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Rambam - 3 Chapters a Day

Kiddush HaChodesh - Chapter Twelve, Kiddush HaChodesh - Chapter Thirteen, Kiddush HaChodesh - Chapter Fourteen

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Kiddush HaChodesh - Chapter Twelve

1

The mean distance traveled by the sun in one day - i.e., in twenty-four hours - is 59 minutes and 8 seconds; in symbols 59' 8".1 Thus, in ten days, it travels 9 degrees, 51 minutes and 23 seconds,2 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 explained3 - 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.4

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.5 [This is what concerns us,] for our sole desire in these calculations is to know [when the moon] will be sighted.6

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.7

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".

א

מהלך השמש האמצעי ביום אחד שהוא כ"ד שעות נ"ט חלקים ושמנה שניות. סימנם כ"ד נט"ח. נמצא מהלכה בעשרה ימים תשע מעלות ונ"א חלקים וכ"ג שניות. סימנם טנ"א כ"ג. ונמצא מהלכה במאה יום צ"ח מעלות ושלשה ושלשים חלקים ונ"ג שניות, סימנם צ"ח ל"ג נ"ג. ונמצא שארית מהלכה באלף יום אחר שתשליך כל ש"ס מעלות כמו שביארנו. רס"ה מעלות ול"ח חלקים ונ' שניות. סימנם רס"ה לח"ן. ונמצא שארית מהלכה בעשרת אלפים יום. קל"ו מעלות וכ"ח חלקים וכ' שניות. סימנם קל"ו כ"ח כ'. ועל הדרך הזה תכפול ותוציא מהלכה לכל מנין שתרצה. וכן אם תרצה לעשות סימנין ידועים אצלך למהלכה לשני ימים ולשלשה ולארבעה עד עשרה תעשה. וכן אם תרצה להיות לך סימנין ידועים מוכנין למהלכה לכ' יום ולל' ולמ' עד מאה תעשה. ודבר גלוי הוא וידוע מאחר שידעת מהלך יום אחד. וראוי הוא להיות מוכן וידוע אצלך מהלך אמצע השמש לכ"ט יום ולשנ"ד יום שהן ימי שנת הלבנה בזמן שחדשיה כסדרן. והיא הנקראת שנה סדורה.שבזמן שיהיו לך אמצעיות אלו מוכנין יהיה החשבון הזה קל עליך לראיית החדש. לפי שכ"ט יום גמורים מליל הראיה עד ליל הראיה של חדש הבא וכן בכל חדש וחדש אין פחות מכ"ט יום לא יותר. שאין חפצנו בכל אלו החשבונות אלא לדעת הראיה בלבד. וכן מליל הראיה של חדש זה עד ליל הראיה לאותו החדש לשנה הבאה שנה סדורה או שנה ויום אחד. וכן בכל שנה ושנה. ומהלך השמש האמצעי לכ"ט יום כ"ח מעלות ול"ה חלקים ושניה אחת. סימנן כ"ח ל"ה א'. ומהלכה לשנה סדורה שמ"ח מעלות ונ"ה חלקים וט"ו שניות סימנן שמ"ח נ"ה ט"ו:

2

There is one point in the orbit of the sun around the Earth - and similarly, in the orbits of the remainder of the seven stars [around the Earth] - when [the sun or] that star will be furthest removed from the Earth.8 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.9 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".10

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.11 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.12

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 months13 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,14 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.

