For someone whose intellectual work was to change the course of history, Leonardo’s schooling would have been decidedly basic—for the simple reason that there was nothing else available. In the late twelfth century, education throughout Europe was in the hands of the monasteries and the cathedrals. The curriculum, if it can be so called, comprised little more than learning to read and write, and to write numbers using the Roman system. Leonardo may also have been taught some practical geometry.1
He would have attended school between ten and twelve years of age. The school was most likely in the cathedral in Pisa. It would have had no desks or chairs; the pupils—all boys—would have sat cross-legged on the floor. Instruction was mostly oral, and the students learned by rote, with the teacher first reciting a phrase and the class then chanting it in unison. Any writing was done Roman style, by scratching a wax tablet with a bone stylus, using the smooth side of the stylus as an eraser, both to correct errors and to clean the surface for further use. Almost certainly, Leonardo would have found computation using Roman numerals tedious. Arithmetic, particularly multiplication by repeated addition, was more speedily done with a counting board (abacus).
After Leonardo had finished his preliminary instruction, his further education in mathematical matters would likely have been in a fondaco run by one of his father’s friends, where he would have learned the systems of measurement and money and the use of an abacus. The fondaco—the name derived from the Arabic funduq—was a business establishment in whose front customers and merchants would discuss merchandise, prices, and politics, while in the rear the bookkeeper kept the accounts. Some of the larger ones provided traveling merchants with accommodation and a place to store goods. They were also where government taxes were levied.
The Italian monetary system at the time was like the system used in England until 1970. The lowest-denomination coins were denari (pennies), twelve denari made a solidus (shilling), and twenty solidi made a libra (pound). The bookkeeper’s abacus was divided into seven horizontal rows. The bottom row was used for denari, the next one up represented solidi, the third was for librae, the fourth denoted multiples of twenty librae, the fifth hundreds of librae, the sixth thousands, and a counter in the seventh and final row represented ten thousand librae. Some bookkeepers used different colored counters to indicate in-between numbers; for example, a red counter in the libra column might have represented five librae, with a black counter being used for a single libra.
Young people became bookkeepers by serving an apprenticeship. At first they would simply stand and watch the master at work, then they would graduate to standing alongside the expert, handing him the counters from bowls placed next to the counting board, and finally they would reach the stage when, under the watchful eye of the master, they could perform some of the computations themselves. Yet even in the hands of an expert, the counting board, like any form of abacus, was cumbersome and provided no permanent record of the calculation.
When he was about fourteen years of age, Leonardo would have left the fondaco and most likely traveled with an older merchant, a form of apprenticeship system common in those days. Around that time his father summoned him to Bugia. No one knows exactly when he made this voyage. In the introduction to Liber abbaci, he later 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.”
Bugia had originally been a minor Roman colony called Saldae. In the eleventh century, the Berbers revived it, and it rapidly grew to be one of the most important Islamic ports on the Barbary Coast. In the middle of the twelfth century, the Pisans ousted their Genoese rivals and established their own trading port there. Through Bugia, the Pisans exported European goods to North Africa and brought to Europe various Eastern luxury items, including silks, spices, and two commodities the city was particularly famous for: a fine grade of beeswax and a high-quality leather.
Leonardo’s father left Pisa to take up his diplomatic post in Bugia sometime between 1180 and 1185, and most likely sent for his son a year or so after his arrival. Leonardo would have begun his journey from Porto Pisano, in all probability setting sail in the spring or early summer. Few vessels put to sea during the fall or winter, when a severe Mediterranean storm could make any voyage hazardous. Pisan ships that made port in Spain or Africa in the fall would have to remain there until the next spring. Ships generally departed on a Monday evening. On the Sunday before leaving, Leonardo probably went to church to pray for a safe journey, followed by a farewell feast with his family. Then, early on the Monday morning, he and the other travelers would have assembled by the Church of Saint Paul on the Arno and mounted horses for the ride to Porto Pisano. Many of his fellow passengers would have been pilgrims heading for the Holy Land.
Sea journeys were always risky. In addition to the possibility of running into a storm—even in the summer—there was always a chance of being attacked by pirates. The Mediterranean swarmed with privateers, originating both from Muslim North Africa—the infamous Barbary Coast—as well as from the Italian ports of Venice, Genoa, and Pisa itself. Italian privateers were financed by groups of shareholders, just as the peaceful merchant vessels were. Although shareholders stipulated that privateers should attack only vessels of enemy countries, once they were on the high seas many captains succumbed to the temptation of a healthy profit and overlooked such restrictions.
Italian merchants had recently started to take advantage of a novel scheme to protect themselves against possible losses of a vessel and its cargo. For a price, a group of wealthy investors would promise to cover any financial loss. The origin of insurance protection in twelfth-century Italy is reflected in our use of the word “policy”, which comes directly from the Italian word polizza, meaning “promise”.
Tunis was the ultimate destination of most voyages. A few ships made the journey by sailing down the Italian coast and then directly across the Mediterranean, but most took a circuitous route that offered the greatest opportunities for trading: first, west to Spain, then south across the Straits of Gibraltar to Morocco, then eastward toward Tunis along the North Africa coast. Each ship’s itinerary was fixed in advance by agreement among the merchants whose goods were being carried. Most likely Leonardo’s ship took the circuitous route. The two-thousand-mile (ca. three-thousand-kilometer) journey to Bugia would have taken approximately two months. For much of the journey the ship would have stayed close to land. Not only did this make navigation simpler and more reliable; there was a greater margin of safety. If a storm struck, the captain could take his vessel close in to shore for shelter.
