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 
occurs three times, ‘We have it in the Porisms that..suggests 
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 npoSeSeucTaL, 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 Arith 
metica 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 
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