A History of Mathematics From Mesopotamia to Modernity

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Understanding the‘ScientificRevolution’ 141


history—because it focuses not on the work in its context with its proper connexions so much as
on its place in an attempted genealogy; and in this instance the case can only be established:



  1. by blurring the very important distinctions between the scientific aims pursued in (say) the
    fourteenth and seventeenth centuries;

  2. by ignoring the lack of evidence of any transmission line (say from Oresme to Descartes).


Historians are rarely (never?) free of presuppositions, but many of them are now studying the
medieval tradition for its own sake, as a particular historical tradition within mathematics. Much
of the medieval work which was supposedly important for Galileo does not seem to have featured in
his reading; and although there was undoubtedly a lively argument in progress about mathematics
and its certainty in sixteenth-century Italian universities (conducted very much along Aristotelian
lines, what is more)^6 it contributed much less to the shaping of mathematicalpracticethan the two
sources which Descartes identified—Greek geometry and Arabic algebra.
However, there was an alternative tradition, almost independent, with at least as much influence;
that of the often very low-level practical calculators who were needed to teach the sons of mer-
chants. Again we could compare the situation in Abbasid Baghdad, and again there seems to be an
important difference: that in the Islamic world skilled mathematicians such as ab ̄u-l Waf ̄a’ wrote
with such schools in mind, while the Western tradition seems to have been at a more basic level.
The works produced by such schools in Italy (where they were probably most important) has been
studied in detail by Warren van Egmond (1980). The texts are referred to as ‘abbacus books’; the
title is misleading, since what we call an abacus, or counting-frame, was never used. The original
text, and the most serious is one which you will often find referred to in histories, Leonardo of
Pisa’sLiber abbaciof the early thirteenth century. Leonardo was an unusually good mathematician
whose distinguishing points are that he worked outside the university; that he had the good fortune
to spend several years in the Arab world with his father, a Pisan merchant; and that he saw an
opportunity to spread the useful things which he had learned, particularly the use of Hindu–
Arabic numbers and algebra, to the practical men among whom he spent his later life. He was,
in the context of the time, an intelligent student from a ‘backward’ country who received a good
education in what was then the metropolis (North Africa), and did what he could with it when
he returned.
The immediate influence of theLiber abbaciseems to have been the diffusion among the Italian
merchants—who had an eye for what was directly useful, as the university men did not—of the
most elementary parts of the Islamic tradition. We could, then, contrast two separate ‘borrowings’
from the world of Islam: the translations of learned works, Greek and Arabic, in the universities on
the one hand, and the adoption of Indian numbers and simple algebra in the cities. These elements
were taught in ‘abbacus schools’ using books, usually simplified versions of Leonardo’s book and
often in Italian to make them more accessible. On a smaller scale, similar works were produced in
the various languages of western Europe—English, French, German, and of course their number
increased dramatically after the invention of printing in the mid-fifteenth century. Van Egmond
claims that ‘nearly all the educated men of the renaissance gained their basic mathematical educa-
tion in schools such as these, including, for example, such notables as Leonardo da Vinci and Niccolò
Machiavelli’ (van Egmond 1980, p. 8); a German printed version finds its way into that must-have



  1. This not very enlightening controversy is documented in Rivka Feldhay’s article: ‘The use and abuse of mathematical entities:
    Galileo and the Jesuits revisited’, in Machamer (ed). (1998, pp. 80–145).

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