"The first day of the dwarves'
New Year ... is as all should know the first day of the last moon
of Autumn on the threshold of Winter."
The first day of the Dwarves' year was calculated according to the last new moon of autumn (that is, the new moon that occurs within two weeks of 6 October, on a modern calendar). Not every Dwarves' new year was a Durin's Day, though: Thorin says 'We still call it Durin's Day when the last moon of Autumn and the sun are in the sky together.' (ibid). Only a Dwarvish new year where this occurs is technically a Durin's Day.
The Mysterious Calendar of the Dwarves
Dwarvish New Years
A selection of modern Dwarvish new years: all dates are shown in the modern (Gregorian) calendar
Browse through Appendix D of The Lord of the Rings, and you'll discover in great detail how the calendars of Hobbits, Men and Elves worked. Strangely, though, Tolkien has nothing to say there about Dwarves: the only clues we have to their calendar are from Thorin's words quoted in The Hobbit.
As so often with Tolkien, though, we can deduce an awful lot from one sentence. It's clear that the Dwarves had a lunar calendar. Our modern calendar, and all those described in Appendix D, are solar calendars: they're based on the Sun's annual motion through the sky. There is an alternative, though: it's possible to design a calendar based on the phases of the moon, and the Dwarves must have used a calendar of this kind.
A lunar calendar works by breaking down time into periods of roughly four weeks, during which the moon changes from new to full to new again. This approach has one big advantage that solar calendars lack - the face of the moon acts like a gigantic 'sundial' marking out the passage of each 'lunar month'. You can see this in action in Sam's reaction after the Fellowship leave Lórien, and the Moon has an unexpected phase: 'either it's out of its running, or I'm all wrong in my reckoning' (The Fellowship of the Ring II 9 The Great River). Sam thinks of the moon in much the same way as a watch or clock, as have many peoples throughout history.
Regulating a calendar by the moon instead of the sun might be useful from one perspective, but it also has a big disadvantage. A lunar year (twelve lunar months) is only 354 days long: eleven days short of a solar year. For every solar year that passes, the calendar will 'slip' back by more than a week. This is much more than a technical inconvenience, because the passage of seasons follows the solar year, not the lunar. Any given date will 'shift' by six months every seventeen years or so - midsummer becomes midwinter in less than a generation.
The Dwarves seem to have taken a direct approach to this problem - they reset their calendar every single year. When Thorin says that their year starts on 'the first day of the last moon of Autumn', what he means is that the Dwarves find a point where the lunar and solar (seasonal) calendars coincide, and restart their calendar from that point.
Perhaps the most curious outcome of this is that Dwarvish years have a variable number of months, sometimes twelve, and sometimes thirteen. Tolkien doesn't mention this 'leap month' (which would occur quite often) directly, but it must have occurred2.
Everything we've said here follows unavoidably from Thorin's few comments in The Hobbit. What we can't say, of course, is whether Tolkien realized these astronomical consequences. He obviously had a very good understanding of how calendars work: the fictional calendars in Appendix D are beautifully constructed, and in some ways an improvement on our own. The ancient idea of exactly aligning a calendar with the solstices and equinoxes, for example, is the basis for many of Tolkien's calendars - the modern Gregorian system is 'out' by ten days or so.
It seems hard to believe that Tolkien would expend such effort on the development of calendars for Hobbits, Men and Elves, and yet forget the Dwarves completely. Perhaps he took a look at their calendar and decided it was too awkward to describe in detail, but if he did, he has left us no evidence.
It isn't clear whether Durin's Day was observed by all Dwarves, or just by the Longbeards, the clan founded by Durin.
Tolkien may not detail the Dwarvish calendar directly, but there is some circumstantial evidence that he considered its operation. Specifically, we can see an analogue in the Jewish calendar, which is also lunar in form, and is also regulated against the solar calendar by the frequent insertion of an extra month.
This is a more important point than it may at first appear, because Tolkien drew certain cultural and historical parallels between his fictional Khazâd and the Jewish peoples. It is known, for example, that the sounds of Khuzdul, the language of the Dwarves, were in part derived from Hebrew. Bearing this in mind, it's tempting to imagine that the Dwarves' lunar calendar might represent another of these parallels.
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