QUESTION: The Mishnah (8b) states that Beis Din does not make less than four Chaser months (months of 29 days) in a year, "and it is not seen more than eight." What do these words mean? Rav Huna explains that these words mean that Beis Din does not make more than eight Malei months (months of 30 days) in a year. The reason for this is that if Beis Din would make nine full months, then the new moon for the month of Tishrei (for Rosh Hashanah) would appear on a Wednesday, while Rosh Hashanah itself would occur on Shabbos. People would laugh at this arrangement and mock the Chachamim, saying that they do whatever they please with the calendar without reason.

The Gemara asks that the same concern should apply to a year with eight Malei months. It also looks strange for the new moon to appear two days before Rosh Hashanah! The Gemara explains that the only year in which there are eight full months is a year that follows a leap year (a year with an extra month of Adar) in which the extra month was Chaser (29 days long), effectively giving an extra day's leeway for establishing Rosh Hashanah. When the new moon appears only one day before Rosh Hashanah, no one will say that the Chachamim are manipulating the calendar unreasonably.

RABEINU ELCHANAN (cited by the SHITAH MEKUBETZES #1) poses the following question: Why does the Gemara need to answer that the previous year was a leap year? The Gemara should answer simply that the Mishnah is discussing a year that follows a year of eight Chaser months and four Malei months. If Beis Din makes this year with the opposite arrangement -- eight Malei months and four Chaser months, then the year will be perfectly balanced!

(a) RABEINU ELCHANAN answers that there will never be a year that has four Malei months and eight Chaser months, unless a correction needs to be made due to a problem from the previous year. Therefore, there will never be a "credit" of four days from the previous year enabling Beis Din to make a year with eight Malei and four Chaser months.

(b) Rabeinu Elchanan suggests an alternative answer and says that, in general, the Chachamim preferred to avoid establishing a year in which the number of Malei months is three more than the number of Chaser months, even when there is a need to do so. The Chachamim feared that this type of year would begin to be established even in situations when it is not necessary. The only situation in which such a year is established is when the preceding year was a leap year, and thus no one will mistakenly think that this type of year should be established on a regular basis. (Y. MONTROSE)

QUESTION: Rav Huna (end of 8b) explains that Beis Din does not make more than eight Malei months (months of 30 days) in a year (see previous Insight). The reason for this is that if Beis Din would make nine full months, then the new moon for the month of Tishrei (for Rosh Hashanah) would appear on a Wednesday, while Rosh Hashanah itself would occur on Shabbos. People would laugh at this arrangement and mock the Chachamim, saying that they do whatever they please with the calendar without reason.

The Gemara asks that the same concern should apply to a year with eight Malei months. It also looks strange for the new moon to appear two days before Rosh Hashanah! The Gemara explains that the only year in which there are eight full months is a year that follows a leap year (a year with an extra month of Adar) in which the extra month was Chaser (29 days long), effectively giving an extra day's leeway for establishing Rosh Hashanah. When the new moon appears only one day before Rosh Hashanah, no one will say that the Chachamim are manipulating the calendar unreasonably.

RABEINU ELCHANAN (cited by the SHITAH MEKUBETZES) has difficulty with this calculation. If Beis Din is worried about the new moon appearing three days before Rosh Chodesh, this will happen with a year that has eight Malei months as well, because the month of Elul is always Chaser, and thus the eight Malei months must occur before Elul, effectively causing the new moon of Elul to appear two and a half days before Rosh Chodesh Elul! Why is Beis Din not concerned that people will notice this problem and mock the Chachamim?

ANSWER: The TOSFOS HA'ROSH answers that the concern that people will mock the Chachamim applies only before Rosh Hashanah, when everyone is thinking about the arrival of Rosh Hashanah. At other times, such as Rosh Chodesh Elul, people will not take notice of the discrepancy (as long as Rosh Hashanah occurs in the proper time). (Y. MONTROSE)

QUESTION: The Beraisa states that the days of the week on which Rosh Hashanah falls in two consecutive years are always four days apart. (For example, if Rosh Hashanah occurs on Sunday this year, then it will occur on Thursday next year.)

RASHI (DH ha'Ika) mentions that the exact figure is four days, eight hours, and 876 Chalakim (48 minutes and 40 seconds; there are 1,080 Chalakim in one hour, and thus one Chelek equals 3 1/3 seconds). How does Rashi arrive at this figure?

ANSWER: RASHI (8b, DH d'Im Ken) points out that the length of a lunar month is 29 days, 12 hours, 40 minutes, and 73 Chalakim (or 29 days, 12 hours, and 793 Chalakim). Since there are twelve lunar months in the year, the number of days in the lunar year is:

M x 12 = 354 days, 9516 Chalakim (or 354 days, 8 hours, and 876 Chalakim).

