Sanctification of the New Month

The months of the year are lunar months, as [implied by Numbers 28:14]: "... the burnt offering of the month when it is renewed," and [Exodus 12:2]: "This month shall be for you the first of months." [Concerning this verse,] our Sages commented: The Holy One, blessed be He, showed Moses in the vision of prophecy an image of the moon and told him, "When you see the moon like this, sanctify it." The years we follow are solar years, as [implied by Deuteronomy 16:1]: "Keep the month of spring."

How much longer is a solar year than a lunar year? Approximately eleven days. Therefore, [to correct the discrepancy between the lunar and the solar calendars,] when these additional days reach a sum of 30 - or slightly more, or slightly less - an additional month is added, causing the year to include thirteen months. This is called a full year. [This is necessary,] because it is impossible to have a year with twelve months and an odd number of days, as [implied by Numbers 28:14]: "... of the months of the year." [On this verse, our Sages commented:] "You count the months of a year, but not the days of a year."

The moon becomes hidden and cannot be seen for approximately two days - or [slightly] less or slightly more - every month: approximately one day before its conjunction with the sun at the end of the month, and one day after its conjunction with the sun, [before] it is sighted in the west in the evening. The first night when the moon is sighted in the west after being hidden is the beginning of the month. Afterwards, 29 days are counted from that day. If the moon is sighted on the night of the thirtieth [day], the thirtieth day will be Rosh Chodesh [of the following month]. If it is not sighted, Rosh Chodesh will be on the thirty-first day, and the thirtieth day will be included in the previous month. There is no need [to sight] the moon on the thirty-first night; whether or not [the moon] is sighted [the new month begins that night]. For there are no lunar months longer than thirty days.

When there are twenty-nine days in a month, [because] the moon was sighted on the thirtieth night, the month is called chaseir, ["lacking"]. If the moon is not sighted and the previous month has thirty days, the month is called me'ubar ["pregnant"] or malei ["full"]. When the moon is sighted on the thirtieth night, the moon is said to have been sighted at the appropriate time. When the moon is sighted on the thirty-first night, but not on the thirtieth night, the moon is described as having been seen on the night of its fullness.

The [establishment of Rosh Chodesh] based on the sighting of the moon is not the province of every individual, as is the Sabbath [of the weekly cycle]. [In the latter instance,] everyone counts six days and rests on the seventh day. [The sanctification of the new month,] by contrast, has been entrusted to the court. [The new month does not begin] until it has been sanctified by the court, and it is the day that they establish as Rosh Chodesh that is Rosh Chodesh. [This is implied by the verse,] "This month will be for you...," - i.e., the testimony [concerning the new month] will be entrusted to you.

The [High] Court would make calculations in a manner resembling the calculations of the astronomers, who know the location of the stars and their paths [in their orbits]. They would perform careful research to determine whether or not they would be able to sight the moon at the appropriate time - i.e., the thirtieth night. If [the judges] determined that it was possible to sight [the moon], they would sit waiting for witnesses [to come and testify] throughout the entire thirtieth day. If witnesses came, and [the court] examined their testimony according to law, and verified the truth [of their statements], the court would sanctify [the new month]. If [the moon] was not sighted, and witnesses did not come, they would complete the thirtieth day, thus making the month full. If, according to their calculations, [the judges] knew that it was impossible for the moon to be sighted, they would not sit [in session] on the thirtieth day, nor would they await [the arrival] of witnesses. If witnesses came, they would know that they are false witnesses, or that clouds appeared to them in a form resembling the moon, but it was not the real moon.

It is a positive commandment of the Torah for the court to calculate and determine whether or not the moon will be sighted, to examine witnesses until the moon can be sanctified, and to send forth [messengers] to inform the remainder of the people on which day Rosh Chodesh was observed, so that they will know the day [on which to celebrate] the festivals [as implied by Leviticus 23:2]: "that you will pronounce as days of holy convocation," and [as implied by Exodus 13:10]: "And you shall observe this statute in its appointed season."

The calculations and the establishment of the months and the leap years is carried out only in Eretz Yisrael [as implied by Isaiah 2:3]: "For out of Zion will emerge the law, and the word of G-d [will emerge] from Jerusalem." If a great sage who received semichah in Eretz Yisrael left for the diaspora without leaving a colleague of equal stature in Eretz Yisrael, he may make calculations, establish the monthly calendar, and institute leap years in the diaspora.If, however, it becomes known to him that a sage of his stature has arisen in Eretz Yisrael - and surely, if a sage of greater stature has arisen in Eretz Yisrael - it is forbidden for him to establish [the monthly calendar] and [institute] leap years in the diaspora. If he transgresses, and [attempts to structure the calendar in this manner], his actions are of no consequence.

Only two kosher men who are suitable to testify about any single thing are valid for testimony of the month. But women and slaves are like all other invalid witnesses and can't testify. A father and son that saw the moon go to court to testify. Not because testimony of the month is valid with with relatives but rather so that if one of them is found to be invalid for being a thief or such like from the list of other invalid witnesses, the second can join with another and testify. And anyone who is invalid to testify by the words of the scribes, even though he is valid according to the Torah, is invalid for testimony of the month.

It is a law of the Torah that we are not meticulous with testimony of the month. For even if they sanctified the month via witnesses and they were found to be zomemin (witnesses who are known to have lied based on the testimony of other witnesses that proves that they couldn't have been able to witness what they claim to have seen) for this testimony, it is sanctified. As such, originally, they would accept testimony from every Israelite man, for every Israelite has a presumption that he is kosher until you know that he is invalid. From when the heretics corrupted and they'd hire men to testify that they'd seen and they hadn't seen, they amended that the court would only accept testimony of the month from witnesses that the court recognised were kosher and that they'd interrogate and investigate the testimony.

As such, if the court do not know the witnesses that saw the moon, the men of the city that it was seen in send with the witnesses that saw the moon other witnesses that justify them to the court and inform them that these are kosher and afterwards they are accepted.

The court calculates with the methods that the astronomers calculate with and know when the moon is seen in this month whether it will be north of the sun or southwards and whether it will be wide or short and in which direction the tips of its horns will point. And when the witnesses come to testify, we check them; "How did you see it? In the north or in the south? In which direction were its horns pointing? How high was it with the vision of your eyes and how wide was it?" If their words are found consistent with that which is known by calculation, we accept them, and if their words are not found consistent, we don't accept them.

If the witnesses said "we saw it in water" or "in clouds" or "in frost", or they saw part of it in the sky and part of it in clouds or in water or frost, this isn't a sighting and we don't sanctify based on this sighting. One says "I saw it with my eyes at a height of about two floors" and the second says "at about a height of three floors", they join up [to form valid testimony]. One says "about three floors" and the second says "about five floors", they don't join up. And we do join up one of them with a second that testifies like him or with a single floor between them.

If they said "we saw it without intention and once we turned our attention to it and intended to see it to testify we didn't see it again", this isn't testimony and we don't sanctify based on it. Perhaps clouds gathered and they appeared like the moon and they vanished and went from them. If the witnesses said "we saw it on the 29th day of the month in the morning in the east before the sun would rise and we saw it in the evening in the west on the night of the 30th", they are believed and we sanctify based on this sighting for it was seen at its time. But the sighting that they said they saw in the morning we don't relate to for we are not responsible the morning sighting and it is known that the clouds gathered and appeared to them like the moon. And similarly, if they saw it at its time and it was not seen on the night of its pregnancy the 31st night, they are believed for we are only responsible for the 30th night sighting alone.

How do we accept testimony of the month? Everyone who is fitting to testify that he saw the moon comes to court. And the court brings them all into one place and makes or them a large feast in order that the people will be accustomed to coming. And the pair that came first we check first with the checks we have said. We bring in the greater and ask him. If his words are consistent with the calculation, we bring his fellow. If we find their words consistent, their testimony is established. And we ask all the other pairs preliminary matters, not because we need them but rather so that they shouldn't leave with upset spirit so they will be accustomed to come.

And afterwards, after the testimony has been established, the head of the court says "it is sanctified" and all the people respond after him "it is sanctified, it is sanctified". And we don't sanctify the month except with three judges. And we don't perform calculations except with thee judges. And we only sanctify a month that is seen at its time. And we only sanctify in the day and if they sanctified it at night, it is not sanctified. Even if the court and all of Israel saw it and the court didn't say "it is sanctified" until it became dark on the night of the 31st, or they investigated the witnesses and the court didn't manage to say "it is sanctified" until it became dark on the night of the 31st, they don't sanctify it and it will be a pregnant month one with 30 days and the first of the month will only be on the 31st even though it was visible on the 30th night. For it is not the sighting that fixes, rather it is the court saying "it is sanctified" that fixes.

If the court itself saw it at the end of the 29th day, if there still hasn't emerged a star of the night of the 30th, the court says "it is sanctified, it is sanctified", for it is still day. And if they saw it on the night of the 30th after two stars emerged, on the next day, we sit two judges with one of them that saw it at night and the two others testify before the three judges and the three sanctify it.

A court which sanctifies the month, whether by accident, whether they were mistaken, whether they were forced, it is sanctified and everyone is obligated to fix the festivals on the day they sanctified. Even though one knows that they made a mistake, he is obligated to rely on them for the matter is only given over to them and the one who commanded to keep the festivals is the one who commanded to rely on them as it is said (Leviticus 23:2) "that you call them…".

Witnesses that see the month, if there is between them and the location of the court a distance of a night and day's journey or less, they go and testify. And if there is between them more than this, they don't go for their testimony after the 30th day doesn't help since the month has already been made pregnant (into a 30 day month).

Witnesses that saw the month go to the court to testify, even on Shabbat, as it is said (Leviticus 23:4) "that you call them at their appointed time" and anywhere it is said "appointed time" the Shabbat is pushed off. Thus, we only violate the Shabbat for the first of the month of Nissan which contains the festival of Passover and for the first of the month of Tishrei which contains the festivals of the Day of Atonement and Tabernacles in order to fix the festivals which are described with the term "appointed time". And in a time when the Holy Temple exists, we violate the Shabbat for all the months because of the additional sacrifice of every first of the month which pushes off the Shabbat.

Just as the witnesses that saw the month desecrate the Shabbat, so too the witnesses who justify them to the court desecrate with them if the court doesn't recognise the ones who saw. And even if the one who informs the court of them is a single witness, he goes with them and desecrates in case he will find another and join up with him.

[If] the witness that saw the month on the night of Shabbat was ill, we ride him on a donkey and even on a bed. And if there is an ambush on the way, the witnesses take weapons in their hands. And if it was a far way, they they take food in their hands. And even if they saw it large and visible to all, they shouldn't say "just as we saw it, others saw it and we don't need to desecrate the Shabbat", rather anyone who sees the month and will be fitting to testify and there will be between him and the location where the court is fixed a night and a day's [journey] or less has a commandment upon him to desecrate the Shabbat and to go and testify.

At first, they'd accept testimony of the month the entire 30th day. One time, the witnesses delayed in coming until the late afternoon and they ruined [the service] in the Temple and didn't know what to do; whether they should do the afternoon elevation offering, perhaps the witnesses would come and it is impossible to bring up the additional [offering] of the day [of the first of the month] after the afternoon continual offering. The court arose and amended that they would only accept witnesses of the month until [the time of] minchah in order that there would be sufficient time in the day to bring up the additional [offering] and afternoon continual [offering] and their libations.

And if minchah arrived and witnesses did not come, they'd do the afternoon continual [offering]. And if witnesses came from minchah an onwards, they treat that day as holy and the morrow as holy and they bring the additional [offering] on the morrow, since they didn't sanctify it after minchah. From when the Temple was destroyed, Rabban Yochanan ben Zakkai and his court amended that they'd accept testimony of the month the entire day and even if witnesses came on the 30th day at the end of the day close to sunset, they accept their testimony and sanctify only the 30th day.

When the court makes the month pregnant (30 days) because witnesses didn't come the entire 30th day, they'd go up to a prepared place and they'd make a feast on the 31st day which is the first of the month. And they don't go up to there at night, rather at first light, before sunrise, and they don't go up to this feast with less than 10 [people]. And they only go up with grain bread and legumes and they eat these at the time of the feast. And this is the meal of the commandment of impregnating the month that is said in every place.

