290 François charette
students of the Greeks, and, although relatively very little was known
around 1900, their works revealed – even to the least interested historians
such as Tannery – a sophisticated mathematical practice within which
demonstration, closely following the Greek model, played an important
role. Yet, it was thought, this assimilation of Greek mathematical thinking
and practice did not instil much progress among them; the Arabs’ contri-
butions, however extensive and honest they may have been, did not bring
about any major breakthrough, nothing comparable to what would happen
in sixteenth- and seventeenth-century Europe. For the ‘Arabs’ were implic-
itly considered as immature custodians of a higher knowledge, who could
not properly deal with it; being mere imitators of a foreign tradition, they
were unable to reach the critical level beyond which real progress could
have been initiated. For Zeuthen, the Eastern Arabs had been unable to
emancipate themselves from the rigid geometrical approach of the Greek;
and the Western Arabs, who supposedly did liberate themselves from this
approach, still remained ‘too reverential’ toward the Greeks. 57
On the other hand, the Indians, in spite of the supposed laxness and
lack of rigour of their mathematical practice and the primitiveness of the
few demonstrations revealed in their works, had indeed achieved results
superior to those of the Arabs in arithmetic and algebra. Some authors even
went as far as comparing this ‘Indian’ style of mathematical practice with
the intuitive works of certain modern mathematicians, probably because it
was realized that absolute rigour had not played a fundamental role in early
modern Europe. 58 Few believed that a stringent axiomatic–deductive system
was a necessary condition for mathematical discovery. Nevertheless, histo-
rians of mathematics unanimously insisted that it was precisely the lack of
a logical, rigorous system of mathematical thought similar to the Greek one
that prohibited any further progress in India. Imagination alone could at
best generate haphazard discoveries. Th e refusal of systematic rationality to
the Orientals was saved whenever one encountered anything ingenious in
their mathematics by explaining it through their having recourse to ‘tricks,
dodges’ ( Kunstgriff e ), as in the case of al-Bīrūnī’s solution of the chessboard
problem. 59 For Tannery, theory provided with demonstration was the57 Zeuthen 1896 : 314.
58 Günther ( 1908 : 127) says of Hero that he is a sort of ‘antique Euler’. Hankel ( 1874 : 202–3)
compares Bhāskara’s numerical methods, especially in his solution of the so-called Pell
equation, with those of modern mathematicians. On the relatively unimportant role of
mathematical proofs (compared to the concern for ‘exactness of contructions’) in early
modern Europe, see Bos 2001 : 8.
59 Cantor 1894 : 713–14.