Contents Cover Title Page Chapter 0. Your Days Are Numbered Chapter 1. A Bridge of Numbers Chapter 2. A Child of Pisa Chapter 3. A Mathematical Journey Chapter 4. Sources Chapter 5. Liber abbaci Chapter 6. Fame Chapter 7. The Fibonacci Aftermath Chapter 8. Whose Revolution? Chapter 9. Fibonacci’s Legacy—in Stone, Parchment, and Rabbits Acknowledgments Notes Footnotes Bibliography A Note on the Author By the Same Author Imprint CHAPTER 0 Your Days Are Numbered TRY TO IMAGINE A DAY WITHOUT numbers. Never mind a day; try to imagine getting through the first hour without numbers: no alarm clock, no time, no date, no TV or radio, no stock market report or sports results in the newspapers, no bank account to check. It’s not clear exactly where you are waking up either, for without numbers modern housing would not exist. The fact is, our lives are totally dependent on numbers. You may not have “a head for figures,” but you certainly have a head full of figures. Most of the things you do each day depend on and are conditioned by numbers. Some of them are obvious, like the ones listed above; others govern our lives behind the scenes. The degree to which our modern society depends on numbers that are hidden from us was made clear by the worldwide financial meltdown in 2008, when overconfident reliance on the advanced mathematics of futures predictions and the credit market led to a total collapse of the global financial system. How did we—as a species and as a society—become so familiar with and totally reliant on these abstractions our ancestors invented just a few thousand years ago? As a mathematician, I had been puzzled by this question for many years, but for most of my career as a university professor of mathematics, the pressures of discovering new mathematics and teaching mathematics to new generations of students did not leave me enough time to look for the answer. As I grew older, however, and came to terms with the unavoidable fact that my abilities to do original mathematics were starting to wane a bit—a process that for most mathematicians starts around the age of forty (putting mathematics in the same category as many sporting activities)—I started to spend more time looking into the origins of the subject I have loved with such passion since I made the transition from “It’s boring” to “It’s unbelievably beautiful” around the age of sixteen. For the most part, the story of numbers was easy to discover. By the latter part of the first millennium of the Current Era, the system we use today to write numbers and do arithmetic had been worked out—expressing any number using just the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and adding, subtracting, multiplying, and dividing them by the procedures we are all taught in elementary school. (Units column, tens column, hundreds column, carries, etc.) This familiar way to write numbers and do arithmetic is known today as the Hindu-Arabic system, a name that reflects its history. Prior to the thirteenth century, however, the only Europeans who were aware of the system were, by and large, scholars, who used it solely to do mathematics. Traders recorded their numerical data using Roman numerals and performed calculations either by a fairly elaborate and widely used fingers procedure or with a mechanical abacus. That state of affairs started to change soon after 1202, the year a young Italian man, Leonardo of Pisa—the man whom a historian many centuries later would dub “Fibonacci”—completed the first general purpose arithmetic book in the West, Liber abbaci, that explained the “new” methods in terms understandable to ordinary people (tradesmen and businessmen as well as schoolchildren).1 While other lineages can be traced, Leonardo’s influence, through Liber abbaci, was by far the most significant and shaped the development of modern western Europe. Leonardo learned about the Hindu-Arabic number system, and other mathematics developed by both Indian and Arabic2 mathematicians, when his father brought his young son to join him in the North African port of Bugia (now Bejaïa, in Algeria) around 1185, having moved there from Pisa to act as a trade representative and customs official. Years later, Leonardo’s book not only provided a bridge that allowed modern arithmetic to cross the Mediterranean, but also bridged the mathematical cultures of the Arabic and European worlds, by showing the West the algebraic way of thinking that forms the basis of modern science and engineering (though not our familiar symbolic notation for algebra, which came much later). What Leonardo did was every bit as revolutionary as the personal computer pioneers who in the 1980s took computing from a small group of “computer types” and made computers available to, and usable by, anyone. As with those pioneers, most of the credit for inventing and developing the methods Leonardo described in Liber abbaci goes to others, in particular Indian and Arabic scholars over many centuries. Leonardo’s role was to “package” and “sell” the new methods to the world. Not only did the appearance of Leonardo’s book prepare the stage for the development of modern (symbolic) algebra and hence modern mathematics, it also marked the beginning of the modern financial system and the way of doing business that depends on sophisticated banking methods. For instance, Professor William N. Goetzmann of the Yale School of Management, an expert on economics and finance, credits Leonardo as the first to develop an early form of present-value analysis, a method for comparing the relative economic value of differing payment streams, taking into account the time value of money. Mathematically reducing all cash flow streams to a single point in time allows the investor to decide which is the best, and the modern version of the present- value criterion, developed by the economist Irving Fisher in 1930, is now used by virtually all large companies in the capital budgeting process.1 THE ONLY PIECE of the story of numbers that was missing was an account of Leonardo himself and, apart