ב"ה

# Rambam - 3 Chapters a Day

## Kiddush HaChodesh - Chapter Nine, Kiddush HaChodesh - Chapter Ten, Kiddush HaChodesh - Chapter Eleven

Classes on 3 Chapters Rambam

## Kiddush HaChodesh - Chapter Nine

1

[There is a difference of opinion among] the Sages of Israel concerning the length of a solar year. Some Sages1 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.2 There is also a difference of opinion among the wise men of Greece and Persia concerning this matter.3

א

שְׁנַּת הַחַמָּה יֵשׁ מֵחַכְמֵי יִשְׂרָאֵל שֶׁאוֹמְרִים שֶׁהִיא שְׁסַ''ה יוֹם וּרְבִיעַ יוֹם שֶׁהוּא שֵׁשׁ שָׁעוֹת. וְיֵשׁ מֵהֶן שֶׁאוֹמְרִים שֶׁהוּא פָּחוֹת מֵרְבִיעַ הַיּוֹם. וְכֵן חַכְמֵי יָוָן וּפָרַס יֵשׁ בֵּינֵיהֶן מַחֲלֹקֶת בְּדָבָר זֶה:

2

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.4

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.

ב

מִי שֶׁהוּא אוֹמֵר שֶׁהִיא שְׁסַ''ה יוֹם וּרְבִיעַ יוֹם. יִשָּׁאֵר מִכָּל מַחֲזוֹר שֶׁל תְּשַׁע עֶשְׂרֵה שָׁנָה שָׁעָה אַחַת וְתפ''ה חֲלָקִים כְּמוֹ שֶׁאָמַרְנוּ. וְיִהְיֶה בֵּין תְּקוּפָה לִתְקוּפָה אֶחָד וְתִשְׁעִים יוֹם וְשֶׁבַע שָׁעוֹת וַחֲצִי שָׁעָה. וּמִשֶּׁתֵּדַע תְּקוּפָה אַחַת בְּאֵי זֶה יוֹם בְּאֵיזוֹ שָׁעָה הִיא תַּתְחִיל לִמְנוֹת מִמֶּנָּה לַתְּקוּפָה הַשְּׁנִיָּה שֶׁאַחֲרֶיהָ וּמִן הַשְּׁנִיָּה לַשְּׁלִישִׁית עַד סוֹף הָעוֹלָם:

3

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 - 642.5

ג

תְּקוּפַת נִיסָן הִיא הַשָּׁעָה וְהַחֵלֶק שֶׁתִּכָּנֵס בּוֹ הַשֶּׁמֶשׁ בְּרֹאשׁ מַזַּל טָלֶה. וּתְקוּפַת תַּמּוּז הֱיוֹת הַשֶּׁמֶשׁ בְּרֹאשׁ מַזַּל סַרְטָן. וּתְקוּפַת תִּשְׁרֵי הֱיוֹת הַשֶּׁמֶשׁ בְּרֹאשׁ מַזַּל מֹאזְנַיִם. וּתְקוּפַת טֵבֵת הֱיוֹת הַשֶּׁמֶשׁ בְּרֹאשׁ מַזַּל גְּדִי. וּתְקוּפַת נִיסָן הָיְתָה בַּשָּׁנָה הָרִאשׁוֹנָה שֶׁל יְצִירָה לְפִי חֶשְׁבּוֹן זֶה קֹדֶם מוֹלַד נִיסָן בְּשִׁבְעָה יָמִים וְתֵשַׁע שָׁעוֹת וְתרמ''ב חֲלָקִים, סִימָנָהּ ז''ט תרמ''ב:

4

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].6 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.7 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,8, and then group the entire sum [into days and hours].9

Afterwards, subtract seven days, nine hours, and 642 units [from this sum]10 and divide the remainder into lunar months of 29 days, 12 hours, and 793 units.11 [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.12 The summer solstice will always take place at either 7:30 PM, 1:30 AM, 7:30 AM, or 1:30 PM.13 The autumnal equinox will always take place either at nine or at three o'clock, either in the day or the night.14 The winter solstice will always take place either at 10:30 PM, 4:30 AM, 10:30 AM, or 4:30 PM.15

