These perfect numbers are also attested by Euphorion in the Mopsopia, when he says: `...equal to his (their) limbs, with the result that they are called perfect'.
He then defines and exemplifies these three types of number (8.12-15); remarks on the rarity of perfection in numerology as in everything else, for only one such number is to be found among the tens, the hundreds, the thousands, and the ten-thousands; and observes that perfect numbers end in 6 and 8 alternately (8.16-20).(6) Elias now relaxes the turgidly elementary maths: enter Euphorion (8.21).
The first half of the line must explain why some numbers are called `perfect'; [GREEK TEXT NOT REPRODUCIBLE IN ASCII] supplies the required sense, and it hardly matters that it is not the word [Elias] himself uses.(7) If it is right, Euphorion could be echoing the standard definition of perfect numbers given in Euclid, El.
The one thing Nicomachus does not mention is Euphorion's line.
It is far from clear that Euphorion is referring to a text of Euclid himself, rather than that he is paraphrasing a mathematical cliche worded in this way before Euclid laid it down.
The date of Euclid is irremediably obscure; the consensus tends to be that his floruit was around 300 B.C., and he is associated with a school in Alexandria.(16) Euphorion is not known to have had any direct connection with Alexandria,(17) though he certainly knew Alexandrian poetry.
400 B.C.), nor to hold that, if the concept was current in the Academy of Euclid's and Euphorion's generations, it owed its existence there to the thought of Pythagoras' later followers.
Euphorion's Suda biography tells us that this was the title of one of the poet's [GREEK TEXT NOT REPRODUCIBLE IN ASCII] so, if it was the name of a whole poetry book, it could have contained poems with their own individual titles); it adds that it dealt with Attica.
[GREEK TEXT NOT REPRODUCIBLE IN ASCII] seems to recall Euclid's [GREEK TEXT NOT REPRODUCIBLE IN ASCII] punning on [GREEK TEXT NOT REPRODUCIBLE IN ASCII] and [GREEK TEXT NOT REPRODUCIBLE IN ASCII] could be `songs' or `limbs', but the latter is much easier to understand with [GREEK TEXT NOT REPRODUCIBLE IN ASCII] and [GREEK TEXT NOT REPRODUCIBLE IN ASCII] and [GREEK TEXT NOT REPRODUCIBLE IN ASCII] are also linked in the sense of body-parts in anatomical contexts.(34) We saw, too, that Nicomachus and, following him,[Elias], exemplified hyperperfect and defective numbers by analogies with bodies which had too many or too few parts; so it seems possible that Euphorion himself used some sort of analogy between perfect numbers and body parts.
If Euphorion's god was cut into a perfect number, can we reconcile the seven of the rhapsodies with a Euclidean perfect number?
This is unusual: Titans and Giants are normally only endowed with the standard number of hands, and one wonders why the Orphic poet or his source made the change A third possibility, of course, is that the context in Euphorion is not about dismemberment at all, but (perhaps) concerns medicine--healing, or the sort of subject which might have justified the line's inclusion in a medical compilation.