Fibonacci and the Algorists

The Student from Pisa

Leonardo of Pisa (Fibonacci, c. 1170 – c. 1240s) is the single most important carrier of Islamic mathematics into European practice, and he says so himself. His father directed the Pisan trading colony at Bugia (Béjaïa, in modern Algeria), and Leonardo relates in the prologue of the Liber Abaci that he was sent there as a boy and instructed in the nine Indian figures, then pursued the art in his travels around the Mediterranean, Egypt, Syria, Greece, Sicily, Provence, finding it superior to all other methods.

Liber Abaci (1202, revised 1228)

Despite the name ("Book of Calculation"; abacus here means calculation generally, not the bead-frame), the book is a comprehensive treatise on Hindu-Arabic arithmetic and algebra:

• The numerals, place value, and zero (zephirum, from Arabic sifr), taught to Latin readers systematically.
• The full algorithm suite, including checks (casting out nines) and the Maghrebi-style fraction bar he learned in North Africa (see al-Hassar).
• Commercial mathematics: exchange, barter, alloying, partnership, interest, the mathematics of the merchant revolution.
• Algebra in the manner of al-Khwarizmi and, extensively, Abu Kamil, many problems are taken over directly.
• The rabbit problem whose answer sequence (1, 1, 2, 3, 5, 8, ...) now carries his name, a minor exercise in the book, immortal by accident. (Comparable recurrence patterns were known earlier in Indian prosody; the name "Fibonacci sequence" is 19th-century, coined by Lucas.)

His other works extend the program: Practica Geometriae (1220, Archimedean and Abu-Kamil-style measurement, citing the Banu Musa's Verba filiorum tradition), Flos and the Liber Quadratorum (1225, number theory of real originality, solving problems posed at Frederick II's court, including a cubic he shows has no rational or simple-surd root and then approximates to high sexagesimal precision, a very Arabic performance).

The Abacus Schools

The Liber Abaci seeded the scuole d'abaco of the Italian cities: vocational schools where merchants' sons learned Hindu-Arabic computation, commercial arithmetic, and some algebra from vernacular libri d'abaco (hundreds survive). For three centuries this culture trained the numerate class of the commercial revolution, double-entry bookkeeping's arithmetic substrate, and kept algebra alive as a practical craft. It is exactly in this milieu that the cubic equation was finally solved arithmetically (Scipione del Ferro, c. 1515, a Bologna abacus-tradition professor; then Tartaglia; published in Cardano's Ars Magna, 1545), completing the program Khayyam had declared open four centuries earlier.

Resistance and Victory of the Numerals

Adoption was not instant: some city statutes (a Florentine guild rule of 1299 is the standard example) restricted the new numerals in official ledgers, on anti-fraud grounds (a 0 is easier to alter into a 9 or 6 than Roman numerals are to falsify), and Roman numerals persisted in registers for generations. Printing settled the matter; by the sixteenth century the victory of algorism over the counting board was a stock subject for allegorical woodcuts (Gregor Reisch's Margarita Philosophica, 1503, shows Boethius computing happily with Hindu-Arabic numerals while Pythagoras glowers over a counting board).

Assessment

Fibonacci is the proof that transmission is not photocopying: he absorbed the Arabic tradition as a living participant (probably from Maghrebi teachers of the Ibn al-Yasamin generation), then wrote for his own civilization's needs. Through him and the abacus schools, the arithmetic and algebra of Baghdad, Cairo, and Bugia became the daily mathematics of Florence and Venice, and thence of the world.

Sources: L. E. Sigler, Fibonacci's Liber Abaci: A Translation into Modern English (2002); Katz; W. Van Egmond's studies of the abacus-book corpus; R. Franci & L. Toti Rigatelli on the abacus schools; Devlin, The Man of Numbers (popular).