ב

נקודה אחת יש בגלגל השמש וכן בשאר גלגלי השבעה כוכבים. בעת שיהיה הכוכב בה יהיה גבוה מעל הארץ כל מאורו. ואותה הנקודה של גלגל השמש ושאר הכוכבים חוץ מן הירח סובבת בשוה. ומהלכה בכל שבעים שנה בקירוב מעלה אחת. ונקודה זו היא הנקראת גובה השמש. מהלכו בכל עשרה ימים שניה אחת וחצי שניה שהיא ל' שלישיות. נמצא מהלכו בק' יום ט"ו שניות. ומהלכו באלף יום שני חלקים ושלשים שניות.ומהלכו בעשרת אלפים יום כ"ה חלקים. ונמצא מהלכו לתשעה ועשרים יום ארבע שניות ועוד. ומהלכו בשנה סדורה נ"ג שניות. כבר אמרנו שהעיקר שממנו התחלת חשבון זה הוא מתחלת ליל חמישי שיומו שלישי לחדש ניסן משנת תתקל"ח וארבעת אלפים ליצירה. ומקום השמש במהלכה האמצעי היה בעיקר הזה בשבע מעלות ושלשה חלקים ול"ב שניות ממזל טלה. סימנן ז"ג ל"ב. ומקום גובה השמש היה בעיקר זה בכ"ו מעלות מ"ה חלקים ושמנה שניות ממזל תאומים. סימנם כ"ו מ"ה ח'. כשתרצה לידע מקום השמש במהלכה האמצעי בכל זמן שתרצה. תקח מנין הימים שמתחלת יום העיקר עד היום שתרצה. ותוציא מהלכה האמצעי באותן הימים מן הסימנין שהודענו. והוסיף הכל על העיקר ותקבץ כל מין עם מינו. והיוצא הוא מקום השמש במהלכה האמצעי לאותו היום. כיצד הרי שרצינו לידע מקום השמש האמצעי בתחלת ליל השבת שיומו ארבעה עשר לחדש תמוז משנה זו שהיא שנת העיקר. מצאנו מנין הימים מיום העיקר עד תחלת היום זה שאנו רוצים לידע מקום השמש בו מאה יום. לקחנו אמצע מהלכה לק' יום שהוא צ"ח ל"ג נ"ג והוספנו על העיקר שהוא ז"ג ל"ב. יצא מן החשבון מאה וחמש מעלות ול"ז חלקים וכ"ה שניות. סימנן ק"ה ל"ז כ"ה. ונמצא מקומה במהלך אמצעי בתחלת ליל זה במזל סרטן בט"ו מעלות בו ול"ז חלקים ממעלה ט"ז. והאמצעי שיצא בחשבון זה פעמים יהיה בתחלת הלילה בשוה. או קודם שקיעת החמה בשעה. או אחר שקיעת החמה בשעה. ודבר זה לא תחוש לו בשמש בחשבון הראיה. שהרי אנו משלימים קירוב זה כשנחשוב לאמצע הירח. ועל הדרך הזאת תעשה תמיד לכל עת שתרצה ואפילו אחר אלף שנים. שתקבץ כל השארית ותוסיף על העיקר יצא לך המקום האמצעי. וכן תעשה באמצע הירח ובאמצע כל כוכב וכוכב. מאחר שתדע מהלכו ביום אחד כמה הוא ותדע העיקר שממנו תתחיל. ותקבץ מהלכו לכל השנים והימים שתרצה ותוסיף על העיקר ויצא לך מקומו במהלך אמצעי. וכן תעשה בגובה השמש תוסיף מהלכו באותם הימים או השנים על העיקר יצא לך מקום גובה השמש לאותו היום שתרצה. וכן אם תרצה לעשות עיקר אחר שתתחיל ממנו חוץ מעיקר זה שהתחלנו ממנו בשנה זו. כדי שיהיה אותו עיקר בתחלת שנת מחזור ידוע. או בתחלת מאה מן המאות. הרשות בידך. ואם תרצה להיות העיקר שתתחיל ממנו משנים שעברו קודם עיקר זה לאחר כמה שנים מעיקר זה הדרך ידועה. כיצד היא הדרך. כבר ידעת מהלך השמש לשנה סדורה ומהלכה לכ"ט יום ומהלכה ליום אחד. ודבר ידוע שהשנה שחדשיה שלמים היא יתרה על הסדורה יום אחד. והשנה שחדשיה חסרין היא חסרה מן הסדורה יום אחד. והשנה המעוברת אם יהיו חדשיה כסדרן תהיה יתרה על השנה הסדורה שלשים יום. ואם יהיו חדשיה שלמים היא יתרה על הסדורה ל"א יום. ואם יהיו חדשיה חסרין היא יתרה על הסדורה כ"ט יום. ומאחר שכל הדברים האלו ידועים תוציא מהלך אמצע השמש לכל השנים והימים שתרצה ותוסיף על העיקר שעשינו. יצא לך אמצעה ליום שתרצה משנים הבאות. ותעשה אותו היום עיקר. או תגרע האמצע שהוצאת מן העיקר שעשינו ויצא לך העיקר ליום שתרצה משנים שעברו. ותעשה אותו אמצע העיקר. וכזה תעשה באמצע הירח ושאר הכוכבים אם יהיו ידועים לך. וכבר נתבאר לך מכלל דברינו שכשם שתדע אמצע השמש לכל יום שתרצה מימים הבאים כך תדע אמצעה לכל יום שתרצה מימים שעברו

Footnotes
1.

