On this day, in 2941 T. A., there was the last new moon in Rivendell before that crucial midsummer's eve. The calculations presented in "The Moon in 'The Hobbit'" show that this is the latest possible date of the June new moon that allows for both a broad silver crescent on midsummer's eve and for the very young moon seen on Durin's Day.
In the notes published by John Rateliff, Tolkien implicitly assumes 26 June as the date of this new moon but based on an assumed lunar period of 29 days, rather than 29.5.
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