Devlin's Angle

October 2002

The 800th birthday of the book that brought numbers to the west

Liber abaci, the book that gave numbers to the western world, is exactly 800 years old this year.

Numbers are so ubiquitous in modern life that it is easy to take them entirely for granted - to fail to notice how indispensable they are. Yet think how different life would be without them. How would we measure our height, weight, or wealth? How would we measure temperature or speed, or keep track of time or record the date? How would we pay for goods, or receive payment for our labor? How would we measure out and weigh groceries? What method would we use to "number" the pages of a book? What would take the place of telephone numbers or postal codes or street addresses? And these are just a few of the more visible uses of numbers. Beneath all of modern science, technology, medicine, business, and commerce lie oceans of numbers and mathematics.

But it wasn't always this way. In fact, the way we write numbers today, using just the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the methods we use to compute with them are less than two thousand years old. And for much of that period this system was unknown in the western world.

Prior to the advent of the modern way to write numbers, the most common system in use was the one invented by the Romans. The Roman numeral system, still found today in certain specialized circumstances, began with the simplest number system of all: the tally system, where you simply make a vertical mark to record each item in a collection: I, II, III, IIII, IIIII, etc.

This becomes hard to read once you have more than four or five items to count, so the Romans introduced a few additional symbols: V for five, X for ten, L for fifty, C for a hundred, and M for a thousand. For example, using this system, the number one thousand two hundred and seventy eight (1,278) can be written as MCCLXXVIII. This works out as M + C + C + L + X + X + V + I + I + I, or in modern notation 1000 + 100 + 100 + 50 + 10 + 10 + 5 + 1 + 1 + 1, which sums to 1278.

The order in which the symbols M, C, L, X, V, I are written does not matter, whereas the order in which we write the digits in the number 1278 very definitely does make a difference. In technical language, the Roman numeral system was not positional. (Later variants of the Roman system did have a positional element.) Much of the power and efficiency of our modern system comes from its positional nature.

Roman numerals are fine for recording numbers, and for doing simple additions and subtractions, which meant they were adequate, if somewhat cumbersome, for commerce and trade. But multiplication and division are not at all easy, and there is no way the Roman system could form the basis for any scientific or technical work. (Merchants and accountants used a physical abacus to do arithmetical computations.)

Then, in 1202, a young Pisan scholar called Leonardo wrote a book, Liber abaci ("The Book of Calculation"), in which he described a remarkably efficient new way to write numbers and do arithmetic that he had learned from Arab traders and scholars while traveling through North Africa. They, in turn, had picked it up from the Indians, who had developed it over many hundreds of years in the early part of the first millennium.

Leonardo was born in 1175 AD in Pisa (we assume), and died in 1250, presumably in Pisa. His full name is Leonardo Pisano (Leonardo of Pisa), but he is better known today as Fibonacci, a name that probably arose as a contraction of the Latin filius Bonacci (son of Bonacci). There is no evidence that Leonardo ever referred to himself this way. The name seems to have been given to him by later scholars. Leonardo did sometimes refer to himself as "Bigollo," which was a Tuscan dialect term meaning traveler.

Fibonacci's father, Guilielmo (William) Bonacci, was a Pisan merchant, who (from around 1192) held a diplomatic post in North Africa. Guilielmo was based in Bugia (later Bougie and now Bejaia), a Mediterranean port in Northeastern Barbary (now Algeria). Bugia lay at the mouth of the Wadi Soummam, near Mount Gouraya and Cape Carbon. Guilielmo's main duties were to represent the merchants of the Republic of Pisa in their dealings with the customs. At that time, Pisan merchants traded extensively there and elsewhere. (By the end of the twelfth century, the struggle between the Papacy and the Holy Roman Empire had left many Italian cities independent republics. Some of them, most notably Genoa, Venice, and Pisa, had become major maritime traders.)

Fibonacci traveled widely in Barbary with his father, and was later sent on business trips to Egypt, Syria, Greece, Sicily, and Provence. He seems to have learned much of his mathematics in Barbary. In particular, it was there that he observed the Arab merchants using a remarkable system for writing numbers and doing arithmetic.

After Leonardo ended his travels and returned to Pisa in 1200, he wrote (in Latin) a number of mathematics books, only some of which have survived to this day. His first book, and by far the most famous, was Liber abaci. In it Fibonacci described the Hindu-Arabic numerals and the place-valued decimal system for expressing numbers that we use today, and gave detailed instructions on how to compute with them (a process that became known as algorism, which subsequently led to the modern word algorithm). Fibonacci himself always referred to the numerals as "Hindu"; later writers introduced the term "Hindu-Arabic", and even "Arabic".

Liber abaci was a big book. The English language translation, which has just been published, runs to 672 pages. The first chapter begins:

"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated."