At first, when the court would sanctify the month, they'd burn bonfires on the tops of the mountains in order to inform those who were far away. Once the Cutites ruined it, for they'd light bonfires in order to cause the people to err, they amended that messengers would go out and they'd inform the public. And these messengers do not desecrate the holidays and there is no need to say Shabbat, for we don't desecrate Shabbat to establish it [the month], rather only to sanctify it.

For 6 months the messengers would go out. For Nissan because of Passover. And for Av because of the fast (the 9th of Av). And for Elul because of Rosh Hashanah, in order that they'd sit in anticipation on the 30th day of Elul, if they'd be informed that the court had sanctified the 30th day, they'd treat that day alone as holy, and if they weren't informed, they'd keep the 30th day holy and the 31st day holy until the messengers of Tishri would come to them. And for Tishri because of fixing the festivals. And for Kislev because of Chanukah. And for Adar because of the Purim. And in the time when the Holy Temple exists, they even go out for Iyar because of Passover Minor.

The messengers of Nissan and the messengers of Tishri only go out on the day of the first of the month after the sun will rise, [waiting] until after they have heard from the mouth of court "it is sanctified". And if the court sanctifies at the end of the 29th day as we have said and they heard from the court "it is sanctified", they go out in the evening. And the messengers of the other 6 months can go out in the evening after the moon has been seen, even though the court is yet to sanctify the month. Since the month has been seen, they go out, for tomorrow the court will certainly sanctify it.

Every place that the messengers reach, they'd make the festivals a one day holiday as is written in the Torah. And the distant places that the messengers wouldn't reach would make two days because of the doubt, for they wouldn't know which day was the day that the court fixed the first of the month.

There are places that the messengers of Nissan would reach them and the messengers of Tishri would not reach them. And according to the law, they ought to make Passover one day of holiday, for the messengers reached them and they knew on which day the first of the month was fixed. And they ought to make the holiday of the festival of Tabernacles two days, for the messengers didn't reach them. And in order not to distinguish between the festivals, the sages amended that any place that the messengers of Tishri didn't reach would make two days [of holidays], even the holiday of Ceasing (Shavuot).

And how much is between the messengers of Tishri and the messengers of Nissan? Two days. For the messengers of Tishri didn't travel on the 1st of Tishri because it is a holiday and not on the 10th of it because it is the Day of Atonement.

The messengers do not need to be two, rather even one is believed. And not only a messenger alone but even a merchant from the rest of the populace that comes on his way and says "I heard from the mouth of the court that they sanctified the month on a particular day" is believed and they fix the festivals by his mouth. For this matter is something which is prone to be revealed and a single kosher witness is believed for it.

[If] the court sat all of the 30th day and witnesses didn't come, and they awoke at first light and impregnated the month as we have explained in this chapter. And after 4 or 5 days, distant witnesses come and they testify that they saw the month at its time which is the night of the 30th. And even if they come at the end if the month. They intimidate them very threateningly and they hassle them with questions and they bother them with checks and they are meticulous with their testimony and the court attempts to not sanctify this month since its name of pregnant has gone out.

And if the witnesses stand with their testimony and it is found consistent and the witnesses are known and intelligent men and the testimony has been appropriately investigated. They sanctify it and retract and count for that month from the 30th day since the moon was seen on its night.

And if the court need to leave this month pregnant as it was before these witnesses came, they leave it. And this is what they said "we impregnate the month for necessity". And there are amongst the great sages those who argue on this matter and say we never impregnate the month for necessity; since witnesses came, we sanctify and don't intimidate them.

It appears to me that there is only an argument amongst the sages for the months other than Nissan and Tishri. Or with witnesses of Nissan and Tishri that came after the festivals had passed. For what has been done has been done and the time of the offerings and the time of the festivals has passed. But if the witnesses came in Nissan and Tishri before half of the month, we accept their testimony and we don't intimidate them at all. For we don't intimidate witnesses that testify about the month that it was seen at its time to impregnate it.

But we intimidate witnesses who's testimony has become ruined and the matter is tending such that the testimony will not stand and the month will become impregnated. We intimidate them in order that the testimony will stand and the month will be sanctified at its time. And similarly, if witnesses come to disprove the witnesses that saw it at its time before the court have sanctified it, we intimidate the disproving witnesses until their disproof doesn't stand and the month is sanctified at its time.

A leap year is a year in which another month has been added. And we only ever add an Adar and make two Adars, an Adar Rishon (first Adar) and an Adar Sheini (Second Adar). And why do we add this month? Because of the spring time, so that the Passover will fall at that time, as it is said (Deuteronomy 16:1) "Guard the month of the spring", that this month should be in the spring. And if it wasn't for the addition of this month, Passover would sometimes come in the days of the sun (the summer) and sometimes in the days of the rain (the winter).

For three indicators we make the year a leap [year]. For the epoch and for the spring and for the fruit of the trees. How? The court calculates and knows; if the epoch of Nissan (when the sun enters Aries) will be on the 16th of Nissan or after this time, they make the year into a leap [year]. And they make that Nissan an Adar Sheini in order that the Passover will be in the spring time. And on this indicator, we rely and make leap [years] and we are not concerned about another indicator.

And similarly, if the court saw that the spring has yet to arrive, rather it is still dark. And the fruit of the trees that tend to sprout around Passover have not sprouted. We rely on these two indicators and make the year a leap [year]. So that the spring will be present to offer up from it the waved omer (an offering of grain) on the 16th of Nissan. And in order that the fruit will sprout normally throughout the spring time.

And we rely on three lands with respect to the spring. On the land of Yehuda and on the Jordan Bank and in the Galil. And if the spring arrived in two of these lands and in one it didn't, we don't make it a leap [year]. And if it arrived in one of them and it didn't arrive in two, we make it a leap [year], if the fruit of the trees is yet to sprout. And these are the main things that we make leap [years] for in order that the years will be solar years.

And there exist other things that the court makes leap [years] for due to necessity. And these are they: Because of the roads that are not mended and the people are unable to go up [to Jerusalem], we make the year a leap [year] until the rain stops and the roads are mended. And because of the bridges which were demolished and the rivers are found blocking and preventing the people or they are endangering themselves and dying, we make the year a leap [year] until they mend the bridges. And because of the passover ovens that were destroyed in the rain and they don't have place to roast their passover [offerings], we make the year a leap [year] until they will have built the ovens and they will have dried. And because of the Israelite exiles that have been uprooted from their place and are yet to have arrived in Jerusalem, we make the year a leap [year] in order that they will have enough time to arrive.

But we don't make the year a leap [year], not for the snow and not for the cold and not for the Israelite exiles that are yet to be uprooted from their place. And not for the ritual impurity. For example, were the majority of the congregation or the majority of the priests ritually impure, we don't make the year a leap [year] in order that they will have enough time to purify themselves and do [the offerings] in purity, rather they do in impurity. And if they made the year a leap [year] because of the impurity, it is a leap [year].

There are things for which we don't make a leap [year] at all but we make them a support for a year that needs to become a leap [year] because of the epoch or because of the spring and the fruits of the trees. And these are they: Because of the goats and sheep that are yet to be born or which are few. And because of the chicks that haven't flown. We don't make a leap [year] because of these, in order that the goats and sheep should be accessible for the passover [offerings] and the chicks accessible for the Re'iah [offering] or for those who are obligated in a bird offering. But we make them a support for the year.

How do we make them a support for the year? We say "this year needs making into a leap [year] because of the epoch that has dragged" or "because of the spring and the fruit of the trees that are yet to arrive and further for the goats are small and the chicks are delicate."

We only make the year a leap [year] with those invited to. How? The head of the high court says to so-and-so and so-and-so from the Sanhedrin (the high court) "you are invited to location so-and-so where we will calculate and see and know whether this year needs making into a leap [year] or doesn't need it. And those who were invited alone are the ones that make it into a leap [year]. And with how many do we make it a leap [year]? We start with three judges from the great Sanhedrin, from those that were ordained. Two said "let's not sit and let's not see whether making a leap [year] is necessary or if not" and one says "let's sit and check", the individual is nullified in his minority. Two said "let's sit and check" and one says "let's not sit", we add two from the those invited and debate the matter back and forth.

Two say "it requires making into a leap [year]" and three say "it doesn't require it". The two are nullified in their minority. Three say "it requires making into a leap [year]" and two say "it doesn't require making into a leap [year]", they add two from those invited and they debate back and forth with seven. If they all conclude to make a leap [year] or not to make a leap [year], they do as they do as they concluded. And if they argue, they go after the majority, whether to make a leap [year] or whether not to make a leap [year]. And the head of the high court, who is the head of the seating of the seventy-one, needs to be amongst the seven. And if they concluded with three to make a leap [year], it is a leap [year] so long as the prince is with them or he wants. And with making the year a leap [year], we start from the side. And to sanctify the month, we start with the greatest.

We don't sit for making the year a leap [year] either a king or a kohen gadol (high priest). A king because of his soldiers and his wars, perhaps his opinion will be be biased because of them to make a leap [year] or not a leap [year]. And a kohen gadol because of the cold, perhaps his opinion will be biased to not make a leap [year] in order that Tishrei will not come in the cold days and he immerses on the Day of Atonement five immersions.

[If] the head of the high court, and he is the one called prince, is on a far journey, we don't make it a leap [year] except under the condition of if the prince wants it. [If] he comes and wants, it is a leap [year], [if] he doesn't want, it is not a leap [year]. And we don't make a leap [year] except in the land of Yehudah for the Shechina (divine presence) is there as it said (Deuteronomy 12:5) "for his dwelling you will seek". And if they made it a leap [year] in the Galil, it is a leap [year]. And we only make it a leap [year] in the day. And if they made it a leap [year] at night, it is not a leap [year].

The court is able to calculate and to know which years will be leap [years] at any time they want, even several years. But they don't say "year such-and-such is a leap [year]", rather after Rosh Hashanah they say "this year is a leap [year]". And this matter is [only] because of pressure but at a time when there is no pressure, we don't inform that it is a leap [year after Rosh Hashanah], rather in Adar they say "this year is a leap [year] and the coming month is not Nissan but rather Adar Sheini". [If] they said before Rosh Hashanah "this entering year is a leap [year]", it is not a leap [year] with this statement.

[If] the 30th day of Adar arrived and they were yet to make the year a leap [year], they don't make it a leap [year] at all. For that day is fitting to be the first if the month of Nissan and once Nissan has entered and they haven't made it a leap [year], they are not able to make a leap [year]. And if they made it a leap [year] on the 30th day of Adar, it is a leap [year]. [If] witnesses come after they made it a leap [year] and they testify on the moon, they sanctify the month on the 30th day and it will be the first of the month of Adar Sheini. And if they sanctify it before they make the year a leap [year], they can no longer make a leap [year] for we don't make leap [years] in Nissan.

We don't make the year a leap [year] in a year of famine, for everyone runs to the granaries to eat and to live and it is impossible to add for them time to forbid the new [produce] (which will only become permissible on the 16th of Nissan). And we don't make a leap [year] in the shvi'it (the seventh year of the agricultural cycle), for everyone's hand is controlling the undergrowth, and we won't find [grain] for the omer offering and the two loaves [offering]. And they were used to making a leap [year] on the eve of shvi'it.

It seems to me that this that the sages said "we don't make a leap [year] in years of famine and shvi'it", is that we don't make a leap [year] because of the necessity of the roads and the bridges and such like. But if the year was fitting to be made a leap [year] because of the epoch or because of the spring and the fruit of the trees, we always make a leap [year] at any time.

When the court makes the year a leap [year], they write letters for all the distant places and they inform them that it has been made a leap [year] and for what reason it was made a leap [year]. And with the language of the prince they are written. And it says to them "it is known to you that I and my colleagues have agreed and we have added on this year such and such". [If] they want, 29 days, [if] they want, 30 days. For the leap [year] month, the court has the authority to add a full [month] or a lacking [month] for the people who are far away when they inform them. But they themselves go according to the sighting, if full or if lacking.

