1 Grk “Since” or “Because.” This begins a long sentence that extends through v. 4. Because of the length and complexity of the Greek sentence and the tendency of contemporary English style to use shorter sentences, the Greek sentence has been divided up into shorter English sentences in the translation.
2 This is sometimes translated “narrative,” but the term itself can refer to an oral or written account. It is the verb “undertaken” which suggests a written account, since it literally is “to set one’s hand” to something ( ἐπιχειρέω). “Narrative” is too specific, denoting a particular genre of work for the accounts that existed in the earlier tradition. Not all of that material would have been narrative.
3 Or “events.”
4 Or “have been accomplished.” Given Luke’s emphasis on divine design (e.g., Luke 24:43–47) a stronger sense (“fulfilled”) is better than a mere reference to something having taken place (“accomplished”).

In other versions

Inasmuch as many have undertaken to compile a narrative of the things that have been accomplished among us,

Many have undertaken to draw up an account of the things that have been fulfilled among us,

Many have undertaken to compile a narrative about the events that have been fulfilled among us,

Many people have set out to write accounts about the events that have been fulfilled among us.

Since many have attempted to compile an account concerning the events that have been fulfilled among us,

Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us,

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