1. When [the new moon] was sanctified [based on the testimony of witnesses regarding] the sighting of the moon, the court would calculate the time of the conjunction of the sun and the moon in an exact manner, as the astronomers do.¹The term conjunction refers to the point when the sun, the moon and the earth are positioned in that order in a direct line. Therefore, as seen from the earth, the moon does not reflect the light of the sun. [When the sun and the moon have the same latitude (see Chapter 16) - i.e., when they are in the same plane - a conjunction is the cause of a solar eclipse. Ordinarily, however, there is a difference in latitude, and an alignment of this nature does not cause an eclipse.] [This enabled them] to know whether or not the moon would be sighted.

The first level of these calculations represent approximations of the time of the conjunction, and their accuracy is not great. This approximation of the time of the conjunction is calculated according to the mean rates of movements.²Our translation differs from the standard published text of the Mishneh Torah, and is based on the version found in authoritative manuscripts and early printings.

The term mean rate of movement refers to the average movement of the sun or the moon in angular degrees over a particular period. To explain: It was easy to calculate the number of conjunctions between one solar eclipse and another. Afterwards, this number would be multiplied by 360 (the number of degrees in a circle) and then divided by the number of years, months, days, or hours (depending on the mean one wanted to reach) that had passed between the two eclipses.

In fact, however, the sun - and to a much greater extent, the moon - would deviate from this mean rate of movement - i.e., the position in which they are located in the heavens differs from the position that would be reached by calculating the mean rate of progress. As is explained in the succeeding chapters, there are various ways of correcting and adjusting these mean calculations so that the actual position of these celestial bodies can be determined. The time of the conjunction as calculated in this manner is referred to as the molad.

The essentials of the calculations that are used when a court to sanctify [the new moon based on the testimony of witnesses of] the sighting [of the moon] does not exist - i.e., the calculations we use today - are referred to as ibbur. To explain:

2. Day and night are constantly considered a twenty-four hour composite, [on the average:] twelve [hours] of daylight and twelve [hours] of night. An hour can be divided into 1080 units. This number was chosen because it can be divided in half, into fourths, eighths, thirds, sixths, ninths, and tenths.³The only integer that cannot be divided into this sum is seven. The commentaries have noted that the number 360 also can be divided into all the fractions mentioned by the Rambam and question why he did not use this smaller figure. Each of these fractions contains many of these units.

3. According to this calculation, [the interval] between one conjunction of the moon and the sun and the subsequent conjunction according to their mean movement is twenty-nine full days, twelve hours of the thirtieth day, and 793 units of the thirteenth hour. This is the interval between one conjunction and the next, [and thus,] the length of a lunar month.

4. [Accordingly,] an [ordinary] lunar year, which includes twelve of these months, would include three hundred fifty-four days, eight hours, and eight hundred seventy-six units.

A leap year, which would include thirteen of these months, would include three hundred eighty-three days, twenty-one hours, and five hundred eighty-nine units.

A solar year is three hundred sixty-five days and six hours.⁴See Chapter 9, Halachah 1, which explains that there are two views concerning whether this is an approximation or an exact figure. See also Chapter 10, Halachah 6. Thus, a solar year exceeds an [ordinary] lunar year by ten days, twenty-one hours, and two hundred and four units.

5. When the days of a lunar month are counted in groups of seven, according to the weekly cycle, there is a remainder of one day, twelve hours, and 793 units (in numerical terms, 1 - 12 - 793). This is the remainder for a lunar month.

Similarly, when the days of a lunar year are counted in groups of seven, according to the weekly cycle, there is a remainder of four days, eight hours, and 876 units (in numerical terms, 4 - 8 - 876). This is the remainder for an ordinary lunar year. The remainder for a leap year will be five days, twenty-one hours, and 589 units (in numerical terms, 5 - 21 - 589).

6. When you know the time of the conjunction [of the sun and the moon] for any particular month, and add 1 - 12 - 793, you will arrive at the time of the conjunction of the following month. Thus, you will be able to determine on which day of the week and at which hour it will take place, and how many units of that hour will have passed.⁵As mentioned above, the Rambam is speaking about an average figure. Accordingly, this figure alone is not sufficient for the calculations of when the moon can be sighted. It is useful for structuring the fixed calendar, as explained in this and the following chapter.

7. What is implied? If the conjunction [of the moon and the sun] for the month of Nisan takes place on Sunday, five hours and 107 units after sunrise (in numerical terms 1 - 5⁶Kinat Eliyahu suggests that the Rambam should have stated 1 - 17, since five hours after sunrise is seventeen hours after the beginning of the day. - 107), by adding the remainder for a lunar month, 1 - 12 - 793, you will be able to determine that the conjunction for the month of Iyar will take place on Tuesday night, five hours and 900 units after nightfall (in numerical terms, 3 - 5 - 900). One may follow this same method [of calculation] month after month for eternity.

8. Similarly, if you know the time of the conjunction for a particular year and you add its remainder - either the remainder of an ordinary year or the remainder of a leap year - to the time of the conjunction, you will determine the time of the conjunction of the following year. This method [of calculation] may be followed year after year for eternity.

