Yule

The establishment of the Yuledays in the Shire Calendar cannot date from earlier than the establishment of the Shire. The King's Reckoning is the main basis of the Shire Calendar, but it has no Yule (its equivalent days are simply known as Mettarë and Yestarë, 'last-day' and 'first-day'). So, it would seem that the Hobbits introduced this usage themselves, apparently based on a traditional feast held at this time of year.

Géol is a very old word, and its meaning is difficult to discover with any certainty. Some sources suggest that it may ultimately carry the meaning 'renewal' or 'rebirth'. This makes sense if Yule marks the end of one year and the beginning of another, but it isn't clear how reliably we can treat this intrepretation.

It is surely no coincidence that these are the two possible dates of the Winter Solstice, meaning that the Shire Calendar is arranged so that it begins and ends on the shortest days of the year.

In devising the Hobbits' calendar, Tolkien seems to have disregarded the complication that the Solstice isn't a fixed event (it moves on the calendar very slowly over the centuries). This is the reason, for instance, that we celebrate our modern 'Yuletide' about four days after the Solstice itself - because the date was settled by the Romans more than 2,000 years ago, when 25 December actually was the shortest day. The Hobbits created their calendar much further in the past than that, and for them the Winter Solstice would have fallen as much as a month later than it does for us, in late January. So, from a technical perspective, it could be said that the Shire Calendar starts roughly thirty days before it truly should.