The city Leonardo arrived in was one of the most important ports in North Africa, and its Arab traders ventured 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. It was part of the Maghreb, a region in North Africa that today comprises Morocco, Algeria, Tunisia, Libya, and Mauritania, but in Leonardo’s times the name referred to the much smaller part of that region lying between the high Atlas Mountains and the Mediterranean Sea. The Maghreb was united as a single political entity during the first years of Arab rule, in the early eighth century, and again for several decades under the Berber Almohads from 1159 to 1229. At other times the ties had been primarily through trade and cultural exchange.
Leonardo’s father would most likely have lived in the sizable Italian community near the harbor. Most of the business activity was centered on the fondaco. The Pisans had signed treaties with the various cities they traded with, governing issues of legal jurisdiction, safe conduct, and access to and from the fondaco, the first of which had been signed on July 2, 1133, with Alibibn Yusof, the king of Morocco and Tlemcen. Since Guilielmo brought his son over to prepare him for his future, we can be sure that he encouraged Leonardo to spend a lot of his time in the fondaco. The Arabs viewed mathematics in a very practical manner, as something to be used by traders, land surveyors, and engineers, and wrote texts for those professional people, so Guilielmo could well have seen the Hindu-Arabic system as a powerful new tool that would benefit his son.
Much of what we know about Leonardo’s time in Bugia comes from the brief prologue with which be began Liber abbaci. The first part describes the approach his book takes.13
You, my Master Michael Scott,14 most great philosopher, wrote to my Lord15 about the book on numbers which some time ago I composed and transcribed to you;16 whence complying with your criticism, your more subtle examining circumspection, to the honor of you and many others I with advantage corrected this work. In this rectification I added certain necessities, and I deleted certain superfluities. In it I presented a full instruction on numbers close to the method of the Indians, whose outstanding method I chose for this science. And because arithmetic science and geometric science are connected, and support one another, the full knowledge of numbers cannot be presented without encountering some geometry, or without seeing that operating in this way on numbers is close to geometry; the method is full of many proofs and demonstrations which are made with geometric figures.2 And truly in another book that I composed on the practice of geometry17 I explained this and many other things pertinent to geometry, each subject to appropriate proof. To be sure, this book looks more to theory than to practice.18 Hence, whoever would wish to know well the practice of this science ought eagerly to busy himself with continuous use and enduring exercise in practice, for science by practice turns into habit; memory and even perception correlate with the hands and figures, which as an impulse and breath in one and the same instant, almost the same, go naturally together for all; and thus will be made a student of habit; following by degrees he will be able easily to attain this to perfection. And to reveal more easily the theory I separated this book into xv chapters,19 as whoever will wish to read this book can easily discover. Further, if in this work is found insufficiency or defect, I submit it to your correction.
At this point, the prologue changes direction, as Leonardo recounted how he came to learn this remarkable new calculating method, thereby providing the only autobiographical information we have about its author. Why he included this is unknown. Like mathematicians before and after him, Leonardo cared little for the history of the discipline. Mathematics is eternal, and exactly when something new is discovered and by whom is of secondary importance. Mathematicians admire those who make great discoveries, but their interest is generally in what is discovered, not in who got there first. Nevertheless, Leonardo presumably realized that the invention his book described was a monumental one, and at the back of his mind may have lurked the notion that one day people would wonder how this great Hindu invention found its way from the Muslim scholars and merchants who had held it for many centuries to the practical trading men of northern Europe. In any event, he broke with tradition and inserted an all-too-brief summary of the part he played in the story.
As my father was a public official away from our homeland in the Bugia customshouse established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learned from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily, and Provence, and their various methods, to which locations of business I traveled considerably afterwards for much study, and I learned from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method.20 Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle geometric art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum.21 If, by chance, something less or more proper or necessary I omitted, your indulgence for me is entreated, as there is no one who is without fault, and in all things is altogether circumspect.
As Pisa’s trading representative in Bugia, Leonardo’s father would have had the task of maintaining relations with the Muslim authorities, safeguarding the rights of the fondaco, keeping records of the goods passing through, and overseeing the proper levying of taxes—activities that would surely have required that Guilielmo was fluent in Arabic. This supposition is borne out by a surviving account from the funduq in Damascus in 1183 that refers to the “Christian clerks of the customs” who “write in Arabic, which they also speak.”3 It is reasonable to assume the same was true elsewhere in the Arabic-speaking regions where the Italians did business. In bringing his son to join him to complete his professional education, Guilielmo presumably intended that Leonardo not only learn the Arabs’ marvelous new ways of doing arithmetic but also master their language.
There is no way to know for certain whether Leonardo actually did learn to read Arabic, but the evidence suggests so, and this is the accepted view of historians today.4 With a mastery of Arabic, Leonardo would have been able to broaden his mathematical knowledge well beyond what he could observe in the Bugian marketplace. Among the Arabic scholars, teachers, and students who were known to have moved between the cities of the Maghreb in the late eleventh and early twelfth centuries were several mathematicians: al-Hassār moved from Ceuta to Marrakech and then to Spain, ibn ‘Aqnū n moved from Marrakech to Seville and then back to Marrakech, ibn al-Mun’im was born in al-Andalus but worked in Marrakech, and the Andalusian al-Qurashi worked first in Seville and then in Bugia. As a result their mathematical works, and any they had copies of, were presumably circulating between all six cities and would thus have been available in Bugia, where Leonardo could have had access to them.