## Devlin's Angle |

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.

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.*

Devlin's Angle is updated at the beginning of each month.