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.16 Add one day and six hours17 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.18 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.19

ד

דֶּרֶךְ חֶשְׁבּוֹן הַתְּקוּפָה כָּךְ הִיא. תֵּדַע תְּחִלָּה כַּמָּה מַחֲזוֹרִין שְׁלֵמִים מִשְּׁנַת הַיְצִירָה עַד מַחֲזוֹר שֶׁתִּרְצֶה. וְקַח לְכָל מַחֲזוֹר מֵהֶן שָׁעָה אַחַת וְתפ''ה חֲלָקִים. קַבֵּץ כָּל הַחֲלָקִים שָׁעוֹת וְכָל הַשָּׁעוֹת יָמִים וְתִגְרַע מִן הַכּל שִׁבְעָה יָמִים וְתֵשַׁע שָׁעוֹת וְתרמ''ב חֲלָקִים. וְהַשְּׁאָר תּוֹסִיף אוֹתוֹ עַל מוֹלַד נִיסָן שֶׁל שָׁנָה רִאשׁוֹנָה מִן הַמַּחֲזוֹר. יֵצֵא לְךָ בְּאֵי זוֹ שָׁעָה וּבְכַמָּה בַּחֹדֶשׁ תִּהְיֶה תְּקוּפַת נִיסָן שֶׁל אוֹתָהּ הַשָּׁנָה מִן הַמַּחֲזוֹר. וּמִמֶּנָּה תַּתְחִיל לִמְנוֹת אֶחָד וְתִשְׁעִים יוֹם וְשֶׁבַע שָׁעוֹת וּמֶחֱצָה לְכָל תְּקוּפָה וּתְקוּפָה. וְאִם תִּרְצֶה לֵידַע תְּקוּפַת נִיסָן שֶׁל שָׁנָה זוֹ שֶׁהִיא שְׁנַת כָּךְ וְכָךְ בַּמַּחֲזוֹר שֶׁאַתָּה עוֹמֵד בּוֹ. קַח לְכָל הַמַּחֲזוֹרִין הַשְּׁלֵמִים שָׁעָה וְתפ''ה לְכָל מַחֲזוֹר. וּלְכָל הַשָּׁנִים הַגְּמוּרוֹת שֶׁשָּׁלְמוּ מִן הַמַּחֲזוֹר עֲשָׂרָה יָמִים וְכ''א שָׁעוֹת וְר''ד חֲלָקִים לְכָל שָׁנָה וְקַבֵּץ הַכּל. וְתִגְרַע מִמֶּנּוּ ז' יָמִים וְט' שָׁעוֹת וְתרמ''ב חֲלָקִים. וְהַשְּׁאָר תַּשְׁלִיכֵם חָדְשֵׁי הַלְּבָנָה כ''ט יוֹם וְי''ב שָׁעוֹת וְז' מֵאוֹת וְצ''ג חֲלָקִים. וְהַנִּשְׁאָר פָּחוֹת מֵחֹדֶשׁ הַלְּבָנָה תּוֹסִיף אוֹתוֹ עַל מוֹלַד נִיסָן שֶׁל אוֹתָהּ הַשָּׁנָה. וְתֵדַע זְמַן תְּקוּפַת נִיסָן שֶׁל אוֹתָהּ הַשָּׁנָה בְּכַמָּה יוֹם בַּחֹדֶשׁ הִיא וּבְכַמָּה שָׁעָה. תְּקוּפַת נִיסָן לְפִי חֶשְׁבּוֹן זֶה אֵינָהּ לְעוֹלָם אֶלָּא אוֹ בִּתְחִלַּת הַלַּיְלָה אוֹ בַּחֲצִי הַלַּיְלָה אוֹ בִּתְחִלַּת הַיּוֹם אוֹ בַּחֲצִי הַיּוֹם. וּתְקוּפַת תַּמּוּז לְעוֹלָם אֵינָהּ אֶלָּא אוֹ בְּז' שָׁעוֹת וּמֶחֱצָה אוֹ בְּשָׁעָה אַחַת וּמֶחֱצָה בֵּין בַּיּוֹם בֵּין בַּלַּיְלָה. וּתְקוּפַת תִּשְׁרֵי לְעוֹלָם אֵינָהּ אֶלָּא אוֹ בְּט' שָׁעוֹת אוֹ בְּג' שָׁעוֹת בֵּין בַּיּוֹם בֵּין בַּלַּיְלָה. וּתְקוּפַת טֵבֵת לְעוֹלָם אֵינָהּ אֶלָּא אוֹ בְּי' שָׁעוֹת וּמֶחֱצָה אוֹ בְּאַרְבַּע שָׁעוֹת וּמֶחֱצָה בֵּין בַּיּוֹם בֵּין בַּלַּיְלָה. אִם תִּרְצֶה לֵידַע בְּאֵי זֶה יוֹם מִימֵי הַשָּׁבוּעַ וּבְאֵי זוֹ שָׁעָה תִּהְיֶה הַתְּקוּפָה. קַח שָׁנִים גְּמוּרוֹת שֶׁעָבְרוּ מִשְּׁנַת הַיְצִירָה עַד שָׁנָה שֶׁתִּרְצֶה וְהַשְׁלֵךְ הַכּל כ''ח כ''ח וְהַנִּשְׁאָר יוֹתֵר מִכ''ח קַח לְכָל שָׁנָה יוֹם אֶחָד ו' שָׁעוֹת. וְקַבֵּץ הַכּל וְהוֹסֵיף עָלָיו ג' וְהַשְׁלֵךְ הַכּל ז' ז'. וְהַנִּשְׁאָר מִן הַיָּמִים וּמִן הַשָּׁעוֹת תַּתְחִיל לִמְנוֹת מִתְּחִלַּת לֵיל אֶחָד בְּשַׁבָּת וְלַאֲשֶׁר יַגִּיעַ הַחֶשְׁבּוֹן בּוֹ תִּהְיֶה תְּקוּפַת נִיסָן. וְלָמָּה מוֹסִיפִין שְׁלֹשָׁה לְפִי שֶׁתְּקוּפָה רִאשׁוֹנָה שֶׁל שְׁנַת הַיְצִירָה הָיְתָה בִּתְחִלַּת לֵיל רְבִיעִי:

