A Pregnant Year שָּׁנָה מְעֻבֶּרֶת

They testified that the year may be intercalated throughout the whole of Adar, whereas they used to say: only until Purim. They testified that the year may be intercalated conditionally. There was such a case with Rabban Gamaliel who went to receive permission from the governor in Syria and he delayed in coming back; and they intercalated the year on condition that rabban gamaliel should approve; and when he came back he said: I approve, and the year was intercalated.

(2) How much longer is a solar year than a lunar year? Approximately eleven days.5The Rambam speaks in more precise figures in Chapter 6. At present, he is speaking in general terms to give an outline of the how the Jewish calendar is structured. Therefore, [to correct the discrepancy between the lunar and the solar calendars,] when these additional days reach a sum of 30 - or slightly more, or slightly less - an additional month is added, causing the year to include thirteen months. This is called a full year.
[This is necessary,] because it is impossible to have a year with twelve months and an odd number of days, as [implied by Numbers 28:14]: "... of the months of the year." [On this verse, our Sages6Megillah 5a. commented:] "You count the months of a year, but not the days of a year."

(1) A leap year is a year that includes an additional month.1This month is added to juxtapose the lunar calendar upon which the months are based, with the solar calendar, which governs the years.The extra month that is added is always Adar,2The Mechilta (commenting on Exodus 12:2 states that just as when a month is made full the addition is made at the end of the month, so too, when a year is made full the addition is made at the end. (The year is considered as beginning in Nisan and ending in Adar.) and thus the year contains two Adarin: the first Adar and the second Adar.3Based on Hilchot Nedarim 10:6, it appears that the Rambam considers the first Adar to be the additional month of the leap year. The Tur and the Ramah (Orach Chayim 427:1) differ and consider the second Adar to be the additional month.
In practice, Purim is always celebrated in the second Adar (Hilchot Megillah 1:12). There is a difference of opinion with regard to whether to commemorate birthdays, yahrzeits, and the like that took place in Adar of an ordinary year in the first or the second Adar of a leap year. The accepted custom in the Ashkenazic community is to commemorate them in the first Adar of a leap year (Kitzur Shulchan Aruch 221:3).
Why is this month added? Because of the season of spring, so that Pesach will fall then, as [implied by Deuteronomy 16:1]: "Take heed the month of spring"4Based on this verse, Rav Sa'adiah Gaon considers the adjustment of the yearly calendar to be a separate mitzvah. The Rambam, by contrast, considers it one dimension of the mitzvah of establishing the calendar. [This command can be interpreted to mean, "Take heed] that this month falls in the spring season. Were the month [of Adar] not to be added [from time to time], there are times when Pesach would fall in the summer, and times [when it would fall] in the rainy season.

(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).

(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.

(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.

“But My covenant I will keep with Isaac, [whom Sarah shall bear you at this designated time [lamo’ed] next year]” – Rabbi Huna in the name of Rabbi Idi – that year was a leap year.22The verse is expounded to mean that Sarah will bear Isaac at the next festival [mo’ed]. The longest interval between festivals is the six months between Sukkot and Passover, but that is not enough time for a pregnancy to come to term. The Midrash explains that that year was a leap year, so there was a seven-month interval between those holidays, long enough for a viable pregnancy.

(24) “She opened it and saw the child [yeled], and behold, a boy [na’ar] crying. She had compassion for him and she said: This is from the children of the Hebrews” (Exodus 2:6).
Rabbi Ḥanina bar Pappa said: That day was the twenty-first of Nisan. The ministering angels said before the Holy One blessed be He: ‘Master of the universe, shall he who is destined to recite a song at the sea on this date be stricken by water on this day?’ Rav Aḥa bar Ḥanina said: That day was the sixth of Sivan. The ministering angels said before the Holy One blessed be He: ‘Shall he who is destined to receive the Torah on Mount Sinai on this day be stricken on this day?’ They raised an objection: On the seventh of Adar Moses was born. According to the opinion of the one who says on the sixth of Sivan he was cast into the Nile, it works out well, as from the seventh of Adar until the sixth of Sivan are the three months that he was hidden. However, according to the one who says it was on the twenty-first of Nisan, it does not work out well. But it is not a difficulty, as that year was a leap year; most of the first month, most of the last month, and the entire middle month are considered three months.56Moses was born on the seventh day of the first Adar. Therefore, from his birth until the twenty-first of Nisan there were twenty-three days of the first Adar, the entire second Adar, and twenty-one days in Nisan.