J.R.R. Tolkien>
The Lord of the Rings /
The Silmarillion
Durin's Day is New Year's day in the Calendar of the Dwarves. Since the actual date of the New Year varied from year to year, the dwarves marked each time the last new moon of Autumn as the beginning of a new year. The beginning of a new year, however, did not need to begin with Durin's Day. Only when the last new moon and the sun can be seen at the same time is the new year called a Durin's Day.
These are the dwarvish new years on our calendar:
*October 14, 2004 is the only definite Durin's day because not only do the sun and the moon appear in the sky together, there is also a (partial) solar eclipse.