What is the Hebrew calendar? How is it calculated? How is the Hebrew calendar different from the Gregorian or Muslim calendar? Why is a Muslim actually younger than he states he is? Why is a Gregorian month pointless? How is a year measured? When was the observation-based calendar exchanged for a mathematically computed one? Why did the change occur? How many years were determined by the first method? What shortcuts can help calculate every Hebrew date (without using computer programs or 150-year calendars)? Rabban Gamliel and Rabbi Yehoshua were famously disputed on certain calendar calculations. How was the dispute resolved? How are leap-years determined? How do we know if the month will be 30 or 29 days long? What is Rabbi Nachshon’s iggul, and how can it be identified? Of this and more, in the following article.

**The First Mitzva**

On the first pasuk of the Torah, “In the beginning G-d created the heaven and the earth” (Bereshis 1:1) Rashi explains that the Torah should have really begun with the first mitzva of establishing a calendar: “This month is to be for you the first month, the first month of your year” (Shemos 12:2). While the Book of Bereshis provides the background and setting for the mitzvos, the first mitzva is where the Torah, as a book of laws, really begins.

In this week’s article we will provide a short overview of the basic rules upon which the Hebrew calendar is based, as well as a historic outline of how the modern Hebrew calendar came to be.

**Introduction**

To understand our calendar, we must first become familiar with several terms.

Time is measured in minutes, hours, days, months, and years. While these terms seem to be universal concepts, there are actually different ways of calculating them. The same term can mean one thing in one calendar, while meaning something completely different in another.

Passage of time can be measured according to the sun – which would result in a solar calendar; the moon – which gives us the lunar calendar; and a combination of the two, which gives us the lunisolar calendar (as is the Hebrew calendar).

The year, in the Gregorian calendar is based on measuring the time it takes for the sun to return to the same position vs. planet earth, while earth goes through the four seasons – fall, winter, spring and summer. The Torah divides the year to six seasons based on the agricultural cycles: planting, gathering, cold, heat, summer fruit, winter. This helps humans know what weather to expect in each season; to count the planting and harvest cycles, and, to forecast which stars and constellations will appear in the night sky. This information was crucial for farmers, sailors, and even fortune tellers.

The month in the Hebrew calendar expresses the moon’s cycle. Every full moon wanes until it disappears completely, and then is “reborn”, and continues to grow to a full moon — until it wanes again. Knowing when the moon would be full and when it would disappear was also essential for ancient populations.

A solar year is slightly more than 12 moon cycles. Therefore, when the excess builds up to the size of a month, the Hebrew calendar adds another month to the year, and the year becomes a leap year (as is this year).

The lunar year, on which the Islamic Calendar is based, counts 12 lunar months before beginning a new year. The Muslim year, while accurately counting the months, says nothing about the seasons or earth’s position in the solar system, stars, or constellations. There is no regularity in the Muslim calendar, and months can fall in a different season every year.

The solar year, on which the Gregorian calendar is based, tells us all about the seasons, stars, and constellations, but says nothing about the moon. The new crescent or full moon can occur at any point in the month, as a month in the Gregorian calendar is simply a division oof the 365 days of the year to 12. The Chinese calendar took it a step further, doing away with months altogether, counting only the days of the year.

The Hebrew calendar is one that combines both calendars. While the months always begin on Rosh Chodesh when the new moon is first sighted, the seasons remain fixed, and the holiday of Pesach will always be in the spring. This result is achieved by adding the leap month – the additional month of Adar I, seven times in a 19-year cycle.

As a result, when an adherent of the Islamic calendar declares he is 33 years old, he is actually only 32 years old, because he is missing 12 leap months.

**Establishing the Hebrew Calendar**

When Moshe Rabbenu was told the first mitzva of creating the Hebrew calendar, the year was 2448 years AM (Anno Mundi — from Latin “in the year of the world”). Who measured and calculated the years before Moshe Rabbenu? Yalkut Shimoni (Bo, 189) writes: “Rabbi Simon explains the following pasuk: ‘You have done great things, You, O Lord my G-d. Your wonders and Your thoughts are for us’ (Tehilim 40:6): For the first 2448 years until the Jewish people left Egypt those calculations were done by Hashem, Who made them and consecrated the months. Once they [the Jewish people] left Egypt, the responsibility was given to them.”

