Rosh Chodesh Calculation

Q. In the Jewish calendar, months are either 29 or 30 days in length. Why is it that for a 29 day month (known as a choser), we observe only one day Rosh Chodesh, while for a 30 day month (which is referred to as a moleh) we observe 2 days Rosh Chodesh?
A. This question was posed to Rabbeinu Yeshaya ha’Rishon (b. 1180) [Teshuvos Ha’Rid 32]. He writes that one might suggest that we observe two days of Rosh Chodesh because of sfeika d’yoma (doubts in the calendar), which is the reason we observe two days of Yom Tov. However, this cannot be correct, for if so, we would observe two days Rosh Chodesh every month. Rather, the reason is as follows.

The Jewish months follow the lunar cycle which is approximately 29½ days. In theory, each month on the calendar should be 29½ days. However, this cannot be done, because a month must consist of integral numbers of days, and not fractions.

For this reason, we are required to alternate months between 29 and 30 days in length, so that on average, the cycle is 29½ days. On a moleh (30-day month), Rosh Chodesh should begin in the middle of the 30th day when the new moon appears. However, this cannot be done, because we may not split the holiness of a day in half. If part of the day is holy, then the entire day must be holy.

Rather, in such a month we designate the 31st day as Rosh Chodesh. However, in recognition of the fact that the new moon really appeared in the middle of the 30th day, and the latter half of the 30th day was fitting to have been declared Rosh Chodesh, the Rabbis instituted that the 30th day should be observed as Rosh Chodesh as well.

Rabbeinu Yeshaya points out that the custom of observing two days of Rosh Chodesh has very early sources and is alluded to in Sefer Shmuel I (20:27).
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