Since the sun travels throughout the entire 360° sphere over the course of a solar year, and a year is slightly longer than 365 days, the daily distance the sun travels is slightly less than one degree - more precisely, 59 minutes, 8 seconds and 19.8 thirds. Although the Rambam does not mention the thirds in this figure, he includes them in his subsequent calculations.

2.

When performing simple multiplication, the sum appears to be three seconds less. These three seconds have been added because of the inclusion of the multiples of the thirds, as mentioned in the previous note. Similarly, in subsequent calculations the Rambam also adds the multiples of the thirds.

3.

See Chapter 11, Halachah 10.

4.

See Chapter 8, Halachah 6, which explains that a year in which all the months follow in order, one full and one lacking, is referred to as a regular year.

5.

A lunar month is slightly longer than 29 days. Therefore, potential witnesses endeavor to sight the moon in the heavens on the night between the twenty-ninth and thirtieth days.

6.

Indeed, many of the subsequent calculations mentioned by the Rambam may be accurate only on the first night of the month and may not be accurate on the subsequent nights.

7.

The one day is added when both the months of Marcheshvan and Kislev are full. The commentaries raise the question why the Rambam does not mention the possibility of the year being lacking a day, as occurs when Marcheshvan and Kislev are both lacking.

8.

As stated in Chapter 11, Halachah 13, the Earth is not in the exact center of the orbits of the sun, the moon, or the other five planets. Therefore, there is one point in their orbits where they are furthest removed from the Earth. The knowledge of the location of this point is significant in calculating the true position of the sun, as will be explained in the following chapter.

9.

As the Rambam mentions in Hilchot Yesodei HaTorah 3:3, not only do the sun and the stars move in their orbits, the orbits themselves move in the heavens. This movement can be seen most clearly by charting the movement of the apogee, the point in the orbit furthest from the Earth. The movement of the sun's orbit and similarly, that of the other stars, is relatively slow. The moon's orbit, by contrast, is moving at a much faster pace, as mentioned in the notes on Chapter 14, Halachah 1.

10.

Since more than 800 years have passed since the composition of the Mishneh Torah, the apogee of the sun has moved approximately twelve degrees and is presently located in the constellation of Cancer.

11.

Since, as explained in the previous chapter, the mean distance does not represent the place where the sun can actually be seen in the sky, there will be a slight discrepancy. The mean position represents the sun's position at 6 PM. During the summer months, the sun will reach that position before sunset, and during the winter months, it will reach that position after sunset.

12.

See the conclusion of Chapter 14.

13.

I.e., both Marcheshvan and Kislev. See Chapter 8, Halachot 6-10, for the ground rules regarding the determination of when a year is regular, when its months are full, and when they are lacking.

14.

See Chapter 6, Halachah 11, which relates that seven of the years in a nineteen-year cycle are leap years, and states which of these years will be leap years.

Kiddush HaChodesh - Chapter Thirteen

1

[The following method should be used] if you wish to know the true position1 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.2 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.3

א

אם תרצה לידע מקום השמש האמתי בכל יום שתרצה. תוציא תחלה מקומה האמצעי לאותו היום על הדרך שביארנו. ותוציא מקום גובה השמש. ותגרע מקום גובה השמש ממקום השמש האמצעי והנשאר הוא הנקרא מסלול השמש:

2

[The next step is] to calculate the angular distance of the course of the sun.4 If the angular distance of the course is less than 180 degrees, one should subtract5 the angle [determined by the] course6 from the sun's mean position. If the angular distance of the course is more than 180 degrees, one should add7 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.

