According to Thorin:
“The first day of the dwarves’ New Year,” said Thorin, “is as all should know the first day of the last moon of Autumn on the threshold of Winter. We still call it Durin’s Day when the last moon of Autumn and the sun are in the sky together. But this will not help us much, I fear, for it passes our skill in these days to guess when such a time will come again.”
This is confusing.
On every dwarves’ New Year the sun and moon will be very close together and therefore in the sky together.
But Bilbo actually sees a new crescent moon:
Soon he saw the orange ball of the sun sinking toward the level of his eyes. He went to the opening and there pale and faint was a thin new moon above the rim of Earth.
This at least establishes that the story is not considering the day when the moon is entirely invisible. On the following day the moon indeed appears as a thin crescent above the sun, and will so appear every dwarves’ New Year. But the crescent will be so close to the sun that the sun’s light will blot it out until the sun begins to redden and dim at sundown. The reader is supposed to understand that the tradition is about the sun and moon being
visibly in the sky together.
If the sky is clear at sundown, and depending on the amount of haze in the air, and depending on how far the new crescent moon is from the moment of complete invisibility, and depending on latitude, and depending on the eyesight of the observer, the crescent moon may or may not be visible.
The crescent moon may be barely seen by one person and not seen by a person standing close by.
Presumably the dwarvish year was computed by a mixture of calculation and observation so that the calendar could be kept working in case of overcast. When working with a full moon or one near full the problems of whether a moon was actually seen would be greatly minimized. Then the dwarvish New Year could be calculated. And if at sundown observers saw the crescent moon before the sun had completely set, then it was Durin’s Day.
But then comes the problem of what is “completely set”. Assuming a seascape horizon to the west, does it still count if the crescent can be glimpsed when the sun has half set? What if the sun has three-quarters set? And what if the viewer is in a mountain valley? It is really impossible to tell how often Durin’s Day might occur without more information than Tolkien gives.
The dwarves must have had rules which Tolkien does not tell. All we know is that on this particular day the crescent moon was sufficiently far from the setting sun as seen from the valley on the Lonely Mountain that there was no question that it was Durin’s Day.
The dwarves and Bilbo would of course be able to tell when the dwarves’ New Year will come by noting the size of the decreasing crescent of the moon and be right within about two or three days. Yet though finding the door is the chief concern of the dwarves and Bilbo, unbelievably none give the matter of the dwarves’ New Year a single thought, so far as we are told, until Bilbo sees the thrush cracking a snail just before sundown.
Stupid, stupid, stupid dwarves! Stupid, stupid, stupid Bilbo!
Still this is a nice change-of-pace chapter without any enemies and sets everything up for the forthcoming explosion.