**1 **[There is a difference of opinion among] the Sages of Israel concerning the length of a solar year. Some Sages 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. There is also a difference of opinion among the wise men of Greece and Persia concerning this matter.

**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.

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.

**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]. 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. 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**, and then group the entire sum [into days and hours].

Afterwards, **subtract seven days, nine hours, and 642 units** ~~[from this sum] and divide the remainder into ~~ and subtract [from the sum] all full lunar months consisting of 29 days, 12 hours, and 793 units. **lunar months**~~[The number of days, hours, and units that] remain [after all the complete] lunar months [have been calculated]~~ **The remainder consists of less then a month and will 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. The **summer solstice** will always take place at either ~~7:30 PM~~ 1:30 AM - [7:30 night], ~~1:30 AM~~ 7:30 AM - [1:30 day], ~~7:30 AM,~~ 1:30 PM - [7:30 day] or ~~1:30 PM~~ 7:30 PM - [1:30 night]. The **autumnal equinox** will always take place either at nine or at three ~~o'clock~~ hours, either in the day or the night. The **winter solstice** will always take place either at ten and half or at four and half hours, either in the day or the night ~~10:30 PM, 4:30 AM, 10:30 AM, or 4:30 PM~~.

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. Add one day and six hours for each year remaining. Total the sum [of the hours and the days], and then add three days. Afterwards, ~~divide~~ subtract the days ~~into~~ by 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. The result will be [the day and the time] on which the vernal equinox will occur.

**5 **What is implied? If a person desires to know the day and the time of the vernal equinox of the year 4930 ~~after~~ [to the] creation, [the following procedure should be used:] [That number] should be ~~divided~~ consistently subtracted by 28, until a remainder of one year is left, 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: 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, and then add seven days to this sum in the present time. ~~Divide the sum by thirty~~ Subtract from the sum portions of 30 days until you have less then 30 days, 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. If the year in question is a leap year, begin counting from Rosh Chodesh of the second 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.

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~~ subtract by portions of 30 days until you have less then 30 days, 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 days - or in a most unusual case - four days. 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.

Your donation will help us continue to promote the project for the benefit of the general public.