5

What is implied? If a person desires to know the day and the time of the vernal equinox of the year 4930 after creation,20 [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.

ה

כֵּיצַד. הֲרֵי שֶׁרָצִינוּ לֵידַע תְּקוּפַת נִיסָן שֶׁל שְׁנַת תְּשַׁע מֵאוֹת שְׁלֹשִׁים וְאַרְבַּעַת אֲלָפִים לַיְצִירָה. כְּשֶׁתַּשְׁלִיךְ הַכּל כ''ח כ''ח תִּשָּׁאֵר שָׁנָה אַחַת. תִּקַּח לָהּ יוֹם אֶחָד וְשֵׁשׁ שָׁעוֹת וְתוֹסִיף עָלָיו ג' נִמְצֵאת תְּקוּפַת נִיסָן בְּלֵיל חֲמִישִׁי שֵׁשׁ שָׁעוֹת בַּלַּיְלָה. וּכְשֶׁתּוֹסִיף עָלֶיהָ שֶׁבַע שָׁעוֹת וּמֶחֱצָה הֲרֵי תִּהְיֶה תְּקוּפַת תַּמּוּז בְּשָׁעָה וּמֶחֱצָה מִיּוֹם ה'. וּכְשֶׁתּוֹסִיף עָלֶיהָ שֶׁבַע שָׁעוֹת וּמֶחֱצָה תִּהְיֶה תְּקוּפַת תִּשְׁרֵי בְּט' שָׁעוֹת מִיּוֹם ה'. וּכְשֶׁתּוֹסִיף עָלֶיהָ שֶׁבַע שָׁעוֹת וּמֶחֱצָה תִּהְיֶה תְּקוּפַת טֵבֵת בְּד' שָׁעוֹת וּמֶחֱצָה מִלֵּיל ז'. וּכְשֶׁתּוֹסִיף עָלֶיהָ שֶׁבַע שָׁעוֹת וּמֶחֱצָה תִּהְיֶה תְּקוּפַת נִיסָן הַבָּאָה בִּתְחִלַּת יוֹם ו'. וְעַל דֶּרֶךְ זוֹ עַד סוֹף הָעוֹלָם תְּקוּפָה אַחַר תְּקוּפָה:

