Abbacus School

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 ofWestern Europe, where it
stayed alive until c. 1960.