Leonardo then goes on to present a large collection of problems designed to provide exercise in using the new number system. Some of the problems were of a practical nature, aimed at merchants: problems about the price of goods, calculation of profits, and conversions between different currencies. Others were more like the word problems you find in modern algebra texts, including the famous rabbit problem that led to the number sequence that today bears his name: the Fibonacci sequence. (Many of the 175,000 hits you get when you do a web search on the name "Fibonacci" are to the Fibonacci sequence. I discussed the rabbit problem and the Fibonacci sequence in this column in March 1999. Click here.)

The book also contains a geometric explanation of the rules for solving quadratic equations, but the main focus is on the arithmetic problems.

The first edition of Liber abaci appeared in 1202. Of course, in those days, books were produced by hand. No copies of that edition are known to exist today. Fibonacci prepared a second edition in 1228, which carried a preface stating that "... new material has been added from which superfluous had been removed ...".

The earliest complete printed copy of the 1228 edition is one printed by Baldassarre Boncompagni in Rome in the period 1857-1862.

Fibonacci wrote a number of other books, three of which have, along with the 1228 edition of Liber abaci, survived to this day:

Practica geometriae, published in 1220, contained a large collection of geometry problems, based in large part on Euclid's Elements, together with a lot of practical trigonometric problems aimed at surveyors.

Flos ("The Flower"), published in 1225, is largely devoted to algebra, and contains Fibonacci's solutions to a series of problems posed to him in a contest organized for the emperor Frederick II.

Liber quadratorum ("The book of squares"), published in 1225, is a book on advanced algebra and number theory, and is Fibonacci's most mathematically impressive work, revealing his substantial mathematical abilities. It deals mainly with the solution of various kinds of equations involving squares, generally with more than one variable, where the solutions have to be whole numbers - the very kind of problem that led Fermat to pose his famous problem, eventually solved by Andrew Wiles in 1994.

Among the lost works are Di minor guisa, a book on commercial arithmetic, and a commentary on Book X of Euclid's Elements, which contained a numerical treatment of irrational numbers, which Euclid had dealt with geometrically.

The title Liber abaci is sometimes mistranslated as "book of the abacus", but it is more accurately rendered as "book of calculation", since, not only did it say nothing about using an abacus, it described methods that eliminated the need for such a device. It is sometimes spelt with two c's: Liber abacci.

Liber abaci was not the first book written in Europe to describe the new numeral system. For example, the ninth century Arabic mathematician Al-Khwarizmi wrote one such exposition, which, from around 1140 onwards, several scholars translated into Latin and other western European languages. Nor did Liber abaci achieve the popularity of some later, more elementary expositions, such as De Algorismo, written by the thirteenth century English scholar John of Halifax (also known as Sacrobosco). The eleven chapters of Halifax's text dealt with topics such as addition, subtraction, multiplication, division, square roots and cube roots. Liber abaci did, however, turn out to be the most influential exposition. The reason was that those other translations were written for, and read only by, academic scholars. Interested solely in the benefits of the system within mathematics, they did not see its significance to the commercial world. In contrast, when Leonardo wrote Liber abaci, he did so for the merchants. He took pains to explain the concepts in a way that those highly practical men could understand, presenting many examples from everyday commercial life.

Fibonacci's expository writings made him something of a celebrity. As is still the case today when accomplished mathematicians and scientists excel at exposition, Fibonacci's skill as a writer - his ability to reach out to the layperson - came to overshadow his very significant abilities as a mathematician, and it was only long after his death that the full range of his mathematical accomplishments was finally recognized. Today, he is regarded as the greatest number theorist during the entire 1300 year period between Diophantus in the fourth century A.D. and Fermat in the 17th century.

And yet, for all his fame then and now, we know remarkably little about Leonardo the man. We do know that he became a favorite guest of the Holy Roman emperor, Frederick II, who was a great lover of learning and scholarship, with a particular interest in mathematics and science. After Fibonacci returned to Pisa in 1200, he corresponded with some of the scholars at Frederick's court, among them Michael Scott, the court astrologer, and Theororus, the court philosopher. It was through them that the emperor came to hear of this talented mathematician. When the court met in Pisa in 1225, another court scholar who knew Fibonacci, Dominicus Hispanus, suggested to Frederick that he invite the Pisan to come to the court to demonstrate his mathematical prowess.

The court scholar Johannes of Palermo presented Fibonacci with a number of mathematical challenges, which the latter solved. Fibonacci wrote up three of his solutions under the title Flos, which he proudly presented to Frederick.

Our knowledge of Fibonacci's travels to Africa come from a brief passage he wrote in Liber abaci:

"When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms."

