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Book Review The Man of : Fibonacci’s Revolution Reviewed by John Hannah

The Man of Numbers: Fibonacci’s Arithmetic Revolution Keith Devlin 5 Walker and Company, 2011 US$25.00, 192 pages ISBN-10: 0802778127 8 ISBN-13: 978-0802778123 1 1 The Fibonacci numbers are among the most popu- 3 lar topics in —appealing to students, 2 mathematics enthusiasts, and research mathe- maticians as well. As Devlin points out in the book Figure 1. A tiling using successive Fibonacci under review (p. 143), their simple recurrence numbers. relation

an+1 = an + an−1 makes them an ideal context in which school most famous book Liber Abbaci, Fibonacci offers children can practice arithmetic in the decimal a few precious details of his life: place-value system. They also lend themselves to When my father, appointed by his attractive pictorial representation, either through homeland, held the post of pub- tilings of the plane (Figure 1) or through their lic notary in the custom-house of connection to the . Bejaia [on the coast of modern By the time students come to university, text- ] for the Pisan merchants book authors can assume familiarity with the frequenting it, he arranged for me Fibonacci numbers, using them as a basis for to come to him when I was a boy friendly examples in a diverse range of topics, and, because he thought it would including linear , numerical methods, and be useful and appropriate for me, engineering mathematics. he wanted me to spend a few days But who was Fibonacci? What did he actually there in the abbaco school, and do? Where does his work fit into the story of the to be taught there. Here I was in- development of mathematics? troduced to that art [the abbaco] What we know about Fibonacci as a person by the wonderful kind of teaching can be stated quite briefly. The name Fibonacci that used the nine figures of the In- is actually more of a nickname, conferred on him dians. Getting to know the abbaco six hundred years later by the historian Guillaume pleasedmefarbeyondall elseandI Libri. His real name, Leonardo Pisano, tells us set my mind to it, to such an extent that he came from in northern . In the that I learnt, through much study dedication at the start of the 1228 version of his and the cut and thrust of disputa- tion, whatever study was devoted John Hannah is a senior lecturer in mathematics at the Uni- to it in Egypt, Syria, Greece, Sicily versity of Canterbury in New Zealand. His email address is and Provence, together with their [email protected]. different methods, in the course of DOI: http://dx.doi.org/10.1090/noti844 my subsequent journeys to these

May 2012 Notices of the AMS 661 places for the sake of trade. (Trans- 404–405], but also extended treatments of topics lation by Burnett in [2, p. 87].) that we might now see as early developments in Fibonacci’s reputation led to an invitation to a solving systems of linear equations [12] or early mathematical challenge at the court of the Holy acknowledgment of negative solutions [9]. Even Roman emperor Frederick II (he wrote up some of when he uses methods that we now see as obso- the problems and their solutions in another book lete, such as simple false position, we recognize Flos) but the only other official acknowledgment a fellow mathematician as he seeks proofs for of his life that we know about is a decree of techniques that seem to have been discovered the Republic of Pisa in 1240 awarding an annual experimentally [3]. salary of twenty pounds plus expenses for services With so much raw material, we can be fairly con- rendered. fident that we know what mathematics Fibonacci Although we know only a little about Fibonacci’s actually did, even if further study is needed for us life, we do at least seem to have a significant to fully appreciate it. Fibonacci’s role in the devel- fraction of his mathematical output [13, p. 605]. opment of mathematics is harder to describe with His most substantial writings are any confidence, whether we are interested in his dependence on potential sources or his influence • Liber Abbaci, an extensive account of com- on those who followed him. Our understanding of putation in the decimal system, along with these influences is handicapped by the medieval commercial arithmetic, problem solving custom of not acknowledging sources, so that and second-degree algebra, writers borrowed freely from whichever available • Practica Geometriae, a book about practi- manuscripts seemed relevant to their topic. Thus cal and theoretical , and to identify Fibonacci’s sources we have to fall back • Liber Quadratorum, , on comparisons of his texts with earlier texts that described by Vogel as “a first-rate scien- share common