1. [There is a difference of opinion among] the Sages of Israel concerning the length of a solar year. Some Sages¹Shemuel (Eruvin 56a). maintain that it is 365 days and 1/4 of a day - i.e., six hours. Others maintain that it is slightly less than that figure.²Rav Ada (Ibid.). His opinion is discussed in the following chapter. There is also a difference of opinion among the wise men of Greece and Persia concerning this matter.³Ptolemy and Albatani, astronomers whose opinions were valued by the Rambam, from Greece and Arabia respectively, both maintain that the length of a solar year is less than 365 days and six hours. There is, however, a difference between the figures each of them suggests. According to contemporary science, the length of a tropical solar year is 365 days, 5 hours, 48 minutes and 45 1/2 seconds. It is decreasing at the rate of 0.530 second per century.

2. According to the opinion that [a solar year] is [exactly] 365 and 1/4 days, there will be a remainder of one hour and 485 units after every nineteen-year cycle, as we mentioned.⁴Chapter 6, Halachah 10.

Between the start of each of the successive seasons of the year, there will be ninety-one days and seven and one-half hours. When you know the date and the hour of the beginning of one season, you can calculate [the beginning of] the following season by [adding the above amount]. Similarly, you can calculate the beginning of the following season, and continue forever.

3. The equinox of Nisan (spring) [takes place] at the hour and the unit when the sun enters the beginning of the constellation of Aries. The solstice of Tammuz (summer) [takes place] when the sun is located in the beginning of the constellation of Cancer. The equinox of Tishrei (autumn) [takes place] at the hour and the unit when the sun enters the beginning of the constellation of Libra. The solstice of Tevet (winter) [takes place] when the sun is located in the beginning of the constellation of Capricorn.

According to this calculation, in the first year of creation the vernal (spring) equinox took place seven days, nine hours, and 642 units before the conjunction of the month of Nisan, in numbers, 7 -9 - 642.⁵This figure is based on the following principles: The times of the seasons are calculated according to the opinion of Rabbi Yehoshua, who maintains that the world was created in Nisan. Although the lunar calendar is calculated according to the opinion of Rabbi Eliezer, who maintains that the world was created in Tishrei, our Sages did not see a contradiction in interrelating the two conceptions, as we will continue to explain.

The conjunction of the month of Tishrei is the hour of man's creation, the fourteenth hour of Friday, the sixth day of creation. Based on this figure, by subtracting six times 1 day, 12 hours, and 793 units (the remainder of a lunar month), we can calculate that the conjunction of the month of Nisan - six months before creation - took place on Thursday, nine hours and 642 units after nightfall.

The vernal equinox is calculated according to the conception that the world was created in Nisan. The sun was created in the first hour of the fourth day of creation. This is considered the first vernal equinox. Thus, the autumnal equinox (half a year later) took place on Wednesday, three hours after daybreak, one day and twenty-three hours before the conjunction of Tishrei.

Since there is a difference of five days, ten hours, and 642 units between six months according to the lunar calendar, and half a year according to the solar calendar, it follows that the conjunction of Nisan was seven days, nine hours, and 642 units after the vernal equinox (Perush).

4. The method of calculating [the beginning of] the seasons can be explained as follows. First, it is necessary to calculate the number of [nineteen-year] cycles that have passed until the [nineteen-year] cycle in question. Afterwards, add one hour and 485 units for every [nineteen-year] cycle. Afterwards, group all the units into hours, and all the hours into days. [Once a] total [has been reached], subtract seven days, nine hours, and 642 units [from it].⁶For the calculations will be based on Rosh Chodesh Nisan, which took place this amount of time after the vernal equinox. Add the remainder to [the time of] the conjunction of Nisan in the first year of the [nineteen-year] cycle in question, and you will be able to know the hour and the date of the the vernal equinox of the first year of this cycle.⁷A significant point arises from these statements, when one calculates the progression of the date and time of the vernal equinox by adding one hour and 485 units for each nineteen-year cycle. It follows that within the entire six millennia of the world's existence, the vernal equinox will have advanced approximately fifteen days. Thus, from the standpoint of the solar calendar, it will always be possible for Pesach, the fifteenth of Nisan, to occur in the spring. From this date, you can calculate [the beginnings of] all the subsequent seasons [by] adding ninety-one days and seven and one- half hours for every season.