Everything we have said about fixing the first of the month by sight and making the year a leap [year] because of the time or for necessity is only done by the Sanhedrin (High Court) that is in the land of Israel or an ordained court in the land of Israel that has been given permission from the Sanhedrin. For such was said to Moses and Aaron, (Exodus 12:2) "this month is for you the first month", and according to tradition they learnt, man from man from Moses our master, that such is the explanation of the matter: this testimony will be given over to you and to all who stand after you in your place. But at a time when the Sanhedrin doesn't exist in the land of Israel, we don't fix months and we don't make leap years except with the calculations that we calculate today.

And this matter is a law of Moses from Sinai. That at a time that there is a Sanhedrin, we fix by the sighting, and when the Sanhedrin doesn't exist, we fix by this calculation that we calculate today and we don't relate to the sighting. Rather, sometimes the day we fix with this calculation will be the day of sighting, or a day before, or a day after. And that it will be after the sighting by a day is a wonder (i.e. very rare), and [this is so] in lands that are to the west of the land of Israel.

And from when did all of Israel start to calculate with this calculation? From the end of the sages of the Talmud, at the time that the land of Israel was made desolate and there didn't remain there a fixed court. But in the days of the sages of the Mishnah and similarly in the days of the sages of the Talmud, until the days of Abaye and Rava, they'd rely on the fixing of the land of Israel.

When the Sanhedrin existed and they fixed based on the sighting, the children of the land of Israel (those who lived in Israel) and all the places that would be reached by the messengers of Tishrei would make the holidays one day only. And the remaining distant places that the messengers of Tishrei wouldn't reach would make two days due to doubt, for they wouldn't know the day that the children of the land of Israel fixed the month on.

In this time that the Sanhedrin doesn't exist and the court of the land of Israel fixes on this calculation, it should have been according to the law that in all places, they would only make one day holiday, even the distant places that are in the outside of the land [of Israel], like the children of the land of Israel. For everyone relies and fixes based on one calculation. But it is an amendment of the sages that they be careful with the custom of their fathers that is in their hands.

Therefore, any place that the messengers of Tishri didn't arrive at when the messengers went out will make two days, even at this time, as they did in the times when the children of the land of Israel would fix based on the sighting. And the children of the land of Israel in this time make one day according to their custom for they never made two days. It is found, the second [day of] holiday that we make in the exile at this time originates from the words of the scribes that amended this matter.

[Regarding] the holiday of Rosh Hashanah at the time when they fix based on the sighting, the majority of the children of the land of Israel would make it two days out of doubt. For they would not know on which day the court fixed the month for the messengers don't go out on the holiday.

And not only such but even in Jerusalem itself, which is the place of the court, many times they'd make the holiday of Rosh Hashanah two days. For if the witnesses didn't come all of the 30th day, they'd treat that day that the witnesses were anticipated as holy and the morrow holy. And since they'd make it two days and even at the time of the sighting, [as such] they amended that even the children of the land of Israel would always make it two days at this time that we fix based on the calculation. Behold, you have learnt that even the second [day of] holiday of Rosh Hashanah at this time originates from the words of the scribes.

Making one day of holiday is not dependent on the closeness of the place. How? If there will be a place, between it and between Jerusalem a journey of 5 days or less so that with certainty it would have been possible that the messengers would arrive there, we don't say that the people of this place make one [day] holiday. For who says to us that the messengers would go out to this place? Perhaps the messengers wouldn't go out to this place because there wasn't an Israelite there, and after they returned to fix based on the calculation, Israelites settled there that are obligated to make two days. Or because there was a siege on the way, like the way that was between Yehuda and Galil in the days of the sages of the Mishnah. Or because there were non-jews blocking the messengers to pass between them.

And were the matter to depend on the closeness of the place, all the children (inhabitants) of Egypt would make one day. For it is possible that the messengers of Tishrei would arrive there. For there is only between Jerusalem and Egypt via Ashkelon a journey of 8 days or less, and similarly the majority of Syria. Behold, you have learnt that the matter is not dependent on the place's being close.

It is found that the root of this matter on this way is such: Any place that has between it and between Jerusalem a journey greater than 10 complete days makes two day always like their custom from before. For the messengers of every single Tishrei would only reach a place that has between it and between Jerusalem a journey of 10 days or less. And any place that has between it and between Jerusalem a journey equal to 10 days or less, [such] that it is possible that the messengers of Tishrei would reach it, we see if that place was from [a part of] the land of Israel that had in it [people of] Israel at the time of the sighting in the second conquering [of the land], like Usha and Shaphram and Lod and Yavneh and Nov and Tiberius and such like, they make only one day alone. And if that place was from Syria, like Tyre and Damascus, and Ashkelon and such like, or from the outside of the land [of Israel], like Egypt and Amon and Moav and such like, they do like the custom of their fathers in their hands, if one day, one day, and if two days, two days.

A place that has between it and between Jerusalem 10 days or less than 10, and it is [in] Syria or the outside of the land [of Israel], and they don't have a custom. Or it is a newly formed city in the desert of the land of Israel. Or a place that [people of] Israel dwells now. They make two days like the custom of the majority of the world. And every second day holiday is from the words of the scribes and even the second day of the holiday of Rosh Hashanah which everyone makes at this time.

This that we calculate at this time, every single one in his city, and we say "the first of the month will be on day so-and-so and the holiday will be on day so-and-so, it is not with our calculation that we fix it and we do not rely on it. For we don't make years leap [years] and we don't fix months in the outside of the land [of Israel]. And we only rely on rely on the calculation of the children of the land of Israel and their fixing. And this that we calculate is only to reveal the matter. Since we know that on this calculation they rely, we calculate to know the day that the children of the land of Israel fixed it, whichever day it is. And via the fixing of the children of the land of Israel, it is it that the first of the month or the holiday will be. Not because of the calculation that we calculate.

When [the new moon] was sanctified [based on the testimony of witnesses regarding] the sighting of the moon, the court would calculate the time of the conjunction of the sun and the moon in an exact manner, as the astronomers do. [This enabled them] to know whether or not the moon would be sighted. The first level of these calculations represent approximations of the time of the conjunction, and their accuracy is not great. This approximation of the time of the conjunction is calculated according to the mean rates of movements. The time of the conjunction as calculated in this manner is referred to as the molad. The essentials of the calculations that are used when a court to sanctify [the new moon based on the testimony of witnesses of] the sighting [of the moon] does not exist - i.e., the calculations we use today - are referred to as ibbur. To explain:

Day and night are constantly considered a twenty-four hour composite, [on the average:] twelve [hours] of daylight and twelve [hours] of night. An hour can be divided into 1080 units. This number was chosen because it can be divided in half, into fourths, eighths, thirds, sixths, ninths, and tenths. Each of these fractions contains many of these units.

According to this calculation, [the interval] between one conjunction of the moon and the sun and the subsequent conjunction according to their mean movement is twenty-nine full days, twelve hours of the thirtieth day, and 793 units of the thirteenth hour. This is the interval between one conjunction and the next, [and thus,] the length of a lunar month.

[Accordingly,] an [ordinary] lunar year, which includes twelve of these months, would include three hundred fifty-four days, eight hours, and eight hundred seventy-six units. A leap year, which would include thirteen of these months, would include three hundred eighty-three days, twenty-one hours, and five hundred eighty-nine units. A solar year is three hundred sixty-five days and six hours. Thus, a solar year exceeds an [ordinary] lunar year by ten days, twenty-one hours, and two hundred and four units.

When the days of a lunar month are counted in groups of seven, according to the weekly cycle, there is a remainder of one day, twelve hours, and 793 units (in numerical terms, 1 - 12 - 793). This is the remainder for a lunar month. Similarly, when the days of a lunar year are counted in groups of seven, according to the weekly cycle, there is a remainder of four days, eight hours, and 876 units (in numerical terms, 4 - 8 - 876). This is the remainder for an ordinary lunar year. The remainder for a leap year will be five days, twenty-one hours, and 589 units (in numerical terms, 5 - 21 - 589).

When you know the time of the conjunction [of the sun and the moon] for any particular month, and add 1 - 12 - 793, you will arrive at the time of the conjunction of the following month. Thus, you will be able to determine on which day of the week and at which hour it will take place, and how many units of that hour will have passed.

What is implied? If the conjunction [of the moon and the sun] for the month of Nisan takes place on Sunday, five hours and 107 units after sunrise (in numerical terms 1 - 5 - 107), by adding the remainder for a lunar month, 1 - 12 - 793, you will be able to determine that the conjunction for the month of Iyar will take place on Tuesday night, five hours and 900 units after nightfall (in numerical terms, 3 - 5 - 900). One may follow this same method [of calculation] month after month for eternity.

Similarly, if you know the time of the conjunction for a particular year and you add its remainder - either the remainder of an ordinary year or the remainder of a leap year - to the time of the conjunction, you will determine the time of the conjunction of the following year. This method [of calculation] may be followed year after year for eternity. The first conjunction from which we begin, the conjunction of the first year of creation, was on Monday night, 5 hours and 204 units after nightfall (in numerical terms, 2 - 5 - 204). This is the starting point for these calculations.

In all the calculations to determine the time of the conjunction, when the remainder [of one period] should be added to another remainder, [the following principles should be adhered to:] When a sum of 1080 units is reached, it should be counted as an hour, and added to the number of the hours. When a sum of twenty-four hours is reached, it should be counted as a day, and added to the number of days. When the number of days is greater than seven, [all multiples of] seven should be subtracted from the sum, and the remainder be focused on. For the purpose of our calculations is not to know the number of days, but rather to know on which day of the week, and at what hour and after how many units will the conjunction take place.

[The fixed calendar is structured in] a nineteen-year cycle, including seven leap years and twelve ordinary years. This is called a machzor. Why was this [structure] chosen? Because when you total the number of days in twelve ordinary years and seven leap years together with their hours and their units, counting all [sums of] 1080 units as an hour, [all sums of] twenty-four hours as a day, and adding them to the number of days, the total will equal nineteen solar years, each of these years being 365 days and six hours. The difference between the days of the solar calendar [and the lunar calendar] will be only one hour and 485 units (in numerical terms, 1 - 485).

Thus, in such a [nineteen-year] cycle, the months are lunar months, and the years are solar years. The seven leap years in each cycle should be the following: The third year of the cycle, the sixth year, the eighth year, the eleventh year, the fourteenth year, the seventeenth year, the nineteenth year (in numbers, 3, 6, 8, 11, 14, 17, 19).

When you add the remainders of each of the twelve ordinary years, [the remainder of each year] being 4 - 8 - 876, and the remainders of the seven leap years, [the remainder of each year] being 5 - 21 - 589, and then divide the entire sum in groups of seven, there is a remainder of two days, sixteen hours, and 595 units (in numerical terms, 2 - 16 - 595). This is the remainder of a [nineteen-year] cycle.

When you know the time of the conjunction of the beginning of a [nineteen-year] cycle, by adding 2 - 16 - 595 to it you will be able to determine the beginning of the next [nineteen-year] cycle, and similarly all the [subsequent nineteen-year] cycles for eternity. As stated above, the conjunction [marking] the beginning of the first [nineteen-year] cycle took place on 2 - 5 - 204. [The expression,] the conjunction of a year refers to the conjunction of the month of Tishrei for that year.

Using the above method, it is possible to know the conjunction [marking] the beginning of any particular year, or any particular month, whether for the years that have passed or for the years to come. What is implied? One should take the number of years that have passed until Tishrei of the [desired] year and group them in nineteen- year cycles. Thus, one will be able to determine the number of nineteen-year cycles that have passed and the number of years that have passed within the [nineteen-year] cycle that has not been completed [until the desired year]. One should add 2 - 16 - 595 for each cycle, 4 - 8 - 876 for every ordinary year of the cycle that has not been completed, and 5 - 21 - 589 for every leap year [of the cycle that has not been completed]. One should then add together the entire sum, calculating [the groups of 1080] units as hours, the [groups of 24] hours as days, and the groups of seven days [as weeks]. [By adding] the remainder of the days, hours, and units [to 2 - 5 - 204], one can determine the time of the conjunction of the desired year.

