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 Abfi’l Wafa al-Bùzjànl (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 care 
fully 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 (1) come, (a) deter 
minate equations of the second degree, and (b) indeterminate 
equations of the first degree. Diophantus does indeed promise 
to show how to solve the general quadratic ax 2 + hx + c = 0 so 
far as it has rational and positive solutions ; the suitable place 
for this would have been between Books I and II. 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.