wrote, was that none of them came anywhere near the Indian method.

It is worth being precise about what he did and did not do. He did not invent the numerals, and he never claimed to. He learned them abroad from teachers he names as his teachers, and his achievement was to see what they were worth and to write the book that would carry them west.

Why this matters

The most consequential piece of mathematics in medieval Europe arrived through a trading post, in the education of a boy sent to join his father at work.

A merchant's son, taught abroad, convinced he had found something better. What would you ask him?

Ask Fibonacci

• “What did your teachers at Bugia actually show you?”
• “Why did you travel so far to compare methods of counting?”
• “How did merchants in Pisa keep their accounts before you?”
• “What convinced you the Indian method beat all the others?”
• “Did your father intend you to become a mathematician?”

Chapter 2 · The Book of Calculation

Nine figures, one sign for nothing, and several hundred problems about money

1202 · Pisa

The Liber Abaci opens with a sentence that changed European arithmetic: there are nine figures, and with them and with the sign zephirum, any number whatever may be written.

What follows is not a treatise. It is a handbook for people with cargo to value and partners to pay. There are chapters on converting between the coinages of different cities, on barter, on dividing profits among investors according to what each put in, on weights and measures that change at every port, on the rule of three, and on extracting roots. Every method arrives attached to a worked case with real money in it.

His fractions do not look like ours. He writes them with a bar — which is itself part of what he brought west — but often as a sum of parts each having one on top, and he places the fraction to the LEFT of the whole number rather than the right. There are no decimals anywhere; decimal fractions were three centuries away.

The book was aimed squarely at merchants, and merchants are who took it up. The reasoning was not aesthetic. You cannot easily multiply two Roman numerals, and you cannot check the result. With the new figures you could do both before the ship sailed.

Why this matters

It is a rare case of a mathematical advance spreading because it was commercially useful, long before it was academically respectable.

A textbook of trade that changed how a continent counts. What would you ask him?

Ask Fibonacci

• “Why is the sign for nothing so important?”
• “How do you write a fraction, and why to the left of the whole number?”
• “What is the rule of three, and why does a merchant need it every day?”
• “Why is your book full of money rather than theorems?”
• “What can you do with the new figures that you cannot do with Roman ones?”

Chapter 3 · The Rabbits

One problem among hundreds, and the only one anybody remembers

1202 · Pisa

Somewhere in the middle of the Liber Abaci, among the problems about cargo and coinage, there is one about rabbits. A pair is put in an enclosure; each pair produces a new pair from its second month onwards; how many pairs are there after a year?

Working it month by month gives 1, 1, 2, 3, 5, 8, 13, 21 and onwards, each number the sum of the two before it. Fibonacci sets the problem, works the answer, and moves on. He does not name the sequence, does not claim to have discovered it, and gives no sign of thinking it more interesting than the exchange problems either side of it.

Two corrections are worth making. The sequence was not new: Indian scholars studying the patterns of long and short syllables in poetic metre — Pingala, Virahanka, Hemachandra — had described exactly this rule centuries before he was born. And the name is not his. The French mathematician Édouard Lucas attached it to him in the nineteenth century, some six hundred years after his death.

Fibonacci himself would have pointed you at the Liber Quadratorum, his book on square numbers, which is a far more original piece of mathematics and which almost nobody reads.

Why this matters

A clear case of how fame in the history of ideas attaches to the memorable rather than the important — and of how easily an earlier tradition gets written out.

The problem he is famous for is the one he cared least about. What would you ask him?

Ask Fibonacci

• “What was the rabbit problem actually meant to teach?”
• “Did you know Indian scholars had described that sequence before you?”
• “How do you feel about a sequence being named after you six centuries later?”
• “What is in the Liber Quadratorum that you think deserves more attention?”
• “Is the rabbit model a realistic description of actual rabbits?”

Chapter 4 · Three Centuries of Resistance

Why the better system took so long to win

1202 – c. 1500 · Pisa · Florence · Europe

It would be tidy if Europe had read the Liber Abaci and switched. It did not. The Hindu-Arabic numerals took roughly three hundred years to displace Roman ones in European accounting, and the resistance was not simply stupidity.