If you desire to know [the time and the date of] the vernal equinox of a particular year within a given [nineteen-year] cycle, [the following procedure should be used:] Add one hour and 485 [units] for every [nineteen-year] cycle. For each complete year that has passed within the [nineteen-year] cycle [under discussion], add ten days, twenty-one hours, and 204 units,⁸The difference between a lunar year and a solar year, as stated in Chapter 6, Halachah 4., and then group the entire sum [into days and hours].⁹I.e., calculate every group of 1080 units as an hour, and every group of 24 hours as a day.

Afterwards, subtract seven days, nine hours, and 642 units [from this sum]¹⁰By making this subtraction, one bases the calculation on the conjunction of the month of Nisan, and not on the time of the first vernal equinox, which preceded that conjunction by this number of days and hours. and divide the remainder into lunar months of 29 days, 12 hours, and 793 units.¹¹Until this point in the calculation, the Rambam has not taken into consideration the existence of leap years. He does this now by grouping the remainder into months and subtracting the complete months. The number of complete months subtracted represents the number of leap years that have passed in the nineteen-year cycle. [The number of days, hours, and units that] remain [after all the complete] lunar months [have been calculated] should be added to [the day and the time of] the conjunction of Nisan in that year. [In this manner,] you will be able to determine the date and the time of the vernal equinox of the year desired.

According to this calculation, the vernal equinox will always take place either at nightfall, at midnight, at daybreak, or at noon.¹²There are 30 hours between the time of the equinox (or solstice) of one year and the next. Since the first vernal equinox took place at nightfall between Tuesday and Wednesday, the second vernal equinox took place at midnight between Wednesday and Thursday, the third at daybreak on Friday, and the fourth at noon on the Sabbath. Similarly, in subsequent years, the time of the equinox will continue to advance in six (i.e., 30) hour intervals according to such a pattern. The summer solstice will always take place at either 7:30 PM, 1:30 AM, 7:30 AM, or 1:30 PM.¹³There is a difference of seven and a half hours between the time of the vernal equinox and the time of the summer solstice. Since the first vernal equinox took place at nightfall, the first summer solstice took place at 1:30 AM. Afterwards, the time of the summer solstice advances in six- (i.e., 30-) hour intervals every year, in a manner parallel to the progression of the vernal equinox, as described in the previous note. The autumnal equinox will always take place either at nine or at three o'clock, either in the day or the night.¹⁴The first autumnal equinox took place at 9 AM. Afterwards, the time of the equinox has advanced in six-hour intervals, as explained. The winter solstice will always take place either at 10:30 PM, 4:30 AM, 10:30 AM, or 4:30 PM.¹⁵The first winter solstice took place at 4:30 PM. Afterwards, the time of the solstice has advanced in six-hour intervals, as explained.

If you desire to know the day of the week and the hour of the equinox, [the following procedure should be used:] Count the number of complete years that have passed from the year of creation until the desired year, and divide them into groups of twenty- eight.¹⁶This number is chosen because after twenty-eight years, the equinox takes place on the same day of the week and the same hour as it did originally. This figure can be calculated as follows: 1 and 1/4 days (the difference between the time of the equinox in two successive years) times 28 equals 35 days. Thirty-five days are five full weeks.

Based on this calculation, it is each twenty-eight years that the sun returns to its original position at the time of creation. To commemorate this occurrence, a special blessing, Birkat HaChamah, is recited. (See Hilchot Berachot 10:18.) Add one day and six hours¹⁷The difference in the time of the equinox from one year to the next. for each year remaining. Total the sum [of the hours and the days], and then add three days. Afterwards, divide the days into groups of seven. The remainder of the days and the hours should be added to the time of nightfall on the first day of the week.¹⁸I.e., the night between the Sabbath and Sunday. The result will be [the day and the time] on which the vernal equinox will occur.

Why is it necessary to add three days? Because the first equinox of the year of creation took place at the beginning of the fourth day.¹⁹I.e., the night between Tuesday and Wednesday. By making this addition, it is possible for these calculations to start from the beginning of the week.