ב

ותראה כמה מעלות הוא מסלול השמש. אם היה המסלול פחות מק"פ מעלות. תגרע מנת המסלול ממקום השמש האמצעי. ואם היה המסלול יותר על ק"פ מעלות עד ש"ס תוסיף מנת המסלול על מקום השמש האמצעי. ומה שיהיה אחר שתוסיף עליו או תגרע ממנו הוא המקום האמתי:

3

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.8

ג

ודע שאם יהיה המסלול מאה ושמונים בשוה או ש"ס בשוה. אין לו מנה אלא יהיה המקום האמצעי הוא המקום האמתי:

4

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.9

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.

ד

וכמה היא מנת המסלול. אם יהיה המסלול עשר מעלות. תהיה מנתו כ' חלקים. ואם יהיה כ' מעלות תהיה מנתו מ' חלקים. ואם יהיה ל' מעלות תהיה מנתו נ"ח חלקים. ואם יהיה מ' מעלות תהיה מנתו מעלה אחת וט"ו חלקים. ואם יהיה נ' מעלות תהיה מנתו מעלה אחת וכ"ט חלקים. ואם יהיה ס' מעלות תהיה מנתו מעלה אחת ומ"א חלקים. ואם יהיה ע' מעלות תהיה מנתו מעלה אחת ונ"א חלקים. ואם יהיה פ' מעלות תהיה מנתו מעלה ונ"ז חלקים. ואם יהיה צ' מעלות תהיה מנתו מעלה אחת ונ"ט חלקים. ואם יהיה ק' מעלות תהיה מנתו מעלה אחת ונ"ח חלקים. ואם יהיה ק"י תהיה מנתו מעלה אחת ונ"ג חלקים. ואם יהיה ק"כ תהיה מנתו מעלה אחת ומ"ה חלקים. ואם יהיה ק"ל תהיה מנתו מעלה אחת ול"ג חלקים. ואם יהיה ק"מ תהיה מנתו מעלה אחת וי"ט חלקים. ואם יהיה ק"נ תהיה מנתו מעלה אחת וחלק אחד. ואם יהיה ק"ס תהיה מנתו מ"ב חלקים. ואם יהיה ק"ע תהיה מנתו כ"א חלקים. ואם יהיה ק"פ בשוה אין לו מנה כמו שביארנו אלא מקום השמש האמצעי הוא מקומה האמתי:

5

[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.10

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.

ה

היה המסלול יותר על ק"פ מעלות. תגרע אותו מש"ס מעלות ותדע מנתו. כיצד הרי שהיה המסלול ר' מעלות. תגרע אותו מש"ס תשאר ק"ס מעלות. וכבר הודענו שמנת ק"ס מעלות מ"ב חלקים. וכן מנת המאתים מ"ב חלקים:

6

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.

ו

וכן אם היה המסלול ש' מעלות. תגרע אותו מש"ס ישאר ס'. וכבר ידעת שמנת ס' מעלות מעלה אחת ומ"א חלקים. וכן היא מנת הש' מעלות. ועל דרך זו בכל מנין ומנין:

7

[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.11

ז

הרי שהיה המסלול ס"ה מעלות. וכבר ידענו שמנת הששים היא מעלה אחת ומ"א חלקים. ומנת הע' היא מעלה אחת ונ"א חלקים. נמצא בין שתי המנות י' חלקים. ולפי חשבון המעלות יהיה לכל מעלה חלק אחד. ויהיה מנת המסלול שהוא ס"ה מעלה אחת ומ"ו חלקים:

8

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.

ח

וכן אילו היה המסלול ס"ז היתה מנתו מעלה אחת ומ"ח חלקים. ועל דרך זו תעשה בכל מסלול שיהיה במניינו אחדים עם העשרות. בין בחשבון השמש בין בחשבון הירח:

9

[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,12 105° 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.

