Ile or ala? : the unit size of a Roman ile

[D B Campbell]

Arrian Tactica ¶ 18.3: αἱ δὲ δύο ταραντιναρχίαι ἱππαρχία, δώδεκα καὶ πεντακοσίων ἱππέων, ἥντινα Ῥωμαῖοι εἴλην καλοῦσιν.
So which is it: ala or ile? Does the greek text not say: ‘eilen’? And is this translated as an ala? It seems that the Perseus website equates eilen with ile.

The internet is wonderful, but sometimes you just need a good book.
Your Perseus link is the online Liddell & Scott, but here's what Liddell & Scott (8th edn.) actually says:
   
So you can see that eilê is just the Ionian version of Doric Greek ilê. In fact, Karl Müller, one of the early editors of Arrian (1846), even emended all instances of eilê to read ilê, to avoid confusion; Herscher (1885) followed him, but I think De Voto used the Roos (1907) text, which doesn't.

It seems that an ile is also a troop of 64 cavalrymen, but it also seems that Macedonian Armies had ile of greater unit size. So the question is, what is the unit size of a Roman ile?

What Arrian says is that there are 64 cavalrymen in a (Hellenistic) ilê (as in Polyb. 6.25.1, where it is equated to the Latin turma), but the unit of 512 cavalrymen is what the Romans call an ilê (i.e. an ala). Köchly (1855) even emended the text to read ἥντινα Ῥωμαῖοι ἄλην καλοῦσιν. But I think ilê is close enough to ala for us to see what Arrian means here.