Let’s carry on looking at the Greek of Ephraim Graecus, Beatitudines aliae capita XX. I apologise if it’s a bit dull, but it’s useful to me. Into section 3:
γ’. Μακάριος ὃς γέγονεν ἐπὶ τῆς γῆς ὡς ἄγγελος οὐράνιος καὶ μιμητὴς τῶν Σεραφίμ, ἁγνοὺς ἔχων καθ’ ἑκάστην τοὺς λογισμούς.
Traversari’s modern translation (which helps quite a bit in sorting out the sense):
Beatus, qui in terra est tamquam Angelus coelestis, & imitator factus Seraphim, castas assidue cogitationes habet.
(I.e. Blessed [is he], who on earth is like an angel of heaven, & has become an imitator of the Seraphim, [and] continually has pure thoughts.)
As before, we start with “Μακάριος ὃς”, “Blessed [is he] who“, and we expect a verb. This time we’re not getting a verb in participle form, but instead a normal main verb, a 3rd person perfect indicative active, “γέγονεν”, “he has become”. The next bit is simple; ἐπὶ τῆς γῆς, meaning “upon the earth”.
Then we get ὡς, meaning “as, like”. Alright, Traversari tipped me off; so I hunted around until I found an excuse for it! But it still fits. Next ἄγγελος οὐράνιος, i.e., like a heavenly angel. Finally “καὶ μιμητὴς τῶν Σεραφίμ”, “and an imitator of the Seraphim”.
So the first clause means:
Blessed [is he], who has become, on earth, like a heavenly angel and an imitator of the seraphim…
Nothing unusual here.
But the rest gets messy.
ἁγνοὺς | ἔχων | καθ’ ἑκάστην | τοὺς λογισμούς.
The object of this clause is the accusative plural, ἁγνοὺς τοὺς λογισμούς = “pure thoughts”.
In truth, I’m not sure that I would have recognised λογισμος as “thought”, from Liddell and Scott. I got the idea from Traversari; but I see that even in Lampe’s Lexicon of Patristic Greek (p.806), meaning 1 is “argument, faith based”. But meaning 2 is indeed “thought”, thankfully.
ἔχων = a present participle, “having”.
But what on earth is “καθ’ ἑκάστην”? From googling I find that it appears in Hebrews 3:13, where καθ’ means “each”, and “hekastos” is an adjective meaning “every”, but not as a phrase. However I find “καθ’ ἑκάστην ἡμέραν” and “καθ’ ἑκάστην τὴν ἡμέραν” both rendered as “every day” in Mastronarde’s Introduction to Attic Greek p.159.
This is in fact the meaning. I find in the 1826 A new Greek and English Lexicon by James Donnegan, p.292, in the middle of the entry for hekastos the following entry:
καθ’ ἑκάστην (ἡμέραν understood), every day.
This Traversari has rendered as “continually”. So we end up with
Blessed [is he], who has become, on earth, like a heavenly angel and an imitator of the seraphim, having pure thoughts every day”.
That was harder work than it should have been!