(M = the length of a month)

Next, in order to find the difference in days between one year and the next, one must divide the length of the year into weeks and find the remainder:

354 days and 9516 Chalakim / 7 = 50 weeks, 4 days, 8 hours, and 876 Chalakim.

QUESTION: The Gemara points out that an ordinary lunar year is 354 days long, and a solar year is 365 days long.

If the lunar year is only 354 days long, then why does it not fall steadily behind the solar calendar (such that, after several years, the winter months should occur in the summer, and the summer months in the winter)?

ANSWER: According to the figures mentioned by the Gemara, one can easily understand why the lunar calendar, which is established based on a nineteen-year cycle, does not fall behind the solar calendar:

1. 7 years of the 19-year lunar cycle are leap years, in which there are 13 lunar months instead of 12 (years 3, 6, 8, 11, 14, 17, and 19).

2. A solar year is 365 days and 6 hours (a quarter of a day). Let this sum be "S." (See Insights to Ta'anis 10:2, and Insights to Eruvin 56:1.)

3. A lunar month is 29 days, 12 hours, and 793 Chalakim. Let this sum be "L."

4. 19 x S = 6939 days, 18 hours. This is the length of 19 solar years.

5. ((L X 13) X 7) + ((L X 12) X 12) = 6939 days, 16 hours, 33 minutes, and 3 1/3 seconds. This is the length of the 19-year lunar cycle. (This equation expresses the sum of the 7 lunar years with 13 months, and 12 lunar years with 12 months, in the 19-year cycle).

6. Accordingly, the difference between the 19-year lunar cycle and 19 solar years is only 1 hour, 26 minutes, and 56 and 2/3 seconds every 19 years. This difference that occurs every nineteen years is negligible.

(To put this difference into perspective, according to the Gemara's calculations of the lengths of the solar and lunar years, it will take approximately 9,440 years for the lunar year to fall 30 days behind the solar year. By that time, Mashi'ach certainly will have arrived and the Sanhedrin will have been reestablished, and the lunar year will be adjusted each year as the Sanhedrin sees fit.)

QUESTION: RASHI (in the first DH Sheneihem Chaseirin) mentions that a formula to calculate when Rosh Hashanah will occur is "A'T B'SH G'R." What does this Siman mean?

ANSWER: The TUR, in his discussion of the Jewish calendar (OC 428), writes that when one knows on which day of the week the first day of Pesach begins in any given year, he can determine the day of the week on which all of the other festivals will occur during the coming year. The mnemonic "A'T B'SH" expresses the formula for this calculation.

The term "A'T B'SH" refers to a method of linguistic extrapolation whereby each letter at one end of the alphabet is replaced with its corresponding letter at the other end of the alphabet (e.g. Alef with Tav, Beis with Shin, and so on).

Each day of Pesach is assigned a letter, beginning with Alef, so that the first day of Pesach is assigned the letter Alef, the second day Beis, the third day Gimel, and so on.

As the Tur demonstrates, the first day of Pesach (Alef) always occurs on the same day of the week as the festival whose name begins with the letter Tav (which corresponds to the letter Alef in "A'T B'SH") -- Tish'ah b'Av (Tish'ah b'Av is called a "Mo'ed," see Ta'anis 29a).

The second day of Pesach (Beis) always occurs on the same day of the week as the festival that begins with the letter Shin -- Shavuos.

The third day of Pesach (Gimel) always occurs on the same day of the week as the festival that begins with the letter Reish (which corresponds to Gimel in "A'T B'SH") -- Rosh Hashanah.

The fourth day (Dalet) corresponds to Kaf, which refers to Keri'as ha'Torah -- the festival on which the new cycle of reading the Torah begins, Simchas Torah. (This Siman applies only in Chutz la'Aretz. In Eretz Yisrael, Simchas Torah occurs on the same day of the week as the third day of Pesach.)

Finally, the fifth day of Pesach (Heh) corresponds to the letter Tzadi, for it occurs on the same day of the week as the day of Tzom -- the Fast of Yom Kippur.

The Tur adds that the sixth day (Vav) corresponds to the letter Peh, which stands for Purim, the festival established by the Rabanan. The day on which the sixth day of Pesach occurs is the same day on which the previous Purim occurred.

Later commentators have added to the Tur's list, pointing out that the seventh day of Pesach (Zayin) occurs on the same day of the week of the day represented by "Ayin," which stands for "Etz" ("tree") -- Tu b'Shevat (this Siman applies only in non-leap years). Alternatively, it refers to Tu b'Av, the day on which the cutting of the wood for the Mizbe'ach in the Beis ha'Mikdash was completed, which was celebrated as a festive day (see Ta'anis 30b-31a).