from a few scholarly articles, of the nature of his book. History has relegated him to an occasional footnote. Indeed, his name is known today primarily in connection with the Fibonacci numbers, a sequence of numbers that arises from the solution to the rabbit problem,3 one of many whimsical challenges he put in Liber abbaci to break the tedium of the hundreds of practical problems that dominate the book. Part of the reason Leonardo has been overlooked, whereas comparable figures like Copernicus, Galileo, and Kepler were not, may be that to most laypersons science seems to serve a greater purpose than mathematics. Another reason why generations may have overlooked Leonardo is that the change in society brought about by the teaching of modern arithmetic was so pervasive and all-powerful that within a few generations people simply took it for granted. There was no longer any recognition of the magnitude of the revolution that took the subject from an obscure object of scholarly interest to an everyday mental tool. Compared with Copernicus’s conclusions about the position of Earth in the solar system and Galileo’s discovery of the pendulum as a basis for telling time, Leonardo’s showing people how to multiply 193 by 27 simply lacks drama. The comparative neglect of Leonardo has no doubt been caused by two other factors. Very little was recorded about his life, discouraging biographies. And Leonardo was more a salesperson of modern arithmetic rather than its inventor. The mathematical advances he described in Liber abbaci were developed by others, and others also wrote books describing those mathematical ideas. In the world of scientific biography, the inventor tends to get the glory. But inventions —an idea, a theory, a process, a technology—need to be made accessible to the world. The personal computer on which I write these words, with its familiar windows, mouse-controlled pointer, and the like, was invented by brilliant teams of researchers at the Stanford Research Institute and the Xerox Palo Alto Research Center in the 1970s, but it was put into everyone’s hands by a few pioneering entrepreneurs. The computer revolution would undoubtedly have happened anyway, just as we would have figured out the motion of the planets had Kepler not lived, and gravity without Newton. But the likes of Apple Computer’s Steve Jobs and Microsoft’s Bill Gates will always be linked to the rise of the personal computer, and in this way Leonardo should be linked to the rise of modern arithmetic. WHAT LEONARDO BROUGHT to the mathematics he learned in Bugia and elsewhere in his subsequent travels around North Africa were systematic organization of the material, comprehensive coverage of all the known methods, and great expository skill in presenting the material in a fashion that made it accessible (and attractive) to the commercial people for whom he clearly wrote Liber abbaci. He was, to be sure, a highly competent mathematician—in fact, one of the most distinguished mathematicians of medieval antiquity—but only in his writings subsequent to the first edition of Liber abbaci in 1202 did he clearly demonstrate his own mathematical capacity. Following the appearance of Liber abbaci, the teaching of arithmetic became hugely popular thoughout Italy, with perhaps a thousand or more handwritten arithmetic texts being produced over the following three centuries. Moreover, the book’s publication, and that of a number of his other works, brought Leonardo fame throughout Italy as well as an audience with the Holy Roman Emperor Frederick II. Since the Pisan’s writings were still circulating in Florence throughout the fourteenth century, as were commentaries on his works, we know that his legacy lived on long after his death. But then Leonardo’s name seemed to be suddenly forgotten. The reason was the invention of movable-type printing in the fifteenth century. Given the Italian business world’s quick adoption of the new arithmetic, not surpisingly the first mathematics text printed in Italy was a fifty-two-page textbook on commercial arithmetic: an untitled, anonymous work known today as the Aritmetica di Treviso (Treviso Arithmetic), after the small town near Venice where it was published on December 10, 1478. Soon afterward, Piero Borghi brought out a longer and more extensive arithmetic text, printed in Venice in 1484, that became a true bestseller, with fifteen reprints, two in the 1400s and the last one in 1564. Filippo Calandri wrote another textbook, Pitagora aritmetice introductor, printed in Florence in 1491, and a manuscript written by Leonardo da Vinci’s teacher Benedetto da Firenze in 1463, Trattato d’abacho, was printed soon afterward. These early printed arithmetic texts were soon followed by many others. Though Liber abbaci was generally assumed to be the initial source for many, if not all, of the printed arithmetic texts that were published, only one of them included any reference to Leonardo.4 Luca Pacioli, whose highly regarded, scholarly abbacus book Summa de arithmetica, geometria, proportioni et proportionalità (All that is known about arithmetic, geometry, proportions, and proportionality) was printed in Venice in 1494, listed Leonardo among his sources, and stated, “Since we follow for the most part Leonardo Pisano, I intend to clarify now that any enunciation mentioned without the name of the author is to be attributed to Leonardo.” The general absence of creditation was not unusual; citing sources was a practice that became common much later, and authors frequently lifted entire passages from other writers without any form of acknowledgment. But without that one reference by Pacioli, later historians might never have known of the great Pisan’s pivotal role in the birth of the modern world. Yet Pacioli’s remark was little more than a nod to history, for a reading of the entire text shows that the author drew not from Liber abbaci itself but from sources closer to his own time. There is no indication he had ever set eyes on a copy of Liber abbaci, let alone read it. His