In his 1906 introduction to Lane's Nights, Stanley Lane-Poole outlines some of the essential truths about the 1001 Nights in such a compact and succinct way that they still hold up as one of the best today. Despite no mention of Nabia Abbott (her finding was in the late 1940s) Lane-Poole's introduction sums up nicely the few truths one can speak of when talking about the 1001 Nights.
Even his statement that Galland "discovered the Arabic original," while problematic, still seems to be the case. Galland's manuscript remains the oldest (apart from the few scraps of story that Abbott found) collection of the 1001 Nights.
I included paragraph breaks for easier reading online although this selection is all one paragraph:
"Few story-books have enjoyed the popularity of the "Arabian Nights." For two centuries children have marvelled at the wonders of the Eastern fairyland and older readers have delighted in the picture of mediaeval Arab life which the "Thousand and One Nights" unfold. It is singular that so little should be known of the origin and history of a book so renowned.
We know that Galland discovered the Arabic original and published his French translation in 1704-1707, but of the date and place and manner of its composition we know scarcely anything.
The native critics of Arabic literature paid small attention to a collection of romances which appeared to them (as one of them wrote) only as a "corrupt book of silly tales." A work called the "Thousand Nights" or "Thousand and One Nights" is referred to twice in the tenth century by Arabic historians, who say it was translated from a Persian story-book called the "Thousand Tales"; but we do not possess this Persian book, and have no means of determining how far it corresponded with the "Arabian Nights" as we know it, or how far the "Thousand and One Nights" of the tenth century resembled the book which Galland and Lane and others translated.
All that we can safely deduce from these meagre references is that the framing scheme of the "Nights" - of the jealous Sultan who kills a wife every morning till Shahrazad keeps him interested in her stories - was borrowed from the Persian "Thousand Tales," and that a few of these tales were apparently identical with some which are included in the "Arabian Nights.""
The Arabian Nights' Entertainments. Translated by Edward William Lane, Edited by Stanley Lane-Poole. London, G. Bell and Sons, LTD., 1925.