Talmud - Arachin 9a ערכין ט׳ א׳

then the new moon of Tishrei would arrive three days before Rosh HaShana. A lunar month is twenty-nine and a half days, and therefore a twelve-month period should have six full months of thirty days and six deficient months of twenty-nine days. If there were nine full months, this would cause subsequent months to start three days after the new moon. This in turn would lead people to deride the rabbinical court and accuse it of declaring the new month whenever it wanted, without regard for the astronomical facts. The Gemara explains its question: Now, too, when there are eight full months in one year, the new moon of Tishrei will arrive two days before Rosh HaShana, which will also cause people to disparage the court.

The Gemara answers: It is as Rav Mesharshiyya says, with regard to a different question: This is referring to a case where the year before was a leap year. Here, too, one can answer that it is a case where the year before it was a leap year, and the extension of the year was accomplished with an added month of twenty-nine days. Accordingly, remove a full, thirty-day month from the calculation of the current year corresponding to the extra deficient, twenty-nine-day month of the previous year, and the result is that only seven thirty-day months have an impact on the appearance of the new moon relative to the following Rosh HaShana. The Gemara asks: But even in a year with only seven thirty-day months the new moon would still precede Rosh HaShana by one day. The Gemara answers: People do not pay attention to a difference of one day, and therefore they will not deride the court.

Until this point the Gemara has discussed Rav Huna’s interpretation of the mishna, that the statement: It did not seem appropriate to establish more than eight, means that the Sages did not wish to establish more than eight full months in one year. The Gemara presents an alternative opinion. Ulla says: The mishna means that it did not seem appropriate to the Sages to establish more than eight deficient, twenty-nine-day months. And the mishna is saying what the reason is for the previous statement in the mishna, as follows: What is the reason that no fewer than four full, thirty-day months may be established during a year? It is because it did not seem appropriate to the Sages to establish more than eight deficient, twenty-nine-day months, so that the shortest possible year would comprise four full months and eight deficient months.

The Gemara asks: What is different about nine deficient months, in that the Sages did not deem it appropriate to have so many deficient months in one year? Apparently, the Sages were concerned that if so, i.e., if there were nine deficient months, the new moon of Tishrei would appear three days after Rosh HaShana. But now, too, if there are eight deficient months in one year, the new moon will appear two days after Rosh HaShana.

The Gemara answers: It is as Rav Mesharshiyya says with regard to a different question: This is referring to a case where the year before was a leap year. Here, too, one can answer that it is a case where the year before it was a leap year, and the added month was a thirty-day month. Accordingly, remove a deficient, twenty-nine-day month from the calculation of the current year corresponding to the extra full, thirty-day month of the previous year, and the result is that only seven deficient months have an impact on the appearance of the new moon relative to the following Rosh HaShana.

The Gemara asks: But even in a year with seven deficient months there is still one day between Rosh HaShana and the appearance of the new moon, and therefore people will deride the court. The Gemara answers: People will assume that the new moon was sighted by witnesses and reported to the court, and they will say: The new moon was visible yesterday but we did not pay attention and did not notice it.

Traduction : The William Davidson digital edition of the Koren Noé Talmud, with commentary by Rabbi Adin Even-Israel Steinsaltz.

Arachin 9a explores calendar calculations for full and deficient months, addressing concerns about the new moon's timing relative to Rosh HaShana and court credibility.