006 - הִלְכוֹת קִדּוּשׁ הַחֹדֶשׁ פֵּרֶק ו

The Rambam begins by describing how, in the times when the new month was established based on the ראייה—the sighting of the new moon—Beis Din would perform very precise astronomical calculations. They would determine the exact moment when the הירח—moon—and the חמה—sun—would align, meaning the moon would be directly under the sun. At that moment, the moon is completely dark from our perspective, as the sun's light is above it and the visible side is black. This precise alignment is what they needed to know.

This exact point of alignment is referred to as the מולד—the new moon. The Rambam notes that this calculation was done בדקדוק הרבה—with great precision—just as the אצטגנינים—astrologers or stargazers—would do. The reason for this was to determine whether, when witnesses came to testify about seeing the new moon, it was actually possible for them to have seen it. In other words, if witnesses arrived on the night of the 30th and claimed to have seen the new moon, the Beis Din needed to know if it was astronomically possible for the moon to have been visible at that time.

However, the Rambam explains that there are two types of calculations. The first is this highly precise calculation, but there is also a more general, approximate calculation. This second type is called החשבון שמחשבין בקירוב—an approximate calculation. Here, we do not know the exact moment of conjunction, but rather use the מהלך האמצעי—the average movement. This is not the true, precise position of the moon, but an average, which is close enough for calendrical purposes. This average calculation is what is referred to as the מולד—the new moon.

In our times, when we no longer have a Beis Din to sanctify the month based on witnesses, we rely on this average calculation. The Rambam calls this עיבור—the calculation of the months. This is the method we use today to determine the calendar. So, to summarize: there is the precise calculation used by Beis Din when they sanctified the month by sighting, and there is the average calculation, עיבור, which is what we use now. The Rambam will now explain how, once you know the מולד of a given month, you can calculate the מולד of the next month, and so on, using this average cycle.

The Rambam now explains the basic units of time used in these calculations. He states that the יום והלילה—the day and the night—are always considered to be 24 hours. This means that, for the purposes of the calendar, we use fixed hours of 60 minutes each, not variable-length hours as sometimes used in other halachic contexts.

Each day is divided into 12 hours of daytime and 12 hours of nighttime, making a total of 24 hours. But the Rambam goes further: each hour is divided into אלף ושמונים חלקים—1,080 parts. This is not the 60 minutes we are used to, but a much finer division, allowing for more precise calculations.

Why did Chazal choose to divide the hour into 1,080 parts? The Rambam explains that this number is highly divisible: it can be split into halves, quarters, eighths, thirds, sixths, ninths, fifths, tenths, and more. This makes it very convenient for calculations, as you can always get a whole number when dividing by these fractions, with no remainder. The commentators point out that 1,080 can even be divided by 27 and 54, which 360 (another commonly used number) cannot. This divisibility is the main reason for choosing 1,080 as the base unit for parts of an hour in these astronomical calculations.

Now the Rambam describes the length of the lunar month according to this average calculation. From the moment the הירח—moon—and the חמה—sun—align (as calculated by this method), until they align again in their average movement, is a fixed period.

This period is 29 days, 12 hours, and 793 parts (out of 1,080 per hour). In other words, from the beginning of the 30th day, you count 12 hours and then 793 parts of the next hour. This is the time between one מולד and the next מולד—the average lunar month.

This cycle—29 days, 12 hours, and 793 parts—is the basis for all further calculations in the Jewish calendar. It is the standard length of a lunar month as used for determining the מולד and, therefore, for setting the months and years.

The Rambam now turns to the length of the lunar year. If you take twelve of these lunar months—each being 29 days, 12 hours, and 793 parts—the total comes to 354 days, 8 hours, and 876 parts. This is the length of a regular lunar year.

If the year is a leap year, with thirteen months, you multiply the monthly cycle by thirteen. The result is 383 days, 21 hours, and 589 parts. This is the length of a leap year in the lunar calendar.