ט

כיצד הרי שרצינו לידע מקום השמש האמתי בתחלת ליל השבת י"ד יום לחדש תמוז משנה זו. תוציא אמצע השמש תחלה לעת הזאת. וסימנו ק"ה ל"ז כ"ח כמו שביארנו. ותוציא מקום גובה השמש לעת הזאת. יצא לך סימנו פ"ו מ"ה כ"ג. ותגרע מקום הגובה מן האמצעי. יצא לך המסלול י"ח מעלות ונ"ב חלקים ושתי שניות. סימנם י"ח נ"ב ב'. ואל תקפיד בכל מסלול על החלקים אלא אם יהיו פחות משלשים אל תפנה אליהם. ואם היו שלשים או יותר תחשב אותם מעלה אחת ותוסיף אותה על מנין מעלות המסלול. לפיכך יהיה מסלול זה י"ט מעלות ותהיה מנתו על הדרך שביארנו ל"ח חלקים:

10

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 approximately13 thirty [or more], they should be considered a minute, and added to the sum of the minutes.

י

ולפי שהמסלול הזה היה פחות מק"פ. תגרע המנה שהיא ל"ח חלקים מאמצע השמש ישאר ק"ד מעלות ונ"ט חלקים וכ"ה שניות. סימנם ק"ד נ"ט כ"ה. ונמצא מקום השמש האמתי בתחלת ליל זה במזל סרטן בט"ו מעלות בו פחות ל"ה שניות. ואל תפנה אל השניות כלל לא במקום השמש ולא במקום הירח ולא בשאר חשבונות הראיה. אלא חקור על החלקים בלבד. ואם יהיו השניות קרוב לשלשים עשה אותם חלק אחד והוסיפו על החלקים:

11

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,14 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.

יא

ומאחר שתדע מקום השמש בכל עת שתרצה. תדע יום התקופה האמיתי כל תקופה שתרצה. בין תקופות הבאות אחר עיקר זה שממנו התחלנו. בין תקופות שעברו משנים קדמוניות

Footnotes
1.

As stated in Chapter 11, the true position of the sun refers to the position at which it is seen in the heavenly sphere. The difference between the sun's true position and its mean position stems from the fact that the Earth is not located at the exact center of the sun's orbit.

2.

The method through which both these figures can be calculated is stated in the previous chapter.

3.

I.e., the path the sun has traveled in its orbit from the apogee until it reached its present position.

4.

The course of the sun is an arc extending from the mean position of the sun to its apogee. The angular distance of the course is derived by drawing straight lines from the mean position and the apogee to the center of the sun's orbit.

If the apogee of the sun were located at O°, the angular distance of the course and the sun's mean position would be the same. Since, however, the apogee also moves within the heavenly sphere, there is a variance between these two figures.

5.

Before the sun reaches the perigee, the point in its orbit that is closest to the Earth, its true position will always be less than its mean position. Hence, the angle referred to as the angle [determined by the] course must be subtracted from its mean position to arrive at the true position.

6.

The angle [determined by the] course refers to the extent of the deviation between the position of the sun that can be observed in the sky and its mean position. The manner of determining this figure is described in Halachah 4.

To express these concepts in geometric terms: The sun's true position represents the angle at which it can be found in the sphere of the heavens of which the Earth is the center (c). When the angle that is called the angle [determined by the] course (a) is added to this angle, the sum is equivalent to the angle of the course of the sun (b).

How is this figure derived? Refer to the accompanying diagram: The sum of the angles of the triangle a, c, and d equals 180°, and the angles b and d equal 180°. Hence, b equals c a.

Thus, the true position of the sun is equivalent to its mean position minus the figure referred to as the angle [determined by the] course. Thus, when the course is less than 180 degrees, the sun's true position is always a small amount less than its mean position.

7.

When the sun passes the perigee, its true position will always be greater than its mean position. Hence, the angle referred to as the angle [determined by the] course must be added to its mean position to arrive at the true position.

Why is this so? Refer to the following diagram: d refers to the true position of the sun, b to its mean position and a to the angle referred to as the angle [determined by the] course. E to its true position minus 180° and f refers to its mean position minus 180°. a + f + c equals 180. E + c equals 180. Thus, a + f equals e. Hence, the mean position plus the angle [determined by the] course will be equal to the true position.

8.

I.e., when the sun is at the apogee or perigee, there will be one straight line between the Earth (the center of the heavenly sphere), the center of the sun's orbit, and the actual position of the sun.

9.

