After you have determined this arc [of sighting], consider it. Know [these rules]: If the arc of sighting is nine degrees or less, it is impossible for the moon to be sighted anywhere in Eretz Yisrael. If the arc of sighting is more than fourteen degrees, it is impossible for it not to be seen and openly revealed throughout Eretz Yisrael.

If the arc of sighting is between the beginning of the tenth degree and the end of the fourteenth degree, [the following procedure should be followed:] One should consider the arc of sighting in relation to the first longitude and determine whether or not the moon will be seen from the limits prevalent [at that time]. They are referred to as the sighting limits.

The following are the sighting limits: If the arc of sighting is between nine and ten degrees, or more than ten [degrees], and the first longitude is thirteen degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between ten and eleven degrees, or more than eleven [degrees], and the first longitude is twelve degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between eleven and twelve degrees, or more than twelve [degrees], and the first longitude is eleven degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between twelve and thirteen degrees, or more than thirteen [degrees], and the first longitude is ten degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between thirteen and fourteen degrees, or more than fourteen [degrees], and the first longitude is nine degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted. This concludes the sighting limits.

What is implied? If we were to consider the arc of sighting for Friday night, the second of Iyar of this year, we would calculate the arc of sighting to be eleven degrees and eleven minutes, as you have already determined. Since the arc of sighting is between ten and fourteen degrees, it is necessary to consider it in relation to the first longitude. It has already been established that on this night, the [first] longitude is ten degrees and twenty seven minutes. Since the first longitude is greater than eleven [degrees], we can be assured that the moon will be sighted on that night according to the limits established. Similar procedures should be followed [on every occasion when it is necessary to consider] the arc of sighting in relation to its first longitude.

It is well-known and obvious that although the calculations indicate that the moon should be sighted on [a particular] night, its sighting is [only] probable. It is, however, also possible that it will not be sighted, because it is covered by clouds, because the place [from where it could be sighted] is in a valley, or because there is a tall mountain in the west, blocking [view of] the people in the place [from where it could be sighted], and it will be as if they are in a valley. For the moon will not be able to be sighted by a person in a low place, even when [its crescent] is large. Conversely, it will be possible for a person on a high and lofty mountain to sight [the moon], even though [its crescent] is very small. Similarly, sighting it will be possible for a person dwelling on the seashore, or a person travelling on a ship in the Mediterranean Sea, even though [its crescent] is very small.

Similarly, in the rainy season, on a clear day, the visibility of the moon will be greater than it is in the summer. For on a clear day in the rainy season, the air is clear and the heavens appear more brilliant, because there is no dust clouding the air. In the summer, by contrast, the air is like smoke, because of the dust. Therefore, the moon will appear small.

When the two limits [mentioned] with regard to the arc of sighting and the first longitude are short, [the crescent of] the moon will be small, and sighting the moon will be possible only from a very high altitude. If the arc of sighting and the first longitude are long, and [several] degrees were added to their minimal limits, the moon will appear large. Its size and the extent of its visibility will increase according to the length of the arc [of sighting] and the first longitude.

Therefore, the court should always have its attention focused on the following two matters: a) the season when [the moon] was sighted, and b) the place [where the witnesses were located]. [The judges] should ask the witnesses, "Where were you when you saw the moon?" For if the arc of sighting was short, and according to the calculations there is only a limited opportunity to see the moon, [more care is taken regarding their testimony]. For example, the arc of sighting was nine degrees and five minutes, and the first longitude was exactly thirteen degrees, and witnesses came, [claiming] that they saw [the moon]: If this occurs in the summer, or if the [witnesses] were located in a low place, we suspect [the veracity of] their [testimony] and subject them to much cross-examination. In the rainy season, or in a very high place, [under such circumstances, the moon] would surely be sighted unless clouds obscured it.

[Let us examine a hypothetical situation:] Witnesses saw the new moon at the appropriate time. They came [to the court], and gave testimony. The court accepted their [testimony] and sanctified this first month. Afterwards, they counted twenty-nine days from the day that was sanctified. On the night of the thirtieth, the moon was not sighted, either because sighting it was impossible, or because it was covered by clouds. The court waited the entire thirtieth day, as we have explained, [but] witnesses did not arrive. [Therefore,] they added a day to the month, and thus, Rosh Chodesh of the second month was on the thirty-first day, as explained.

