(Previous Section: Calculating in Medieval Europe.)
Al-Khwˆarizmˆı’s text was written in the ninth century for use by merchants as well as scholars and astronomers (Devlin, Man 16). In the West, however, only educated men read the Latin translations of his works. Therefore, medieval Italian merchants who had not had the benefit of formal education were largely (if not completely) unaware of the Hindu-Arabic numerals and how to apply them for easier calculation. To remedy this situation, the Italian Leonardo Pisano described methods in his book Liber Abaci which were intended, from the outset, to be mastered by merchants and other businessmen as well as scholars (Devlin, Finding 80).
As was common practice at the time, Leonardo as an author borrowed ideas, methods, and explanations (sometimes the very words) of knowledgeable experts preceding him. Leonardo acknowledged that he was aware of Al-Khwˆarizmˆı’s books and commentaries on them; he had possibly even studied them. He made “no claims of originality in Liber Abaci, although he did so in another of his books, Liber Quadratorum” (Devlin, Finding 81).
Some people confuse Leonardo of Pisa with Leonardo da Vinci, polymath of the Renaissance, but the famous inventor, scientist, and painter of the “Mona Lisa” was born in Vinci, between and Florence, in 1452, about 200 years after the death of Leonardo of Pisa. All that we know for certain about Leonardo Pisano is contained in a few sentences he wrote about himself in the 1228 edition of his famous Liber Abaci (Horadam). Presented here is the paragraph in English, translated from Latin by Richard E. Grimm in “The Autobiography of Leonardo Pisano” (1973):
“After my father’s appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and, in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business, I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid’s geometric art, I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.”
This exceptionally brief glimpse of the man’s life provides provocative clues about “his personality and the mathematical quality of his mind.” These few sentences reveal that he was a man with great intellectual curiosity who was excited by mathematical scholarship and valued it so much that he wished to share his knowledge with the working class, not just scholars. The words with which he concludes his introduction to Liber Abaci show that he is open-minded, willing and even eager to learn new ideas, receptive to constructive criticism, and modest despite his genius. He was not only revered by readers, students, and authorities of his day, but modern scholars also conclude that he was a man well worthy of respect; some even develop a “warmth of feeling for the modest humility of the man” (Horadam).
In increasing numbers, serious mathematics scholars today are beginning to recognize that Leonardo Pisano may have been the “most noteworthy mathematical genius of the Middle Ages,” and definitely the most influential of all medieval writers in promoting the Hindu-Arabic numerals to European scholars. (Smith and Karpinski).
His given name was Leonardo, so his full name was Leonardo of Pisa, or Leonardo Pisano in Italian, and he was born about the year 1175. He is better known by the nickname that apparently was given him by the math historian Guillaume Libri in 1838, six-and-a-half centuries after his birth: Fibonacci [pronounced fib-on-arch- ee] which is short for filius Bonacci, is the name Leonardo ascribed himself in Liber Abaci (Livio; Knott).
An abbreviation of the Latin phrase, “filius Bonacci,” Fibonacci means “the son of Bonaccio.” Fi’-Bonacci, then, is a general name like the English names of Robin- son and John-son or Bronson (Brown’s son) (Knott, Smith and Karpinski). But (in Italian) Bonacci is also the plural of Bonaccio; therefore, two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as his family name (as in “the Smiths” for the family of John Smith).
Fibonacci referred to himself in writings as “Bonacci” and “Bonaccii” as well as “Bonacij.” It was common in medieval Pisa to mix spoken Italian with written Latin; there remains uncertainty about which is the most correct spelling of what may be considered his surname. However, it is certain he did not use “Fibonacci” when referring to himself (Knott).
