The Man of Numbers: Fibonacci’s Arithmetic Revolution

Visitors to the Camposanto, the sacred burial ground of Pisa, so sadly damaged by Allied bombers in the Second World War, find all manner of monuments lining the walls, from Roman sarcophagi to statues of the illustrious citizens of later centuries. One of these, recently cleaned up and restored, dates from 1863 and it commemorates one of Pisa’s heroes, the mathematician Leonardo da Pisa, often known as Fibonacci (a corruption of Filius Bonacci, the son of Guiglielmo Bonacci). Although no contemporary depiction of him survives, he springs from the sculptor’s imagination with classical features, a cowled head and a long tunic. He lived from around 1170 to 1250.

Keith Devlin, Walker & Company, 2011.

Fibonacci is best known today for his famous mathematical puzzle of the breeding rabbits. Shut up a pair of rabbits in an enclosure, assume that the doe will give birth to a pair of baby rabbits every month and that these two will be up and breeding a pair a month within a month. Genetically impossible of course, but the numbers can be built up into a sequence of 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on. These ‘Fibonacci’ numbers reappear in all kinds of strange places, not least as the typical number of petals in a flower.

Fibonacci was not the first to work out this sequence and Keith Devlin is not nearly as interested in it as in Fibonacci’s contribution to the commercial revolution of 13th and 14th century Italy. These years were the age of trading breakthroughs for the Italians as they captured new routes and filled them with finished grain in return for raw materials. The Venetians strengthened their position immensely after the Fourth Crusade of 1204 had allowed them to lay their hands on ports across the Mediterranean. Florence expanded fast after 1200. Yet there was a bottleneck in the commercial background. There were experts who could manipulate an abacus as quickly as one might operate a calculator today, but the final answers were always written in Roman numerals. As soon as complex issues arose, concerning how to divide profits or change money between coins of different alloys, the system just broke down. The Arab traders, on the other hand, were using a system they had adopted from India. It comprised nine numbers, each with a single-digit symbol, 8 for VIII, for instance, and, crucially, a zero, which was recognized as a number in its own right. The man who transferred the system into Italy was, Devlin argues in this entertaining book, none other than Fibonacci.

Fibonacci’s father had been posted by Pisa to the port of Bugia, in modern Algeria, where he acted as the Pisan go-between with the Berbers. Leonardo, still a boy, went with him. He must have picked up Arabic, as he tells how he talked to merchants from Egypt and Syria, and he soon grasped the superiority of their calculations and became obsessed by them. In 1202 he published a mammoth 600-page manuscript, the Liber Abaci, ‘The Book of Calculation’. It was the first time that the system had been spelled out fully and aimed directly at Italian merchants. Everything from how to divide profits and measure land to dealing in currency exchange was covered with a myriad examples to show how each kind of calculation could be made.

No copies of the 1202 manuscript survive but there are some of the second edition of 1228. By this time, Fibonacci was famous. He had been summoned to meet the formidable Holy Roman Emperor, Frederick II, who had set him three problems which he triumphantly solved. He wrote up the answers in his Liber quadratorum and this work has helped confirm him as the finest mathematician of the Middle Ages.

By the end of the 13th century there were a mass of simpler books of calculation and schools were teaching the system. In Florence in 1343, between 1,000 and 1,200 boys were working in abbaci schools. Until recently, however, historians have not been able to link Fibonacci directly to the introduction of the new system. The workbooks used by students and merchants did not appear to overlap with anything in Liber Abaci. Perhaps the system had come in at a less erudite level and slowly infiltrated the Italian cities. Yet later writers often named Fibonacci as the man who introduced arithmetic and algebra to Europe.

Devlin shows how the question has been resolved. Fibonacci must have realized that the huge manuscript of the Liber Abaci and another text he wrote for merchants on geometry, which was scarcely less large, were too much for the ordinary merchant to master. So he wrote a much shorter and simple text, now lost, and this can now be directly linked to the manuals to be found in the schools some decades later. Fibonacci was truly ‘the man of numbers’, both at a sophisticated level in algebra, but on the market floor.

