412 14. EQUATIONS AND ALGORITHMSIf x > 0 and a > b, then 1 < rn = (x + a)/(x + b) < a/6. That is, the given ratio
must be larger than 1 and less than the ratio of the larger number to the smaller.1.4. The significance of the Arithmetica. The existence of Arabic manuscripts
of Diophantus' treatise shows that his work was known to the Muslim mathemati-
cians of the Middle Ages. Sesiano (1982, pp. 9-20) discusses the extent to which
a number of Islamic and Byzantine mathematicians were influenced by his work or
commented on it. He comments (p. 9) that, "There is nothing to suggest that the
Egyptian Abu-Kamil had any direct (or even indirect) knowledge of Diophantus'
Arithmetica, although the problems in his Algebra dealing with indeterminate anal-
ysis are perfectly Diophantine in form and the basic methods are attested to in the
Arithmetica." In contrast, the Diophantine connection is clear in the case of the
eleventh-century mathematician al-Karkhi, (also known as al-Karaji, 953-1029),
whose Fakhri has many points of contact with Diophantus. Tracing the influ-
ence of Diophantus, however, is more difficult. Jacob Klein (1934-36, p. 5), citing
nineteenth-century work of Tannery and others, says that "the special influence of
the Arithmetic of Diophantus on the content, but even more so on the form, of this
Arabic science is unmistakable if not in the Liber Algorismi of Al-Khowarizmi
himself, at any rate from the tenth century on." In a treatise on algebra published
in the late sixteenth century, the engineer mathematician Rafael Bombelli stated
that, although it had been agreed up to his time that algebra was an invention
of the Muslims, he was convinced, after reading the work of Diophantus, that the
invention should be ascribed to the latter.
At the very least, Diophantus used equations and developed a symbolism for
handling algebraic expressions, and that, in the long run, was an important inno-
vation. As two prominent Russian historians of science say:
Diophantus was the first to deduce that it was possible to formulate
the conditions of a problem as equations or systems of equations;
as a matter of fact, before Diophantus, there were no equations at
all, either determinate or indeterminate. Problems were studied
that we can now reduce to equations, but nothing more than that.
[Bashmakova and Smirnova 1997, p. 132]1.5. The view of Jacob Klein. In several places in Part 2 and in the present
part we have used without comment the "standard view" among historians of a
contrast between logistike and arithmetike in the science and philosophy of ancient
Greece, logistike being counting or computation and arithmetike being the study
of the theoretical properties of numbers. A different point of view is contained in
the extended essay by Jacob Klein (1934-36). Klein maintains that even the word
arithmos itself has been misinterpreted, that Euclid and Diophantus did not have
in mind cardinal numbers in the abstract, but used the word arithmos to mean a
set or collection. As he says (p. 7), "arithmos never means anything other than
'a definite number of definite objects.'" He goes on to say (p. 19) that for Plato
"'arithmetic' is, accordingly, not 'number theory,' but first and foremost the art
of correct counting."^4 In particular, Klein denies that Euclid was thinking about
(^4) Such may well be the case. If so, that is unfortunate for Plato's reputation. Neugebauer (1952, p.
146) offers the opinion that "Plato's role has been widely exaggerated. His own direct contributions
to mathematical knowledge were obviously nil.. .The often adopted notion that Piato 'directed'
research fortunately is not borne out by the facts."