A history of Greek mathematics - Wilbourhall.org

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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
the Indians and Arabs have been drawn from
attributed to
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
'
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
1
Diophantus, ed. Tannery, vol. ii, p. xx.
2 lb., p. xviii.
much later authors.