citation of Leonardo reflects the fact that, at the time, the Pisan was considered the main authority, whose book was the original source of all the others. Despite the great demand for mathematics textbooks, Liber abbaci itself remained in manuscript form for centuries, and therefore inaccessible to all but the most dedicated scholars.5 Not only was it much more scholarly and difficult to understand than many other texts; it was very long. Over time it became forgotten, as people turned to shorter, simpler, and more derivative texts. That one mention in Pacioli’s Summa was the only clue to Leonardo’s pivotal role in the dramatic growth of arithmetic.6 It lay there, unnoticed, until the late eighteenth century, when an Italian mathematician called Pietro Cossali (1748– 1815) came across it when he studied Summa in the course of researching his book Origine, transporto in Italia, primi progressi in essa dell-algebra (Origins, transmission to Italy, and early progress of algebra there).2 Intrigued by Pacioli’s brief reference to “Leonardo Pisano”, Cossali began to look for the Pisan’s manuscripts, and in due course learned from them of Leonardo’s important contribution. In his book, published in two volumes in 1797 and 1799, which many say is the first truly professional mathematics history book written in Italy, Cossali concluded that Leonardo’s Liber abbaci was the principal conduit for the “transmission to Italy” of modern arithmetic and algebra, and that the new methods spread first from Leonardo’s hometown of Pisa through Tuscany (in particular Florence), then to the rest of Italy (most notably Venice), and eventually throughout Europe.3 As a result, Leonardo Pisano, famous in his lifetime then completely forgotten, became known—and famous—once again. But his legacy had come extremely close to being forever lost. The lack of biographical details makes a straight chronicle of Leonardo’s life impossible. Where and when exactly was he born? Where and when did he die? Did he marry and have children? What did he look like? (A drawing of Leonardo you can find in books and a statue of the man in Pisa are most likely artistic fictions, there being no evidence they are based on reality.) What else did he do besides mathematics? These questions all go unanswered. From a legal document, we know that his father was called Guilichmus, which translates as “William” (the variant Guilielmo is also common), and that he had a brother named Bonaccinghus. But if Leonardo’s fame and recognition in Italy during his lifetime led to any written record, it has not survived to the present day. Thus a book about Leonardo must focus on his great contribution and his intellectual legacy. Having recognized that numbers, and in particular powerful and efficient ways to compute with them, could change the world, he set about making that happen at a time when Europe was poised for major advances in science, technology, and commercial practice. Through Liber abbaci he showed that an abstract symbolism and a collection of seemingly obscure procedures for manipulating those symbols had huge practical applications. The six-hundred-page book Leonardo wrote to explain those ideas is the bridge that connects him to the present day. We may not have a detailed historical record of Leonardo the man, but we have his words and ideas. Just as we can come to understand great novelists through their books or accomplished composers through their music—particularly if we understand the circumstances in which they created—so too we can come to understand Leonardo of Pisa. We know what life was like at the time he lived. We can form a picture of the world in which Leonardo grew up and the influences that shaped his ideas. (In that we are helped by the survival to this day, largely unchanged, of many of the streets and buildings of thirteenth-century Pisa.) And we know how numbers were used prior to the appearance of Liber abbaci, and how the book changed that usage forever. CHAPTER 1 A Bridge of Numbers LIBER ABBACI TRANSLATES AS “Book of calculation”. The intuitive translation “Book of the abacus” is both incorrect and nonsensical, inasmuch as Leonardo’s book showed how to do arithmetic without the need for any such device as an abacus.7 The distinction is reflected in Leonardo’s spelling. The Latin and Italian word abbacus was used in medieval Italy from the thirteenth century onward to refer to the method of calculating with the Hindu-Arabic number system. The first known written use of the word abbacus with this spelling and meaning was in fact in the prologue of Leonardo’s book. Thereafter, the word abbaco was widely used to describe the practice of calculating. A maestro d’abbaco was a person who was proficient in arithmetic. In fact, abbaco still has that as its primary (preferred) meaning in contemporary Italian.8 Medieval authors did not usually give their works titles. The name we use today for Leonardo’s book comes from his opening statement: Here begins the Book of Calculation Composed by Leonardo Pisano, Family Bonacci, In the Year 1202 In later writings, he also referred to the work as Liber numerorum and, in the dedicatory letter for his book Flos, as his Liber maior de numero. In chapter 5 of another of his books, De practica geometrie, written between the publication of the two editions of Liber abbaci, he used the title Liber abbaci again. “Since at the beginning of the treatise I had promised to discuss how to find cube roots, a topic to which I gave special attention in Liber abbaci, I rewrote the material for a regular chapter here.”1 In addition to its appearance in the opening statement, the word abbaci (the Latin genitive of abbacus) occurs in Liber abbaci three other times: in the prologue, where Leonardo described how he pursued studio abbaci “for some days” in Bugia; in chapter 12, when he stated he would treat a questionibus abbaci; and toward the end of the book, when he explained that his numerical determination of the approximate square root of 743 was done secundum abbaci materiam.