Here are the conversions I used from the Shire calender to our present Gregorian calendar. I used Shire Reckoning because Bilbo Baggins is our viewpoint character and he is the character who is alleged to have first recorded the account. I began by deciding that Midyear's Day would coincide with June 21 (generally the same day as the Summer Solstice). Unfortunately, that conflicts with Tolkien's own conversion of our New Year's Day "more or less to the Shire January 9." I have decided to ignore the descrepency and chalk up the New Year's Day conversion to an error on Tolkien's part. Here is the table that I have devised:
CONVERSION OF SHIRE RECKONING TO THE GREGORIAN CALENDAR
Afteryule = December 22 to January 20
Solmath = January 21 to February 19
Rethe = February 20 to March 21*
Astron = March 22* to April 20*
Thrimidge = April 21* to May 20*
Forelithe = May 21* to June 19*
1 Lithe (Midsummer Eve) = June 20*
Midyear's Day = June 21*
Overlithe = June 21 (only on leap years)**
2 Lithe = June 22
Afterlithe = June 23 to July 22
Wedmath = July 23 to August 21
Halimath = August 22 to September 20
Winterfilth = September 21 to October 20
Blotmath = October 21 to November 19
Foreyule = November 20 to December 19
1 Yule = December 20
2 Yule = December 21
* minus one day on leap years with the exception of years at the end of a century. However neither TA 2941 nor 2942 were leap years (example leap years: TA 2940; 2944; 3004; 3016).
** Overlithe would have been skipped in years that ended a century (such as TA 2900 or 3000).
Now, here is the exchange between Elrond and Thorin in the book:
We find another clue while Bilbo is sitting on the Doorstep on the first day of the last week of autumn:"Then what is Durin's Day?" asked Elrond.
"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."
So this would have been the first waxing crescent of the moon following the New Moon. This should also be the dwarves' New Year and Durin's Day.If [Bilbo] lifted his head he could see a glimps of the distant forest. As the sun turned west there was a gleam of yellow upon its far roof, as if the light caught the last pale leaves. Soon he saw the orange ball of the sun sinking towards 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.
To determine the actual date, we need to know when was the end of autumn and the beginning of winter. Our own calendar doesn't help much; Tolkien tells us in Appendix D of The Lord of the Rings (The Calendars) that the seasons were loosely defined rather than observed strictly as periods of three-months. The Summer Solstice (approximately) marked both Midsummer and Mid-year's Day in Middle-earth. This tells us that the Winter Solstice (ocurring at the end of the Yule season on approximately 2 Afteryule) represented the middle of winter, not the beginning. From Appendix D we also learn that the month of WInterfilth was originally the last month of the year for the hobbits before they adopted the calendar of Numenor. Both winter and their new year began with the start of Blotmath. The Men of Dale, and likely the Dwarves of Erebor, followed a similar practice.
We can futher narrow the range by determining the earliest and latest possible dates for Durin's Day. Thorin and Company reached Esgaroth on 22 Halimath, also the date of Bilbo's birthday (September 12 on our Gregorian calendar). The Company spent more that two weeks at Lake-town; when they left, it took three days to reach Lonely Mountain. Figuring in the time spent changing camps and searching the Mountain for the Secret Door, Durin's Day couldn't have been much before 22 Winterfilth.
On the other side of things, we know that Bilbo and Gandalf spent Yule at Beorn's house and spent the remainder of winter with Beorn. They had to travel about 400 miles over rough terrain to reach Beorn's house from Erebor; even mounted, it would have taken them more than one month to get there. That eliminates all of Foreyule. The travelers left several days after the Battle of Five Armies, twenty-two days after the demise of Smaug. So Durin's Day could not be later than early in the month of Blotmath. If we equate the start of winter with the beginning of Blotmath then Durin's Day could not have fallen later than 30 Winterfilth.
Using the Shire Reckoning, the first day of the last full month of autumn would have been 22 WInterfilth (assuming we are correct about the start of winter). The phases of the moon bear this out. If Midsummer's Eve (1 Lithe, or June 20 for us) was a broad crescent moon as Tolkien described (he did not specify a waxing nor waning crescent) then Durin's Day should have been on or near either 21 Winterfilth (October 11) if waxing, or 4 Winterfilth (September 24) if waning. 4 Winterfilth is impossible as it falls before the Company's departure for Erebor. We can then assume that the Mid-year moon was a waxing crescent moon and we can estimate that Durin's Day fell on 22 Winterfilth in accordance with the Shire calendar.
I am ready to call the twenty-second day of Winterfilth as Durin's Day. On our calendar that would be October 12.
For us in the present era, this means that the dwarves' New Year would fall in the range of September 21 to October 20 on the Gregorian Calendar. The next Dwarven New Year will be on October 14, 2015. A study of moonrises/moonsets and sunrises/sunsets would tell us which years would include a Durin's Day.