The time of the conjunction of a year determined through the above method is the conjunction of Rosh Chodesh Tishrei. By adding 1 - 12 - 793 to this figure, one can determine the conjunction of Marcheshvan, and by adding 1 - 12 - 793 to [the conjunction of] Marcheshvan, one can determine the conjunction of Kislev. Similarly, one can determine the conjunction of all subsequent months for eternity.

We never set the New Month of Tishrei according to this calculation on the first day toward Sabbath (Sunday), the fourth day toward Sabbath (Wednesday), or the day before Sabbath (Friday). And a menmonic for them is "ADU" (1,4,6). Instead, when the molad (new moon time) for Tishrei is on one of these days, we set the New Month on the day after it. How so? Supposing the molad was on the first day toward Sabbath, we set the New Month of Tishrei on the second day (Monday). And if the molad was on the fourth day toward Sabbath, we set the New Month on the fifth day (Thursday). And if the molad was on the sixth day toward Sabbath, we set the New Month on the seventh day (Saturday).

Similarly, if the conjunction [for the month of Tishrei] takes place at noon or after noon, Rosh Chodesh should be established on the following day. What is implied? When the conjunction takes place on Monday, six hours after daybreak or later, Rosh Chodesh is established on Tuesday. If, however, the conjunction takes place before noon, even if only a single unit prior, Rosh Chodesh is established on the day of the conjunction, provided that day is neither Sunday, Wednesday, nor Friday.

When the conjunction takes place at noon or after noon, and [Rosh Chodesh Tishrei would be] postponed to the following day - if that following day is either Sunday, Wednesday, or Friday, Rosh Chodesh is postponed again, and is established on the third day after the conjunction. What is implied? If the conjunction takes place on the Sabbath after noon (in numbers, 7 - 18), in such a year Rosh Chodesh should be established on Monday. Similarly, if the conjunction takes place on Tuesday at noon or after noon, Rosh Chodesh should be established on Thursday.

In an ordinary year, when the conjunction [of the month] of Tishrei falls on the night of the third day, nine hours and 204 units (in numbers, 3 - 9 - 204) or more after nightfall, Rosh Chodesh is postponed, and instead of being established on Tuesday, it is established on Thursday.

A similar situation [may arise] in a year that follows a leap year: If the conjunction for Tishrei takes place on Monday, three hours and 589 units or more after daybreak (in numbers, 2 - 15 - 589), Rosh Chodesh is not established on Monday, but on Tuesday.

If, however, the conjunction of an ordinary year occurs [even] one unit earlier - i.e., were it to be 3 - 9 - 203 or earlier - [Rosh Chodesh] should be established on Tuesday, [rather than postponed until Thursday as mentioned above]. Similarly, if the conjunction of a year following a leap year occurs [even] one unit earlier - i.e., were it to be 2 - 15 - 588 or earlier - [Rosh Chodesh] should be established on Monday. Thus, the way to determine [the day on which] Rosh Chodesh Tishrei will be established according to these calculations is as follows: One should first determine the day [of the week], the hour of the day - or night - and the number of units of the hour when the conjunction takes place. The day of the conjunction will be the day of Rosh Chodesh, except in the following instances: a) [The conjunction] takes place on Sunday, Wednesday, or Friday; b) The conjunction takes place at noon or after noon; c) In an ordinary year, [the conjunction] takes place on the night of the third day, after 204 units of the tenth hour have passed, or later [that day]; d) In an ordinary year that follows a leap year, the conjunction takes place on Monday past 589 units of the fourth hour after daybreak has passed or later [that day]. If the conjunction occurs in one of these four instances, [Rosh Chodesh] is not established on the day of the conjunction, but rather on the day that follows, or on the day following that, as explained.

Why is [Rosh HaShanah] not established [on the day of the conjunction] when it falls on Sunday, Wednesday, or Friday? Because these calculations determine the conjunction of the sun and the moon only according to their mean [rate of] progress, and do not [necessarily] reflect the true position [of the sun and the moon in the celestial sphere], as explained. Therefore, they instituted that [on] one day [Rosh Chodesh] would be established and on the following day it would be postponed, so that they would ascertain the day when the true conjunction takes place. What is implied? [When according to our calculations, the conjunction occurs on] Tuesday, we establish [Rosh Chodesh]. [When it occurs] on Wednesday, we postpone it. [When it occurs] on Thursday, we establish [Rosh Chodesh]. [When it occurs] on Friday, we postpone it. [When it occurs] on the Sabbath, we establish [Rosh Chodesh]. [When it occurs] on Sunday, we postpone it. [When it occurs] on Monday, we establish [Rosh Chodesh].

This same principle, that the calculations are based on the mean rate of progress, is also the motivating factor for the other four reasons for the postponement [of Rosh Chodesh]. As proof of this, there are times when [according to the calculations] the conjunction takes place on Tuesday, and [Rosh Chodesh] is postponed until Thursday, and yet the moon will not be seen Thursday night, nor even Friday night. This indicates that the true conjunction of the sun and the moon did not take place until Thursday.

A lunar month is twenty-nine and one half days, and 793 units, as we have explained. It is impossible for Rosh Chodesh to begin in the middle of the day - i.e., that a portion of the day would be part of the previous month and a portion of the day would be part of the following month - as [implied by Numbers 11:20]: "For a month of days...." According to the Oral Tradition, this was interpreted [to mean], "You count the days of a month; you do not count the hours [of a month]."

Therefore, some lunar months are established as lacking [a day], and others as full. A month that is lacking has only twenty- nine days, even though a lunar month is several hours longer. A full month is thirty days, even though a lunar month is several hours shorter. In this manner, the months will be calculated according to complete days, not according to hours.

If a lunar month were exactly twenty-nine and a half days [long], the years [would be divided evenly] into full and lacking months, and there would be exactly 354 days to a lunar year. Thus, there would be six full months and six lacking months. It is the units that exist in every month that exceed the half day - which ultimately add up to hours and days - that cause certain years to have more lacking months than full months, and other years to have more full months than lacking months.

According to this reckoning, the thirtieth day of the month is always established as Rosh Chodesh. If the month is lacking, the thirtieth day will be Rosh Chodesh of the coming month. If the month is full [the coming month will have two days that are Rosh Chodesh]. The thirtieth day will be Rosh Chodesh, since a portion of it is [fit to be] Rosh Chodesh. [Nevertheless,] it will be counted as the completion of the previous month, which was full. The thirty-first day also will be Rosh Chodesh, and the reckoning [of the days of the coming month] will start from it. It is the day established [as Rosh Chodesh]. Thus, according to this calculation, there are some months that have only one day Rosh Chodesh, and other months that have two days Rosh Chodesh.

The following is the order of the full and lacking months according to [our] fixed calendar: Tishrei is always full. Tevet is always lacking. From Tevet on, there is one full month and one lacking month in sequence. What is implied? Tevet is lacking; Shevat is full; Adar is lacking; Nisan is full; Iyar, lacking; Sivan, full; Tammuz, lacking; Av, full; Elul, lacking. In a leap year, the first Adar is full, and the second Adar is lacking.

Two months remain: Marcheshvan and Kislev. Sometimes they are [both] full; sometimes they are [both] lacking; and sometimes Marcheshvan is lacking and Kislev is full. A year in which both of these months are full is called a year of complete months. A year in which both these months are lacking is called a year of lacking months. And a year in which Marcheshvan is lacking and Kislev is full is called a year whose months [proceed] in order.

The way to know whether the months of a year will be lacking, will be complete, or will [proceed] in order [can be explained] as follows: First, determine the day on which Rosh HaShanah will fall in the year about whose months you desire to know, as explained in Chapter 7. Then determine the day on which Rosh HaShanah will fall in the year that follows. Afterwards, count the number of days between them without including the day on which Rosh HaShanah falls in either of these years. If there are only two days between them, the months of the year will be lacking. If there are three days between them, the months of the year will proceed in order. And if there are four days between them, the months of the year will be complete.

When does the above apply? When the year in question is an ordinary year. When, however, [the year in question] is a leap year [different rules apply]: If there are only four days between the day on which [Rosh HaShanah] is established [in the leap year] and the day on which it will be established in the following year, the months of the year will be lacking. If there are five days between these [two days], the months of the year will proceed in order. And if there are six days between them, the months of the year will be complete.

What is implied? If we desire to know the order of the months of the present year, and [we know the following]: Rosh HaShanah falls on Thursday; it is an ordinary year; and in the following year Rosh HaShanah falls on Monday, there are three days between them and the months of the year proceed in order. If Rosh HaShanah falls on Tuesday in the following year, the months of the year will be complete. If Rosh HaShanah falls on the Sabbath in the present year, and on Tuesday in the following year, the months of the year will be lacking. Similar concepts should be applied regarding the calculation [of the order of the months] of a leap year, as was explained.

There are certain indications upon which one can rely, so that one will not err regarding the calculation of the order of the months of a year. These principles are based on the fundamental principles of the fixed calendar and the determination of the days on which Rosh HaShanah will be established and those that will cause it to be postponed, as we explained previously. Whenever Rosh HaShanah is celebrated on a Tuesday, [the months of] the year will [proceed] in order. [This applies regardless of whether the year] is an ordinary year or a leap year. Whenever Rosh HaShanah is celebrated on the Sabbath or on a Monday, [the months of] the year will never [proceed] in order. [This applies regardless of whether the year] is an ordinary year or a leap year. [The following rules apply when] Rosh HaShanah falls on a Thursday. If the year is an ordinary year, it is impossible for its months to be lacking. If it is a leap year, it is impossible for its months to proceed in order.

[There is a difference of opinion among] the Sages of Israel concerning the length of a solar year. Some Sages maintain that it is 365 days and 1/4 of a day - i.e., six hours. Others maintain that it is slightly less than that figure. There is also a difference of opinion among the wise men of Greece and Persia concerning this matter.

According to the opinion that [a solar year] is [exactly] 365 and 1/4 days, there will be a remainder of one hour and 485 units after every nineteen-year cycle, as we mentioned. Between the start of each of the successive seasons of the year, there will be ninety-one days and seven and one-half hours. When you know the date and the hour of the beginning of one season, you can calculate [the beginning of] the following season by [adding the above amount]. Similarly, you can calculate the beginning of the following season, and continue forever.

The equinox of Nisan (spring) [takes place] at the hour and the unit when the sun enters the beginning of the constellation of Aries. The solstice of Tammuz (summer) [takes place] when the sun is located in the beginning of the constellation of Cancer. The equinox of Tishrei (autumn) [takes place] at the hour and the unit when the sun enters the beginning of the constellation of Libra. The solstice of Tevet (winter) [takes place] when the sun is located in the beginning of the constellation of Capricorn. According to this calculation, in the first year of creation the vernal (spring) equinox took place seven days, nine hours, and 642 units before the conjunction of the month of Nisan, in numbers, 7 -9 - 642.

The method of calculating [the beginning of] the seasons can be explained as follows. First, it is necessary to calculate the number of [nineteen-year] cycles that have passed until the [nineteen-year] cycle in question. Afterwards, add one hour and 485 units for every [nineteen-year] cycle. Afterwards, group all the units into hours, and all the hours into days. [Once a] total [has been reached], subtract seven days, nine hours, and 642 units [from it]. Add the remainder to [the time of] the conjunction of Nisan in the first year of the [nineteen-year] cycle in question, and you will be able to know the hour and the date of the the vernal equinox of the first year of this cycle. From this date, you can calculate [the beginnings of] all the subsequent seasons [by] adding ninety-one days and seven and one- half hours for every season. If you desire to know [the time and the date of] the vernal equinox of a particular year within a given [nineteen-year] cycle, [the following procedure should be used:] Add one hour and 485 [units] for every [nineteen-year] cycle. For each complete year that has passed within the [nineteen-year] cycle [under discussion], add ten days, twenty-one hours, and 204 units,, and then group the entire sum [into days and hours]. Afterwards, subtract seven days, nine hours, and 642 units [from this sum] and divide the remainder into lunar months of 29 days, 12 hours, and 793 units. [The number of days, hours, and units that] remain [after all the complete] lunar months [have been calculated] should be added to [the day and the time of] the conjunction of Nisan in that year. [In this manner,] you will be able to determine the date and the time of the vernal equinox of the year desired. According to this calculation, the vernal equinox will always take place either at nightfall, at midnight, at daybreak, or at noon. The summer solstice will always take place at either 7:30 PM, 1:30 AM, 7:30 AM, or 1:30 PM. The autumnal equinox will always take place either at nine or at three o'clock, either in the day or the night. The winter solstice will always take place either at 10:30 PM, 4:30 AM, 10:30 AM, or 4:30 PM. If you desire to know the day of the week and the hour of the equinox, [the following procedure should be used:] Count the number of complete years that have passed from the year of creation until the desired year, and divide them into groups of twenty- eight. Add one day and six hours for each year remaining. Total the sum [of the hours and the days], and then add three days. Afterwards, divide the days into groups of seven. The remainder of the days and the hours should be added to the time of nightfall on the first day of the week. The result will be [the day and the time] on which the vernal equinox will occur. Why is it necessary to add three days? Because the first equinox of the year of creation took place at the beginning of the fourth day.

