For (the sanctification of) two months, Nissan and Tishrei, [witnesses who saw the new moon] desecrate the Sabbath [in order to go to apprise beth-din thereof]. For, for them (Nissan and Tishrei) messengers go out to Suria [to apprise the exile of the day of their designation, for which reason they are of special significance. By Torah law, the Sabbath may be desecrated for (the sanctification of) all (of the months); but the rabbis forbade it, for the institution of the festivals (in their proper times) is not dependent upon them. But these two were left in the province of Torah law, for all of the festivals are dependent upon them.] And when the Temple stood, they desecrated the Sabbath for all of them for the sake of the [Rosh Chodesh] offering, [that it be sacrificed in its proper time.]
Bartenura on Mishnah Rosh Hashanah
מחללין את השבת – witnesses who saw the [New] Moon, in order to walk and to inform the Jewish court.
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English Explanation of Mishnah Rosh Hashanah
Introduction
As we stated in the introduction, in mishnaic times they set the calendar each month based on the testimony of witnesses who would come to the central court and testified that they had seen the new moon. Our mishnah teaches that on two months in order to testify that they had seen the new moon the witnesses may even profane Shabbat. We shall explain what is special about these two months below. What is critical is that the rabbis perceived it to be so critical that these witnesses come on time that they even allowed Shabbat to be profaned. Shabbat might need to be profaned if, for instance, the witnesses had to travel beyond the Shabbat border limit (2000 cubits outside of the city) in order to get to Jerusalem to testify.
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Bartenura on Mishnah Rosh Hashanah
שבהם שלוחי ב"ד יוצאים לסוריא – to inform the Diaspora the day of its establishment. Therefore, they considered, that from the Torah, they would violate [Shabbat and Holy Days] for all of them. But the Rabbis prohibited it because the establishment of the [correct days for the] Festivals is not dependent upon them. But for these two months, they relied upon Torah law, for all the Festivals are dependent upon them (i.e., Nisan and Tishrei).
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English Explanation of Mishnah Rosh Hashanah
On account of two months they profane Shabbat: on account of Nissan and Tishri, for on those months messengers go forth to Syria and in them the dates of the festivals are fixed. There are two months on which witnesses may come to testify even if they need to profane Shabbat Nissan and Tishri. This is because on those days the dates of the holidays, Yom Kippur, Sukkot, Pesah and Shavuot were set. Once we know when the first day of Nissan is we know when Pesah is, and we know when Shavuot will be as well. Similarly, once we know when the first of Tishri falls, we know when Yom Kippur and Sukkot fall. The messengers would go forth to Syria, meaning to the Diaspora, and let them know when the festivals would fall. Therefore, it was critical that they find out what day Rosh Hodesh was as soon as possible.
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English Explanation of Mishnah Rosh Hashanah
When the Temple stood they used to profane Shabbat for all the months, in order that the sacrifice might be offered on the right day. In the Temple there was a special sacrifice offered on Rosh Hodesh. Hence, it was essential that the witnesses testify on time every month so that the sacrifice would be offered on the correct day. Hence, they could profane Shabbat in order to get to Jerusalem to testify concerning any of the months. .