For comparison, the solar year is 365 days and 6 hours (according to the opinion of Shmuel, which the Rambam follows here). The difference between the solar year and the regular lunar year is 10 days, 21 hours, and 204 parts. This is how much longer the solar year is than the lunar year, and this difference is the reason for the need to add leap years to the Jewish calendar, to keep the lunar and solar years in sync.

Here the Rambam explains how to determine the day of the week for the מולד of the next month or year. The key is to focus on the remainder after dividing by seven, since the week has seven days.

If you take the length of the lunar month—29 days, 12 hours, and 793 parts—and divide it by seven, the remainder is 1 day, 12 hours, and 793 parts. This is the amount you add to the day, hour, and parts of the current מולד to get the next month's מולד. The Rambam gives a mnemonic for this: א' י"ב תשצ"ג—Alef (1), Yud-Beis (12), Tav-Shin-Tzadi-Gimel (793).

Similarly, if you take the length of the lunar year—354 days, 8 hours, and 876 parts—and divide by seven, the remainder is 4 days, 8 hours, and 876 parts. This is the amount you add to the current year's מולד to get the next year's מולד in a regular year. The mnemonic is ד' ח' תתע"ו—Dalet (4), Ches (8), Tav-Tav-Ayin-Vav (876).

For a leap year, the remainder is 5 days, 21 hours, and 589 parts, with the mnemonic ה' כ"א תקפ"ט—Hei (5), Chaf-Alef (21), Tav-Kuf-Pey-Tes (589). By adding these remainders to the current מולד, you can always determine the day, hour, and parts for the next month or year.

The Rambam now gives the practical application: once you know the מולד—the new moon—of any given month, you simply add the remainder from the previous calculation (א' י"ב תשצ"ג: 1 day, 12 hours, 793 parts) to find the מולד of the next month.

This method allows you to determine, for any month, on which day of the week, at what hour, and at how many parts the מולד will occur. All you need is the starting point—the מולד of one month—and the fixed interval to add each time.

To illustrate how this works, the Rambam provides an example. Suppose the מולד ניסן—the new moon of Nissan—was on Sunday (א'), at five hours during the day, and 107 parts. The mnemonic for this is א' ה' ק"ז—Alef (1, Sunday), Hei (5 hours), Kuf-Zayin (107 parts).

To find the מולד אייר—the new moon of Iyar—you add the remainder for a month (א' י"ב תשצ"ג: 1 day, 12 hours, 793 parts). The result is that the מולד אייר will be on Tuesday night (ג'), five hours into the night, and 900 parts. The mnemonic for this is ג' ה' תת"ק—Gimel (3, Tuesday), Hei (5 hours), Tav-Tav-Kuf (900 parts).

This process can be repeated month after month, and even year after year, by adding the appropriate remainders. In this way, you can calculate the מולד for any month or year, indefinitely.

The Rambam now explains how, once you know the מולד—the molad, or mean conjunction—for a given year, you can calculate the molad for the following year. If you know the molad for this year, you simply add the remainder of the year to it. If it is a regular year, you add the remainder for a regular year; if it is a leap year, you add the remainder for a leap year. This gives you the molad for the next year.

This process can be repeated year after year, indefinitely, "עד סוף העולם"—until the end of time. The Rambam emphasizes that the starting point for all these calculations is the very first molad, which took place in the first year of creation. That initial molad occurred on the night of Monday, five hours into the night, and 204 parts. The mnemonic for this is ב' ה' ר"ד—Monday, five hours, 204 parts. This is the foundational molad from which all subsequent calculations begin.

So, by knowing the initial molad and the amount to add each year—depending on whether the year is regular or a leap year—you can determine the molad for any subsequent year, simply by continuing this process year after year.

The Rambam now details how to handle the calculations when adding up the remainders from year to year. In all these computations, whenever you add up the remainders—whether it's hours or parts—there are certain thresholds where you convert one unit to the next.

For example, when the total number of חלקים—parts—reaches 1080, that completes one hour. You then add that hour to the total number of hours. Similarly, when the hours reach 24, that completes a day, and you add that day to the total number of days.