What is implied? If a person desires to know the day and the time of the vernal equinox of the year 4930 after creation, [the following procedure should be used:] [That number] should be divided by 28, leaving a remainder of one year, thus producing the figure of one day and six hours. By adding three days to this figure, it can be determined that the vernal equinox will take place on the night of the fifth day at midnight. By adding seven and one-half hours to this figure, it can be determined that the summer solstice will take place on Thursday, an hour and one half after daybreak. By adding seven and one-half hours to this figure, it can be determined that the autumnal equinox will take place on Friday, at nine hours after daybreak. By adding seven and one-half hours to this figure, it can be determined that the winter solstice will take place on the night of the sixth day, four and one half hours after nightfall. Similarly, by adding seven and one-half hours to this figure, it can be determined that the vernal equinox of the following year will take place on Friday, at daybreak. In this manner, it is possible to calculate [the time of the beginning of all] the seasons forever.

[The following procedure should be used] if one desires to know the date of the month on which the vernal equinox will fall this year: First, determine the day of the week on which [the equinox] will fall. Then determine the day [of the week] on which Rosh Chodesh of Nisan will fall, and how many complete years have passed within the nineteen-year cycle. Add eleven days for every year, and then add seven days to this sum in the present time. Divide the sum by thirty, and begin counting the remainder of days from Rosh Chodesh Nisan. If the date coincides with the day of the week on which the equinox falls, this is sufficient. If not, add one, two, or three days to this number until you reach the day [of the week] on which the equinox falls. If the year in question is a leap year, begin counting from Rosh Chodesh of the second Adar. When a day is determined through this calculation, the equinox will take place on that date.

What is implied? Should we desire to know the date of the vernal equinox of the year 4930, which is the ninth year of the two- hundred-sixtieth [nineteen-year] cycle, [the following procedure should be used:] We have already determined that Rosh Chodesh Nisan will take place on Thursday, and that the equinox will take place on Thursday. Since this is the ninth year of the [nineteen-year] cycle, there are eight complete years [to take into consideration]. When eleven days are added for every year, we reach a sum of 88. When seven is added, the total will be 95. When this number is divided by 30, there will be a remainder of five. When we add five days to Rosh Chodesh Nisan, which is Thursday, we reach Monday. Since we know that the equinox will not fall on Monday, but rather on Thursday, we continue adding days until Thursday, the day of the equinox. Thus, we can determine that this year the vernal equinox will take place on the eighth of Nisan. A similar process can be followed [to determine the date of the equinox] every year.

Although we said that one should continue to add days until one reaches the day of the week on which the equinox takes place, one should never have to add more than one, two, or three days - or in a most unusual case - four days. If you find it necessary to add any more days than this, know that you have made an error in your calculations, and you should recalculate carefully.

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 units (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, and divide the remainder by the length of a lunar month. 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. [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.

As stated in the laws mentioned previously, the court made precise calculations and knew whether or not the [new] moon would be visible. Accordingly, we are assured that anyone with a proper spirit and heart, who desires words of wisdom and probes to grasp the mysteries, will wish to know the methods of calculation used to determine whether or not the [new] moon would be visible on a particular night.

There are many differences of opinion among the sages of the nations of the previous eras who studied astronomy and mathematics, with regard to these methods of calculation. Great wise men have blundered regarding these matters. Concepts were hidden from them and doubts arose [in their minds]. There are those who have made many calculations, but have not been able to find the correct approach to determine when the moon becomes visible. Rather, they plunged into the mighty waters, to return with merely a potsherd in their hands.

Over the course of history, through much research and investigation, several sages have discovered the proper methods of calculation. We also possess traditions regarding these principles that we have received from the sages, and proofs that were not written in texts that are of common knowledge. For these reasons, I have considered it proper to explain a method of calculation that will be available for anyone whose heart spurs him to approach the task and perform it.

A person should not regard these calculations lightly, because they are not required in the present age, for these methods are indeed abstract and deep matters. They constitute the mystery of the calendar, which was known [only] to great sages, who would not convey these matters to [most] other people, but only to ordained and perceptive [sages]. The calendar that is employed in the era when there is no court to determine [the months according to the testimony of] witnesses, and which we use at present [is], by contrast, [a simple matter that] can be appreciated even by school children in three or four days.

A wise man of the gentile nations or a sage of Israel who studied Greek wisdom may meditate on the methods of calculation I have used to determine the appearance of the moon and may detect a slight approximation [and imprecision] with regard to certain matters. He should not presume that we have overlooked this point and were not aware that there was an approximation regarding that matter. Instead, he should assume that whenever we were not exact, it was because our mathematical calculations proved that [this inaccuracy] did not affect the knowledge of the time when the moon would become visible, and thus it was not significant. Therefore, we were not precise regarding this matter.

Similarly, should a person see that [our use of] one of the methods leads to a minor inadequacy that is inappropriate for this method of computation, [he should realize] that this was intentional. For this method produced an advantage from another perspective that will produce a correct result - [albeit] through approximate calculations - without requiring lengthy computations. Thus, a person who is not practiced in such matters will not be flustered by complex computations that are of no avail with regard to the visibility of the moon.

The [following] fundamental principles must be known by a person as a prelude to all astronomical computations, whether for the purpose of determining the visibility [of the moon] or for other purposes: The heavenly sphere is divided into 360 degrees [and twelve constellations]. Each constellation includes thirty degrees, beginning with the constellation of Aries the ram. Every degree contains sixty minutes, every minute sixty seconds, and every second sixty thirds. You may continue and divide into further fractions to the extent that you desire.

Therefore, were you to calculate that a particular star's position in the heavenly sphere is seventy degrees, thirty minutes and forty seconds, you would know that this star is located in the constellation of Gemini the twins, in the middle of the eleventh degree. For the constellation of Aries includes thirty degrees, and the constellation of Taurus the bull includes thirty degrees. Thus, there remain ten and one half degrees of the constellation of Gemini, plus forty seconds of the next degree.

Similarly, were you to calculate that a particular star's position in the heavenly sphere is 320 degrees, you would know that this star is located in the constellation of Aquarius the water bearer, in its twentieth degree. The same applies to all other calculations. The order of the constellations is the following: Aries the ram, Taurus the bull, Gemini the twins, Cancer the crab, Leo the lion, Virgo the virgin, Libra the balance, Scorpio the scorpion, Sagittarius the archer, Capricorn the goat, Aquarius the water-bearer, Pisces the fishes.

In all calculations, when you collect fractions or add numbers, each integer should be added to its kind, the seconds to the seconds, the minutes to the minutes, and the degrees to the degrees. When calculating seconds, they should be grouped in sets of sixty [or less]. Whenever sixty seconds are reached, they should be considered a minute and added to the sum of the minutes. When calculating minutes, they should be grouped in sets of sixty [or less]. Whenever sixty minutes are reached, they should be considered a degree and added to the sum of the degrees. When calculating degrees, they should be grouped in sets of 360. If a sum above 360 is reached, the remainder after 360 has been subtracted is the figure that is of consequence.

In all computations, whenever you desire to subtract one number from another, should the second number be greater than the first number, even if it is merely one minute greater, it is necessary to add 360 degrees to the first number so that it is possible to subtract the [greater] number from it.

What is implied? When it is necessary to subtract two hundred degrees, fifty minutes and forty seconds - in symbols 200° 50' 40" - from one hundred degrees, twenty minutes and thirty seconds - in symbols 100° 20' 30" - [one should follow this procedure]: [First,] one adds 360 to 100, producing a sum of 460. Afterwards, one begins to subtract the seconds. Since it is impossible to subtract 40 from 30, it is necessary to convert one of the 20 minutes into 60 seconds. When added to 30, this produces a sum of 90. [From the 90] subtract 40, producing a total of 50 seconds. Afterwards, one must subtract 50 minutes from the 19 [remaining], for one of the minutes has already been converted into seconds. Since 50 cannot be subtracted from 19, one must convert a degree into 60 minutes. When this figure is added to 19, it produces a sum of 79. When 50 is subtracted [from 79], a total of 29 minutes remain. Afterwards, one must subtract the 200 degrees from the 459 decrees, for one of the degrees has already been converted into minutes. Thus 259 degrees remain. In symbols [the remainder is] 259° 29' 50". All other subtractions should be performed following a similar method.

The sun, the moon, and the remainder of the seven stars, each proceeds at a uniform speed in its orbit. They are never inclined to heaviness, nor to lightness. Rather, the speed at which they proceed today is the same speed at which they proceeded yesterday. And tomorrow, and indeed on every other day, they will proceed at this speed. Although the orbits in which they all travel encircle the earth, the earth is not at the center of [their orbits].

Therefore, if one measured the progress [of any of these stars] against the sphere that encompasses the world in which the earth is the center - i.e., the sphere of the constellations - its [rate of] progress [would appear to] change. Its rate of progress in the sphere of the constellations on one day could appear less or more than its progress on the previous day or on the following day.

The uniform speed at which a planet, the sun, or the moon progresses is referred to as its mean motion. The progress that [this celestial body appears to make] in the sphere of the constellations that is sometimes greater and sometimes less [than its actual rate of progress] is referred to as its true motion. This determines the true position of the sun or the true position of the moon.

We have already stated that the calculations that we explain in these laws are intended solely to determine the visibility of the [new] moon. Therefore, we have established the starting point from which we will always begin these calculations: the eve of Thursday, the third of Nisan, of the present year, the seventeenth year of the 260th [nineteen-year] cycle - i.e., the year 4938 since creation - which is the year 1489 with regard to contracts, and 1109 years after the destruction of the Second Temple. This is the year that will be referred to as the starting point in these calculations.

Since the sighting of the moon is significant only in Eretz Yisrael as explained, all our calculations are centered on the city of Jerusalem and locations within six or seven days' journey [from it. [In these places,] the moon is frequently sighted, and the people come and give testimony in the court. This location is situated approximately 32 degrees north of the equator, [and the surrounding areas extend] from 29° to 35° [north]. Similarly, in longitude, it is situated approximately 24 degrees west of the center of the populated area, [and the surrounding areas extend] from 21° to 27° [west].

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". Thus, in ten days, it travels 9 degrees, 51 minutes and 23 seconds, 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 explained - 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. 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. [This is what concerns us,] for our sole desire in these calculations is to know [when the moon] will be sighted. 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. 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".

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. 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. 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". 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. 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. 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 months 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, 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.

[The following method should be used] if you wish to know the true position of the sun on any particular day you desire: First, it is necessary to calculate the mean position of the sun through the methods of calculation we have explained. Then calculate the position of the apogee of the sun. Afterwards, subtract the apogee of the sun from the mean position of the sun. The remainder is referred to as the course of the sun.

[The next step is] to calculate the angular distance of the course of the sun. If the angular distance of the course is less than 180 degrees, one should subtract the angle [determined by the] course from the sun's mean position. If the angular distance of the course is more than 180 degrees, one should add the angle [determined by the] course to the sun's mean position. The figure remaining after making this addition or subtraction represents [the sun's] true position.

If the course [of the sun] is an even 180 degrees or an even 360 degrees, there will be no angle [determined by the course to add or to subtract]. Instead, the [sun's] mean position is its true position.

