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

RELATION OF WORKS 451
Relation of the 'Porisms' to the Arithmetica.
Did the Porisms form part of the Arithmetica in its original
form ? The phrase in which they are alluded to, and which
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occurs three times, ' We have it in the Poi^isms that suggests
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that they were a distinct collection of propositions concerning
the properties of certain numbers, their divisibility into a
certain number of squares, and so on ; and it is possible that
it was from the same collection that Diophantus took the
numerous other propositions which he assumes, explicitly or
implicitly. If the collection was part of the Arithmetica, it
would be strange to quote the propositions under a separate
title The Porisms ' ' when it would have been more natural
to refer to particular propositions of particular Books, and
more natural still to say tovto yap irpoSiSeiKrai, or some such
phrase, for this has been proved ' ', without any reference to
the particular place where the proof occurred. The expression
'We have it in the Porisms ' (in the plural) would be still
more inappropriate if the Porisms had been, as Tannery
supposed, not collected together as one or more Books of the
Arithmetica, but scattered about in the work as corollaries to
particular propositions. Hence I agree with the view of
Hultsch that the Porisms were not included in the Arithmetica
at all, but formed a separate work.
If this is right, we cannot any longer hold to the view of
Nesselmann that the lost Books were in the middle and not at
the end of the treatise ; indeed Tannery produces strong
arguments in favour of the contrary view, that it is the last
and most difficult Books which are lost. He replies first to
the assumption that Diophantus could not have proceeded 3
to problems more difficult than those of Book V. 'If the
fifth or the sixth Book of the Arithmetica had been lost, who,
pray, among us would have believed that such problems had
ever been attempted by the Greeks 1
It would be the greatest
error, in any case in which a thing cannot clearly be proved
to have been unknown to all the ancients, to maintain that
it could not have been known to some Greek mathematician.
If we do not know to what lengths Archimedes brought the
theory of numbers (to say nothing of other things), let us
admit our ignorance. But, between the famous problem of the
Gg2
452 DIOPHANTUS OF ALEXANDRIA
cattle and the most difficult of Diophantus's problems, is there
not a sufficient gap to require seven Books to fill if? And,
without attributing to the ancients what modern mathematicians
have discovered, may not a number of the things
attributed to the Indians and Arabs have been drawn from
Greek sources? May not the same be said of a problem
solved by Leonardo of Pisa, which is very similar to those of
Diophantus but is not now to be found in the Arithmetica 1
In fact, it may fairly be said that, when Chasles made his
reasonably probable restitution of the Porisms of Euclid, he,
notwithstanding that he had Pappus's lemmas to help him,
undertook a more difficult task than he would have undertaken
if he had attempted to fill up seven Diophantine Books with
numerical problems which the Greeks may reasonably be
supposed to have solved.'
It is not so easy to agree with Tannery's view of the relation
of the treatise On Polygonal Numbers to the Arithmetica.
According to him, just as Serenus's treatise on the sections
of cones and cylinders was added to the mutilated Conies of
Apollonius consisting of four Books only, in order to make up
a convenient volume, so the tract on Polygonal Numbers was
added to the remains of the Arithmetica, though forming no
part of the larger work. 2 Thus Tannery would seem to deny
the genuineness of the whole tract on Polygonal Numbers,
though in his text he only signalizes the portion beginning
with the enunciation of the problem Given a number, to find
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in how many ways it can be a polygonal number ' as ' a vain
attempt by a commentator ' to solve this problem. Hultsch,
on the other hand, thinks that we may conclude that Diophantus
really solved the problem. The tract begins, like
Book I of the Arithmetica, with definitions and preliminary
propositions ; then comes the difficult problem quoted, the
discussion of which breaks off in our text after a few pages,
and to these it would be easy to tack on a great variety of
other problems.
The name of Diophantus was used, as
were the names of
Euclid, Archimedes and Heron in their turn, for the purpose
of palming off the compilations of much later authors.
1
Diophantus, ed. Tannery, vol. ii, p. xx.
2 lb., p. xviii.