[edit] More content about Yoma arichta and establishment of two day chag

Has to be sorted...

Rosh Hashanah is the only holiday that falls out on Rosh Chodesh.

Since there was no possible way of knowing if witnesses would arrive during the first or second day, both days were observed as Rosh Hashana to avoid accidentally

First day had to be observed regardless since witnesses could come until the last minute before it ended and then the holiday would have started retroactively from the night before and there was no way of knowing this until the day was over.

Many times the witness would only arrive during the day time, which meant the night before was retroactively holy.

This led to two issues. Firstly if the witnesses would arrive on either of the two possible days during later hours, it would mean the holiday of Rosh Hashana had already started the night before. In order to avoid the possibility of desecrating Rosh Hashana, Bais Din and the nation would start observing the holiday from first night that had a possibility of becoming 'Alef Tishrei'.

Some years however, witnesses of the new moon would not arrive until the second day. This meant that 'Alef Tishrei', the day that was just observed as Rosh Hashana, was a regular mundane day and was really the last day of Elul. Since there was no way of knowing when witnesses would arrive, there was no choice but to observe both days to avoid any possible desicaration of Rosh Hashana.