For the first 1671 years after Moshe Rabbenu was first given the responsibility for consecrating the months and setting up the calendar, the year depended upon three parameters:

- Calculating the year. Leap years were not fixed, but rather determined every year by a delegation of elders who decided if the year should be made a leap year, or not. The sages were permitted to include various considerations, provided the holiday of Pesach would always fall in the spring. (For example, the sages might decide not to make a Shmitta year a leap year so the Shmitta would not be any longer than necessary. Ben Pazi cautioned them to ensure Hoshana Raba would not fall on Shabbos.).
- Calculating the months. The Great Assembly (Sanhedrin) would convene in Jerusalem every month and calculate when the new moon would appear.
- Receiving witnesses. After calculating when the new moon could possibly be sighted in Eretz Yisroel, the Sanhedrin waited for witnesses to appear and testify seeing it. Based on this testimony, the Sanhedrin would consecrate the new month, and the previous one would have 29 days. If, according to their calculations, the new moon could not be sighted, or the day was cloudy and no witnesses could appear, the month would have 30 days, and Rosh Chodesh would be on the following day.

**The Contemporary Hebrew Calendar**

Only the Sanhedrin is authorized to determine months and years. Therefore, in the year 4119, when the Land of Israel was destroyed, and soon no permanent court would be left, Hillel II instituted the complete computed calendar, consecrating all the months until Moshiach’s arrival.

What rules govern the computed calendar?

To understand how it works we must first learn several terms:

**Year Types**

Every year has three identifying features: 1) The day of the week on which Rosh Hashana falls (which obviously depends upon the previous year), 2) the number of months in the year — if it will be a leap year or not, and 3) the number of days in Cheshvan and Kislev – 29 or 30 days.

A **meu’beret** year (a leap year) is a year in which a second month of Adar is added, called Adar I. Adar I always has 30 days.

A **simple** year – a year with twelve months, and only one month of Aadar.

A **chaserah** year (“deficient” or “incomplete”) is 353 or 383 days long. In this year, both Cheshvan and Kislev have 29 days.

A **ke’sidrah** year (“regular” or “in-order”) is 354 or 384 days long (depending on if it is a leap year or not). In this year, Cheshvan has 29 days while Kislev has 30 days.

A **shlemah** year (“complete” or “perfect”, also “abundant”) is 355 or 385 days long because both Cheshvan and Kislev have 30 days.

A **simple** year can be chaserah (D – ח) – 353 days, ke’sidrah (R – כ) — 354 days, shlemah (C – ש) — 355 days. A **leap** year (with the addition of Adar I which always has 30 days) has: chaserah — 383 days, ke’sidrah — 384, and shlemah — 385 days.

**Yearly Formula**

Every year is commonly marked by a formula which expresses the combination of the four factors — if the year is simple or a leap year; the day of the week on which Rosh Hashana falls, if it is ke’sidrah, shlemah, or chaserah (R-regular, C-complete, D-deficient), and what day of the week will the first day of Pesach fall on.

There are 14 valid combinations – 7 for a simple year and 7 for leap years: (The day Rosh Hashana falls in the week is on the right, while the first day of Pesach in on the left).

common years leap years

5R7 | 5C3 |

7C3 | 7D3 |

2C5 | 2D5 |

3R5 | 3R7 |

2D3 | 2C7 |

7D1 | 7C5 |

5C1 | 5D1 |

Some combinations are rather common, such as the 5R7 which occurs in 18% of the years (1 year in 5), while the 5C1, and the 5D1 are quite rare, occurring only approximately 3% of the years — about once in thirty years.

**Halachic Hours**

Halacha divides the 24-hour day into fixed hours, equal to 1⁄24 of a day. Each hour is divided into 1080 chalakim (parts). A part is 3+1⁄3 seconds (1⁄18 minute).