6

[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:21 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,22 and then add seven days to this sum in the present time.23 Divide the sum by thirty,24 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.25 If the year in question is a leap year, begin counting from Rosh Chodesh of the second Adar.26 When a day is determined through this calculation, the equinox will take place on that date.

ו

אִם תִּרְצֶה לֵידַע בְּכַמָּה יוֹם בַּחֹדֶשׁ תִּהְיֶה תְּקוּפַת נִיסָן שֶׁל שָׁנָה זוֹ. תֵּדַע תְּחִלָּה בְּאֵי זֶה יוֹם מִימֵי הַשָּׁבוּעַ תִּהְיֶה וּבְאֵי זֶה יוֹם יִקְבְּעוּ נִיסָן שֶׁל שָׁנָה זוֹ וְכַמָּה שָׁנִים גְּמוּרוֹת עָבְרוּ מִן הַמַּחֲזוֹר. וְתִקַּח לְכָל שָׁנָה אַחַד עָשָׂר יוֹם וְתוֹסִיף עַל סִכּוּם הַיָּמִים ז' יָמִים בִּזְמַנִּים אֵלּוּ. וְהַשְׁלֵךְ הַכּל ל' ל' וְהַנִּשְׁאָר פָּחוֹת מִל' תַּתְחִיל לִמְנוֹתוֹ מֵרֹאשׁ חֹדֶשׁ נִיסָן. אִם יַגִּיעַ לְיוֹם הַתְּקוּפָה מוּטָב וְאִם לָאו הוֹסֵיף יוֹם אוֹ שְׁנֵי יָמִים אוֹ שְׁלֹשָׁה יָמִים עַל הַמִּנְיָן עַד שֶׁיַּגִּיעַ לְיוֹם הַתְּקוּפָה. וְאִם תִּהְיֶה הַשָּׁנָה מְעֵבֶּרֶת תַּתְחִיל לִמְנוֹת מֵרֹאשׁ חֹדֶשׁ אֲדָר שֵׁנִי וְלַיּוֹם שֶׁיַּגִּיעַ הַחֶשְׁבּוֹן בְּאוֹתוֹ הַיּוֹם מִן הַחֹדֶשׁ תִּהְיֶה הַתְּקוּפָה:

7

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.27

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.

ז

כֵּיצַד. הֲרֵי שֶׁרָצִינוּ לֵידַע בְּכַמָּה בַּחֹדֶשׁ תִּהְיֶה תְּקוּפַת נִיסָן שֶׁל שְׁנַת תתק''ל. שֶׁהִיא שָׁנָה תְּשִׁיעִית מִמַּחֲזוֹר ר''ס. מָצָאנוּ רֹאשׁ חֹדֶשׁ נִיסָן נִקְבָּע בָּהּ בַּחֲמִישִׁי וּתְקוּפַת נִיסָן בַּחֲמִישִׁי. וּלְפִי שֶׁהָיְתָה שָׁנָה זוֹ תְּשִׁיעִית לַמַּחֲזוֹר יִהְיוּ הַשָּׁנִים הַגְּמוּרוֹת שְׁמוֹנֶה. כְּשֶׁתִּקַּח לְכָל שָׁנָה מֵהֶן י''א יוֹם יִהְיוּ כָּל הַיָּמִים פ''ח. תּוֹסִיף ז' הֲרֵי הַכּל צ''ה. תַּשְׁלִיךְ הַכּל ל' ל' נִשְׁאֲרוּ ה' יָמִים. כְּשֶׁתַּתְחִיל לִמְנוֹת ה' יָמִים מֵרֹאשׁ חֹדֶשׁ נִיסָן שֶׁהָיָה בַּחֲמִישִׁי יַגִּיעַ הַחֶשְׁבּוֹן לְיוֹם שֵׁנִי. וּכְבָר יָדַעְנוּ שֶׁאֵין הַתְּקוּפָה בְּשֵׁנִי בְּשַׁבָּת אֶלָּא בַּחֲמִישִׁי. לְפִיכָךְ תּוֹסִיף יוֹם אַחַר יוֹם עַד שֶׁתַּגִּיעַ לַחֲמִישִׁי שֶׁהוּא יוֹם הַתְּקוּפָה. נִמְצֵאת תְּקוּפַת נִיסָן בְּשָׁנָה זוֹ בְּיוֹם שְׁמִינִי מֵחֹדֶשׁ נִיסָן. וְעַל הַדֶּרֶךְ הַזֹּאת תַּעֲשֶׂה בְּכָל שָׁנָה וְשָׁנָה:

8

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 days28 - or in a most unusual case - four days.29 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.

ח

זֶה שֶׁאָמַרְנוּ תּוֹסִיף יוֹם אַחַר יוֹם עַד שֶׁתַּגִּיעַ לְיוֹם הַתְּקוּפָה. לְעוֹלָם לֹא תִּהְיֶה צָרִיךְ לְהוֹסִיף אֶלָּא יוֹם אֶחָד אוֹ ב' אוֹ ג'. וּפֶלֶא גָּדוֹל הוּא שֶׁתִּהְיֶה צָרִיךְ לְהוֹסִיף אַרְבָּעָה יָמִים. וְאִם מָצָאתָ שֶׁאַתָּה צָרִיךְ לְהוֹסִיף עַל זֶה תֵּדַע שֶׁטָּעִיתָ בַּחֶשְׁבּוֹן וְתַחֲזֹר וְתַחְשֹׁב בְּדִקְדּוּק:

Footnotes
1.

Shemuel (Eruvin 56a).

2.

Rav Ada (Ibid.). His opinion is discussed in the following chapter.

3.

Ptolemy and Albatani, astronomers whose opinions were valued by the Rambam, from Greece and Arabia respectively, both maintain that the length of a solar year is less than 365 days and six hours. There is, however, a difference between the figures each of them suggests. According to contemporary science, the length of a tropical solar year is 365 days, 5 hours, 48 minutes and 45 1/2 seconds. It is decreasing at the rate of 0.530 second per century.

4.

Chapter 6, Halachah 10.

5.

This figure is based on the following principles: The times of the seasons are calculated according to the opinion of Rabbi Yehoshua, who maintains that the world was created in Nisan. Although the lunar calendar is calculated according to the opinion of Rabbi Eliezer, who maintains that the world was created in Tishrei, our Sages did not see a contradiction in interrelating the two conceptions, as we will continue to explain.

The conjunction of the month of Tishrei is the hour of man's creation, the fourteenth hour of Friday, the sixth day of creation. Based on this figure, by subtracting six times 1 day, 12 hours, and 793 units (the remainder of a lunar month), we can calculate that the conjunction of the month of Nisan - six months before creation - took place on Thursday, nine hours and 642 units after nightfall.

The vernal equinox is calculated according to the conception that the world was created in Nisan. The sun was created in the first hour of the fourth day of creation. This is considered the first vernal equinox. Thus, the autumnal equinox (half a year later) took place on Wednesday, three hours after daybreak, one day and twenty-three hours before the conjunction of Tishrei.

Since there is a difference of five days, ten hours, and 642 units between six months according to the lunar calendar, and half a year according to the solar calendar, it follows that the conjunction of Nisan was seven days, nine hours, and 642 units after the vernal equinox (Perush).

6.

For the calculations will be based on Rosh Chodesh Nisan, which took place this amount of time after the vernal equinox.

7.

A significant point arises from these statements, when one calculates the progression of the date and time of the vernal equinox by adding one hour and 485 units for each nineteen-year cycle. It follows that within the entire six millennia of the world's existence, the vernal equinox will have advanced approximately fifteen days. Thus, from the standpoint of the solar calendar, it will always be possible for Pesach, the fifteenth of Nisan, to occur in the spring.

8.

