1. The mean distance traveled by the sun in one day - i.e., in twenty-four hours - is 59 minutes and 8 seconds; in symbols 59' 8".¹Since the sun travels throughout the entire 360° sphere over the course of a solar year, and a year is slightly longer than 365 days, the daily distance the sun travels is slightly less than one degree - more precisely, 59 minutes, 8 seconds and 19.8 thirds. Although the Rambam does not mention the thirds in this figure, he includes them in his subsequent calculations. Thus, in ten days, it travels 9 degrees, 51 minutes and 23 seconds,²When performing simple multiplication, the sum appears to be three seconds less. These three seconds have been added because of the inclusion of the multiples of the thirds, as mentioned in the previous note. Similarly, in subsequent calculations the Rambam also adds the multiples of the thirds. in symbols 9° 51' 23". In one hundred days, it travels 98 degrees, 33 minutes and 53 seconds, in symbols 98° 33' 53".

The remainder [of the degrees] traveled [by the sun] over the course of one thousand days - after all the multiples of 360 have been subtracted, as explained³See Chapter 11, Halachah 10. - is 265 degrees, 38 minutes and 50 seconds, in symbols 265° 38' 50". The remainder [of the degrees] traveled [by the sun] over the course of ten thousand days is 136 degrees, 28 minutes and 20 seconds, in symbols 136° 28' 20".

In this manner, one can multiply [the mean distance of a day] and calculate the distance [traveled] by the sun over any number of days. Similarly, if one would like to make pre-calculated figures for the mean distance for two days, for three days, for four days, up to ten days, one may do so. Similarly, if one desires to make pre-calculated figures for the mean distance for twenty days, for thirty days, for forty days, until one hundred days, one may do so. These figures become evident once one knows the mean distance for a single day.

It would be proper for one to know and have prepared the mean distances traveled by the sun in 29 days, and in 354 days, [the latter] being the number of days in a lunar year when the months follow a regular pattern. This is called a regular year.⁴See Chapter 8, Halachah 6, which explains that a year in which all the months follow in order, one full and one lacking, is referred to as a regular year.

When you have these figures prepared, it will be easy to calculate the visibility of the moon. For there are 29 full days from the night when the moon was sighted in one month to the night that it may be sighted in the following month. Similarly, each and every month, there will be a difference of 29 days [between the nights on which the moon may be sighted], no more and no less.⁵A lunar month is slightly longer than 29 days. Therefore, potential witnesses endeavor to sight the moon in the heavens on the night between the twenty-ninth and thirtieth days. [This is what concerns us,] for our sole desire in these calculations is to know [when the moon] will be sighted.⁶Indeed, many of the subsequent calculations mentioned by the Rambam may be accurate only on the first night of the month and may not be accurate on the subsequent nights.

Similarly, [the difference in the sun's position] between the night when the moon will be sighted in a particular month one year and the night when it will be sighted [in that month] the following year will be that of a regular year, or that of a regular year plus one day.⁷The one day is added when both the months of Marcheshvan and Kislev are full. The commentaries raise the question why the Rambam does not mention the possibility of the year being lacking a day, as occurs when Marcheshvan and Kislev are both lacking.

The mean distance traveled by the sun in one month is 28 degrees, 35 minutes and one second, in symbols 28° 35' 1". The distance it travels over the course of a regular [lunar] year is 348 degrees, 55 minutes and 15 seconds, in symbols 348° 55' 15".

2. There is one point in the orbit of the sun around the Earth - and similarly, in the orbits of the remainder of the seven stars [around the Earth] - when [the sun or] that star will be furthest removed from the Earth.⁸As stated in Chapter 11, Halachah 13, the Earth is not in the exact center of the orbits of the sun, the moon, or the other five planets. Therefore, there is one point in their orbits where they are furthest removed from the Earth. The knowledge of the location of this point is significant in calculating the true position of the sun, as will be explained in the following chapter. With the exception of the moon, that point in the orbit of the sun and, similarly, in the orbit of the other planets rotates in a uniform pattern, traveling about one degree in seventy years.⁹As the Rambam mentions in Hilchot Yesodei HaTorah 3:3, not only do the sun and the stars move in their orbits, the orbits themselves move in the heavens. This movement can be seen most clearly by charting the movement of the apogee, the point in the orbit furthest from the Earth. The movement of the sun's orbit and similarly, that of the other stars, is relatively slow. The moon's orbit, by contrast, is moving at a much faster pace, as mentioned in the notes on Chapter 14, Halachah 1. This point is referred to as the apogee.

Accordingly, in ten days, the apogee of the sun travels one and a half seconds - i.e., [a second and] thirty thirds. Thus, in one hundred days, [the apogee] travels fifteen seconds. In one thousand days, it travels two minutes and thirty seconds, and in ten thousand days, 25 minutes. In twenty-nine days, it travels four seconds and a fraction. In a regular year, it travels 53 seconds.

