We have now covered all the singular occurrences of arche, noting the most complex Greek case to be the accusative, to which the John 1:1 arche, has no connection. The most important thing to remember is that John, like most of the New Testament authors, **drop** the definite article when referring to THE beginning. What distinguishes John slightly from the others is the sheer extent to which he does this (lots and lots). The usefulness of a rule is directly proportionate to its power to predict. Yet to try and come up with some kind of rule requires also looking briefly at the plural.
Orchai (plural nominative), archais (plural dative) and Archas (plural accusative) occur in total 10 times. We have no occurrences of the plural genitive in the New Testament. In only one of the ten occurrences, is there an implied definite article as we can observe in the singular, and these are nearly all from Pauline epistles and used to imply "principalities" or "rulers", which we have already seen negates the implied definite article even in the singular. A new and somewhat obscure meaning pokes its nose into the discussion thanks to Luke: two of the four occurrences of archais are anarthrous and mean "corners". OK!
So what of the tenth plural that **does** drop the "THE" in Greek, while preserving its articular meaning/emphasis? This is an important point, because it really helps us shed light on the Hebrews writer's methodology. Here is the verse Hebrews 1:10, which is a direct quotation from the LXX, that we reference as Psalm 102:25 (in the Septuagint this is 101:26):
In [the] beginning[s], Lord, you laid the foundations of the earth, and the heavens are the work of your hands.
Unfortunately, neither the NIBC commentary I have been mainly using, written by Donald Hagner, nor any of the online commentaries, make any effort to explain why a plural lurks beneath this "beginning". My brief forays into the word show that the LXX translators had probably understood from a significant number of Hebrew scriptural sayings (and their own usage of Hebrew as bilingual translators), lə·p̄ā·nîm usually refers to "formerly" or even "the former times" (plural, see Ruth 4:7). This may mean that the English translation here is a bit misleading.
But the key point is that while the Hebrews writer has consistently shown that he does not write like John and the others (who do drop the definite article), here is the one time that we have him writing arche simply by quoting the Old Testament in the version he is most familiar with - the Greek (LXX) version. Hebrews is recognised as being probably the most polished Greek in the New Testament. The writer is careful and all of his quotations are precise. So when he copies the sacred Scriptures into his letter, it seems a fair assumption that he submits his own grammatical preference to what he considers supremely authoritative. In Hebrews 1:10, therefore, the Hebrews writer suspends his preference for explicitly writing the definite article in honour of precise LXX quotation.
Hurrah, now having analysed all occurrences of arche, we will be ready to state our rule!