Bonacci may be a kind of nickname meaning “lucky son” (literally, “son of good fortune”) which certainly seems representative of the fortuitous circumstances of his life, what little is known of it (Knott). Finally, Mario Livio explains in The Golden Ratio that the nickname Fibonacci can also mean “son of good nature” (Livio 92). Given that he was called “beloved” by his countrymen in an official proclamation in the year 1241, this last explanation for the meaning of his name, “Bonacci,” may be most plausible of all. Leonardo’s humility is evident in the introduction to Liber Abaci, when he says, “If by chance I have omitted anything more or less proper or necessary, I beg forgiveness, since there is no one who is without fault and circumspect in all matters” (Livio 94). Occasionally he signed his name as Leonardo Bigollo since he was one who had studied in foreign lands and, in Tuscany, bigollo means “a traveler” (O’Connor and Robertson; Smith and Karpinski). Interestingly, the term may also mean “good-for-nothing” (O’Connor and Robertson) or “man of no importance” (in the Venetian dialect) (Livio 93). These are similar to yet another meaning of the word, which is “blockhead,” so some have thought Leonardo might have adopted the term as a humble way to identify himself or that it “may have been applied to him by the commercial world or possibly by the university circle,” and subsequently taken upon himself as a challenge “to prove what a blockhead could do” (Smith and Karpinski). The historian of mathematics, Guillaume Libri, appears to have been the first to use the name “Fibonacci” when referring to Leonardo; this was found in a footnote in his 1838 book Histoire des Sciences Mathematique en Italie (History of the Mathematical Sciences in Italy) (Livio 93). Since most modern authors now call Leonardo Pisano “Fibonacci,” one should be prepared to see any of the above variations of his name when reading about him. Contemporary writers identify Leonardo’s father as Guglielmo Bonacci (William), and mention that he was a kind of customs officer in the present-day Algerian port city of Béjaïa, formerly known as Bugia or Bougie, where wax candles (still called “bougies” in French) were exported to France (Devlin, Man 9). A legal document provides the name of one brother, Bonaccinghus, but nothing further is known about the rest of Leonardo’s family (Livio; Smith and Karpinski).Leonardo says his father was a kind of “state official;” it is unclear what the position entails precisely. By some accounts, Guglielmo was elected the consul for Pisa. This was a highly prestigious public office so, if this was in fact the case, most likely William was a wealthy merchant himself before he became the representative for the Pisan merchants who were trading in Bugia (O’Connor and Robertson).
Pisans exported European goods to North Africa through Bugia and brought various Eastern luxury items to Europe, including silks, spices, and a fine grade of beeswax useful for candles, and high-quality leather (Devlin, Man 39). At the close of the twelfth century, Bugia was a center of African commerce sheltered by Mount Gouraya at the mouth of the Wadi Soummam near Cape Carbon. Being a commercial agent of one of the most important Islamic ports on the Barbary Coast concerned with taxation of trade between Pisa and North Africa, Fibonacci’s father would have spent a lot of time monitoring trade transactions in the bustling harbor. He must have been fluent in Arabic since he was expected to maintain relations with the Muslim authorities, safeguard the rights of the fondaco (trading post), keep records of the goods passing through, and oversee the proper levying of taxes (Devlin, Man 41). Of course, Guglielmo would have recognized the value of accounting skills and surely believed “the future would be bright for people who understood numbers thoroughly” (“Fibonacci” Famous). Thus, Leonardo was summoned by his father and taken as a child (Devlin says he was likely no more than fifteen years old) to the Arab port city on the North African coast for the furtherance of his mathematics (accounting) education under the tutelage of Moorish masters (Devlin, Finding 13; Livio; Smith and Karpinski).
Pisa at the dawn of the thirteenth century was in the midst of a “golden age” of commercial, religious, and intellectual prosperity (Smith and Karpinski). She was a busy port and a major Mediterranean trading hub for the importing and exporting of merchandise from both inland and overseas (Livio 93).
“When Leonardo was growing up, a new, heavily fortified city wall was being constructed, to protect the city from attack both by Muslims – this was the time of the Crusades – and by rival Italian cities as part of the ongoing political struggle” between the empire and the papacy (Devlin, Finding 57). Rather than having a detrimental effect, this interurban warfare contributed to the stimulation of commercial activity (Smith and Karpinski). In addition, the city’s many naval fleet victories secured the profitable expansion of trading territory throughout the Mediterranean and also into Syria, significantly enriching the government coffers. At this time, Pisa’s military and economic dominance in the Mediterranean rivaled those of Genoa and Venice (“Editors”).
In addition to city walls, the thriving economy inspired the people of Pisa to begin building an impressive cathedral complex, the Field of Miracles, comprising a cathedral, a baptistery, a bell tower and a cemetery. The 179-foot bell tower, begun in 1173, is known as the Leaning Tower of Pisa and is universally recognized as an unofficial emblem of Italy.
Leonardo would have encountered a commercial frenzy every time he accompanied his merchant father to the customs houses or in bustling streets beside the crowded River Arno. Such activity required unceasing measurements of merchandise quantities and price negotiations; as a customs officer, Leonardo’s father would need to calculate import tax levies and audit ships’ manifests. He must have himself been just one of the many scribes and stewards recording inventories, orders and transactions. Prices were recorded in librae (pounds), solidi (shillings), and denarii (pennies); scribes entered the values in long columns, using Roman numerals, and used abaci to perform the calculations (Devlin, Finding 62).
