Spring 2003

Unraveling the Mystery of Calendars and Particularly the Jewish Calendar


Ed Cohen


Before I talk about the Hebrew (Jewish) calendar, I will say a few words about calendars in general. In fact, I'll divide this talk into 3 parts: (1) calendars in general; (2) ancient Hebrew calendar; (3) modern Hebrew calendar.

Also, instead of BC and AD, most Jewish writers now use BCE (before the common era), and CE (the common era). I will use those terms, which are already used by those in Biblical studies.


According to astrophysicists (although the Orthodox Jews or fundamentalists may not agree), the universe was started about 13+ billion years ago (with what is called the "Big Bang"--the latest scientific evidence) and the Earth was formed about 4+ billion years ago. Whether we (the people) are a lucky accident or not is quite a debatable subject (and the evolutionists and biologists are still arguing about it). And whether there are other similar like people on other planets is also quite debatable, but still undetermined. In our galaxy, we have 9 known planets going around the Sun, which was formed about 6 billion years ago. It should burn for another 7 billion years (however just 1.5 billion with the same brightness). There are over 60 moons around the planets, of which we have one, as does Pluto. Mercury and Venus are the only planets without moons. I heard last month that several more moons around Jupiter were discovered.


A tropical year is the length of time that the Earth takes to make one revolution around the Sun between successive equinoxes (in other words = 1 year). It equals 365.2422 mean solar days, or 365 days, 5 hours, 48 minutes, and 46 seconds. It is decreasing at a rate of 0.53 (or about half) a second per century. It is the basis of the modern calendar (Gregorian or civil) used in most countries of the world. It took the place of the Julian (Julius Caesar) calendar in 1582CE in Catholic countries; England and its colonies in 1752CE. The Gregorian calendar was accomplished by two mathematicians--astronomers: Livius (Giglio) and Clavius. [Note that we have approximately 1/4 day left over to make a leap year. That would be fine if the year were 365 days and 6 hours, which I pointed out that it wasn't.


A synodic month is the time between two successive conjunctions of the Moon and the Sun, as viewed from the Earth. The mean synodic month = 29.5306 days, or 29 days, 12 hours, 44 minutes, and 2.78 seconds. You just have to remember that these two numbers (Earth going around the Sun, Moon going around the Earth) are not nice to deal with and will have to be dealt with with respect to calendars.


Farming was the first occupation: people had to eat. Farmers had to know about the seasons; therefore there were attempts at calendars. The first calendars were done by the Egyptians and Babylonians. There were 3 types of calendars (that were figured out): solar (by the sun alone); lunar (by the moon alone); and luni-solar (by a mixture of the sun and moon). The Egyptian was solar: it had 12 months with 30 days each + 5 extra days = 365 days. Since the year was approximately 365 1/4 days, their calendar lost 1 day every 4 years, and 1 year every 4 x 365 + 4 (for the leap years) = 1464 years; thus moving backwards through all the seasons, starting over again in about 1464 years. Our civil (or Gregorian) calendar is much like theirs, but more refined.


The Muslims, whose calendar started in 622CE (Mohammed died in 632CE), had a lunar calendar. As I noted previously the moon goes around the earth in 29 days and a number of hours and minutes, which is not very agreeable. They had 12 months in their year with 30, 29, 30, 29, etc. days totaling 354 days with leap years (one extra day at the end) in 11 years out of 30. This totaled 30 x 354 + 11 = 10,631 days in 30 years; the actual moon rotation being 10631.012 days; therefore being off by 1 day in about 2500 years--a very accurate calendar. Of course, losing 10-12 days every year put the calendar out of season very fast; but that did not seem to bother them, except that Ramadan, the fast month, could appear in every season, and when it was hot that made it very unpleasant. The Jews, like the Babylonians, had a luni-solar calendar. Mohammed did not want to have a calendar like the Jews: that was why, I believe, he chose to have a strictly lunar calendar. I am now going to discuss the luni-solar calendar of the Jews; i.e, a calendar that is influenced by both the moon and the sun.


What I would like to point out before we go more deeply into the Jewish calendar is the following: no matter what kind of calendar we use: solar, lunar, or luni-solar, the elements (or God) have not given us a reasonable type of calendar because the sun and moon do not present us with "nice" times of revolution: earth revolving about the sun = 365days+; moon revolving about the earth = 29days+.


However, all these calendars had 12 months because of the nature of the moon; but the number of days were different: solar = 365 or 366 days; lunar = 354 or 355 and luni-solar = 353, 354, 355, 383, 384, 385 for the Hebrew, as will be explained.


(2): According to the Bible (i.e., the Old Testament or the 5 books of Moses, Torah, Chumash or the Pentateuch, Septuagint--the Greek words), in Exodus, God told the Jews to leave Egypt on the 15th of NISAN, which is the start of Passover. Moses was told also that NISAN (Aviv in the Bible) was the first month, and every new month requires a special blessing. The actual names that you will see used in the Hebrew calendar come from the Babylonian-Assyrian language. Only four months are mentioned in the Bible and none of them are in use now.