families of problems. On this basis, tific achievement” showing Fibonacci “as it is generally accepted that Fibonacci knew about a major theorist” [13, p. 610]. the algebra of al-Khw¯arizm¯ıand Ab¯uK¯amil [13, Vogel’s Dictionary of Scientific Biography article p. 611] from the ninth and early tenth centuries, [13] gives an excellent, but necessarily brief, but that it is unlikely that he knew Al-Karaj¯ı’s work overview of all these works. However, until quite in the late tenth century. Indeed, his knowledge of recently, it has been quite difficult for English- Arabic mathematics seems to be over two hundred speaking readers to get firsthand experience of years out of date [8]. Similar comparisons of Fi- Fibonacci’s works. The books have come down to bonacci’s texts with later texts have led to the view us in a handful of Latin manuscripts that lay ig- that, in his introduction of decimal arithmetic and nored in European libraries until being published its applications to commercial life, he became the by Boncompagni in the nineteenth century [1]. teacher of the masters of computation and survey- English translations of the above three books have ors over the next two or three hundred years [13, appeared only in the last twenty-five years [7], p. 612]. However, sharing families of problems [10], and [11]. need not indicate dependence on one another, Unfortunately, even with these translations now but merely dependence on some other common available, there are still significant hurdles facing source, and so Høyrup has even been able to argue the curious mathematician (or lay reader) wanting that Fibonacci may have been recording an already to know what Fibonacci actually did. Foremost existing culture, rather than starting a new one among these is the lack of any symbolism. Prob- [6]. On the other hand, Fibonacci’s more advanced lems and solutions are presented entirely in words. work on was essentially lost, with Moreover, this isn’t always in the most elegant Eng- comparable material not appearing in European lish, as some translators eschew style in favor of mathematics until the rediscovery of Diophan- an almost literal word-for-word correspondence tus in the late fifteenth century and Fibonacci’s between the original Latin and the final English. own writings not becoming well known until their Thus, even when Fibonacci is doing something we publication by Boncompagni in the nineteenth might find familiar, like using an efficient elimina- century [1]. tion method to solve a linear problem involving Clearly there is material enough here for several four unknowns [11, pp. 415–417], it can be dif- books about Fibonacci and, of course, there are ficult to understand what the problem is asking several different audiences such books could be or which method Fibonacci is using, and the cal- aimed at. In the introduction to his book (Chapter culation takes two pages of solid text. But there 0: Your days are numbered), Devlin pitches his tent are lots of interesting things to be found, not just in the popular market, seeking readers who are isolated examples like the famous rabbit problem well-educated lay people rather than, say, expert that gives rise to the Fibonacci numbers [11, pp. mathematicians or historians. He reminds us about

662 Notices of the AMS Volume 59, Number 5 the ubiquity of numbers in modern life and about placing Pisa in a geographic and political context the decimal representation of numbers that we might have been a bit easier if Devlin had included now take for granted. It is good to stop occasionally one or two maps.2 An assertion here that Fibonacci to think about things we take for granted, and wrote Liber Abbaci primarily for merchants (p. 35) this book is going to do that for the decimal may mislead some readers, although Devlin cor- system. Where did this system come from? Or rects himself later in the book, admitting that more precisely, how did the decimal system reach some sections were aimed more at scholars (p. 87) the West? Devlin identifies Fibonacci, and more and that the dense pages of Liber Abbaci were particularly his Liber Abbaci, as the key influence too challenging even for some abbacus authors in the development of this aspect of modern (p. 120). culture, ranking him alongside Copernicus and In Chapter 4 Devlin discusses Fibonacci’s possi- Galileo (p. 5) in terms of revolutionizing our ble sources for Liber Abbaci, particularly the initial world. After a brief account of Fibonacci and Liber chapters introducing decimal arithmetic and, at Abbaci, to be expanded later in the book, Devlin greater length, the final section of the last chapter, admits that the lack of appropriate sources makes dealing with algebra. Much of