When the days exceed seven, you subtract groups of seven from the total, because we are only interested in the day of the week, not the total count of days. The leftover after removing all complete weeks tells you on which day of the week, at what hour, and at how many parts the molad will occur. The purpose is not to know the total number of days, but specifically to determine the day of the week, the hour, and the part for the molad.

The Rambam introduces the concept of the machzor—the 19-year cycle. Every 19 years, there are seven leap years and twelve regular years. This cycle is fundamental to the Jewish calendar.

Why do we rely on this 19-year cycle? Because when you add up all the days, hours, and parts from twelve regular years and seven leap years, and you convert every 1080 parts into an hour and every 24 hours into a day, you find that in 19 years, the lunar and solar cycles almost align. Each solar year is 365 days and 6 hours, so over 19 years, the difference between the lunar and solar cycles is only one hour and 485 parts. The mnemonic for this is א' תפ"ה—one hour, 485 parts.

This means that after every 19 years, the lunar and solar years are nearly synchronized, with only a tiny remainder. This is why the 19-year cycle is used for the Jewish calendar, ensuring the months remain aligned with the seasons.

As a result of this 19-year machzor, all the months in the Jewish calendar are lunar months, while the years are based on the solar cycle. The seven leap years in each cycle are specifically set in the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the cycle.

This fixed pattern ensures that the calendar stays balanced. By distributing the leap years in these positions, the months remain in sync with the solar year, preventing the holidays from drifting through the seasons. The Rambam lists these years explicitly, showing the precise structure of the cycle.

The Rambam now explains how to calculate the total remainder for a full 19-year cycle. When you add up the remainders from all twelve regular years, you get ד' ח' תתע"ו—four days, eight hours, and 776 parts. When you add up the remainders from all seven leap years, you get ה' כ"א תקפ"ט—five days, 21 hours, and 589 parts.

When you combine all these and then divide by seven (to remove complete weeks), you are left with two days, 16 hours, and 595 parts. The mnemonic for this is ב' ט"ז תקצ"ה—two days, 16 hours, 595 parts. This is the remainder of the 19-year cycle, and it is the amount you add to the starting molad to find the molad at the beginning of the next cycle.

Once you know the molad at the start of a 19-year machzor, you simply add the remainder of the cycle—ב' ט"ז תקצ"ה—to find the molad at the start of the next cycle. This process can be repeated for each subsequent cycle, "עד סוף העולם"—until the end of time.

The Rambam reminds us that the molad at the start of the first cycle was ב' ה' ר"ד—Monday, five hours, 204 parts. The molad of the year is always the molad of Tishrei for that year. So, by knowing the starting point and the remainder for each cycle, you can calculate the molad for any cycle and any year within it.

The Rambam now gives a practical method for determining the molad of any year or month you wish, whether in the past or future. First, you take the total number of years that have passed since creation and break them into complete 19-year cycles. You count how many full cycles have passed and how many years remain in the current cycle.

For each complete cycle, you add ב' ט"ז תקצ"ה—two days, 16 hours, 595 parts. For each regular year in the incomplete cycle, you add ד' ח' תתע"ו—four days, eight hours, 776 parts. For each leap year, you add ה' כ"א תקפ"ט—five days, 21 hours, 589 parts.

After adding everything together, you convert every 1080 parts into an hour, every 24 hours into a day, and then subtract groups of seven days to find the remainder. The leftover days, hours, and parts tell you the molad of the year you want to know. This method allows you to calculate the molad for any year or month, past or future, using simple arithmetic based on the cycle and the remainders.

The Rambam concludes by explaining how to find the molad for each month within a year. The molad of the year, as calculated above, is the molad of Rosh Chodesh Tishrei. To find the molad for MarCheshvan, you add א' י"ב תשצ"ג—one day, 12 hours, 793 parts—to the molad of Tishrei. To find the molad for Kislev, you add the same amount again to the molad of MarCheshvan.

This process continues for each subsequent month, always adding the same interval—one day, 12 hours, and 793 parts—to the previous month's molad. In this way, you can calculate the molad for every month, one after the other, "עד סוף העולם"—until the end of time.