They began to count twenty-nine days from the day of Rosh Chodesh in the second [month], but the moon was not sighted. If you would say that [again] a day should be added to the month - i.e., the following day would be the thirtieth of the month, and the thirty-first day should be Rosh Chodesh of the third month [an undesirable situation would arise]. For it is possible that the moon would not be sighted on the night of the thirtieth of this month as well. Thus, [hypothetically, we could see an] ongoing [pattern, in which] a day is added and the new month begins after thirty days throughout the year. Thus, in the last month, it would be possible for the moon to be seen on the night of the twenty-fifth or the night of the twenty-sixth. There could be no more ludicrous and demeaning situation than this.

Nor can one say that the [hypothetical situation described] - that the moon is never sighted throughout the year - is an infrequent occurrence [that need not be considered]. The contrary [is true]; it is very likely [to take place]. This often occurs in countries that have long rainy seasons and cloudy [skies]. For the intent is not that the moon will never be seen throughout the year, but that it will not be sighted at the beginning of the month, and will be sighted only afterwards. [In certain months] it will not be sighted, because sighting it is impossible, and in the months when sighting it is possible, it possibly will not be sighted because of the clouds, or because its [crescent] was very small, and no one focused [his attention] on sighting it.

[These difficulties were avoided due to the following] tradition which existed among our Sages, [having been] transmitted from one to another in a chain extending back to Moses our teacher: When the moon was not sighted at the beginning of the months, month after month, the court establishes [the following sequence], one full month with thirty days, one lacking month with twenty-nine days. In this manner, they would calculate and establish one full month and one lacking month - establishing these months, but not sanctifying them. For the sanctification of the months is dependent solely on the sighting [of the moon]. At times they would have a full month follow another full month, or a lacking month follow another lacking month, depending on the results of their calculations.

The intent of their calculations is always that it is possible to sight the moon in the following month at its proper time, or on the night following the added day, but not to sight it beforehand - e.g., on the night of the twenty-eighth. Through the sighting calculations mentioned above, it is possible for you to determine when it will be possible [for the moon to be] sighted, and when it is possible that it will not be sighted. [The court] relies on [similar calculations] and [accordingly,] establishes two full months in succession, or two lacking months in succession. Never should there be fewer than four full months in a year, nor should there ever be more than eight full months. When a full month is established according to calculations, a celebratory feast is also made in honor of the full month, as mentioned previously in Chapter 3.

Whenever you find statements in the Talmud according to which it appears that the court relies on computations [rather than the testimony of witnesses], or that there is [a chain of tradition extending] from Moses at Sinai that this matter has been entrusted to them, and it is their decision whether to make the month lacking or full, and similarly, the fact that, during one year, Rabbi [Yehudah HaNasi] declared nine months as lacking - these and all similar matters are dependent on this principle [and apply at] a time when the moon is not sighted at the appropriate time.

Similarly, the statements of our Sages that a day is added to the month when necessary, applies in an instance when a full month [is declared] according to the calculations, [so that] one month is made lacking, and another month full. They have the authority to declare full months in succession one after the other, and lacking months in succession. [Only] when the moon is not seen at the appropriate time does [the court] declare full months when necessary. When, however, the moon is seen at the appropriate time - its first shining after its conjunction with the sun - it is always sanctified.

All the above concepts apply when there is a court that relies on the testimony of witnesses. In the present era, by contrast, we rely solely on the calculations based on the mean [motion of the sun and the moon] that are [simple and] widespread throughout Israel, as explained [previously] in these laws.

It is explained in the texts of astronomical and geometrical calculations that if the moon is sighted in Eretz Yisrael, it will be sighted in all the lands in the world that are located to the west of Eretz Yisrael and are at the same latitude. If the calculations indicate that the moon will not be sighted in Eretz Yisrael, sighting it is, nevertheless, possible in those lands that are west of Eretz Yisrael and at the same latitude. Therefore, the fact that the moon is sighted in a country that lies west of Eretz Yisrael is not an indication of whether [or not] the moon will be sighted in Eretz Yisrael.