As mentioned, the starting point for all our calculations is the eve of Thursday, the third of Nisan, 4938 years after creation. The position of the sun in terms of its mean distance on this date was 7 degrees, 3 minutes and 32 seconds in the constellation of Aries, in symbols 7° 3' 32". The apogee of the sun at this starting point was 26 degrees, 45 minutes and 8 seconds in the constellation of Gemini, in symbols 26° 45' 8".¹⁰Since more than 800 years have passed since the composition of the Mishneh Torah, the apogee of the sun has moved approximately twelve degrees and is presently located in the constellation of Cancer.

Accordingly, if you desire to know the position of the sun according to its mean distance at any given time, you should calculate the number of days from the starting point mentioned until the particular day you desire, and determine the mean distance it traveled during these days according to the figures given previously, add the entire sum together, accumulating each unit of measure separately. The result is the mean position of the sun on that particular day. For example, if we desired to determine the mean position of the sun at the beginning of the eve of the Sabbath on the fourteenth of the month of Tammuz of the present year, the starting point [for these calculations, we should do the following]: Calculate the number of days from the starting point until the date on which you desire to know the position of the sun. [In this instance,] it is one hundred days. The mean distance the sun travels in one hundred days is 98° 33' 53". We then add that to the starting point, which is 7° 3' 32", and arrive at a total of 105 degrees, 37 minutes and 25 seconds, in symbols 105° 37' 25". Thus, the sun's mean position at the beginning of this night will be 15 degrees and 37 minutes of the sixteenth degree in the constellation of Cancer.

At times, the sun will be located in the mean [position] that can be determined using the above methods of calculation at the beginning of the night, and at times an hour before the setting of the sun, or an hour afterwards.¹¹Since, as explained in the previous chapter, the mean distance does not represent the place where the sun can actually be seen in the sky, there will be a slight discrepancy. The mean position represents the sun's position at 6 PM. During the summer months, the sun will reach that position before sunset, and during the winter months, it will reach that position after sunset. This [lack of definition concerning] the sun's [position] will not be of consequence with regard to calculating the visibility [of the moon], for we will compensate for this approximation when calculating the mean position of the moon.¹²See the conclusion of Chapter 14.

One should follow the same procedure at all times - for any date one desires, even if it is one thousand years in the future. When [the mean distance traveled by the sun] is calculated and the remainder [after all the multiples of 360 have been subtracted] is added to [the figures of] the starting point, you will arrive at the mean position.

The same principles apply regarding the mean position of the moon, or the mean position of any other planet. Once you know the distance it travels in a single day, and you know the starting point from which to begin [calculations], total up the distance it travels throughout as many years or days as you desire, add that to the starting point, and you will arrive at its position according to its mean distance.

The same concepts apply regarding the apogee of the sun. Add to the starting point the distance it travels over the course of days or years, and you will know the position of the apogee of the sun for the day you desire.

Similarly, if you desire to establish another date as the starting point instead of the date which [we have chosen] to begin in this year, [choosing] a year that will be the beginning of a particular nineteen-year cycle, or that will be the beginning of a new century, you may. Similarly, if you would like to use as a starting point a date in the past, before the date given above, or a date many years in the future, the path [to arrive at such a starting point] is well known.

How is this figure to be calculated? We have already established the mean distance traveled by the sun in a regular year, in twenty- nine days, and in a single day. It is known that a year whose months¹³I.e., both Marcheshvan and Kislev. See Chapter 8, Halachot 6-10, for the ground rules regarding the determination of when a year is regular, when its months are full, and when they are lacking. are full is one day longer than a regular year. Similarly, a year whose months are lacking is one day shorter than a regular year. With regard to a leap year,¹⁴See Chapter 6, Halachah 11, which relates that seven of the years in a nineteen-year cycle are leap years, and states which of these years will be leap years. if its months are regular, it will be thirty days longer than a regular year. If its months are full, it will be thirty-one days longer than a regular year. If its months are lacking, it will be twenty-nine days longer than a regular year.

Since these principles are already established, it is possible to calculate the mean distance traveled by the sun for as many years or as many days as you desire, and add it to [the mean position of the sun on the date established previously as] the starting point, and you will be able to determine the mean [position of the sun] for any future date. Afterwards, you can use that date as a starting point.

[Conversely,] you may subtract the mean [distance traveled by the sun over the course of a particular period] from [the mean position of the sun on the date established previously as] the starting point, and you will be able to determine the mean [position of the sun] for any past date. Afterwards, you can use that date as a starting point.

The same principles also apply with regard to the mean position of the moon or any of the other planets, if [their mean positions on any particular date] are known to you. It also should be apparent that just as it is possible to determine the mean position of the sun for any future date, so too, it is possible to determine its mean position for any previous date.