Two proverbs or a double proverb?

[ἑκηβόλος]

This post is me wondering about the LSJ entry for ἀμέλγω:

ἀμέλγω [α^], fut.
A.“-ξω” Theoc.23.25, milk, with acc. of animals milked, “μῆλα . . ὅσσ᾽ ἤμελγε” Od.9.238; ἤμελγεν ὄϊς καὶ μηκάδας αῖγας ib. 244; “βόας” Theoc.4.3: metaph., ἀμέλγεις τοὺς ξένους you drain them of all they have, Ar.Eq.326: > prov.> , ἀ. τὸν τράγον, of wasted labour, > Plb.33.21.1, Luc.Demon.28> :—Med., “ἀ. χροὸς αἷμα” Nic.Al.506: metaph., “ἐκ Σαπφοῦς τόδ᾽ ἀμελγόμενος μέλι τοι φέρω” Lyr.Adesp. 62.
etc.

As presented in LSJ, ἀμέλγειν τὸν τράγον, is a proverb. In the two references, however, it is used in coordination with ὁ τὸ κόσκινον ὑπέχων and (perhaps playing on the polysemy of ὑπόθεσις / ὑποτιθέναι to make a reference to the dialogue) the variant.

@Polybius

ἀλλὰ γὰρ ὀκνῶ μή ποτ᾽ εἰς τὸ περιφερόμενον ἐμπεσὼν λάθω, πότερον ὁ τὸν τράγον ἀμέλγων ἀφρονέστερος ἢ ὁ τὸ κόσκινον ὑπέχων: δοκῶ γὰρ δὴ κἀγὼ πρὸς ὁμολογουμένην ψευδολογίαν ἀκριβολογούμενος καὶ τὸν ἐπιμετροῦντα λόγον εἰσφέρων παραπλήσιόν τι ποιεῖν.
Translation
But I fear that the well-known adage may apply to me unknown to myself: “Which is the greater simpleton, the man who milks a he-goat or he who holds a sieve to catch the milk?” For it may be said of me that by confuting in detail what is confessed to be a lie, and doing so at great length, I am behaving in a very similar manner.

@Lucian

ἰδὼν δέ ποτε δύο τινὰς φιλοσόφους κομιδῇ ἀπαιδεύτως ἐν ζητήσει ἐρίζοντας καὶ τὸν μὲν ἄτοπα ἐρωτῶντα, τὸν δὲ οὐδὲν πρὸς λόγον ἀποκρινόμενον, Οὐ δοκεῖ ὑμῖν, ἔφη, ὦ φίλοι, ὁ μὲν ἕτερος τούτων τράγον ἀμέλγειν, ὁ δὲ αὐτῷ κόσκινον ὑποτιθέναι;
Translation
He once saw two philosophers engaged in a very unedifying game of cross questions and crooked answers. ‘Gentlemen,’ said he, ‘here is one man milking a billy-goat, and another catching the proceeds in a sieve.’

Polybius says it is part of a single phrase that is going around (well-known), and Lucian shows how it can be used, when divided into two parts.

Are there examples of ἀμέλγειν τὸν τράγον besides these two, where it is used on its own?

[mwh]

Seems to me the Lucian implies the Polybian “Which is sillier?” question, so you’d expect the two examples of αφροσυνη to be conjoined. I don’t know if Lucian read Polybius. It’s conceivable, but in view of Polyb’s το περιφερόμενον this is no evidence that he did.