Wednesday, October 05, 2011

The other Leonardo

kw: book reviews, nonfiction, biographies, mathematicians

Three hundred years before Da Vinci, there was Pisano, Leonardo of Pisa, who lived from about 1175 to at least 1240. He became a legend in his own time, then was nearly forgotten except for a peculiar story about rabbits that became attached to his nickname. Through great struggles modern scholars have pieced together the depth and breadth of his influence, which led directly to the changeover throughout Europe from Roman Numerals and the cumbersome methods of calculation they required, to the Hindu-Arabic Digits, modern arithmetic, and algebraic methods that, once symbolic operations were devised to make them more easily understood, continued and advanced to the present day. Yet, because of a nickname based on a few instances in which he signed himself filis Bonacci, or "offspring of Bonacci", he is known today primarily as the originator of the Fibonacci Series, based on the multiplication of hypothetical rabbits.

The amazing biography of Leonardo of Pisa, The Man of Numbers: Fibonacci's Arithmetic Revolution, by Keith Devlin, brings together in a rather small book everything we know about Leonardo, which is, surprisingly, rather little. Nearly everything known with certainty is based on a handful of statements he made about himself: how he was brought by his father to an African trade center, where he learned the Hindu-Arabic system, his travels to learn more from its expert practitioners, and his determination upon returning to Pisa to publicize the methods throughout Italy and beyond.

Much more is inferred from his works themselves: Liber Abbaci (frequently and wrongly spelled Liber Abaci) and De Practica Geometrie in particular. The former book, first released in 1202 and revised in 1228, brings together all the streams of number and calculation that he had researched and learned and devised on his own, into a scholarly text that is in itself a great education in arithmetic and basic algebra. The latter book did much the same for geometric methods. But it was Liber Abbaci that he himself revised into a simpler text (now lost), which was endlessly copied, frequently with some of the geometrical methods of the other book appended, and effected a revolution in, first, mercantile calculations, and later, mathematics in general.

A modern person conversant with algebra would find Liber Abaci to be tough sledding. I contains hundreds of examples, worked out for the reader, but every one is what we call a "word problem", and the solution is also in narrative format that takes a great deal of getting used to. It was not until a few generations after Leonardo's death that symbolic notation was developed into things like the dreaded 3x+4y=12, where in Leonardo's time one had to say, "Twelve are found of the first quantity tripled and four times the second quantity…", except I have used "quantity" to represent special words the mathematicians used for their unknowns. (Note: this is only half a problem. There would be a second set of quantities so one could use cancellation to solve the system.)

While such methods are cumbersome compared to modern algebra, they made things possible that were impossible with Roman numerals. Why, it is really quite difficult just to multiply XXVI by LIII (the answer is MCCCLXXVIII). The notion of putting a line above the I, V, X, L, C, D and M to multiply each by 1,000 came along centuries after Roman numerals fell out of use for everything except cornerstones, clock faces and movie credits. Thus, calculations that produced quantities greater than a few thousand were not attempted.

It was quite a feat of detective work that demonstrates how Leonardo was behind the revolution. Nearly half the book traces the threads that bind the story together. A great part of the problem was that there was little notion of plagiarism. It was commonplace for a new book to be written that consisted largely of extended quotes from other works, entirely without attribution. It is very rare to find a prior author mentioned or cited in a medieval manuscript. So Leonardo of Pisa was cited once or twice in his own lifetime, or just after, and no more than two or three times later, except for that persistent rabbit puzzle.

In case someone doesn't know this: Suppose you have a mated pair of young and fertile rabbits that can bear young once per month, and rabbits of age one month can mate for the first time, a month later to bear their own little rabbits. Suppose further that each litter is exactly two, a male and a female. How many rabbits, total, will you have after one, two, and up to thirty months? You start with 1 pair. A month later you have 2 pair, the original pair plus a newborn pair. The next month you have 3 pair: the original pair, the young pair now old enough to mate, and a pair of newborns from the original pair. And so it goes: 1, 2, 3, 5, 8, 13, 21, 34, 55,… each month's quantity is the sum of the prior two quantities. If you assume immortal rabbits, this goes on without end. This is the Fibonacci series, and I encourage those who find it a new thing to look it up. I'll not go into it here.

Today we live in a world of numbers. By the time we're grown, even if some of us "hated math", we find many kinds of simple calculations second nature. We check grocery receipts to make sure we got correct change; we can keep track of our own bowling scores; we buy gasoline and do a quick division to see how many miles per gallon we are getting. We are used to the numerals 9 8 7 6 5 4 3 2 1, and would be practically crippled without them. It is not too much to say we owe more to this Leonardo, than to the better known one.

No comments: