142 A History ofMathematics
Fig. 1German arithmetic book from Holbein’s ‘The Ambassadors’ (National Gallery, London).catalogue of Renaissance things to own, the list of objects in Holbein’sThe Ambassadors(Fig. 1).
He further claims that the introduction of these schools, around 1300, was commercially driven,
a result of ‘the commercial revolution of the thirteenth century’. This involved the increasing use
of devices—some new, some perhaps adaptations from the Islamic world—such as banks, letters
of credit, and bills of exchange. Later, we find more sophisticated accounting methods, leading
eventually to the famous invention of double entry bookkeeping.
[The reader, if not an accountant, may well wonder what this important development which is
so often referred to is. Briefly, it consists in the practice of entering every event (sale or purchase)
twice, once as a credit and once as a debit; it was in use in Genoa in 1340 (but possibly earlier),
and it was first properly expounded in texts in the fifteenth century, most famously by Fra Luca
Pacioli, still considered as the ‘patron saint’ of accountants. For definitions see de Roover (1937),
‘Aux origines d’une technique intellectuelle: la formation et l’expansion de la comptabilité à partie
double’, inAnnales d’histoire économique et sociale9 (1937). For its relation to the introduction of
the zero, to perspective and much else, see Rotman (1987).]
Unlike the speculations of university men, the textbooks used in abbacus schools were justified
solely by their supposed usefulness. Indeed, they did not even serve the purpose of creating a
privileged caste, as in ancient Babylonia—solving equations was a skill, not a class marker. The
calculating tradition is undoubtedly important, in contributing skills which were not obviously
learned in the more formal context of universities. However, there seems little sign that in two
hundred years the abbacus schools and similar institutions were responsible for innovation. Since
the numerical requirements were relatively simple (no astronomy, for example), the kind of soph-
isticated approaches to number found in ab ̄uK ̄amil, Khayyam, or al-K ̄ash ̄i were not raised. The
contents of the textbooks were often quite basic—the writing of numbers and how to calculate
with them, a little geometry (measuring circles and triangles by approximate formulae); sometimes