If, however, the moon is not sighted on the mountains of the countries located west of Eretz Yisrael that are at the same latitude, the moon surely was not sighted in Eretz Yisrael.

Similarly, if the moon was not sighted in Eretz Yisrael, it certainly was not sighted in those countries that are east of Eretz Yisrael, and at the same latitude. If [the moon] was sighted in Eretz Yisrael, however, [there is no certainty whether it will be sighted] in these easterly lands; it may be sighted, and it may not be sighted. Accordingly, if the moon was sighted in a country located to the east of Eretz Yisrael at the same latitude, the moon was surely sighted in Eretz Yisrael. If, however, the moon was not sighted in a country that lies east [of Eretz Yisrael], that is not an indication that the moon will not be sighted in Eretz Yisrael. On the contrary, it is possible that it will be sighted in Eretz Yisrael.

All the above statements apply when the countries to the west and to the east [of Eretz Yisrael] are at the same latitude as Eretz Yisrael - i.e., they are 30 to 35 degrees north [of the equator]. If they are located in a more northerly position, or in a less northerly position, different principles apply, for they are not parallel to Eretz Yisrael. In the cities that are located to the east and west [of Eretz Yisrael], the statements we made [concerning the sighting of the moon are of abstract interest only], to clarify all the laws regarding the sighting [of the moon] to make the Torah great and glorious. [The intent is not that] the people living in the east or the west should depend on [their] sighting of the moon, or that [their sighting] should be of any consequence whatsoever. Instead, we rely only on the sanctification of the moon [performed] by the court in Eretz Yisrael, as we have explained several times previously.

Since our Sages said that among the questions posed to the witnesses [in order to verify their testimony] was "In which direction was [the crescent of] the moon inclined," I feel that it is appropriate to explain how this factor can be calculated. My statements will not be exact, because [this knowledge] is of no consequence regarding the actual sighting of the moon. The starting point of these calculations is to know the degree of inclination of the constellations' [position].

The orbit of the sun that passes through the center [of the sphere] of the constellations does not pass directly through the center of the earth from east to west. Instead, it is inclined above the equator which passes through the center of the earth, [slightly] to the north and to the south. Half of it is northerly in inclination and half is southerly in inclination.

There are two points at which the orbit of the sun intersects the equator, which passes through the center of the earth: The first point is the beginning of the constellation of Aries, and the second point, opposite to [the first], is the beginning of the constellation of Libra. Thus, there are six constellations [whose positions] are inclined to the north, those between the beginning of the constellation of Aries and the end of the constellation of Virgo. And there are six [whose positions] are inclined to the south, those between the beginning of the constellation of Libra and the end of the constellation of Pisces.

From the beginning of the constellation of Aries, the constellations' [positions] begin to be inclined slightly and diverge from the equator in a northerly direction until the beginning [of the constellation] of Cancer. The beginning [of the constellation] of Cancer is approximately twenty-three and one half degrees north of the equator. [Afterwards,] the constellations begin to proceed toward the equator slowly until the beginning [of the constellation] of Libra, which is positioned on the equator. From the beginning of the constellation of Libra, the constellations' [positions] begin to be inclined slightly and diverge from the equator in a southerly direction until the beginning [of the constellation] of Sagittarius. The beginning [of the constellation] of Sagittarius is approximately twenty-three and one half degrees south of the equator. [Afterwards,] the constellations begin to proceed toward the equator slowly until the beginning [of the constellation] of Aries.

Thus, [in their daily orbits,] the beginning [of the constellation] of Aries and the beginning [of the constellation] of Libra revolve on the equator. Therefore, when the sun is positioned in the beginning [of these constellations], it will not be inclined either to the north or to the south. It will rise due east and set due west, and the daytime and the nighttime [hours] will be equal throughout the world.

Thus, it should be clear to you that each of the degrees [in the sphere] of the constellations is inclined to the north or to the south, and that there is a specific measure to their inclination. The greatest inclination will be no more than approximately twenty-three and half degrees.

