Background: The Abbaco Tradition

The Pamiers text was designed for use in commercial training schools, which flourished in this part of Europe from the 1200s through the 1500s.

These institutions were known as schools of algorisme (“algorithm”) in French, and of abbaco or abaco in Italian (a word with the general sense of “calculation”). They arose first in northern Italy, whose economy was the most vibrant in Europe during this period (Spiesser 2003, pp. 34-35). A banker and official in Florence, Italy, reported that in 1345 at least 1,000 boys in that city alone were receiving instruction in abbaco and algorismo (Biggs 2009, p. 73). Such schools also began to appear in neighboring southern France, and a few in Catalonia (the area around Barcelona, Spain) and coastal North Africa. These four regions of the western Mediterranean had extensive trade and cultural ties with one another at the time, so it isn’t surprising that they shared methods of practical mathematics and its instruction (Høyrup 2006).

Over 400 of the abbaco and algorisme works have survived. The common ancestor of many of them is Liber Abbaci (1202) by Leonardo Fibonacci of Pisa, Italy. He was only a boy, he reports, when his father, a customs official representing Pisan merchants at their trading enclave of Bugia, in what is now Algeria, brought him to the customs house there to be taught Hindu-Arabic numerals and arithmetic (Sigler 2002, pp. 3, 15). Based on place value, the Arab-style calculations could be carried out on an abacus or with pen and paper, in either case using efficient, step-by-step procedures called algorithms. Fibonacci’s book, written in Latin, introduced this new way of reckoning to European scholars. Then the techniques spread with the aid of texts, like that of Pamiers, that were written in languages more accessible than Latin. Maryvonne Spiesser (University of Toulouse), who has extensively studied the abbaco and algorisme treatises of the region, writes that the Pamiers manuscript “founded the tradition of commercial arithmetics in southern France” (2002, p. 297).

Figure 2. A 19^th-century statue of Leonardo Fibonacci in his likely birthplace of Pisa, Italy. The statue, about 15 feet tall, is part of the Fortezza Camp Santo at Giardino Scotto, a fortified garden off a road called Lungarno Fibonacci along the Arno River. (Image made in 2002 by Gordon Wilson, Emeritus Professor of English, Schoolcraft College.)

The boys sent to such abbaco schools were typically 10- to 12-year-olds. Having completed grammar school, they were now training to become merchants, wholesalers, bookkeepers, bankers, or in some cases tax collectors, customs agents, or navigators. Daily instruction was provided by a reckoning master (maestri d’abbaco or maistre d’algorisme); to facilitate his teaching, he would often write his own abbaco text and hire some assistant instructors. The lessons were taught either in the master’s own home or inside a private business, a municipal office, or a religious order. The topics and techniques taught in the abbaco schools were strongly influenced by a tradition that had long thrived within Arab-Islamic culture, that of mu‘āmalāt mathematics, i.e., the mathematics applied to commerce and other aspects of social intercourse. In addition to mathematics, the boys were often given further instruction in reading and writing.

Figure 3. A master teaches his pupils in this drawing from an illuminated manuscript version of Filippo Calandri’s Trattato di Arithmetica, an abbaco treatise originally printed in Florence in 1491. (Source: Biblioteca Riccardiana (Florence, Italy), Ricc. 2669, page 111 verso. Image used by permission; further reproduction is prohibited.) 

In general, the abbaco texts offered practical, simplified treatments in which mathematical techniques were distilled into easy-to-remember rules and algorithms. The focus was on how to carry these out rather than on justifying the theory behind them. At the same time, the books were often innovative in their solutions to particular problems and especially in their pedagogical approach: their presentation was popular, and they introduced the use of illustrations and vernacular languages to the history of mathematics textbooks (Swetz 1987, pp. 292-293).

Typically, such a book would give instruction in these topics:

• Arabic place-value numeration and arithmetic, including basic operations with whole numbers and fractions, proportions, progressions, and extraction of roots
• Units of measurement
• Currency exchange
• Metal-alloy and other alligation (mixture) problems
• Computation of costs, revenues, profits, and losses in both barter and money transactions
• Calculations involving partnerships, taxes, and import duties.

Some of the abbaco texts (but not that of Pamiers) included discussions of more advanced or specialized topics, such as these:

• Algebra. An example is Fibonacci’s Liber Abbaci, mentioned above. As the word “algebra” itself attests, this technique was again one that Fibonacci and others had learned from an Arabic context. The hallmark of this approach was to solve a problem by setting up an equation containing a word or symbol for an unknown quantity, and then manipulating the unknown with the same operations as the known quantities. At the time, this was considered a technique that would be fairly difficult for merchants and tradesmen to master.
• Surveying. An example is Fibonacci’s De Practica Geometriae (1220), which showed how to measure and subdivide variously shaped fields as well as solids (see the critical translation by Hughes 2008).
• Marine navigation. An example is the 440-page manuscript book for mariners written by Michael of Rhodes in the Venetian dialect of Italian in the 1430s (see Dibner Institute / WGBH website).

For more on the abbaco and algorisme tradition, see also the works by Biggs, Davis, Goetzmann, Høyrup, and Swetz in the list of References.