A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

WORKS
44 ( J
Six Books only of the former and a fragment of the latter
survive.
Allusions in the Arithmetica imply the existence of
(3) A collection of propositions under the title of Porisms ;
in three propositions (3, 5, 16) of Book V, Diophantus quotes
as known certain propositions in the Theory of Numbers,
prefixing to the statement of them the words We have '
it in
.'
the Porisms that . .
A scholium on a passage of Iamblichus, where Iamblichus
cites a dictum of certain Pythagoreans about the unit being
the dividing line (fieOopLov) between number and aliquot parts,
says thus Diophantus in the Moriastica .... for he describes
'
as " parts " the progression without limit in the direction of
less than the unit The Moriastica may be a separate work
'.
by Diophantus giving rules for reckoning with fractions ; but
I do not feel sure that the reference may not simply be to the
definitions at the beginning of the Arithmetica.
The seven lost
The Arithmetica.
Books and their place.
None of the manuscripts which we possess contain more
than six Books of the Arithmetica, the only variations being
that some few divide the six Books into seven, while one or
two give the fragment on Polygonal Numbers as VIII. The
missing Books were evidently lost at a very early date.
Tannery suggests that Hypatia's commentary extended only
to the first six Books, and that she left untouched the remaining
seven, which, partly as a consequence, were first forgotten
and then lost (cf. the case of Apollonius's Conies, where the
only Books which have survived in Greek, I-IV, are those
on which Eutocius commented). There is no sign that even
the Arabians ever possessed the missing Books. The Fakhra,
an algebraical treatise by Abu Bekr Muh. b. al-Hasan al-
Karkhi (d. about 1029), contains a collection of problems in
determinate and indeterminate analysis which not only show
that their author had deeply studied Diophantus but in many
cases are taken direct from the Arithmetica, sometimes with
a change in constants; in the fourth section of the work,
1523.2 Q g
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 ; 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.
so