The largest angle determined by the course is when the angular distance of the course itself is 96 degrees - i.e., shortly after the mean position of the sun passes directly above the center of its orbit. After this point is reached, the angle begins to decrease.

Significantly, the rate of the decrease does not correspond exactly to the rate of increase as the angles approach 96 degrees. The reason for this difference is that, as stated above, the position of the Earth is not at the center of the sun's orbit. Hence, at 100 and 80 degrees, although the mean position of the sun has moved an equal distance from the center of its orbit, it has moved different distances from the Earth.

10.

Thus, our computations will be based on the negative of the angle measured previously. To put the Rambam's statements in layman's terms: The angle formed will be the same regardless of whether the mean position of the sun is measured in an increase from O° or a decrease from 360°.

11.

Although the correspondence between the angular length of the course and the angle [determined by the] course is not uniform over a large span, within a span of ten degrees the difference between the actual figure and the approximation arrived at by the Rambam is not of consequence.

12.

Chapter 12, Halachah 2.

13.

The commentaries have questioned the Rambam's use of the word "approximately." Our bracketed additions are made in that light.

14.

The calculation of the equinoxes and solstices is significant with regard to the determination of the calendar, as explained in Chapters 9 and 10. Since the vernal (spring) equinox takes place when, according to the sun's true motion, it enters the constellation of Aries, the date on which that takes place can be calculated for any particular year. Similarly, the summer solstice takes place when, according to the sun's true motion, it enters the constellation of Cancer, and that date can be calculated. Similar concepts apply regarding the autumnal equinox and the winter solstice.

Kiddush HaChodesh - Chapter Fourteen

1

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.1 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".2

א

הירח שני מהלכים אמצעיים יש לו. הירח עצמו מסבב בגלגל קטן שאינו מקיף את העולם כולו. ומהלכו האמצעי באותו הגלגל הקטן נקרא אמצעי המסלול. והגלגל הקטן עצמו מסבב בגלגל גדול המקיף את העולם. ובמהלך אמצעי זה של גלגל הקטן באותו הגלגל הגדול המקיף את העולם הוא הנקרא אמצע הירח. מהלך אמצע הירח ביום אחד י"ג מעלות וי' חלקים ול"ה שניות. סימנם י"ג יל"ה:

2

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]3of its progress in one hundred days will be 237 degrees, 38 minutes and 23 seconds, in symbols 237° 38' 23".4

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".5The 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.

ב

נמצא מהלכו בעשרה ימים קל"א מעלות ומ"ה חלקים וחמשים שניות. סימנם קל"א מה"נ. ונמצא שארית מהלכו בק' יום רל"ז מעלות ול"ח חלקים וכ"ג שניות. סימנם רל"ז ל"ח כ"ג. ונמצאת שארית מהלכו באלף יום רי"ו מעלות וכ"ג חלקים ונ' שניות. סימנם רי"ו כג"ן. ונמצא שארית מהלכו בי' אלפים יום ג' מעלות ונ"ח חלקים וכ' שניות. סימנם ג' נ"ח כ'. ונמצא שארית מהלכו בכ"ט יום כ"ב מעלות וששה חלקים ונ"ו שניות. סימנם כ"ב ונ"ו. ונמצא שארית מהלכו בשנה סדורה שמ"ד מעלות וכ"ו חלקים ומ"ג שניות. סימן להם שד"ם כ"ו מ"ג. ועל דרך זו תכפול לכל מנין ימים או שנים שתרצה:

3

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".6 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".7

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".

ג

ומהלך אמצע המסלול ביום אחד י"ג מעלות ושלשה חלקים ונ"ד שניות. סימנם י"ג גנ"ד. נמצא מהלכו בעשרה ימים ק"ל מעלות ל"ט חלקים בלא שניות. סימנם ק"ל ל"ט. ונמצא שארית מהלכו במאה יום רכ"ו מעלות וכ"ט חלקים ונ"ג שניות. סימנם רכ"ו כ"ט נ"ג. ונמצא שארית מהלכו באלף יום ק"ד מעלות ונ"ח חלקים וחמשים שניות. סימנם ק"ד נח"ן. ונמצא שארית מהלכו בעשרת אלפים יום שכ"ט מעלות ומ"ח חלקים ועשרים שניות. סימנם שכ"ט מח"כ. ונמצא שארית מהלכו בכ"ט יום י"ח מעלות ונ"ג חלקים וד' שניות. סימנם י"ח נג"ד:

