Th e logical Greek versus the imaginative Oriental 289
Indians used to do in their siddhantas. Were they motivated by a concern for the
shortness and corresponding cheapness of their books? 54
Cantor’s judgement was not as severe as Hankel’s, but he was neverthe-
less convinced that the practice of demonstration had almost a character
of exception among Muslim mathematicians. 55 Confronted with a highly
original treatise of practical geometry by Abū al-Wafāʾ (then known only
through a mutilated Persian version which Woepcke had summarized),
Cantor had no recourse to contextualization to explain the absence of
demonstrations – as he did in the case of the geometrical part of the Rhind
Papyrus, where he accepted it precisely because of the practical nature of
the work. For him Abū al-Wafāʾ’s treatise recalled Indian geometry, so that
‘one would almost expect as a proof [of the validity of a particular construc-
tion] the request “look!”, with which Indian geometers are satisfi ed to con-
clude their construction procedures’. 56 Th us for him there was no doubt that
this work was an example of Anschauungsgeometrie , so thoroughly Indian
in style that no deductive demonstrations, even on the part of one of the
best Muslim mathematicians, could be possibly assumed.
Epilogue
For Zeuthen, the breakthrough in the history of mathematics occurred
with the resolution of third-degree equations by means of roots (Cardano
and Tartaglia), an achievement that closes the medieval periodization of
mathematics and announces the rapid advances made thereaft er, modelled
on and inspired by a close study of Greek mathematics. Zeuthen’s periodi-
zation explains why nearly three-quarters of his book (245 pages out of 332
in the fi rst German edition) is devoted to Greek mathematics.
In view of this, it is probably erroneous to assume that Zeuthen and his
colleagues saw the practice of mathematical demonstration as the key to
mathematical progress. Th e Muslims had been competent and respectful
54 ‘Trotz dieser selbst doktrinären Bekannschaft mit der demonstrativen Methode haben
die Araber sich meistens aller Beweise enthalten und Lehrsätze wie Regeln dogmatisch
aneinandergereiht, nicht anders als es die Inder in ihren Siddhanten zu thun pfl egen. Hat sie
dazu die Rücksicht auf Kürze und entsprechend grössere Wohlfeilheit ihrer Bücher bewogen?’
Hankel 1874 : 273.
55 Commenting on an original geometrical problem solved by al-Kūhī in which the latter inserted
a rigorous proof with diorismos, Cantor noted that ‘in general the imitators of the Greeks –
Arabs not excluded – considered [this practice] by no means with the same regularity’ [‘... was
die Nachahmner der Griechen im allgemeinen – die Araber nicht ausgeschlossen – keineswegs
mit gleicher Regelmässigkeit zu beachten pfl egten’]. Cantor 1894 : 705.
56 Cantor 1894 : 700; cf. 709–10.