The Other Leonardo

In Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, Keith Devlin helps rescue a mathematician from 13th-century Italy from near total oblivion, writes Scott McLemee.

April 19, 2017

Picture yourself as a commercial traveler in Venice during a trade fair at the end of the 12th century. Business is booming, so you have to tune out a cacophony of accents and tongues while haggling with a merchant over a lot of well-made boots.

He agrees to a very good price if you buy a half dozen pairs. Jotting down a note to himself about the transaction -- VI shoes at IV gold coins each -- he then moves stone around on his abacus before recording the total price. (His abacus is a board -- the kind used in the Orient, with beads, strings and a frame, are rare but supposedly much more efficient.) You dig the necessary XXIV coins out of your purse and decide to keep one pair of boots for yourself.

Assuming the rest will sell for twice as much when you get back home, what is your profit on the transaction? Enough, you hope, to buy a secondhand abacus and hire someone to tutor the kids. Otherwise, there's a strong chance that you just know it was a good deal but not that you stand to net XVI coins from it.

Within a few years, the whole process of calculating and recording business transactions will change, thanks to one Leonardo of Pisa, who was born circa 1170 and alive as late as 1241. He is not to be confused with the considerably more famous Leonardo, who comes along in the 15th century. Indeed, for long time, there is all too little danger of mixing them up. The mathematician from Pisa's reputation fades into near total oblivion, even as his influence grows, almost exponentially, from one century to the next. For he not only advocated the Hindu-Arabic numerical system so effectively that it was adopted in Europe, but he also provided a comprehensive course of instruction on its use in performing calculations.

The advantages proved considerable. Roman numerals were ill suited for arithmetic (as our marketplace example may suggest) and well-nigh useless for solving the kinds of problems that the Islamic savants knew as al-jabr (algebra). The Hindu-Arabic system was, by contrast, a marvel of efficiency and processing power. Hence the title of Leonardo's enormous treatise Liber abbaci. First available in 1202 and issued in a revised edition in 1228, it was not a manual for using the abacus but rather a method for turning any blank piece of paper into a calculating machine. Conveying Liber abbaci’s impact to the general reader is Keith Devlin’s mission in Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, from Princeton University Press. (Devlin is a senior research scientist at Stanford University and appears on National Public Radio as the Math Guy. Further, Liber abbaci is the title as Devlin gives it, though the doubled B is debatable. A quick JSTOR search shows Liber abaci used about three times as often.)

It turns out that Leonardo of Pisa was not quite erased from the history of mathematics after all. His posthumously bestowed nickname, Fibonacci (“son of Bonacci”) has been affixed to a well-known and much-studied numerical sequence that begins with zero and one and continues with each subsequent term being the sum of the previous two. Like so:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …

This series corresponds to certain patterns in nature; the number of petals on a flower, for example, will tend to be a Fibonacci number. And the longer the sequence goes on, the ratio between each term and its predecessor gets closer to an important constant (phi = 1.618 …) sometimes called the divine proportion or the golden ratio.

Leonardo of Pisa didn't discover the series (Indian mathematicians had been aware of it for centuries), nor did he single it out for particular attention -- and Devlin, for his part, regards it as an injustice of sorts that the greater accomplishment of Liber abbaci should be little known except to historians of mathematics. In 2011 he published a biography, The Man of Numbers: Fibonacci's Arithmetic Revolution, followed the same year by a Kindle ebook original called Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years.

With this third outing, Devlin has taken advocacy for the Italian mathematician's reputation as far as it can go, and then some. Reprising what he's written in the past, he adds the findings of subsequent scholarship on Fibonacci's likely influence on the world of finance and works up entries from his diary into a narrative of research that someone with a red pencil (or two) could have improved a great deal. By the end of the book -- when, apropos of not much in particular, Devlin reprints in full the first article on mathematics he ever published in a newspaper -- it seems clear that Finding Fibonacci has been padded as heavily as a box full of Fabergé eggs.

A fair analogy, I think: parts of the story are priceless, perhaps especially the chapter on how it came to pass that the late Laurence Sigler's translation of Liber abbaci was published in English in 2002 (800 years after the first version appeared in Latin) despite dire and even catastrophic developments that might have spelled the doom even of a project with much wider scholarly audience. Devlin also communicates something distinctive and remarkable about that book: how laboriously the author went about explaining how to write Hindu-Arabic numbers, carefully spacing the digits, lining them up neatly when making calculations … In short, instructing the reader at great and exacting length on skills it is now the job of primary-school teachers to impart.

"It is perhaps inevitable, though to my mind a little sad," Devlin writes in one of his book's best passages, "that the creations that turn out to be the most profound for our lives eventually become so commonplace that we no longer see them for the huge accomplishments they are." True -- when we even notice them at all.


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