Halachah 1 – The Sun’s Average Daily Motion
The Rambam begins by giving the sun’s average movement in the zodiac over different amounts of time.
The sun’s average movement in one full day of twenty-four hours is:
59 minutes and 8 seconds of a degree.
From this, the Rambam provides useful calculation markers:
- In 10 days, the sun moves 9 degrees, 51 minutes, and 23 seconds.
- In 100 days, it moves 98 degrees, 33 minutes, and 53 seconds.
- In 1,000 days, after removing full circles of 360 degrees, the remaining movement is 265 degrees, 38 minutes, and 50 seconds.
- In 10,000 days, the remaining movement is 136 degrees, 28 minutes, and 20 seconds.
The Rambam explains that once one knows the sun’s movement for one day, one can calculate its movement for any number of days. A person may also prepare tables for two days, three days, ten days, twenty days, one hundred days, and so on, in order to make the calculations easier.
He then emphasizes two especially useful values:
- The sun’s mean movement over 29 days is 28 degrees, 35 minutes, and 1 second.
- The sun’s mean movement over a regular lunar year of 354 days is 348 degrees, 55 minutes, and 15 seconds.
These are useful because the calculations of lunar visibility often compare one month to the next and one year to the next. Since the purpose of these calculations is to know whether the new moon can be seen, preparing these average movements makes the process much easier.
Halachah 2 – The Sun’s Apogee and the Starting Point for the Calculation
The Rambam explains that the sun, like the other heavenly bodies, has a point in its orbit where it is farthest from the earth. This point is called the gova, meaning the “height” or apogee.
This point itself slowly moves. For the sun, its apogee moves approximately:
- 1 degree every 70 years
- 1½ seconds every 10 days
- 15 seconds every 100 days
- 2 minutes and 30 seconds every 1,000 days
- 25 minutes every 10,000 days
- 4 seconds in 29 days
- 53 seconds in a regular lunar year
The Rambam then establishes the reference point for these calculations. The calculation begins from the start of Thursday night, which was the third day of Nisan in the year 4938 from Creation. This is the “year of the principle,” the fixed starting point for the astronomy used in these chapters.
At that starting point:
- The sun’s mean position was 7 degrees, 3 minutes, and 32 seconds into Aries.
- The sun’s apogee was 26 degrees, 45 minutes, and 8 seconds into Gemini.
To calculate the sun’s mean position on any later date, one counts the number of days from this starting point, calculates the sun’s mean movement over that number of days, and adds it to the original position. The result is the sun’s average position for that date.
The Rambam gives an example: if one wants to know the sun’s mean position at the beginning of Shabbat night, the fourteenth of Tammuz in that same year, there are 100 days from the starting point. Adding the sun’s 100-day movement to the original position gives a result of 105 degrees, 37 minutes, and 25 seconds, which places the sun in Cancer, 15 degrees and 37 minutes into that sign.
The Rambam notes that this result may not correspond exactly to the beginning of the night, but that small difference does not matter for the moon-visibility calculation, because it will be corrected when calculating the moon’s position.
Finally, he explains that the same method can be used for any date, even thousands of years later. One may also choose a different starting point if desired, such as the beginning of a particular cycle or century. The same method applies not only to the sun, but also to the moon and the other planets, provided one knows their average daily motion and their starting position.
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