What is the angle [determined by the] course? If the course is ten degrees, the [resulting] angle will be 20 minutes. If the course is twenty degrees, the [resulting] angle will be 40 minutes. If the course is thirty degrees, the [resulting] angle will be 58 minutes. If the course is forty degrees, the [resulting] angle will be 1 degree and 15 minutes. If the course is fifty degrees, the [resulting] angle will be 1 degree and 29 minutes. If the course is sixty degrees, the [resulting] angle will be 1 degree and 41 minutes. If the course is seventy degrees, the [resulting] angle will be 1 degree and 51 minutes. If the course is eighty degrees, the [resulting] angle will be 1 degree and 57 minutes. If the course is ninety degrees, the [resulting] angle will be 1 degree and 59 minutes. If the course is one hundred degrees, the [resulting] angle will be 1 degree and 58 minutes. If the course is one hundred ten degrees, the [resulting] angle will be 1 degree and 53 minutes. If the course is one hundred twenty degrees, the [resulting] angle will be 1 degree and 45 minutes. If the course is one hundred thirty degrees, the [resulting] angle will be 1 degree and 33 minutes. If the course is one hundred forty degrees, the [resulting] angle will be 1 degree and 19 minutes. If the course is one hundred fifty degrees, the [resulting] angle will be 1 degree and 1 minute. If the course is one hundred sixty degrees, the [resulting] angle will be 42 minutes. If the course is one hundred seventy degrees, the [resulting] angle will be 21 minutes. If the course is an even one hundred eighty degrees, it has no measure. Instead, its mean position is its true position, as we explained.

[The following procedure should be used] if the course [of the sun] is greater than one hundred eighty degrees: One should subtract the course from three hundred sixty degrees and [calculate the resulting] angle accordingly. What is implied? If the course is 200 degrees, that figure should be subtracted from 360 degrees, leaving a remainder of 160 degrees. Since you already know that the [resulting] angle of a course of 160 degrees is 42 minutes, that same figure will be the [resulting] angle of a course of 200 degrees.

Similarly, if the course was three hundred degrees, one should subtract that figure from three hundred sixty, leaving a remainder of sixty. Since you already know that the [resulting] angle of a course of 60 degrees is 1 degree and 41 minutes, that same figure will be the [resulting] angle of a course of 300 degrees. Similar procedures should be followed in calculating other figures.

[How is the angle determined by the course calculated] when the course is [an intermediate figure - e.g.,] 65 degrees? You already know that the [resulting] angle of 60 degrees is 1 degree and 41 minutes. And you know that the [resulting] angle of 70 degrees is 1 degree and 51 minutes. Thus, there are ten minutes between these [two] measures. Thus, [an increase of] a degree [of the course] will bring an increase of a minute [in the resulting angle]. Thus, the [resulting] angle of a course of 65 degrees will be 1 degree and 46 minutes.

Similarly, if the course was 67 degrees, the [resulting] angle would be 1 degree and 48 minutes. A similar procedure should be followed regarding any course that has both units and tens, both for calculations regarding the sun and for calculations regarding the moon.

[To apply these principles]: Should we desire to know the true position of the sun at the beginning of Friday night, the fourteenth of Tammuz for this present year: First, we should calculate the mean position of the sun for this time, which is, as explained, 105° 37' 25". We should then calculate the apogee of the sun at this time, which is 86° 45' 23". When the apogee is subtracted from the mean position, the remainder, the course [of the sun], will be 18 degrees, 52 minutes and 2 seconds, in symbols 18° 52' 2 ". With regard to the course [of the sun], the minutes are of no consequence. If they are less than thirty, they should be disregarded entirely. If they are more than thirty, they should be considered an additional degree and added to the sum of the degrees. Accordingly, it should be considered as if there are 19 degrees in this course. The [resulting] angle of such a course can be calculated to be 38 minutes in the manner that we explained.

Since the course is less than 180 degrees, the [resulting] angle [of the course], 38 minutes, should be subtracted from the mean position of the sun, leaving a remainder of 104 degrees, 59 minutes and 25 seconds, in figures 104° 59' 25". Thus, the true position of the sun at the beginning of this night will be fifteen degrees less 35 seconds in the constellation of Cancer. One need not pay attention to the seconds at all, neither with regard to the position of the sun, nor with regard to the position of the moon, nor in any other calculations regarding the sighting [of the moon]. Instead, if the number of seconds is approximately thirty [or more], they should be considered a minute, and added to the sum of the minutes.

Since you are able to calculate the location of the sun on any desired date, you will be able to calculate the true date of the equinox or solstice for any equinox or solstice you desire, whether for the equinoxes or solstices that will take place in the future, after the date we established as a starting point, or for the equinoxes or solstices that have taken place in previous years.

There are two mean rates of progress [that are significant] with regard to the moon, for the moon revolves in a small orbit that does not encompass the earth. Its mean progress within this orbit is referred to as the mean within its path. The small orbit [within which the moon revolves] itself rotates in a larger orbit that encompasses the earth. The mean progress of the small orbit within the large orbit that encompasses the earth is referred to as the moon's mean. The rate of progress for the moon's mean in one day is 13 degrees, 10 minutes and 35 seconds, in symbols 13° 10' 35".

Thus, its progress in ten days will be 131 degrees, 45 minutes and 50 seconds, in symbols 131° 45' 50". The remainder [of the sum]of its progress in one hundred days will be 237 degrees, 38 minutes and 23 seconds, in symbols 237° 38' 23". The remainder [of the sum] of its progress in one thousand days is 216 degrees, 23 minutes and 50 seconds, in symbols 216° 23' 50". The remainder [of the sum] of its progress in ten thousand days is 3 degrees, 58 minutes and 20 seconds, in symbols 3° 58' 20". The remainder [of the sum] of its progress in twenty-nine days is 22 degrees, 6 minutes and 56 seconds, in symbols 22° 6' 56".The remainder [of the sum] of its progress in a regular year is 344 degrees, 26 minutes and 43 seconds, in symbols 344° 26' 43". Following these guidelines, you can multiply these figures for any number of days or years you desire.

The distance travelled by the mean within its path in a single day is 13 degrees, 3 minutes and 54 seconds, in symbols 13° 3' 54". Thus, its progress in ten days will be 130 degrees, 39 minutes and no seconds, in symbols 130° 39'. The remainder [of the sum] of its progress in one hundred days will be 226 degrees, 29 minutes and 53 seconds, in symbols 226° 29' 53". The remainder [of the sum] of its progress in one thousand days is 104 degrees, 58 minutes and 50 seconds, in symbols 104° 58' 50". The remainder [of the sum] of its progress in ten thousand days is 329 degrees, 48 minutes and 20 seconds, in symbols 329° 48' 20". The remainder [of the sum] of its progress in twenty-nine days is 18 degrees, 53 minutes and 4 seconds, in symbols 18° 53' 4".

The remainder [of the sum] of its progress in a regular year is 305 degrees, no minutes and 13 seconds, in symbols 305° 13". The position of the moon's mean on Wednesday night, [the third of Nisan, 4938,] the starting point for these calculations, was 1 degree, 14 minutes and 43 seconds, in figures 1° 14' 43", in the constellation of Taurus. The mean within its path at this date was 84 degrees, 28 minutes and 42 seconds, in symbols 84° 28' 42". Since you know the mean rate of progress for the moon's mean, and you know its position on the date of the starting point, you [will be able to calculate] the position of the moon's mean on any date that you desire, as you did with regard to the mean position of the sun. After calculating [the position of] the moon's mean on the beginning of the night that you desire, [the next step in calculating where the moon can be sighted] is to focus on the sun and see the constellation in which it will be located [at that time].

If the sun is located between midway in the constellation of Pisces and midway in the constellation of Aries, the moon's mean should be left without emendation. If the sun is located between midway in the constellation of Aries and the beginning of the constellation of Gemini, 15 minutes should be added to the moon's mean. If the sun is located between the beginning of the constellation of Gemini and the beginning of the constellation of Leo, 30 minutes should be added to the moon's mean. If the sun is located between the beginning of the constellation of Leo and midway in the constellation of Virgo, 15 minutes should be added to the moon's mean. If the sun is located between midway in the constellation of Virgo and midway in the constellation of Libra, the moon's mean should be left without emendation. If the sun is located between midway in the constellation of Libra and the beginning of the constellation of Sagittarius, 15 minutes should be subtracted from the moon's mean. If the sun is located between the beginning of the constellation of Sagittarius and the beginning of the constellation of Aquarius, 30 minutes should be subtracted from the moon's mean. If the sun is located between the beginning of the constellation of Aquarius and midway in the constellation of Pisces, 15 minutes should be subtracted from the moon's mean.

The figure that remains after these additions or subtractions have been made, or when the mean was left without emendation, is the mean of the moon approximately 20 minutes after the setting of the sun for the time when this mean was calculated. This is referred to as the mean of the moon at the time of the sighting.

If you desire to know the true position of the moon on any particular date, first calculate the mean of the moon at the time of the sighting for the desired date. Then calculate the mean of the [moon within its] path and the sun's mean [position] for that date. Subtract the sun's mean from the moon's mean and double the remainder. The resulting figure is referred to as the double elongation.

As mentioned previously, the intent of all the calculations in these chapters is to know how to sight the moon. [In this context, the size of this double elongation is significant.] [To explain:] It is impossible for this double elongation to be less than five degrees or more than 62 degrees on the night the moon is to be sighted. Its measure will never exceed or fall short of these numbers.

Accordingly, it is necessary to contemplate [the length of] this double elongation. If the double elongation is five degrees or near that measure, there is no need to be concerned with an increase, and its measure need not be increased. If [the length of] the double elongation is between six and eleven degrees, one should add one degree to the mean of the [the moon] within its path. If [the length of] the double elongation is between twelve and eighteen degrees, one should add two degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between nineteen and 24 degrees, one should add three degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 25 and 31 degrees, one should add four degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 32 and 38 degrees, one should add five degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 39 and 45 degrees, one should add six degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 46 and 51 degrees, one should add seven degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 52 and 59 degrees, one should add eight degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 60 and 63 degrees, one should add nine degrees to the mean of [the moon] within its path. The mean of [the moon] within its path that results after these additions have been made is referred to as the correct course.

After the angular distance of the correct course is calculated, [the following procedure should be carried out]: If [the result] is less than 180 degrees, the angle of the course should be subtracted from the mean of the moon at the time of the sighting. If [the result] is more than 180 degrees, the angle of the course should be added to the mean of the moon at the time of the sighting. The figure that remains after this addition or subtraction is made is the true position of the moon at the time of sighting.

Know that if the correct course is an even 180 degrees or 360 degrees, there is no angle of the course. Instead, the mean position of the moon at the time of sighting is the true position of the moon at that time.

What is the angle of its course? If the correct course is ten degrees, its angle will be 50 minutes. If the correct course is twenty degrees, its angle will be 1 degree and 38 minutes. If it is 30, its angle will be 2 degrees and 24 minutes. If it is 40, its angle will be 3 degrees and 6 minutes. If it is 50, its angle will be 3 degrees and 44 minutes. If it is 60, its angle will be 4 degrees and 16 minutes. If it is 70, its angle will be 4 degrees and 41 minutes. If it is 80, its angle will be 5 degrees. If it is 90, its angle will be 5 degrees and 5 minutes. If it is 100, its angle will be 5 degrees and 8 minutes. If it is 110, its angle will be 4 degrees and 59 minutes. If it is 120, its angle will be 4 degrees and 40 minutes. If it is 130, its angle will be 4 degrees and 11 minutes. If it is 140, its angle will be 3 degrees and 33 minutes. If it is 150, its angle will be 2 degrees and 48 minutes. If it is 160, its angle will be 1 degree and 56 minutes. If it is 170, its angle will be 59 minutes. If it is an even 180 degrees, [the course] will not have an angle. Instead, as stated above, the moon's mean position will be identical with its true position.

If the correct course is more than 180 degrees, you should subtract it from 360 to obtain its angle, as you did for the course of the sun. Similarly, if the correct course includes units as well as tens, you should [calculate the average increase per degree and add the proportionate amount to the lower figure]. The procedure used to calculate the angle of the course for the course of the sun should be used to calculate the angle of the correct course [of the moon].

