Hilchos Kiddush HaChodesh 12:1
The average distance traveled by the sun in a 24-hour day is 59 minutes and 8 seconds (59, 8). Therefore, in ten days, the sun will travel 9 degrees, 51 minutes and 23 seconds (9, 51, 23). In 100 days, it will travel 98 degrees, 33 minutes and 53 seconds (98, 33, 53). The remainder traveled in 1,000 days, after subtracting all the multiples of 360 is 265 degrees, 38 minutes and 50 seconds (265, 38, 50). The remainder traveled in 10,000 days is 136 degrees, 28 minutes and 20 seconds (136, 28, 20). One can keep multiplying like this in order to determine the distance the sun travels in any number of days. If a person wants, he could calculate the distance in two, three, four days, etc., up to ten days. One could likewise calculated the distance for 20, 30, 40 days, etc., up to 100. These numbers become obvious once you knows the average distance of one day. It would be good to calculate and have handy the average distance the sun travels in 29 days (a lunar month) and in 354 days, which is the length of a regular (i.e., non-leap) lunar year.
With these numbers handy, it will be easy to calculate when the moon will appear. There are 29 full days from the night when the moon appears in one month to the night that it will appear in the next month. Every month, there will be a difference of 29 days, neither more nor fewer. Remember that our only goal right now is to determine when the moon will become visible. The difference in the sun’s position from the night when the moon appears in a given month one year and the night when it will appear the subsequent year is the difference of a regular year, or of a regular year plus one day (if Cheshvan and Kislev are both malei). The average distance traveled by the sun in one month is 28 degrees, 35 minutes and one second (28, 35, 1). The distance it travels in a regular year (of 354 days) is 348 degrees, 55 minutes and 15 seconds (348, 55, 15).
Hilchos Kiddush HaChodesh 12:2
There is a point in the sun’s orbit around the Earth, as well in as the orbits of the other heavenly bodies, in which a body is its farthest from the Earth. Except for the moon, that point in a body’s orbit follows a regular pattern, traveling about one degree in 70 years. This farthest point is called its apogee. Therefore, in ten days, the sun’s apogee will travel 1.5 seconds; in 100 days, 15 seconds; in 1,000 days, two minutes and 30 seconds; in 10,000 days, 25 minutes. In 29 days, it will travel a fraction over four seconds; in a regular year, 53 seconds.
We previously set the starting point for our calculations on Wednesday night (the eve of the fifth day), 3 Nisan in the year 4938 of the Jewish calendar. On this day, the position of the sun based on its average distance was 7 degrees, 3 minutes and 32 seconds in Aries (7, 3, 32). The apogee of the sun at our starting point was 26 degrees, 45 minutes and 8 seconds in Gemini (26, 45, 8). Therefore, if one wants to know the position of the sun based on its average distance at any time, just calculate the number of days from our starting point until that day, then calculate the average distance it traveled in this time based on the procedures we have discussed. Add up the total, figuring each type of unit separately. The result is the average position of the sun on the day in question. For example, if we want to know the average position of the sun at the start of Shabbos eve on 14 Tammuz in the (Rambam’s) current year, we must calculate the number of days from our starting point until this date, which is 100 days. The average distance traveled by the sun in 100 days is 98, 33, 53. We add that to our starting point, which is 7, 3, 32, which gives us a total of 105, 37, 25. Therefore, the average position of the sun at the start of this night will be 15 degrees and 37 minutes in the 16th degree of Cancer. Sometimes the sun will be in the average position that is calculated using this method at the start of the night, sometimes an hour before sunset or an hour after it. This makes no difference in calculating the appearance of the moon because we will compensate for this lack of precision when we calculate the average position of the moon. This process should be utilized for any date, even for one 1,000 years in the future. Once calculated and its remainder added to the starting point, it will yield the sun’s average position.
The same concepts apply to calculating the average position of the moon, or of any other heavenly body. Once we know the distance it travels in one day and we have a starting point from which to start, we just add up the distance traveled for as many years or days as appropriate and add it to our starting point. This will result in the body’s position based on its average distance. These ideas also apply to calculating the sun’s apogee: add the distance it travels over the course of however many days or years to the starting point and you’ll end up with the position of the sun’s apogee for any day you want. If one likes, another date can be set as the starting point, using a year at the start of a preferred 19-year cycle, or at the start of a particular century. One can set a starting point in the past or far in the future, as follows: We already know the average distances traveled by the sun in a regular year, in 29 days, and in one day. We also know that a year whose months are malei is a day longer than a regular year and a year whose months are chaseir is a day shorter. When it comes to a leap year, if the months are regular, it will be 30 days longer than a regular year; if they are malei, 31 days longer and, if chaseir, 29 days longer. With this information, it is possible to calculate the average distance traveled by the sun for as many years or days as one likes. Just add this to our starting point to determine the sun’s position for any future date, which can then be used as a new starting point. Or, subtract the average distance traveled from the starting point to determine the average position for a date in the past, which can then be used as a new starting point. These concepts also apply to calculating the average position of the moon or of other heavenly bodies, provided that their average locations on a given day are known. It should be obvious that we can calculate average positions in the past just as we can calculate them in the future.