**The Molad**

A “new moon” (astronomically called a lunar conjunction and, in Hebrew, molad) is the moment at which the sun and moon have the same ecliptic longitude (i.e. they are aligned horizontally with respect to a north–south line).

At the exact moment of the molad, the moon is still invisible, but as the hours pass, it slowly becomes visible. The period between two molads is a synodic month.

The molad interval, which equals the mean synodic month of ancient times, is 29 days, 12 hours, and 793 “parts” (1 “part” = 1/18 minute = 31/3 seconds) (i.e., 29.530594 days). This “average molad” is remarkably accurate (less than one second from the current true value). In the Talmudic era, when the mean synodic month was slightly shorter than at present, the molad interval was even more accurate, being “essentially a perfect fit” for the mean synodic month at the time. The changes in the month today can be attributed to various natural factors, and while in Talmudic times it was necessary for accepting witnesses who had seen the new moon, Rabbi Yehuda HaNasi determined that all calculation would only be based on the average molad, as determined by Chazal, and not the actual molad. This resolved the issue over which Rabbi Yehoshua disputed with him in the famous ruling that demanded Rabbi Yehoshua appear in Beis Din with his walking stick and pouch on the day he calculated, was Yom Kippur.

**Months**

The Hebrew calendar has two kinds of months – a full month, which contains 30 days, and a chaser (“missing”) month, which contains 29 days. When the previous month has 30 days, we have a double Rosh Chodesh, and when the month is “missing” we only have one day of Rosh Chodesh.

Usually, after every chaser month we’ll have a full month, with three exceptions: Cheshvan and Kislev, which are flexible, and the added month of Adar I in a leap year, that always has 30 days.

The following months will always be full: Tishrei, Shevat, Nissan, Sivan, and Av.

The following month will always be chaser: Tevet, Adar (or Adar II) Iyar, Tamuz, and Elul.

**Leap Years**

As stated above, the discrepancy of nearly 11 days between the 12 synodic months and a solar year leads to adding Adar I, the leap month, every number of years. Rabbi Yehuda Hanasi determined that every 19-year cycle would have 7 leap years: the 3^{rd}, 6^{th}, 8^{th}, 11^{th}, 14^{th}, 17^{th}, and 19^{th}. Therefore, to determine if a year is a regular one or a leap year, all we need to do is divide the Hebrew year by 19. The remainder is the number of years into the next 19-year cycle.

For example, the year 5784, when divided by 19 amounts to 304, and 8/304. This tells us that the cycle began (304+1) in 5777, and the leap years in this cycle are 5779 (year 3); 5782 (year 6); 5784 (year 8); 5787 (year 11); 5790 (year 14); 5793 (year 17); and 5795 (year 19).

**Iggul of Rav Nachshon Gaon**

Every 247 years, or 13 cycles of 19 years, form a period known as an iggul – a circle, or: the Iggul of Rabbi Nachshon. This period is notable in that the precise details of the calendar almost always repeat over this period. This occurs because the molad interval (the average length of a Hebrew month) is 29.530594 days, which over 247 years results in a total of 90215.965 days. This is almost exactly 90216 days – a whole number and multiple of 7 (equalling the days of the week). So over 247 years, not only does the 19-year leap year cycle repeat itself, but the days of the week (and thus the days of Rosh Hashanah and the year length) typically repeat themselves.

While calculating the Hebrew date and day of the week can be easily done on any of the multiple online calendars, Iggul D’Rav Nachshon allows for calculating the Hebrew date even on Shabbos and holidays, using the table that employs this calculation. The table appears in the Tur (OC 428). There, the year’s formula appears in the square related to the current year in the 19-year cycle. Therefore, for the year 5784, for example, we determine what year we are in the 19-year cycle (8). Then, in the 8^{th} line of the column belonging to our year 1 in the cycle (5777) we can find the year’s formula: 7D3 (זחג). Since the 8^{th} year is always a leap year, the table contains a sign reminding users of that detail. Using this table, we can determine that the year 5784 will be a leap year, in which Rosh Hashana falls on Shabbos; a chaserah year, in which both Cheshvan and Kislev have 29 days, and the Pesach Seder will be on Monday night (the Halachic eve of Tuesday).