After 1228, there is only one further known reference to Fibonacci. That is a decree made by the Republic of Pisa in 1240, in which a salary is awarded to "the serious and learned Master Leonardo Bigollo". The salary was given in recognition for the services Fibonacci had given to the city in the form of advice on matters of accounting and for teaching the citizens.

Postscript

Last month, I flew to Italy to attend a conference in Rome. After the meeting was over, I spent a couple of days in Pisa, curious to learn how aware were today's inhabitants of that beautiful city of their famous predecessor - a man who quite literally changed the course of history.

A web search before I left told me that there was a street in Pisa called the Lungarno Fibonacci as well as a statue of the man, athough sources differed as to the statue's location, with one website even claiming that there are two statues. (There are not.)

The street was easy enough to find. It runs alongside the south bank of the Arno River at the eastern end of the city, adjacent to a delightful, if slightly run-down, little park called the Giardino Scotto, named after Leonardo's friend Michael Scott, the astrologer whom I have mentioned already, to whom Leonardo dedicated Liber abaci.

But the statue proved harder to track down. According to some web sites, it was located in the Giardino Scotto, but that is not true. It turns out that it used to be there, but some years ago it was moved. Another web source referred to the statue as being located in a cemetery adjacent to the Piazza dei Miracoli, the beautiful, green-lawned area housing the Cathedral, the Baptistery, and the famous bell tower, the Leaning Tower of Pisa.

Now, there is a cemetery next to the Piazza dei Miracoli: a small Jewish cemetery at the north west corner of the square. But that did not seem to be a very likely location in which to find a memorial to the (presumably) Catholic Leonardo. And indeed, it is not there. So where was this statue?

I walked into the official City of Pisa Information Center at the edge of the square.

"Where is the statue of Leonardo?" I asked the woman sitting behind the desk.

"Leonardo Da Vinci?" the woman replied.

"No, Leonardo of Pisa," I answered.

The good lady, whose English seemed impeccable, looked at me as if I were from another planet. "Leonardo Da Vinci!" she repeated firmly, stressing the words "Da Vinci," clearly intent on correcting me.

"No, Leonardo of Pisa - Fibonacci." I tried to be equally firm.

The information officer clearly thought she was dealing with a complete imbecile. "There is no Leonardo of Pisa," she declared. "There is no such statue here."

It was clearly pointless pursuing this exchange. Leaving the information office somewhat frustrated, I took a second, and more thorough look at one of the tourist information signs posted around the square.

There are, it turns out, not three but four buildings that make up the religious complex of the Piazza dei Miracoli. In addition to the Cathedral, the Baptistery, and the Bell Tower, all begun around the same time in the middle of the twelfth century, there is a fourth building, the Camposanto. Its English name, the information poster said, was Monumental Cemetery. Aha!

The Camposanto was started in 1278, after the other three buildings were essentially completed. Compared to its three sisters, the face this fourth building presents to the outside world is unremarkable. Apart from an ornately carved Gothic tabernacle that rises up above one of the two large metal doorways that open out toward the Cathedral, all the visitor sees from the Piazza is a long, low, clean white stone wall. The Camposanto keeps its more discrete beauty hidden from the outside world, facing inwards, with four cloistered walkways looking onto a long rectangular lawn.

I entered the cemetery through the left-hand door, turned left and walked around the western end. And there, facing me, at the far end in front of the eastern wall, was the imposing statue of Leonardo Fibonacci. (Perhaps it would be more accurate to describe it as a statue to Fibonacci. There is no known contemporary drawing of Leonardo, so the statue may well be a work of pure fiction.)

The statue had started out in the Camposanto. Then it had been moved to the Giardino Scotto - to save it from possible damage during the Second World War, one source told me. (If so, it was a wise move, since the Camposanto was largely destroyed in 1944, and had to be extensively renovated.) After some years, exposure to the riverside weather started to take its toll on the statue, and eventually it was taken away, restored and cleaned, and then returned to its original location, alongside Pisa's other illustrious citizens, where it belongs. Less than fifty yards from the City of Pisa Information Bureau where the lady told me there was no such monument.

Well, the employees in today's City of Pisa information bureau might not know much about Leonardo, but the splendid location the statue occupies indicates that someone in Pisa, at least, recognizes his importance. As well they should.

Happy 800th birthday, Liber abaci.

Request

In my brief time in Pisa, I was unable to find out anything definitive about the history of the statue of Leonardo. Who carved it and when? When exactly was it moved from the Camposanto to the Giardino Scotto and why? When was it renovated and moved back? If any reader knows the answer to any of these questions, or has any other information about the statue, I'd be curious to know. In fact, my interest now piqued, I'd love to hear from anyone with knowledge of Fibonacci other than that available through a routine web search.


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His latest book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time , published by Basic Books.