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".8

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].9

ד

ונמצא שארית מהלכו בשנה סדורה ש"ה מעלות וי"ג שניות בלא חלקים. סימנם ש"ה י"ג. מקום אמצע הירח היה בתחלת ליל חמישי שהוא העיקר לחשבונות אלו במזל שור מעלה אחת וי"ד חלקים ומ"ג שניות. סימנם א' י"ד מ"ג. ואמצע המסלול היה בעיקר זה פ"ד מעלות וכ"ח חלקים ומ"ב שניות. סימנם פ"ד כ"ח מ"ב. מאחר שתדע מהלך אמצע הירח והאמצע שהוא העיקר שעליו תוסיף. תדע מקום אמצע הירח בכל יום שתרצה על דרך שעשית באמצע השמש. ואחר שתוציא אמצע הירח לתחלת הלילה שתרצה התבונן בשמש ודע באי זה מזל הוא:

5

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.10 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.11 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.12 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.13

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.14 If the sun is located between midway in the constellation of Aries and the beginning of the constellation of Sagittarius, 15 minutes should be subtracted from the moon's mean.15 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.16 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.17

ה

אם היתה השמש מחצי מזל דגים עד חצי מזל טלה. תניח אמצע הירח כמות שהוא. ואם תהיה השמש מחצי מזל טלה עד תחלת מזל תאומים. תוסיף על אמצע הירח ט"ו חלקים. ואם תהיה השמש מתחלת מזל תאומים עד תחלת מזל אריה. תוסיף על אמצע הירח ט"ו חלקים. ואם תהיה השמש מתחלת מזל אריה עד חצי מזל בתולה תוסיף על אמצע הירח ט"ו חלקים. ואם תהיה השמש מחצי מזל בתולה עד חצי מזל מאזנים. הנח אמצע הירח כמות שהוא. ואם תהיה השמש מחצי מזל מאזנים עד תחלת מזל קשת. תגרע מאמצע הירח ט"ו חלקים. ואם תהיה השמש מתחלת מזל קשת עד תחלת מזל דלי. תגרע מאמצע הירח ל' חלקים. ואם תהיה השמש מתחלת מזל דלי עד חצי מזל דגים. תגרע מאמצע הירח ט"ו חלקים:

6

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 sun18 for the time when this mean was calculated. This is referred to as the mean of the moon at the time of the sighting.

ו

ומה שיהיה האמצע אחר שתוסיף עליו או תגרע ממנו או תניח אותו כמות שהוא. הוא אמצע הירח לאחר שקיעת החמה בכמו שליש שעה באותו הזמן שתוציא האמצע לו. וזה הוא הנקרא אמצע הירח לשעת הראיה

Footnotes
1.

As mentioned in Chapter 11, the rate of the advance of the sun, the moon, and the other planets does not appear to be uniform. For the sun, the deviation is relatively minor and can be resolved by postulating that the Earth is not at the center of the sun's orbit. The deviations of the moon from its mean rate of advance, however, are larger than that of the sun, and more irregular. (According to modern science, these deviations result from the gravitational pull of the sun and other celestial bodies.)

To resolve this difficulty, some ancient astronomers (Ptolemy and Aristotle, among others) postulated that with regard to the moon, two orbits were involved: One orbit encompassed the Earth, although the Earth was not at its center. Around this orbit existed one (and according to some opinions, more than one) smaller orbit, within which the moon rotated. This smaller orbit is referred to as an epicycle. Because of the moon's position in this smaller orbit, it would appear to be either ahead of or behind the mean position of the center of this orbit.

2.

This refers to the rate of progress that is apparent to an observer on the Earth. In theory, however, this figure is a result of two different motions. The entire orbit of the moon is moving in the heavens. (The orbit of the sun is also moving, as reflected in the movement of the sun's apogee, as mentioned in Chapter 12, Halachah 2. The sun's orbit is moving at a very slow pace, one and a half seconds a day. In contrast, the moon's orbit moves much faster, slightly more than 11 degrees each day. This movement is from east to west, opposite to the movement of the heavenly sphere.)