The difference between a lunar year and a solar year, as stated in Chapter 6, Halachah 4.

9.

I.e., calculate every group of 1080 units as an hour, and every group of 24 hours as a day.

10.

By making this subtraction, one bases the calculation on the conjunction of the month of Nisan, and not on the time of the first vernal equinox, which preceded that conjunction by this number of days and hours.

11.

Until this point in the calculation, the Rambam has not taken into consideration the existence of leap years. He does this now by grouping the remainder into months and subtracting the complete months. The number of complete months subtracted represents the number of leap years that have passed in the nineteen-year cycle.

12.

There are 30 hours between the time of the equinox (or solstice) of one year and the next. Since the first vernal equinox took place at nightfall between Tuesday and Wednesday, the second vernal equinox took place at midnight between Wednesday and Thursday, the third at daybreak on Friday, and the fourth at noon on the Sabbath. Similarly, in subsequent years, the time of the equinox will continue to advance in six (i.e., 30) hour intervals according to such a pattern.

13.

There is a difference of seven and a half hours between the time of the vernal equinox and the time of the summer solstice. Since the first vernal equinox took place at nightfall, the first summer solstice took place at 1:30 AM. Afterwards, the time of the summer solstice advances in six- (i.e., 30-) hour intervals every year, in a manner parallel to the progression of the vernal equinox, as described in the previous note.

14.

The first autumnal equinox took place at 9 AM. Afterwards, the time of the equinox has advanced in six-hour intervals, as explained.

15.

The first winter solstice took place at 4:30 PM. Afterwards, the time of the solstice has advanced in six-hour intervals, as explained.

16.

This number is chosen because after twenty-eight years, the equinox takes place on the same day of the week and the same hour as it did originally. This figure can be calculated as follows: 1 and 1/4 days (the difference between the time of the equinox in two successive years) times 28 equals 35 days. Thirty-five days are five full weeks.

Based on this calculation, it is each twenty-eight years that the sun returns to its original position at the time of creation. To commemorate this occurrence, a special blessing, Birkat HaChamah, is recited. (See Hilchot Berachot 10:18.)

17.

The difference in the time of the equinox from one year to the next.

18.

I.e., the night between the Sabbath and Sunday.

19.

I.e., the night between Tuesday and Wednesday. By making this addition, it is possible for these calculations to start from the beginning of the week.

20.

The commentaries understand this as an indication that this portion of the Mishneh Torah was composed during that year.

21.

The date of the equinox also can be determined by the calculations mentioned in Halachah 4. In this and the following halachot, however, the Rambam offers a simpler calculation, which uses approximations, but ultimately enables one to arrive at the same result.

22.

The Rambam is using an approximation. The difference between a lunar year and a solar year is ten days, twenty-one hours, and 204 units. However, to simplify the calculation, the Rambam rounds off the figure to eleven days.

23.

I.e., in the Rambam's time. The figure of seven days is reached as follows: In the year 4930, 259 nineteen-year cycles had passed. When an hour and 485 units are added for every nineteen-year cycle, a total of 15 days, 15 hours, and 335 units is obtained. Since the first equinox took place more than seven days before the conjunction of Nisan, eight days are subtracted from this figure, leaving a remainder of approximately seven days.

24.

To account for any leap years. Here, too, the Rambam is rounding off the figure; the length of a lunar month is slightly less.

25.

Since the calculation suggested by the Rambam contains several approximations, it may not be exact, and days may have to be added to reconcile the discrepancy.

26.

There will be more than thirty days remaining. Therefore, the reckoning should be made from Rosh Chodesh Adar.

27.

Further calculations are necessary, for in this instance, it is impossible that the equinox will take place on the first of the month, the eighth, or the fifteenth.

28.

I.e., although the calculation mentioned by the Rambam operates using approximations, the difference between these approximations and the actual data will hardly ever exceed three days.

29.

The maximum difference between the approximations employed by the Rambam and the actual data is three and one half days. Thus, it is possible, but highly improbable, that there be a four-day difference.

## Kiddush HaChodesh - Chapter Ten

1

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.

