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 mathe 
maticians 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 Arithmetical: 
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.’ 1 
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 Conics 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 Dio 
phantus 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 pur 
pose of palming off the compilations of much later authors. 
1 Diophantus, eel. Tannery, vol. ii, p. xx. 
- lb., p. xviii.