Fibonacci and the Liber abaci

The story

Doing sums with Roman numerals

Try doing a multiplication, for example 24 times 13, with Roman numerals: XXIV times XIII. It is almost impossible without tricks, because Roman numerals have no value that depends on position and have no zero. To do calculations, in the Middle Ages, merchants used the abacus, a board with pebbles or counters on which the numbers were moved, and they had to write the results in Roman numerals. Calculating was a job for specialists.

Meanwhile, in India, for some centuries, a much more convenient system had been in use, which the Arabs had taken up and spread in their world: ten digits, from 0 to 9, in which each digit is worth more or less according to the place where it stands. The number 222 written this way contains the same symbol three times, but each is worth something different: two hundred, twenty and two. In Europe almost nobody knew it.

A boy from Pisa in Africa

Leonardo was born at Pisa around 1170. His father, Guglielmo Bonacci, was an official who worked for the Pisan merchants at Bougie, in Algeria, a busy port where traders from many countries met. As a boy Leonardo was sent there and learned calculation with the Indian digits from Arab teachers. He was struck by how convenient and precise that method was, compared with the European one.

He then travelled to Egypt, Syria, Greece, Sicily and Provence, for business and for study, and compared the various methods of calculation he came across. When he returned to Pisa he decided to write a book to explain all this to Europeans.

The Liber abaci

The book came out in 1202. It is called “Liber abaci”, that is, “Book of Calculation” (the word “abacus” here does not mean the instrument, but the art of calculating). It is an enormous treatise in Latin, divided into fifteen chapters. At the beginning it explains the nine Indian digits and the sign “0”, which he calls “zephirum”, from the Arabic, from which our words “zero” and “cipher” come. Then it teaches how to do addition, subtraction, multiplication and division with that system, how to work with fractions and how to solve practical problems: how to divide profits among partners, how to change coins, how to calculate interest, how to mix metals.

The book also contains curious problems, such as the one about rabbits: a pair of rabbits becomes adult after a month, and every following month produces a new pair; how many pairs are there after a year? The answer forms a sequence of numbers, 1, 1, 2, 3, 5, 8, 13, 21, in which each number is the sum of the two before. Today it is called the “Fibonacci sequence”, a name given much later, in the 19th century (Fibonacci is a shortening of “filius Bonacci”, son of Bonaccio).

A respected mathematician

The book did not change the world at once: the new digits were looked on with suspicion, because zero seemed easy to falsify, and in Florence, in 1299, the Arte del Cambio, the guild of the money-changers, even banned them from its own account books. But merchants grasped their advantages, and in the “abacus schools” of the Italian towns boys learned practical calculation.

Leonardo was also known at the court of the Emperor Frederick II, who around 1225 tested him with some problems, which he solved. In 1241 the commune of Pisa granted him an annual payment for his work as an adviser on the public accounts. We do not know exactly when he died: perhaps after 1240. In Pisa there is a statue dedicated to him in the Camposanto, near the square of the famous leaning tower.