The objections were partly practical. A Roman numeral is hard to alter without the alteration showing; a 0 can be made into a 6 or a 9 with a stroke of the pen, and a 1 into a 7. Where accounts might end up in court, that mattered. In 1299 the guild of money-changers in Florence forbade the new figures in official books, and other cities did likewise.

There was institutional inertia too. Reckoning had long been done on a counting board with tokens, by people trained in that method, and the written figures made a whole craft redundant. Universities, meanwhile, taught arithmetic as part of a classical curriculum with little interest in commercial calculation.

What carried the day in the end was the abacus schools of the Italian cities, teaching merchants' sons the practical arithmetic of Fibonacci's book, generation after generation, until the men keeping the ledgers had all learned it that way.

Why this matters

A better method does not win because it is better; it wins when enough people have been trained in it and the institutions catch up.

Three centuries between the book and the victory. What would you ask him?

Ask Fibonacci

• “Why would anyone ban the new figures from official accounts?”
• “What happened to the men who reckoned on counting boards?”
• “Did you live to see your figures widely used?”
• “Why did merchants take up your method before scholars did?”
• “What would you say to a money-changer who did not trust a written 0?”

Chapter 5 · A Salary from the Republic

The court of the Emperor, and what was left behind

c. 1225 – c. 1245 · Pisa · Palermo

Fibonacci's later life is thinly documented, but two things are recorded. The first is an appearance at the court of the Emperor Frederick II, whose scholars had heard of him. Johannes of Palermo set him problems as a public test — including finding a number satisfying certain conditions on squares — and Fibonacci solved them. Out of that came the Liber Quadratorum of 1225, on square numbers, which mathematicians generally regard as his most original work.

The second is a document of 1240 in which the Republic of Pisa grants him an annual salary, in recognition of his services to the commune in accounting and in teaching. It is a small record and an unusually clear one: a city paying a mathematician because his arithmetic was useful to it.

He died at some point after that, probably around 1245, and the date is not known precisely. Nor is much else about him personally: no letters survive, and the name Fibonacci — filius Bonacci, son of Bonaccio — was attached to him by later writers rather than used in his own lifetime.

What survives is the arithmetic. Every calculation done in the Western world today is done with the figures his book introduced, learned in schools whose distant ancestors were the abacus schools that taught from it.

Why this matters

The record of his life is thin, which is normal for a medieval figure and is worth noticing — most of what is confidently said about him is inference from his books.

A salary, an imperial examination, and very little else on the record. What would you ask him?

Ask Fibonacci

• “What problems did they set you at Frederick's court?”
• “Why did Pisa decide to pay you a salary?”
• “Why is so little known about your life?”
• “Where does the name Fibonacci actually come from?”
• “If only one of your books survived, which should it be?”

What Fibonacci changed

The Liber Abaci carried the Hindu-Arabic numerals into Latin Europe and, through the abacus schools that taught from it, into the ordinary practice of European commerce. Every calculation performed in the Western world is done with those figures. His book also brought the fraction bar west, and his Liber Quadratorum is a genuine and neglected contribution to the theory of numbers.

A debate that continues

Almost nothing is documented about his personal life, and most confident statements about him are inferences from his books. The sequence named after him was described by Indian scholars of prosody centuries earlier, and the name itself was attached to both the man and the sequence by later writers.

Keep exploring — ask Fibonacci

• “Does it matter that the sequence was known in India first?”
• “Why do useful ideas sometimes take centuries to be adopted?”
• “Which is the better book — the useful one or the original one?”

Related lives

• Muhammad al-Khwarizmi — Who Named Algebra And The Algorithm
• Luca Pacioli — Who Printed The Merchant's Arithmetic
• François Viète — Who Gave The Known Quantities Letters
• Regiomontanus — Who Made Triangles A Subject

Related themes

Place value and zero · Fractions · Ratio and proportion