5. What is implied? If a person desires to know the day and the time of the vernal equinox of the year 4930 after creation,²⁰The commentaries understand this as an indication that this portion of the Mishneh Torah was composed during that year. [the following procedure should be used:] [That number] should be divided by 28, leaving a remainder of one year, thus producing the figure of one day and six hours. By adding three days to this figure, it can be determined that the vernal equinox will take place on the night of the fifth day at midnight.

By adding seven and one-half hours to this figure, it can be determined that the summer solstice will take place on Thursday, an hour and one half after daybreak. By adding seven and one-half hours to this figure, it can be determined that the autumnal equinox will take place on Friday, at nine hours after daybreak. By adding seven and one-half hours to this figure, it can be determined that the winter solstice will take place on the night of the sixth day, four and one half hours after nightfall.

Similarly, by adding seven and one-half hours to this figure, it can be determined that the vernal equinox of the following year will take place on Friday, at daybreak. In this manner, it is possible to calculate [the time of the beginning of all] the seasons forever.

6. [The following procedure should be used] if one desires to know the date of the month on which the vernal equinox will fall this year:²¹The date of the equinox also can be determined by the calculations mentioned in Halachah 4. In this and the following halachot, however, the Rambam offers a simpler calculation, which uses approximations, but ultimately enables one to arrive at the same result. First, determine the day of the week on which [the equinox] will fall. Then determine the day [of the week] on which Rosh Chodesh of Nisan will fall, and how many complete years have passed within the nineteen-year cycle. Add eleven days for every year,²²The Rambam is using an approximation. The difference between a lunar year and a solar year is ten days, twenty-one hours, and 204 units. However, to simplify the calculation, the Rambam rounds off the figure to eleven days. and then add seven days to this sum in the present time.²³I.e., in the Rambam's time. The figure of seven days is reached as follows: In the year 4930, 259 nineteen-year cycles had passed. When an hour and 485 units are added for every nineteen-year cycle, a total of 15 days, 15 hours, and 335 units is obtained. Since the first equinox took place more than seven days before the conjunction of Nisan, eight days are subtracted from this figure, leaving a remainder of approximately seven days. Divide the sum by thirty,²⁴To account for any leap years. Here, too, the Rambam is rounding off the figure; the length of a lunar month is slightly less. and begin counting the remainder of days from Rosh Chodesh Nisan.

If the date coincides with the day of the week on which the equinox falls, this is sufficient. If not, add one, two, or three days to this number until you reach the day [of the week] on which the equinox falls.²⁵Since the calculation suggested by the Rambam contains several approximations, it may not be exact, and days may have to be added to reconcile the discrepancy. If the year in question is a leap year, begin counting from Rosh Chodesh of the second Adar.²⁶There will be more than thirty days remaining. Therefore, the reckoning should be made from Rosh Chodesh Adar. When a day is determined through this calculation, the equinox will take place on that date.

7. What is implied? Should we desire to know the date of the vernal equinox of the year 4930, which is the ninth year of the two- hundred-sixtieth [nineteen-year] cycle, [the following procedure should be used:] We have already determined that Rosh Chodesh Nisan will take place on Thursday, and that the equinox will take place on Thursday.²⁷Further calculations are necessary, for in this instance, it is impossible that the equinox will take place on the first of the month, the eighth, or the fifteenth.

Since this is the ninth year of the [nineteen-year] cycle, there are eight complete years [to take into consideration]. When eleven days are added for every year, we reach a sum of 88. When seven is added, the total will be 95. When this number is divided by 30, there will be a remainder of five.

When we add five days to Rosh Chodesh Nisan, which is Thursday, we reach Monday. Since we know that the equinox will not fall on Monday, but rather on Thursday, we continue adding days until Thursday, the day of the equinox. Thus, we can determine that this year the vernal equinox will take place on the eighth of Nisan. A similar process can be followed [to determine the date of the equinox] every year.

8. Although we said that one should continue to add days until one reaches the day of the week on which the equinox takes place, one should never have to add more than one, two, or three days²⁸I.e., although the calculation mentioned by the Rambam operates using approximations, the difference between these approximations and the actual data will hardly ever exceed three days. - or in a most unusual case - four days.²⁹The maximum difference between the approximations employed by the Rambam and the actual data is three and one half days. Thus, it is possible, but highly improbable, that there be a four-day difference. If you find it necessary to add any more days than this, know that you have made an error in your calculations, and you should recalculate carefully.