Kiddush HaChodesh - Chapter Ten

According to the opinion among the Sages of Israel that a solar year is less than [365 and] one-quarter [days], there is a view that [the length of the solar year] is 365 days, 5 hours, 997 units, and 48 moments. A moment is a seventy-sixth portion of a unit.

According to this reckoning, the difference between a solar year and a lunar year will be 10 days, 21 hours, 121 units, and 48 moments (in numbers, 10 - 21 - 121 - 48). [According to this calculation,] there will be no remainder at all after a [nineteen-year] cycle. Instead, after every [nineteen-year] cycle, a perfect correspondence will be established between the solar years and the combination of ordinary and full lunar years.

According to this calculation, there are ninety-one days, seven hours, 519 units, and thirty-one moments (in numbers, 91 - 7 - 519 - 31). When you know the date and the time of the beginning of any particular season, you can calculate [the date and the time of the beginning of] the subsequent season according to the seasons of the year, in a way resembling the calculations [that follow the opinion that a solar year is 365 and] 1/4 days.

According to this calculation, the vernal equinox of the first year of creation was nine hours and 642 units1 (in numbers, 9 - 642) before the conjunction of the month of Nisan. Similarly, in every first year of a [nineteen-year] cycle, the vernal equinox is nine hours and 642 units before the conjunction of the month of Nisan.

When you know which is the first year of a [nineteen- year] cycle, [you will be able to calculate the beginning of every subsequent season] by adding 91 days, 7 hours, 519 units, and 31 moments for each and every season until the end of the [nineteen- year] cycle.

If you desire to know when the vernal equinox [of a given year] will fall according to this calculation, first determine how many complete years have passed within this [nineteen-year] cycle. For each year, add the remainder of a year 10 [days], 21 [hours], 121 [units], and 48 moments.

[Afterwards,] group all the moments as units, all the units as hours, and all the hours as days, as done when calculating the conjunction. Subtract nine hours and 642 units from the entire sum,2 and divide the remainder by the length of a lunar month.3 The remainder that is less than the length of a lunar month should be added to the time of the conjunction of Nisan of the year in question. The vernal equinox of that year will take place on the moment arrived at according to these calculations.

It appears to me that [the Sages] relied on this calculation [of the length] of the seasons regarding the institution of a leap year, in the era when the High Court held sessions and would institute a leap year because of the time [when the equinox was scheduled to occur] or for other reasons. For this calculation is more accurate than the former one. It shares a greater resemblance to the data explained by the astronomers than the first opinion, which considered a solar year to be 365 and 1/4 days.

Both these calculations that we have explained are approximations, based on the mean rate of progress of the sun, and not on its actual position [in the celestial sphere]. When one considers the actual position of the sun at these times, the vernal equinox will take place approximately two days before the time determined by either of these calculations.4 [This applies both] according to the opinion that [a solar year is] exactly [365 and] 1/4 days, and according to the opinion that [a solar year is] less than [365 and] 1/4 days.