Within this larger orbit revolves the epicycle, the smaller orbit around which the moon revolves. The epicycle is revolving at approximately 24 1/2 degrees a day, from west to east. Thus, an observer on the Earth would see the epicycle as moving 13 degrees and a fraction (i.e., 24 1/2 - 11 1/5) forward (eastward) in the heavenly sphere every day, as the Rambam states.

3.

I.e., after the multiples of 360 have been subtracted.

4.

It appears that the Rambam has added three seconds. This addition was made because the rate of progress also includes three thirds not mentioned in the original figure, but included in this calculation.

5.

On this basis, we can understand why a lunar month is slightly longer than 29 1/2 days. The mean distance traveled by the sun in 29 days is approximately 28 1/2 degrees (Chapter 12, Halachah 1), approximately 6 1/2 degrees more than the remainder of the progress of the moon's mean. This distance (and the additional approximately almost half a degree traveled by the sun during this time) is travelled by the moon's mean in slightly longer than twelve hours on the following day.

6.

This distance is figured east to west, opposite to the movement of the heavenly sphere.

7.

It appears that the Rambam has subtracted seven seconds. This subtraction was carried out because his figure for the rate of progress had been rounded off. In fact, the rate is seven thirds less than the figure mentioned originally. The lack of these thirds was taken into consideration in this calculation.

8.

Although we have followed the standard printed text of the Mishneh Torah and included this paragraph in Halachah 4, it is clearly part of the previous halachah.

9.

As mentioned in Chapter 12, Halachah 2, and notes, the sun does not always reach its mean position at sunset. In the summer, when the days are longer, it reaches its mean position slightly earlier, and in the winter slightly later. In the following halachah, the Rambam states the values that allow us to compensate for these differences.

10.

This corresponds to the month of Nisan, the time of the vernal equinox, when the sun sets at approximately 6 PM. Hence, there is no need to adjust the position of the moon's mean.

11.

This corresponds to the beginning of the summer, when the days are longer. Since the moon is moving slightly more than thirteen degrees per day away from the sun, its rate of progress per hour is thus slightly more than 30 minutes. When the sun's rate of progress per hour - for it is moving (eastward) in the same direction as the moon - is also taken into consideration, it is proper to consider the moon's progress as thirty minutes per hour. Thus, the Rambam is saying that in these months, the sun will set approximately half an hour after 6 PM.

12.

This corresponds to the middle of the summer, the longest days of the year. To compensate for the further delay in the setting of the sun, an additional fifteen minutes should be added to the moon's mean. [It must be noted that the number 30 in our translation is based on authentic manuscripts of the Mishneh Torah. Most of the standard published texts mention 15 minutes in this clause as well.]

13.

At this time of year, the summer days are beginning to become shorter. Hence, an adjustment of only fifteen minutes is necessary.

14.

This corresponds to the month of Tishrei, the time of the autumnal equinox, when the sun sets at approximately 6 PM. Hence, there is no need to adjust the position of the moon's mean.

15.

This represents the beginning of the winter, when the sun sets at an earlier time. Hence, rather than add minutes to the moon's mean, we subtract them.

16.

This period represents the middle of the winter, the shortest days of the year. To compensate for the further precipitance of the setting of the sun, an additional fifteen minutes should be subtracted from the moon's mean. [It must be noted that, in this instance as well, the number 30 in our translation is a deviation from the standard published texts, based on authentic manuscripts of the Mishneh Torah.]

17.

At this point, the days are beginning to get longer. Therefore, only a fifteen-minute adjustment is necessary.

18.

This is the time when the stars begin to appear in Eretz Yisrael.

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The Mishneh Torah was the Rambam's (Rabbi Moses ben Maimon) magnum opus, a work spanning hundreds of chapters and describing all of the laws mentioned in the Torah. To this day it is the only work that details all of Jewish observance, including those laws which are only applicable when the Holy Temple is in place. Participating in the one of the annual study cycles of these laws (3 chapters/day, 1 chapter/day, or Sefer Hamitzvot) is a way we can play a small but essential part in rebuilding the final Temple.
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