The following represent the extent of inclination from the equator vis-a-vis the number of degrees [in the sphere of the constellations]. Beginning with the constellation of Aries: [A point located at] 10 degrees [in the celestial sphere] will be inclined 4 degrees [from the equator]. [A point located at] 20 degrees will be inclined 8 degrees. [A point located at] 30 degrees will be inclined 11 1/2 degrees. [A point located at] 40 degrees will be inclined 15 degrees. [A point located at] 50 degrees will be inclined 18 degrees. [A point located at] 60 degrees will be inclined 20 degrees. [A point located at] 70 degrees will be inclined 22 degrees. [A point located at] 80 degrees will be inclined 23 degrees. [A point located at] 90 degrees will be inclined 23 1/2 degrees.

If the number [of degrees] also includes units, you should calculate [their degree of inclination] by taking an average between the two figures, as was explained with regard to [the determination of the position of] the sun and the moon. What is implied? Five degrees will be inclined 2 degrees. If the number of degrees [in the celestial sphere] is twenty-three, the inclination [from the equator] will be nine degrees. This same pattern should be followed whenever calculating a number that has both units and tens.

Since you know the extent of the inclination of all the degrees from one until ninety, you will be able to calculate the degree of inclination [of the entire celestial sphere] according to the method of calculation explained with regard to the moon's latitude. For if the number is between 90 and 180, the number should be subtracted from 180. If the number is between 180 and 270, 180 should be subtracted from the number. If the number is between 270 and 360, the number should be subtracted from 360. [After these subtractions have been made], you will know the degree of inclination of the remainder, for it is the degree of inclination of the numbers mentioned previously without any addition or subtraction.

If you desire to know the number of degrees by which the moon's position deviates from the equator to the north or to the south, [you should follow this procedure]: Calculate the inclination of the degree [in the celestial sphere] that will be the true position, and see if its inclination is northerly or southerly. Afterwards, calculate the moon's first latitude and see whether it is northerly or southerly. If the moon's latitude and the inclination of the degree [in the celestial sphere where it is located] have the same direction, then they should be added together. If they are in different directions - i.e., one is southerly and one is northerly - the smaller figure should be subtracted from the larger one. The remainder is the [angular] distance of the moon from the equator to the direction in which the larger figure was inclined.

What is implied? Let us say that we are trying to calculate the degree of inclination of the moon from the equator on the night of sighting, the second day of the month of Iyar of this year. You have already established that the angle [in the celestial sphere] at which the moon was located is the nineteenth degree of the constellation of Taurus. Thus, its inclination to the north will be approximately 18 degrees. The latitude of the moon is approximately four degrees to the south. After you subtract the lesser figure from the larger one, the position of the moon will be fourteen degrees to the north of the equator, for the largest figure is the eighteen degrees north [at which the moon's true position was located]. All [of the phases] of this calculation are approximations and are not exact, because they are of no consequence regarding the sighting.

If you desire to know the direction to which the moon will appear to be inclined [when sighted, it is governed by the following principles]: Calculate [the moon's angular] distance from the equator. If it is located on the equator or within two or three degrees to the north or to the south, it will appear due west and its crescent will appear to be pointed due east.

If [the moon] is inclined to the north of the equator, it will appear in the northwest, and its crescent will appear to be pointed southeast.

If [the moon] is inclined to the south of the equator, it will appear in the southwest, and its crescent will appear to be pointed northeast. To the extent of [the moon's] distance [from the equator], its inclination will increase.

Among the other dimensions of the examination of the witnesses is the question: "How high [in the sky did the moon] appear?" This factor can be determined by the arc of sighting. If the arc is short, the moon will appear close to the earth, and when it is long, it will appear high above the earth. The length of the arc of sighting [will determine] the height at which [the moon] will appear above the earth to the witnesses.

Thus, we have explained all the calculations necessary for the sighting [of the moon] and the examination of the witnesses, so that everything will be comprehensible to men of understanding, and they will not lack awareness of any of the Torah's paths. [Therefore,] they will not venture forth in search of it in other texts. "Seek out of the book of God, read it. None of these will be lacking" [Isaiah 34:16].