The first conjunction from which we begin, the conjunction of the first year of creation, was on Monday night, 5 hours and 204 units after nightfall⁷Tosafot, Rosh HaShanah 8 a,b, explains that this follows the view of Rabbi Eliezer (Rosh HaShanah 10b), who states that the world was created in Tishrei. According to this conception, the conjunction for Rosh HaShanah, the day of Adam's creation, was the fourteenth hour of Friday (the second hour after sunrise). Since the first day of creation was the twenty-fifth of Elul, the year prior to that of Adam's creation is also significant. To calculate the conjunction of that year, we subtract the remainder of an ordinary year - 4 days, 8 hours, and 876 units - from six days and fourteen hours. This produces the figure cited by the Rambam. (in numerical terms, 2 - 5 - 204). This is the starting point for these calculations.

9. In all the calculations to determine the time of the conjunction, when the remainder [of one period] should be added to another remainder, [the following principles should be adhered to:] When a sum of 1080 units is reached, it should be counted as an hour, and added to the number of the hours. When a sum of twenty-four hours is reached, it should be counted as a day, and added to the number of days. When the number of days is greater than seven, [all multiples of] seven should be subtracted from the sum, and the remainder be focused on.

For the purpose of our calculations is not to know the number of days, but rather to know on which day of the week, and at what hour and after how many units will the conjunction take place.

10. [The fixed calendar is structured in] a nineteen-year cycle, including seven leap years and twelve ordinary years. This is called a machzor.

Why was this [structure] chosen? Because when you total the number of days in twelve ordinary years and seven leap years together with their hours and their units, counting all [sums of] 1080 units as an hour, [all sums of] twenty-four hours as a day, and adding them to the number of days, the total will equal nineteen solar years, each of these years being 365 days and six hours.

The difference between the days of the solar calendar [and the lunar calendar] will be only one hour and 485 units (in numerical terms, 1 - 485).⁸As mentioned in Halachah 4, each ordinary lunar year is 10 days, 21 hours, and 204 units shorter than a solar year. Thus, the difference between the 12 ordinary lunar years of a nineteen-year cycle and the corresponding solar years is 130 days, 14 hours, and 288 units.

A lunar leap year is 18 days, 15 hours, and 589 units longer than a solar year. Thus, the difference between the seven leap years of a nineteen-year cycle and the corresponding solar years is 130 days, 12 hours, and 883 units. When this sum is subtracted from the figure mentioned in the previous paragraph, the remainder mentioned by the Rambam is reached.

As explained in Chapter 10, Halachah 1, there is another reckoning, which maintains that the nineteen-year cycle produces a more exact interrelation between the two calendars.

11. Thus, in such a [nineteen-year] cycle, the months are lunar months, and the years are solar years. The seven leap years in each cycle should be the following: The third year of the cycle, the sixth year, the eighth year, the eleventh year, the fourteenth year, the seventeenth year, the nineteenth year⁹By structuring the pattern in this manner, an interrelation [albeit not a totally exact one] is established between the lunar and solar calendars in each of the years. After several years in which the number of days of the solar calendar exceeds those of the lunar calendar, a leap year reverses that pattern and establishes an approximate equivalence. In this way, we ensure that the Pesach is always celebrated after the vernal equinox. (in numbers, 3, 6, 8, 11, 14, 17, 19).

12. When you add the remainders of each of the twelve ordinary years, [the remainder of each year] being 4 - 8 - 876, and the remainders of the seven leap years, [the remainder of each year] being 5 - 21 - 589, and then divide the entire sum in groups of seven, there is a remainder of two days, sixteen hours, and 595 units (in numerical terms, 2 - 16 - 595). This is the remainder of a [nineteen-year] cycle.

13. When you know the time of the conjunction of the beginning of a [nineteen-year] cycle, by adding 2 - 16 - 595 to it you will be able to determine the beginning of the next [nineteen-year] cycle, and similarly all the [subsequent nineteen-year] cycles for eternity. As stated above, the conjunction [marking] the beginning of the first [nineteen-year] cycle took place on 2 - 5 - 204. [The expression,] the conjunction of a year refers to the conjunction of the month of Tishrei for that year.¹⁰This point of clarification is necessary, because there are certain halachic matters regarding which Nisan is considered the beginning of the year.

14. Using the above method, it is possible to know the conjunction [marking] the beginning of any particular year, or any particular month, whether for the years that have passed or for the years to come.

What is implied? One should take the number of years that have passed until Tishrei of the [desired] year and group them in nineteen- year cycles. Thus, one will be able to determine the number of nineteen-year cycles that have passed and the number of years that have passed within the [nineteen-year] cycle that has not been completed [until the desired year]. One should add 2 - 16 - 595 for each cycle, 4 - 8 - 876 for every ordinary year of the cycle that has not been completed, and 5 - 21 - 589 for every leap year [of the cycle that has not been completed].

One should then add together the entire sum, calculating [the groups of 1080] units as hours, the [groups of 24] hours as days, and the groups of seven days [as weeks]. [By adding] the remainder of the days, hours, and units [to 2 - 5 - 204], one can determine the time of the conjunction of the desired year.

15. The time of the conjunction of a year determined through the above method is the conjunction of Rosh Chodesh Tishrei. By adding 1 - 12 - 793 to this figure, one can determine the conjunction of Marcheshvan, and by adding 1 - 12 - 793 to [the conjunction of] Marcheshvan, one can determine the conjunction of Kislev. Similarly, one can determine the conjunction of all subsequent months for eternity.