the discussion fo- a straight chronicle of Fibonacci’s life impossible, cuses on al-Khw¯arizm¯ıand his work, and there is and so this book will have to focus mainly on what a good description of the techniques of al-jabr (or he did and his modern legacy. restoration) and al-muqabala (or balancing) that Chapter 1 begins with a digression on titles, constituted algebra in the ninth century and that examining the meaning of abbaci as used by eventually gave algebra the name it now has. Fibonacci and of Fibonacci as a nickname for Chapter 5 begins with a brief summary of Liber Leonardo of Pisa. The main theme of the chapter, Abbaci and then discusses in more detail some though, is a very brief history of number systems, typical problems. It would always be difficult to from prehistoric tally marks through Roman nu- summarize such a large book (over 600 pages) in merals and on to the Hindus’ positional decimal a mere twenty-six pages, but surprisingly little is system. Hindu- reached Europe said about the aspects that are supposed to have long before Fibonacci (p. 21), with a Spanish revolutionized our world: the decimal system and monk, Vigila, reporting the following in 976: its adoption in commercial arithmetic. If Liber Ab- We must know that the Indians baci did indeed change the way merchants worked, have a most subtle talent and all then it must surely be because of the chapters other races yield to them in arith- about arithmetic and about its applications to metic and geometry and the other pricing and barter, to sharing company profits, liberal arts. And this is clear in the and to the value of coins alloyed from copper and 9 figures with which they are able silver (Chapters 1 to 11). These chapters are rich to designate each and every degree in social detail (particularly regarding commodi- of each order (of numbers).1 ties being traded and their ports of origin) but These early appearances of Hindu-Arabic numer- relatively lightweight in terms of mathematical als seemto have been merely as labels for counters content, with the main tool being the proportional on an board. For Devlin, it was Fibonacci thinking that gives rise to the rule of three. At who first realized the practical significance of first sight these chapters consist solely of recipes these nine figures in the world of commerce. to be learned by doing repeated examples, but As already mentioned, we have very little bi- Fibonacci seems to be teaching for understand- ographical information about Fibonacci. But the ing as he follows his recipes by explanations in culture he lived in has left much other evidence the style of Euclid. He also makes links between behind, and Chapters 2 and 3 give a nice de- apparently different methods, drawing parallels, scription of life in Pisa and of the role of Pisa for example, between companies and alloys [11, in Mediterranean commerce during Fibonacci’s p. 242] that might confuse more literal minded lifetime, along with a tentative description of his students. education both in Pisa and in Bejaia. The central The later material in Liber Abbaci (Chapters 12 spread of eight pages of illustrations includes to 15) has always attracted more attention from some color photos of modern Pisa that helped mathematicians, probably because it is here that bring these descriptions to life, for me at least, but we can imagine seeing the origins of modern ideas like negative numbers or symbolic algebra or meth- 1Quoted without source, p. 21. Occasionally Devlin forgets ods for solving systems of linear equations. Devlin to credit sources for his quotes, even though these sources focuses on a few methods which recur throughout would sometimes be quite good “Further Reading” for in- terested readers. In this case a quick search in Google 2This deficiency is of course easily remedied by using found [5]. Google.

May 2012 Notices of the AMS 663 these chapters: the rule of three, the methods of two rather more tangible reminders of Fibonacci: false position and double false position, and the first, a commemorative statue erected in Pisa in direct method (a kind of rhetorical algebra for the nineteenth century and second, the precious linear problems). These are explained in modern surviving manuscript copies of his texts. Perhaps terms but he also lets Fibonacci speak for himself, the sensual highlight of this book is the inclu- with excerpts usually taken from Sigler’s rather sion of three color photos of pages from these stilted translation [11].3 manuscripts. This is as close as we will ever get to Chapter 6 deals with the episode at the court of hearing Fibonacci speak in his own words. Frederick II. In the absence of direct eye witness By the end of the book, Devlin has