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. [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.2 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. 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. 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.
7 What is implied? If the conjunction [of the moon and the sun] for the month of Nisan takes place on Sunday,
five [seventeen] hours and 107 units after sunrise (in numerical terms 1 - 5 - 107 [1 - 17 - 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 nightfall (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).
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 year (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.
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.