What is implied? Should we desire to know the true position of the moon on Friday night, the second of Iyar in the present year - the starting point for these calculations - the number of complete days that have passed from the date that is the starting point until the date on which we desire to know the true position of the moon is 29. One should [first] calculate the mean position of the sun for that night; this is 35 degrees, 38 minutes and 33 seconds, in symbols 35° 38' 33". You should then calculate the mean of the moon at the time of the sighting, which is 53 degrees, 36 minutes and 39 seconds, in symbols 53° 36' 39". Afterwards, calculate the mean of [the moon] within its path for this time, which is 103 degrees, 21 minutes and 46 seconds, in symbols, 103° 21' 46". Then subtract the mean position of the sun from the moon's mean, producing a remainder of 17 degrees, 58 minutes and six seconds. This is the elongation. Doubling this figure produces a double elongation of 35 degrees, 56 minutes and 12 seconds, in symbols 35° 56' 12". Therefore, five degrees should be added to the course, as mentioned. Thus, the correct course will be 108 degrees and 21 minutes. As mentioned above with regard to the sun, the minutes are of no consequence in the calculation of the course.

When calculating the angle for a course of 108, the result is 5 degrees and one minute. Since the correct course is less than 180 degrees, this figure should be subtracted from the moon's mean, leaving a remainder of 48 degrees, 35 minutes and 39 seconds. The seconds should be rounded off and considered to be a minute. Accordingly, the true position of the moon at this time will be 18 degrees and 36 minutes of the nineteenth degree in the constellation of Taurus, in symbols 18° 36'. In a similar manner, it is possible for you to calculate the true position of the moon at the time of sighting for any date that you desire from the beginning of this year that was chosen as the starting point until the end of all time.

The orbit in which the moon revolves [intersects] the orbit in which the sun revolves at an angle, [so that] a portion of [the moon's orbit] is inclined to the north of the sun's orbit and a portion is inclined south of the sun's orbit. There are two points, one opposite the other, at which these orbits intersect. When the moon is at one of these points, it is revolving in the same plane as the sun. As the moon departs from these points, it is proceeding either to the north or to the south of the sun. The point in the moon's [orbit] at which it begins to be inclined to the north of [the plane of] the sun's [orbit] is referred to as the head, while the point [in its orbit] from which it begins to be inclined to the south of [the plane of] the sun's [orbit] is referred to as the tail.* This head revolves at a uniform pace, [proceeding] in opposition to the movement of the sphere of the constellations without increase or decrease - i.e., it moves from [the constellation of] Pisces to [the constellation of] Aquarius. It continuously follows [this pattern].

The mean movement of the head in one day is 3 minutes and 11 seconds. Thus, its movement in ten days is 31 minutes and 47 seconds; its movement in one hundred days is 5 degrees, 17 minutes, and 43 seconds, in symbols 5° 17' 43". In one thousand days, its movement is 52 degrees, 57 minutes and 10 seconds, in symbols 52° 57' 10". The remainder [of the sum] of its progress in ten thousand days is 169 degrees, 31 minutes and 40 seconds, in symbols 169° 31' 40". Thus, the distance it travels in twenty-nine days is 1 degree, 32 minutes and 9 seconds, in symbols 1° 32' 9". Its progress in a regular year is 18 degrees, 44 minutes and 42 seconds, in symbols 18° 44' 42". The mean position of the head on Thursday night [of the present year,] the starting point for these calculations, is 180 degrees, 57 minutes, and 28 seconds, in symbols 180° 57' 28".

If you desire to calculate the position of the head at any given date, [you should follow this procedure:] First, calculate the mean progress of the head as you calculated the mean of the sun and the mean of the moon. [Afterwards,] subtract this mean from 360 degrees, and the remainder will be the location of the head at that time. The tail's position will always be the [place in the moon's orbit] directly opposite it.

What is implied? Let us suppose that we desired to know the location of the head on Friday night, the second of Iyar of this year - the starting point for these calculations. There are 29 complete days between the night of the starting point and the date for which we desire to know the location of the head.

We should then calculate the mean of the head according to the familiar manner, adding its distance traveled in 29 days to the starting point. Thus, the mean of the head is 182 degrees, 29 minutes and 37 seconds, in symbols 182° 29' 37". This mean should be subtracted from 360, leaving a remainder of 177 degrees, 30 minutes and 23 seconds, in symbols 177° 30' 23". This is the location of the head. The seconds are of no consequence. Thus, the position of the head will be 27 degrees and 30 minutes within the constellation of Virgo. The position of the tail will be [directly] opposite it: 27 degrees and 30 minutes within the constellation of Pisces.

There will always be an even half of the celestial sphere between the position of the head and the position of the tail. Therefore, whenever the head is in a particular constellation, the tail will be seven constellations further in the order of constellations, at the same position with regard to degrees and minutes. For example, if the head is ten degrees within a particular constellation, the tail will be ten degrees within the seventh constellation from it.

After having established the position of the head, the position of the tail, and the true position of the moon, consider [these three figures]: If the position of the moon is the same, both in degrees and in minutes, as its head or tail, then the moon will not be inclined to the north or the south. If the position of the moon has passed the head and it is proceeding in the direction of the tail, know that the moon will be inclined to the north of the [plane] of the sun's [orbit]. If the position of the moon is before the tail and it is proceeding in the direction of the head, know that the moon will be inclined to the south of the [plane] of the sun's [orbit].

The inclination of the moon to the north or to the south is referred to as the moon's latitude. If the moon's incline is northerly, it is referred to as a northerly latitude. If the moon's incline is southerly, it is referred to as a southerly latitude. If the moon is positioned at either [the head or the tail], it has no latitude, as explained above.

The moon's latitude will never exceed five degrees, whether to the north or to the south. This is the pattern it follows. [The moon] begins at the head and diverges slightly [from the sun's orbit, as it proceeds on its own orbit]. [The size of] this divergence continues to increase until it reaches five degrees. At this point, [the moon] begins to come slightly closer [to the sun's orbit], until it has no latitude at all when it reaches its tail. [After it reaches the tail, the moon] will again begin to diverge slightly [from the sun's orbit], until this divergence reaches five degrees. It will then begin to approach [the sun's orbit], until ultimately it has no latitude at all.

[The following procedure should be applied] if you desire to determine the latitude of the moon, and to [know] whether it is northerly or southerly: First, calculate the position of the head and the true position of the moon at the desired date. Then subtract the position of the head from the true position of the moon. The remainder is referred to as "the course of the latitude." If the course of the latitude is between one degree and 180 degrees, the latitude of the moon is northerly. If course of the latitude is more than 180 degrees, the latitude of the moon is southerly. If [the course] is an even 180 degrees or an even 360 degrees, the moon does not have any latitude at all. Afterwards, determine the size of the angle of the course of the latitude - i.e., the extent to which the moon is inclined to the north or to the south. This figure is referred to as the moon's southerly latitude or northerly [latitude], as we explained.

How large is the angle of the course of the latitude? If the course of the latitude is ten degrees, its angle will be 52 minutes. If the course is twenty degrees, its angle will be one degree and 43 minutes. If the course is thirty degrees, its angle will be 2 degrees and 30 minutes. If the course is forty degrees, its angle will be 3 degrees and 13 minutes. If the course is fifty degrees, its angle will be 3 degrees and 50 minutes. If the course is sixty degrees, its angle will be 4 degrees and 20 minutes. If the course is seventy degrees, its angle will be 4 degrees and 42 minutes. If the course is eighty degrees, its angle will be 4 degrees and 55 minutes. If the course is ninety degrees, its angle will be 5 degrees.

If [the course of the latitude] has both units and tens, you should [calculate the average increase per degree and add the proportionate amount to the lower figure], as was done with regard to the course of the sun and the course of the moon. What is implied? When the course of the latitude is 53 degrees, [the size of the angle should be determined as follows]. It has already been established that when the course is 50 degrees, its angle is 3 degrees and 50 minutes. When the course is 60 degrees, its angle is 4 degrees and 20 minutes. Therefore, there is a difference of 30 minutes between them, 3 minutes for each degree. Accordingly, [when] calculating the angle for a course of 53 degrees, [the result] will be 3 degrees and 59 minutes. A similar process should be followed with regard to all other figures.

Since you know the angles for all the values of the course of the latitude until ninety degrees, as was mentioned, you will be able to know the angle for all possible values of the course. For if the course is between 90 and 180 degrees, subtract the course from 180 and find the angle for the remainder.

Similarly, if the course is between 180 and 270 degrees, subtract 180 from [the course] and find the angle for the remainder.

Similarly, if the course is between 270 and 360 degrees, subtract [the course] from 360 and find the angle for the remainder.

What is implied? If the course is 150°, subtract 150 from 180, leaving 30. As mentioned previously, the angle [of a course] of 30 [degrees] will be 2 degrees and 30 minutes. Thus, the angle [of a course] of 150 [degrees] also will be 2 degrees and 30 minutes.

If the course is 200°, subtract 180 from 200, leaving 20. As mentioned previously, the angle [of a course] of 20 [degrees] will be 1 degree and 43 minutes. Thus, the angle [of a course] of 200 [degrees] also will be 1 degree and 43 minutes.

If the course is 300°, subtract 300 from 360, leaving 60. As mentioned previously, the angle [of a course] of 60 [degrees] will be 4 degrees and 20 minutes. Thus, the angle [of a course] of 300 [degrees] also will be 4 degrees and 20 minutes. A similar process should be followed with regard to all other values.

[The following procedure should be applied] if one desires to know the latitude of the moon and whether it is either northerly or southerly at the beginning of Friday night, the second of Iyar of this year: It has already been established that the true position of the moon on this night is 18 degrees and 36 minutes within the constellation of Taurus, in symbols 18° 36". [Similarly, it has been established that] the position of the head at that time is 27 degrees and 30 minutes within the constellation of Virgo, in symbols 27° 30'. [To arrive at the latitude,] you must subtract the position of the head from the position of the moon, leaving a course of the latitude of 231 degrees and 6 minutes, in symbols 231° 6'. [As mentioned,] the minutes are of no consequence with regard to the course. Therefore, according to the principles explained in this chapter, the angle of this course will be 3 degrees and 53 minutes. This is the latitude of the moon at the beginning of this night. It is southerly, for the course is larger than 180 degrees.

All the principles we have explained above were intended so that you will be ready and prepared to know [how] to sight [the moon]. When you desire to know [how to sight the moon on a particular night], calculate the true position of the sun, the true position of the moon, and the position of the head [of the moon's orbit] for the time of the sighting [of the moon]. Afterwards, subtract the position of the sun from the position of the moon. The remainder is referred to as the first longitude.

After having determined the position of the head and the moon's position, you will be able to determine the moon's latitude and whether this latitude is northerly or southerly. This figure is referred to as the first latitude. Be careful with regard to [these two symbols,] the first longitude and the first latitude, and have them at hand [for later calculations].

Consider the first longitude: If the figure you arrive at is equal to nine degrees or less, know that it will definitely be impossible for the moon to be sighted on that night throughout Eretz Yisrael; no other calculation is necessary. If the first longitude is more than fifteen degrees, know that the moon will definitely be sighted throughout Eretz Yisrael; no other calculation is necessary. If the first longitude is between nine and fifteen degrees, it will be necessary for you to make further calculations to know whether the moon will be sighted or not.

When does the above apply? When the true position of the moon is located [in the area] between the beginning of the constellation of Capricorn and the end of the constellation of Gemini. If, however, the position of the moon [is located in the range] between the beginning of the constellation of Cancer and the end of the constellation of Sagittarius, and the first longitude is ten degrees or less, know that it will definitely be impossible for the moon to be sighted on that night throughout Eretz Yisrael; no other calculation is necessary. If the first longitude is more than twenty-four degrees, know that the moon will definitely be sighted throughout Eretz Yisrael; no other calculation is necessary. If the first longitude is between ten and twenty-four degrees, it will be necessary to make further calculations to know whether or not the moon will be sighted.

