Abbacus School

Abstract

The abbacus school was a school for artisans’ and merchants’ sons, functioning in northern Italy (thirteenth to sixteenth century). It taught the use of Hindu-Arabic numerals and fundamental commercial arithmetic: the rule of three, monetary and metrological conversions, simple and composite interest, partnership, simple and composite discounting, alloying, the technique of a “single false position,” and, finally, simple area calculation. Topics like the double false position were not part of the curriculum, but they are often dealt with in the abbacus treatises; they probably served to show virtuosity in the competition for employment and pupils.

The manuscripts connected to the abbacus school tradition are of very different character. Some are messy problem collections, some orderly presentations, and a few genuine encyclopediae. Some are produced by mathematically incompetent compilers and some by the best European mathematicians of the age.

Traditionally but mistakenly, the abbacus books are to be derived from Leonardo Fibonacci’s Liber Abbaci and Practica Geometriae. As shown by closer inspection of the texts, they derive from direct inspiration from a broader Mediterranean environment, which had also inspired Fibonacci around 1200. After c. 1330, however, the abbacus tradition had become an autonomous current, no more significantly influenced by the Arabic or Ibero-Provençal world. Influences from Boethian and Euclidean arithmetic, though existing, remained peripheral.

Printing allowed the preparation and spread of the great works of Luca Pacioli, Girolamo Cardano, and Niccolò Tartaglia, as well as more modest books corresponding to the school curriculum. The former group, integrated with the theory of irrationals of Elements X, provided the basis for the renovation of algebra brought about by Viète and Descartes; the latter made possible the spread of abbacus-type teaching of basic applied arithmetic to the whole of Western Europe, where it stayed alive until c. 1960.