Since Leonardo’s father was a prosperous merchant participating in all of these business activities, the enormous power and indispensability of arithmetic surely made an impression upon the young boy (Devlin, Finding 58). At the wharves, Leonardo observed other professions besides merchants and traders. He saw surveyors and engineers and shipbuilders working with math. Pisa built and maintained a fleet of hundreds of naval and commercial ships, so incoming cargo in the harbor consisted often of timbers for building as well as sacks of grain, salt from Sardinia, squirrel skins from Sicily, and goatskins from North Africa. There were shipments of leather, alum, and dyes for the textile manufacturers of Italy and northwest Europe. Crates of spices sailed in from the Far East and barrels of wine were common. Among her many exports, Pisan ships regularly transported “barrels of Tuscan wine and oil, bales of hemp and flax, and bars of smelted iron and silver” (Devlin, Finding 59).
Presumably, Leonardo had on countless occasions watched scribes as they listed prices in Roman numerals and added them up using an abacus. The impracticality of the Roman system for complex applications in commerce and trade (such as currency conversion, or commission calculations) would have been frustratingly apparent to an inquisitive merchant’s son who surely wondered whether there was not a more efficient way of calculating than Roman numerals and an abacus (Devlin, Finding 62 and Man 15).
Medieval churches provided formal education for wealthy citizens, so Leonardo would likely have attended school “between ten and twelve years of age in the cathedral in Pisa” (Devlin, Finding 37). Dr. Thomas O’Shea, scholar, author, and retired educator in British Columbia, shares an excerpt from a memoir written by a gentleman of Pisa born in 1308 which “gives us some idea of how students of that time were prepared for their future in the world of trade and commerce.” It is reasonable to expect Leonardo’s education would have been similar. Olinto Bernardini writes:
“Naturally, it was assumed that my two brothers and I would each assume a place in the family Company. It was also assumed that we would become competent in the mathematical skills used to run businesses in our parts. Therefore, at the age of ten, as was the custom, my father sent me to begin the study of abaco. This course, under Maestro Pietro Cataneo of Pisa, lasted two years, but well before it was over, it had changed my life forever. Abaco schooling normally began with an introduction to the Hindu numbers, and with an explanation of the place value that gave them meaning. But already familiar with this numeration, and well-versed in multiplication and addition facts, I quickly advanced to the next four mute [stages] where students were taught division and fractions. All this I devoured eagerly and before long had even caught up to students who were one year older. From there, the maestro introduced me to the core of abaco learning: the mathematics of business that even my great-grandfather had used in his quest to make a fortune. At this level, that is, in the sixth and seventh mute, we were taught to work with prices, barter, partnerships, alligation, proportions, monetary systems, measurements, interest and discount. Our teacher took great care in imparting this knowledge to us, for he appreciated that he had but two years to prepare us for important positions in business.” The two-thousand-mile (ca. three-thousand kilometer) journey to Bugia would have taken approximately two months, during which time the ship would have stayed close to land for greater protection from the weather and to pull into ports for trading. In these ports Leonardo would have met Arab traders who had ventured in their travel even farther afield than the Italians, “journeying not only around the Mediterranean but to Russia, India, and China, and deep into the interior of Africa” (Devlin, Man 41). Arriving in Bugia, Leonardo would have likely joined his father in the sizable Italian community near the harbor (Devlin, Man 41). Later, in his autobiographical paragraph in Liber Abaci (1228), Leonardo explained that, after leaving Bugia, he travelled extensively around the Mediterranean Sea, touring the cities of the East along the Mediterranean coast, visiting the great markets of Egypt and Asia Minor, Syria, Sicily and Provence, Constantinople and Greece. While he was touring, he met and learned from scholars as well as from merchants, “imbibing a knowledge of the various systems of numbers in use in the centers of trade” (Smith and Karpinski). With a mastery of Arabic, Leonardo would have been able to broaden his mathematical knowledge well beyond what he had been able to observe in the Bugian marketplace (Devlin, Man 46). He wrote:
“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…”
(O’Connor and Robertson)
This philomath inevitably realized that the mathematics system used by Oriental merchants had many advantages over all others. In fact, he admitted (later in the paragraph) that he considered every other mathematical system as almost a mistake compared to the method of the Hindus.
Leonardo sought instruction in the Hindu-Arabic arithmetic system and practiced it carefully. He recognized its superiority over the clumsy Roman numeral system in the West, and accordingly decided to write a book to explain the superior system and its applications to the Italians (McClenon). Ending his travels around the year 1200, the scholar returned to Pisa and proceeded to share the “treasure of knowledge” he had acquired, supplemented by ideas of his own (O’Connor and Robertson; “Biography”). However, Leonardo did not merely copy the works of others; he was a brilliant mathematician in his own right. He was exceptionally skillful at explaining mathematical theories, problems, and solutions in a way that the common reader could understand.
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Paperback: 128 pages
Author: Shelley Allen, M.A.Ed.
Publisher: Fibonacci Inc.; 1st edition (2019)