א

שְׁנַּת הַחַמָּה לְמִי שֶׁהוּא אוֹמֵר שֶׁהִיא פָּחוֹת מֵרְבִיעַ מֵחַכְמֵי יִשְׂרָאֵל. יֵשׁ מִי שֶׁאוֹמֵר שס''ה יוֹם וְחָמֵשׁ שָׁעוֹת וְתתקצ''ז חֲלָקִים וּמ''ח רֶגַע. וְהָרֶגַע אֶחָד מֵע''ו בְּחֵלֶק. וּלְפִי חֶשְׁבּוֹן זֶה תִּהְיֶה תּוֹסֶפֶת שְׁנַת הַחַמָּה עַל שְׁנַת הַלְּבָנָה י' יָמִים וְכ''א שָׁעָה וְקכ''א חֵלֶק וּמ''ח רֶגַע. סִימָן לָהֶן יכ''א קכ''א מ''ח. וְלֹא תִּמְצָא תּוֹסֶפֶת בְּמַחֲזוֹר שֶׁל י''ט שָׁנָה כְּלָל אֶלָּא בְּכָל מַחֲזוֹר מֵהֶם יִשְׁלְמוּ שְׁנֵי הַחַמָּה עִם שְׁנֵי הַלְּבָנָה הַפְּשׁוּטוֹת וְהַמְעֵבָּרוֹת:

2

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.

ב

בֵּין כָּל תְּקוּפָה וּתְקוּפָה לְפִי חֶשְׁבּוֹן זֶה צ''א יוֹם וְז' שָׁעוֹת וְתקי''ט חֲלָקִים וְל''א רֶגַע. סִימָן לָהֶם צ''א ת''ק י''ט ל''א. וּכְשֶׁתֵּדַע תְּקוּפָה מִן הַתְּקוּפוֹת אֵימָתַי הָיְתָה. תַּחְשֹׁב מֵאוֹתוֹ רֶגַע מִנְיָן זֶה וְתֵדַע תְּקוּפָה שֶׁאַחֲרֶיהָ עַל הַדֶּרֶךְ שֶׁבֵּאַרְנוּ בִּתְקוּפַת הַשָּׁנָה שֶׁהִיא רְבִיעַ:

3

According to this calculation, the vernal equinox of the first year of creation was nine hours and 642 units1 (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.

ג

תְּקוּפַת נִיסָן לְפִי חֶשְׁבּוֹן זֶה הָיְתָה בְּשָׁנָה רִאשׁוֹנָה שֶׁל יְצִירָה קֹדֶם מוֹלַד נִיסָן בְּט' שָׁעוֹת וְתרמ''ב חֲלָקִים. סִימָן לָהֶם ט' תרמ''ב. וְכֵן הִיא לְעוֹלָם בְּכָל שָׁנָה רִאשׁוֹנָה שֶׁל כָּל מַחֲזוֹר קֹדֶם מוֹלַד נִיסָן בְּתֵשַׁע שָׁעוֹת וְתרמ''ב חֲלָקִים:

4

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.

ד

כְּשֶׁתֵּדַע תְּקוּפַת נִיסָן שֶׁל שָׁנָה רִאשׁוֹנָה מִן הַמַּחֲזוֹר. תַּחְשֹׁב מִמֶּנָּה צ''א יוֹם וְז' שָׁעוֹת וְתקי''ט חֲלָקִים וְל''א רֶגַע לְכָל תְּקוּפָה וּתְקוּפָה עַד סוֹף הַמַּחֲזוֹר:

5

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,2 and divide the remainder by the length of a lunar month.3 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.

ה

אִם תִּרְצֶה לֵידַע מָתַי תִּהְיֶה תְּקוּפַת נִיסָן לְפִי חֶשְׁבּוֹן זֶה. תֵּדַע תְּחִלָּה שָׁנִים גְּמוּרוֹת שֶׁעָבְרוּ מִן הַמַּחֲזוֹר. וְתִקַּח לְכָל שָׁנָה מֵהֶן תּוֹסֶפֶת. וְהִיא יכ''א קכ''א מ''ח. וְקַבֵּץ כָּל הָרְגָעִים חