The Gemara quotes Shmuel who makes an important statement regarding the length of the year. Shmuel says that each of the four Tekufos (seasons) of the year lasts 91 days and 7 1/2 hours. Accordingly, a year is 365 days and 6 hours long (91 days and 7 1/2 hours X 4). Furthermore, each Tekufah begins 7 1/2 hours later in the day than the time at which the previous Tekufah began.
How does Shmuel's measurement compare with present-day calculations of the length of the solar year?
In the year 3714 (46 BCE), Julius Caesar arranged the first solar calendar, a calendar based on the earth's position in relation to the sun. His calendar is the basis for the one used by the modern world today. Caesar's astronomers advised him that the solar year is exactly 365 1/4 days (365 days and 6 hours), the same as Shmuel's calculation.
This figure, however, is imprecise. The actual length, rounded to the nearest second, is 365 days, 5 hours, 48 minutes, and 46 seconds. Thus, the actual year is about 11 minutes shorter than Shmuel calculated. In practical terms, every 128 years the equinoxes and solstices will arrive about one day (11 minutes X 128) too early on Shmuel's calendar.
Later astronomers revised the length of the year to 365 days, 5 hours, 55 minutes, and 25 seconds, eliminating about half of the 11-minute discrepancy. Based on this new value, Rav Ada, a third century Amora, organized the Jewish lunisolar calendar into the present 19-year cycle, with years 3, 6, 8, 11, 14, 17, and 19 as leap years. This cycle enables the lunar year to average out to the length of the solar year.
(NOTE: As time passed, astronomers realized that the calendar was not reflecting the true length of the year. In order to prevent the summer months from occurring during the winter and vice versa, the astronomers modified the Julian calendar. The new calendar was called the Gregorian calendar. First, they eliminated ten days that had been added to the year over the passage of time due to the miscalculation of the Julian astronomers. (The day after October 4 that year was not October 5, but October 15.) Also, they agreed to deduct three leap years every 400 years. Three out of four centesimal years (for example, the years 1700, 1800, and 1900) would not be leap years, even though they are divisible by four (those years would have no February 29). Only a centesimal year in which the number of centuries is divisible by 4 (such as the years 1600 and 2000) would be a leap year. According to their calculations, the average year was 365 days, 5 hours, 49 minutes and 12 seconds long. This is off by approximately 26 seconds, but is sufficiently accurate for most purposes. The calendar will be ahead by only one day after 3,323 years. No parallel changes were made to the Jewish lunisolar calendar.)
We follow Rav Ada's calculation in our 19-year cycle lunisolar calendar. However, for the Halachos which depend on the Tekufos, such as when to say the prayer of "v'Sen Tal u'Matar" outside of Eretz Yisrael and when to recite Birchas ha'Chamah, we rely on Shmuel's calculation. (Therefore, the prayer of "v'Sen Tal u'Matar" outside of Eretz Yisrael is added to the Shemoneh Esreh from 60 days after the autumnal equinox, and Birkas ha'Chamah is said every 28 years on the day of the vernal equinox. This is why "v'Sen Tal u'Matar" outside of Eretz Yisrael begins on December 4th or 5th, depending on whether or not the following year is a leap year.
Dr. Norman Bloom of Miami, Florida, noted that since we utilize two different solar calendars, the time when Jews outside of Eretz Yisrael begin saying "v'Sen Tal u'Matar" will migrate towards the summer faster than the spring festivals will migrate towards the summer. This means that eventually the time to begin saying "v'Sen Tal u'Matar" will catch up to Pesach, and the first day to say "v'Sen Tal u'Matar" will coincide with the day on which we are supposed to stop saying it! However, we will not encounter this problem for approximately 42,000 years, which will be long after the Ge'ulah will have come and all of the Jews will be living in Eretz Yisrael. (See also Insights to Ta'anis 10:2.)
QUESTION: The Gemara quotes Shmuel's rule of astronomy that the difference between the hour in which one Tekufah falls and the hour in which the next one falls is half an hour ("v'Ein Tekufah Mosheches me'Chavertah Ela Chatzi Sha'ah"). This is because the actual time difference between Tekufos is 91 days and 7 1/2 hours. RASHI (DH v'Ein Tekufah Mosheches) presents a Siman (mnemonic): "Siman l'Chetzyo AZ'CH GY'CH VA'CH TD'CH."
This Siman seems out of context and its meaning is evasive. (In fact, the BACH deletes it from the text of Rashi.) What does it mean?
ANSWER: The MENACHEM MESHIV NEFESH explains that the Siman is indeed out of place. It belongs at the end of the earlier comments of Rashi (DH v'Ein Tekufas Tamuz), where Rashi discusses the hours of the day in which each of the four Tekufos occur. The Menachem Meshiv Nefesh offers another Siman, "AZG'Y VAT'D," as a mnemonic. This shorter Siman means that if Tekufas Nisan occurs in a given year in the first hour (Alef), then Tekufas Tamuz of that year will be in the seventh hour (Zayin), Tekufas Tishrei will be in the third hour (Gimel), and Tekufas Teves will be in the tenth hour (Yud). The next four letters, "VAT'D," represent the hours in which the Tekufos will fall the following year: Tekufas Nisan in the sixth hour (Vav), Tekufas Tamuz in the first hour (Alef), Tekufas Tishrei in the ninth hour (Tes), and Tekufas Teves in the fourth hour (Dales).
However, this Siman gives us only the whole hour in which the Tekufos fall; it does not tell us exactly when during those hours they occur. Some of the Tekufos will start on the half hour, while others will start on the hour. This is the purpose of the next Siman. The words "Siman l'Chetzyo" in our texts of Rashi should read "Siman l'Chatzayos," as they appear in earlier printings, and they mean "a Siman as to which Tekufos occur on the half hour."
This Siman, "AZ'CH GY'CH VA'CH TD'CH," tells us that when Tekufas Nisan falls on the first hour (Alef), then Tekufas Tamuz will occur on the seventh and a half hour (Zayin Chatzi -- seven and a half). Tekufas Tishrei will be on the third hour (Gimel), and Tekufas Teves will occur on the tenth and a half hour (Yud Chatzi). We see that this Siman is exactly the same as the previous one, except that this Siman contains a "Ches" after every second number in order to show that the Tekufah will be on the half hour and not on the hour.