A history of Greek mathematics - Wilbourhall.org

450 DIOPHANTUS OF ALEXANDRIA
between problems corresponding to problems in Dioph. II
and III, are 25 problems not found in Diophantus, but
internal evidence, and especially the admission of irrational
results (which are always avoided by Diophantus), exclude
the hypothesis that we have here one of the lost Books.
Nor is there any sign that more of the work than we possess
was known to Abul Wafa al-Buzjani (a.d. 940-98) who wrote
a commentary on the algebra of Diophantus ' ', as well as
a Book of proofs of propositions used by Diophantus '
in his
work'. These facts again point to the conclusion that the
lost Books were lost before the tenth century.
The old view of the place originally occupied by the lost
seven Books is that of Nesselmann, who argued it with great
ability. 1 According to him (1) much less of Diophantus is
wanting than would naturally be supposed on the basis of
the numerical proportion of 7 lost to 6 extant Books, (2) the
missing portion came, not at the end, but in the middle of
the work, and indeed mostly between the first and second
Books. Nesselmann's general argument is that, if we carefully
read the last four Books, from the third to the sixth,
we shall find that Diophantus moves in a rigidly defined and
limited circle of methods and artifices, and seems in fact to be
at the end of his resources. As regards the possible contents
of the lost portion on this hypothesis, Nesselmann can only
point to (1) topics which we should expect to find treated,
either because foreshadowed by the author himself or as
necessary for the elucidation or completion of the whole
subject, (2) the Porisms; under head (l) come, (a) determinate
equations of the second degree, and (6) indeterminate
equations of the first degree. Diophantus does indeed promise
to show how to solve the general quadratic ax 2 ± bx ± c =
far as it has rational and positive solutions ;
for this would have been between Books I and II.
so
the suitable place
But there
is nothing whatever to show that indeterminate equations
of the first degree formed part of the writer's plan. Hence
Nesselmann is far from accounting for the contents of seven
whole Books ; and he is forced to the conjecture that the six
Books may originally have been divided into even more than
seven Books ; there is, however, no evidence to support this.
1
Nesselmann, Algebra der Griechen, pp. 264-73.