Moses learned from God how to recognize a new crescent (of the moon)--that is, the beginning of the month. In Jerusalem, this was passed on to the 70 judges, called the Sanhedrin. The Sanhedrin knew that the moon appeared again in about 29 1/2 days (not exactly), which was called the molad (birth or conjunction of the new moon). So, e.g., if 2 witnesses came from afar and said they had seen the beginning crescent of the moon in 29 days (and they were examined thoroughly), the month would have 29 days; otherwise, it would have 30 days. The first temple was destroyed in 586BCE. Some of the Jews (including Ezra) were moved to Babylon after the first temple was destroyed. The second temple was destroyed in 70CE. By the middle of the 4th century, Jews were being spread out from Israel, and there had to be a solid method for determining the Hebrew calendar. It had been essentially kept secret by the Sanhedrin. I would like to point out that the calendar was the basis of the Jewish religion; for if the Jews did not know when the holidays were, the religion would fall apart fast.


(3): So, Hillel II (in the middle of the 4th ce. CE, actually 358CE-359CE), breaking away from the secrecy, instituted what is now the formal Hebrew calendar. It is so fascinating that mathematicians and astronomers have tried to become familiar with it also. One of the foremost Jewish scholars, Maimonides (1135CE-1204CE)--an astronomer, philosopher, and physician--wrote a book about it in the 12th century, explaining the early beginnings and the complications of Hillel II. [SHOW 2 BOOKS BY MAIMONIDES.


I must point out that the Hebrew calendar and the Julian / Gregorian calendar were not intertwined in any way until about the 19th century; i.e., [G] [H] was not necessary. The reason for that is that the Jews lived by themselves and had essentially no use for the Gregorian calendar. All the Jews wanted to figure out was when the Jewish holidays occurred--and they only needed the Hebrew calendar for that. Of course, all that changed, especially in the 20th century.


Now, why is the Hebrew calendar luni-solar, instead of just lunar. The reason is that in the Torah, God tells Moses the following:

[Exodus 12:1, 2]: 1"God said to Moses and Aaron in the land of Egypt, saying, 2'This month [Nisan] shall be for you the beginning of the months, it shall be for you the first of the months of the year.'" and then [Deuteronomy 16:1]: 1"You shall observe the month of Springtime and perform the pesach-offering for the Lord, your God, for in the month of Springtime the Lord, your God, took you out of Egypt at night." Somewhere else, it is told that Sukkos or Feast of Tabernacles should be in the Fall. As it says in [Leviticus 23: 33-34], 33"God spoke to Moses saying: 34Speak to the Children of Israel, saying: On the fifteenth day of this seventh month [= Tishrei] is the Festival of Succos, a seven-day period for the Lord." And in [Numbers 29: 12], 12"On the fifteenth day of the seventh month, there shall be a holy convocation for you; you shall do no laborious work; you shall celebrate a festival to the Lord for a seven-day period." This makes the first day of Succos 178 days beyond the first day of Passover, and can only occur, in the Hillel calendar, at the earliest September 19. Now, this makes the calendar solar as well, because the seasons are under the Sun's supervision, not the Moon's.


How does all this come about??? If the year were just lunar, it would be like the Moslem year. In part (2), where the Sanhedrin adjusted the calendar, every so often an extra month was added--which we know as Adar 2. They judged this on how ripe the fruit and grains were; but the main purpose was to get Passover into the Springtime. I think in the Talmud that there were plenty of arguments as to how this was happening; and Maimonides goes into this in his book also.


All of this did not happen in a Jewish vacuum. Greek astronomer, Meton, made a discovery (in 432BCE) that the lunar calendar could be made to match the seasons (of the Sun). This is now called the Metonic cycle and consists of 235 lunar months in 19 years. It was probably known to the Babylonian astronomers long before Meton's time. [19 x 12 + 7 = 235.]


The exact number of leap years was seven: chosen were 3, 6, 8, 11, 14, 17, 19 -- they have Adar 2. E.g., 5757 was the end of a cycle of 19 years [19 | 5757 = 303; we are now in 5763 or the 6th (leap) year of the 304th cycle. The ordinary years have 353, 354, or 355 days. The leap years have 30 more days. One more complication (a Jewish complication) was that Tishrei 1 [Rosh Hashanah, the 7th month] could not fall on Sunday (Hoshana Rabah would come on Saturday), Wednesday (YK would fall on Friday), or Friday (YK would fall on Sunday). This is the bare bones of the Jewish calendar.


I have almost reached the end of my talk. I would like to say how mathematicians come into the picture. In 1802, Karl Friederich Gauss, a noted mathematician, figured out on what date the first day of Passover would fall in the Gregorian (civil) calendar. [The first day that Passover could occur was March 26; the last day was April 25.] He was doing this as an astronomer, one of the many hats he held in his great mathematical career. He did not prove what he did, although he did it correctly. Two other mathematicians did that independently--(1) T.A. Cisa de Gresy in 1818 and Meyer Hamburger in 1895. Gauss's paper was 2 or 3 pages long; the proofs much longer. Adolf Fraenkel, a mathematician of the 20th century, was also involved with the Hebrew calendar, and wrote an article for the Hebrew Encyclopedia (which, I believe, is still there though he died in the '60s). As well, he did two papers on the Hebrew calendar when he was in his teens--(1908, 1909): (1): getting the Passover date into the Muslim date; (2): a general paper of obtaining Muslim dates from Jewish dates and vice-versa. Many papers were written in the 19th century by various mathematicians on the Hebrew calendar in prestigious mathematical journals after Gauss did his Passover paper. Gauss was a Protestant: he did 4 papers on Easter--the only time he made a mistake, I believe.