told a accounts, the description has to rely on averted vi- good story. There isn’t a lot of mathematical sion: who was Frederick and what did he do? How detail, just enough to give a flavor of Fibonacci’s did Fibonacci report the mathematical problems thinking. A math-phobe could skip these bits, and their solutions later on in Flos? For the reader but realistically the target audience is probably interested in Fibonacci’s mathematics, a highlight someone with a good grasp of high school algebra. of this chapter is an extended excerpt (pp. 95– Mathematicians will enjoy the book, but probably 97) showing how he used the direct method to the best people to give it to would be keen solve an indeterminate linear problem in four un- mathematics students, whether at senior high knowns. Without symbols this is heavy going, but school level or at undergraduate level. It will open Devlin offers a symbolic translation that clarifies their eyes to an important part of mathematics the method. It is awe inspiring to see how far me- that we all take for granted. dieval mathematicians could progress using such primitive tools. References The last three chapters of Devlin’s book look at [1] B. Boncompagni, Liber Abbaci, in Scritti di Leonardo how Fibonacci might have influenced later gener- Pisano, Rome, 1857. ations. Chapter 7 looks at the spread of abbacus [2] C. Burnett, Fibonacci’s “method of the Indians”, Bol- 23 texts and abbacus schools in the three centuries lettino di Storia della Scienze Matematiche (2003) 87–97. following Fibonacci’s death, looking for traces of [3] J. Hannah, False position in Leonardo of Pisa’s Liber his influence in terms of shared problems or simi- Abbaci, Historia Mathematica 34 (2007), 306–322. lar syllabi. Chapter 8 reports recent work by Franci [4] R. Franci, Leonardo Pisano e la trattatistica [4] that tries to make a stronger case for direct dell’abaco in Italia nei secoli XIV e XV, Bollettino borrowing from Fibonacci. Unfortunately Franci’s di Storia della Scienze Matematiche 23 (2003) 33–54. argument depends on us knowing the content of [5] T. F. Glick, S. Livesey, and F. Wallis (editors), a long lost shorter version of Liber Abbaci, writ- Medieval Science, Technology, and Medicine: An Encyclopedia, Routledge, 2005. ten by Fibonacci specifically for merchants. Franci [6] J. Høyrup, Leonardo Fibonacci and Abbaco Culture. deduces this content by applying “medieval liter- A Proposal to Invert the Roles, Revue d’Histoire des ary fornesics” (p. 138), a careful comparison of Mathématiques 11 (2005), 23–56. crucial manuscripts from the years immediately [7] B. Hughes (translator), Fibonacci’s De Practica Ge- after Fibonacci’s death. But some scholars remain ometrie, Sources and Studies in the History of sceptical that such a direct link has been proven Mathematics and Physical Sciences, Springer, New [6]. York, 2008. R. Rashed All this might be described as Fibonacci’s hid- [8] , Fibonacci et les mathématiques arabes, in Le scienze alla corte di Federico II. Micrologus 2 (1994), den heritage, his contribution to our modern 145–160. culture that was borrowed without acknowledg- [9] J. Sesiano, The appearance of negative solutions in ment and absorbed into a fabric that we now mediaeval mathematics, Archive for History of Exact take for granted. This includes not just the use Sciences 32, no. 2 (1985), 105–150. of decimal arithmetic and its application to com- [10] L. E. Sigler (translator), The Book of Squares, merce, but also the beginnings of algebra. The final Academic Press, Boston, MA, 1987. chapter looks at Fibonacci’s more visible heritage. [11] L. E. Sigler (translator), Fibonacci’s , Springer, New York, 2002. It begins with a short account of the Fibonacci [12] K. Vogel, Zur Geschichte der linearen Gleichungen numbers and his relatively minor role in their mit mehreren Unbekannten. Deutsche Mathematik 5 story. But the chapter also gives the stories of (1940), 217–240. Available as a reprint in K. Vogel, Kleinere Schriften zur Geschichte der Mathematik, 3 An exception occurs (pp. 77–78) where he seems to have Boethius: Texte und Abhandlungen zur Geschichte preferred my own translation of the archetypal false po- der Exakten Wissenschaften 20 (1988), 1–2, Franz sition problem [3, p. 309]. Naturally I am quite pleased, Steiner Verlag, Stuttgart, pp. 281–304. if this is the case, but it is unacknowledged and it would [13] K. Vogel, Fibonacci, in Dictionary of Scientific Biog- have been nice to see this paper included in the bibliogra- raphy, vol. IV, (C. C. Gillespie, editor), Scribner, New phy for readers who wanted more detail about Fibonacci’s York, 1971, pp. 604–613. treatment of the method.

664 Notices of the AMS Volume 59, Number 5