This is a short book on a man about whom almost nothing is known. Fibonacci sometimes called himself Bigallo, perhaps a Tuscan dialect word for traveller, and he certainly knew his way around the cosmopolitan Mediterranean world of the 13th century. It was perhaps inevitable that the Hindu-Arabic system would have come to Italy in time—it was simply too useful in a complex trading world—but Devlin has certainly shown that Fibonacci deserves the credit for setting in all in motion. This is a readable and enjoyable book and I actually understood the maths!

Reviewed by Charles Freeman, historical consultant to the Blue Guides and author of Sites of Antiquity: 50 Sites that Explain the Classical World. For more on Fibonacci, see here.

The Roman Forum

‘Archaeology often brings to light relics—mysterious foundations, tumbled blocks, a charred sacrificial pit, the decaying stumps of dead houses—fascinating to the scholar but a stunning bore to the simple visitor.’ So wrote Dilys Powell in The Villa Ariadne. Archaeologists can be monomaniacs and their interests are often distressingly narrow. So it was with some anticipation that I took up David Watkin’s The Roman Forum, whose contentions are very clear: ‘Archaeologists have eliminated much evidence of the fascinating post-antique life of the Forum,’ and their labours have made ‘visiting parts of the Forum about as attractive as looking into the hole made in New York on 9/11.’ Ouch! These accusations run like leitmotifs throughout the book, together with the curious conspiracy theory that guidebooks are complicit; that there are things they ‘do not want us to see’.

By David Watkin (Profile Books, hardback 2009; paperback June 2011)

If one works night and day to produce guidebooks, it is difficult not to get on the defensive. The Portico of the Dii Consentes doesn’t ‘turn out’ to be a modern reconstruction. Pay attention to your Blue Guide! It clearly says that it dates from 1858. There is no plot to keep visitors in the dark about the churches of San Lorenzo in Miranda or Santi Luca e Martina. They are just never, ever open. But for Watkin, everything was better in the time of Piranesi. Piranesi, he tells us, with the authority of one who knew him in a former life, recorded the Forum ‘at the last time when it was still a place of poetry, capable of inspiring great painters, writers and thinkers.’ Glum stuff, but the threnody does begin to strike a chord and the aimiable style in which the bad news is delivered soon reels you in. Watkin laments the fact that the Forum has been turned inside out: its surviving churches open away from it, no longer into it; it has been severed from the life of the city and turned into a visitors’ theme park. Up until a very few years ago, entry was free and one could use the Forum as a thoroughfare; Romans going about their daily business could loiter and linger in it. Now you have to queue to be admitted through a turnstile, custodians are bossy and offhand, and no one who is not a tourist (or an archaeologist) ever goes there. The magnificent ‘challenge of the relationship between ancient and modern’ has been obscured.

Despite the underlying crotchetiness, the book is immensely enjoyable. Watkin’s love for the Forum, his breadth of knowledge, and his wistfulness about what might have been (in a Peter Pan world) are ingenuous, impressive and infectious—a beguiling combination. The chapter on the despoliation of the Forum’s monuments in the service of the new St Peter’s is a superb read. No visitor should ever again imagine that the mere march of time had anything to do with it. What Watkin cannot admit, though, is that if archaeologists hadn’t got their hands on the Forum when they did (in the late 19th century), the urban planners certainly would have. And the challenging relationship between ancient and modern would now be as desperate a tussle there as it is at Largo Argentina.

Nevertheless, if you’re travelling to Rome—either for the first or the fiftieth time—I recommend that you get this book. Not only will it add whole layers of meaning to your visit, but it will also force to you answer the following testing questions: if archaeologists are to be banished, who will take their place (and who will pay for it)? What is the point of a desert like the Forum in the centre of a busy and increasingly cramped-feeling capital city? And whose opinion was nearer to the mark: Palladio’s, for whom the Forum offered ‘not the spectacle of ancient glory but rather the possibility of recreating it’; or Pevsner’s, for whom the Forum belongs ‘to the civilisation of Antiquity, not to what we usually mean when we speak of European civilisation’? Watkin loves the Baroque churches that were built over the ancient ruins in the 17th century. But did their architects believe they were ‘recreating’, or did they believe they were moving forward into a ‘modern’ era?

Reviewed by Annabel Barber

Blue Guide Rome (10th edition) has extensive coverage of the Forum, with a map of the site and detailed notes on all its monuments, past, present and conjectural.