These are the [further] calculations [necessary to determine] the sighting of the moon: Consider the constellation in which the moon is located. If it is the constellation of Aries, subtract 59 minutes from the first longitude. If it is the constellation of Taurus, subtract one degree from the longitude. If it is the constellation of Gemini, subtract 58 minutes from the longitude. If it is the constellation of Cancer, subtract 52 minutes from the longitude. If it is the constellation of Leo, subtract 43 minutes from the longitude. If it is the constellation of Virgo, subtract 37 minutes from the longitude. If it is the constellation of Libra, subtract 34 minutes from the longitude. If it is the constellation of Scorpio, subtract 34 minutes from the longitude. If it is the constellation of Sagittarius, subtract 36 minutes from the longitude. If it is the constellation of Capricorn, subtract 44 minutes from the longitude. If it is the constellation of Aquarius, subtract 53 minutes from the longitude. If it is the constellation of Pisces, subtract 58 minutes from the longitude.The remainder after these minutes have been subtracted from the longitude is referred to as the second longitude.

Why are these minutes subtracted? Because the true position of the moon is not the place where the moon will actually be sighted [in the sky]. Instead, there is a [small] difference in both longitude and latitude. This [difference] is referred to as the sighting adjustment. The sighting adjustment for the moon's longitude at the hour of the sighting of the moon should always be subtracted from the longitude, as we explained.

[The latter point is not necessarily true,] by contrast, with regard to the sighting adjustment for the moon's latitude.If the moon's latitude is northerly, we subtract the minutes of the sighting adjustment for the moon's latitude from the moon's first latitude. If, however, the moon's latitude is southerly, we add the minutes of the sighting adjustment for the moon's latitude to the moon's first latitude. The result after the addition or subtraction of these minutes to or from the [moon's] first latitude is referred to as the second latitude.

How many minutes are added or subtracted? If the moon is in the constellation of Aries, 9 minutes. If it is in the constellation of Taurus, 10 minutes. If it is in the constellation of Gemini, 16 minutes. If it is in the constellation of Cancer, 27 minutes. If it is in the constellation of Leo, 38 minutes. If it is in the constellation of Virgo, 44 minutes. If it is in the constellation of Libra, 46 minutes. If it is in the constellation of Scorpio, 45 minutes. If it is in the constellation of Sagittarius, 44 minutes. If it is in the constellation of Capricorn, 36 minutes. If it is in the constellation of Aquarius, 27 minutes. If it is in the constellation of Pisces, 12 minutes.

Since you know the [number of] minutes [for the adjustment] for each constellation, add or subtract them to or from the first latitude, as was explained, and arrive at the second latitude. You already know whether it will be northerly or southerly, and you will know the number of degrees and minutes of this second latitude. Prepare this figure and have it at hand.

Afterwards, you must set aside [for later use in subtracting or adding] a portion of this second latitude, because the moon fluctuates slightly in its orbit. What is the size of the portion you must separate? [This depends on the position of the moon in the celestial sphere.] If the moon is located between the beginning of the constellation of Aries and its twentieth degree - and similarly, [when the moon is located] between the beginning of the constellation of Libra and its twentieth degree - separate two fifths of the second latitude. If the moon is located between the twentieth degree of the constellation of Aries and the tenth degree of the constellation of Taurus - and similarly, [when the moon is located] between the twentieth degree of the constellation of Libra and the tenth degree of the constellation of Scorpio - separate one third from the second latitude. If the moon is located between the tenth degree of the constellation of Taurus and its twentieth degree - and similarly, [when the moon is located] between the tenth degree of the constellation of Scorpio and its twentieth degree - separate one fourth from the second latitude. If the moon is located between the twentieth degree of the constellation of Taurus and its end - and similarly, [if the moon is located] between the twentieth degree of the constellation of Scorpio and its end - separate one fifth from the second latitude. If the moon is located between the beginning of the constellation of Gemini and its tenth degree - and similarly, [if the moon is located] between the beginning of the constellation of Sagittarius and its tenth degree - separate one sixth of the second latitude. If the moon is located between the tenth degree of the constellation of Gemini and its twentieth degree - and similarly, [if the moon is located] between the tenth degree of the constellation of Sagittarius and its twentieth degree - separate one twelfth from the second latitude. If the moon is located between the twentieth degree of the constellation of Gemini and its twenty-fifth degree - and similarly, [if the moon is located] between the twentieth degree of the constellation of Sagittarius and its twenty-fifth degree - separate one twenty-fourth from the second latitude. If the moon is located between the twenty-fifth degree of the constellation of Gemini and the fifth degree of the constellation of Cancer - and similarly, [if the moon is located] between the twenty-fifth degree of the constellation of Sagittarius and the fifth degree of the constellation of Capricorn - do not make any separation, because at this point the moon does not fluctuate at all from its orbit. If the moon is located between the fifth degree of the constellation of Cancer and its tenth degree - and similarly, [if the moon is located] between the fifth degree of the constellation of Capricorn and its tenth degree - separate one twenty-fourth from the second latitude. If the moon is located between the tenth degree of the constellation of Cancer and its twentieth degree - and similarly, [if the moon is located] between the tenth degree of the constellation of Capricorn and its twentieth degree - separate one twelfth from the second latitude. If the moon is located between the twentieth degree of the constellation of Cancer and its end - and similarly, [if the moon is located] between the twentieth degree of the constellation of Capricorn and its end - separate one sixth from the second latitude. If the moon is located between the beginning of the constellation of Leo and its tenth degree - and similarly, [if the moon is located] between the beginning of the constellation of the Aquarius and its tenth degree - separate one fifth of the second latitude. If the moon is located between the tenth degree of the constellation of Leo and its twentieth degree - and similarly, [if the moon is located] between the tenth degree of the constellation of Aquarius and its twentieth degree - separate one fourth from the second latitude. If the moon is located between the twentieth degree of the constellation of Leo and the tenth degree of the constellation of Virgo, and similarly, [if the moon is located] between the twentieth degree of the constellation of Aquarius and tenth degree of the constellation of Pisces - separate one third from the second latitude. If the moon is located between the tenth degree of the constellation of Virgo and its end - and similarly, [if the moon is located] between the tenth degree of the constellation of Pisces and its end - separate two fifths from the second latitude. This portion that is separated from the second latitude is referred to as the circuit of the moon.

Afterwards, go back and consider whether the latitude of the moon is northerly or southerly. If it is northerly, subtract [the adjustment referred to as] the circuit of the moon from the second longitude. If the moon's longitude is southerly, add the circuit of the moon to the second longitude. When does the above apply? When the moon's position is located between the beginning of the constellation of Capricorn and the end of the constellation of Gemini. If, however, the moon's position is located between the beginning of the constellation of Cancer and the end of the constellation of Sagittarius, the opposite is true: If the moon's latitude is northerly, the circuit should be added to the second longitude, and if the moon's latitude is southerly, the circuit should be subtracted from the second longitude. The remainder after the additions or subtractions have been made to the second longitude is referred to as the third longitude. Know that if there is no fluctuation within the circuit, and there is no figure to be separated from the second latitude, the second longitude also will serve as the third longitude, without any decrease or increase.

Afterwards, go back and see the constellation in which the third longitude - i.e., [the amended figure representing] the distance between the sun and the moon - is located: If it is located in the constellation of Pisces or Aries, add one sixth [of its length] to the third longitude. If the [third] longitude is located in the constellation of Aquarius or the constellation of Taurus, add one fifth [of its length] to the third longitude. If the [third] longitude is located in the constellation of Capricorn or the constellation of Gemini, add one sixth [of its length] to the third longitude. If the [third] longitude is located in the constellation of Sagittarius or the constellation of Cancer, leave the third longitude as it is, without making any addition or subtraction. If the [third] longitude is located in the constellation of Scorpio or the constellation of Leo, subtract one fifth [of its length] from the third longitude. If the [third] longitude is located in the constellation of Libra or the constellation of Virgo, subtract one third [of its length] from the third longitude. The figure resulting from these subtractions or additions to the third longitude, or from leaving it without adjustment, is referred to as the fourth longitude. [Afterwards, a further correction is necessary:] Return to the first latitude, and set aside two thirds [of its length]. This is called the correction [resulting from geographic] latitude. Consider whether the latitude of the moon is northerly. If so, add the correction [resulting from geographic] latitude to the fourth longitude. If the latitude of the moon is southerly, subtract the correction [resulting from geographic] latitude from the fourth longitude. The figure resulting from these subtractions or additions to the fourth longitude is referred to as the arc of sighting.

What is implied? For example, let us attempt to determine whether or not it will be possible to sight the moon on Friday night, the second of Iyar of this year: First, it is necessary to determine the true position of the sun, the true position of the moon, and the moon's latitude for this time according to the methods that were disclosed [in the previous chapters]. The result is that the true position of the sun is seven degrees and nine minutes in the constellation of Taurus, in symbols 7° 9'. The true position of the moon is eighteen degrees and thirty-six minutes in the constellation of Taurus, in symbols 18° 36'. The moon's latitude is three degrees and fifty-three minutes, in symbols 3° 53', and it is southerly. This is the first latitude. Afterwards, you should subtract the position of the sun from the position of the moon, arriving at a remainder of eleven degrees and twenty-seven minutes, in symbols 11° 27'. This is the first longitude. Since the moon is located in the constellation of Taurus, the sighting adjustment for the longitude will be one degree. This should be subtracted from the first longitude, producing a second longitude of 10 degrees and twenty-seven minutes, in symbols 10° 27'. The sighting adjustment for the latitude is ten minutes. Since the moon's latitude is southerly, this sighting adjustment of ten minutes should be added to the moon's latitude, producing a second latitude of four degrees and three minutes, in symbols 4° 3'. Since the moon is located in the eighteenth minute of the constellation of Taurus, it is proper to set aside one fourth of the second latitude as the circuit of the moon. Thus, at this time, the circuit of the moon will be one degree and one minute. We pay no attention to the seconds.

Since the latitude of the moon is southerly and the true position of the moon is located between the beginning [of the constellation] of Capricorn and the beginning [of the constellation] of Cancer, it is correct to add the circuit of the moon to the second longitude. Thus, the third longitude will be eleven degrees and twenty-eight minutes, in symbols 11° 28'. Since this longitude is located in the constellation of Taurus, it is fitting to add one fifth to the third longitude - i.e., two degrees and eighteen minutes. Thus, the fourth longitude will be thirteen degrees and forty-six minutes, in symbols 13° 46'. We then go back to the first latitude and separate two thirds of it. Thus, the correction [resulting from geographic] latitude is two degrees and thirty-five minutes. Since the latitude is southerly, the correction [resulting from geographic] latitude should be subtracted from the fourth longitude, leaving a remainder of eleven degrees and eleven minutes, in symbols 11° 11'. This is the arc of sighting on this night. Following this process, you can determine the number of degrees and the number of minutes every night for the moon's sighting, which you desire for all time.

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.

From all the above, you have seen the extent of the calculations, and the additions and the subtractions that required much effort to present a method that comes close [to being exact] without necessitating extremely complicated calculations. [This process is necessary] because the moon has major incongruities in its orbit. In this vein, our Sages said, "'The sun knows the time of its setting'; the moon does not know the time of its setting." Similarly, our Sages said, "At times, its setting is prolonged, and at times, it is hastened." This is reflected in these calculations, where at times it is necessary to add, and at times it is necessary to subtract until we arrive at the arc of sighting. And as explained, at times the arc of sighting is great, and at times it is small.

The rationales for all these calculations, and the reasons why this number is added, and why that subtraction is made, and how all these concepts are known, and the proofs for each of these principles are [the subject] of the wisdom of astronomy and geometry, concerning which the Greeks wrote many books. These texts are presently in the hands of the sages. The texts written by the Sages of Israel in the age of the prophets from the tribe of Yissachar have not been transmitted to us. Nevertheless, since these concepts can be proven in an unshakable manner, leaving no room for question, the identity of the author, be he a prophet or a gentile, is of no concern. For a matter whose rationale has been revealed and has proven truthful in an unshakable manner, we do not rely on [the personal authority of] the individual who made these statements or taught these concepts, but on the proofs he presented and the reasons he made known.

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 G-d, read it. None of these will be lacking" [Isaiah 34:16].

The End of the Laws of the Sanctification of the Month.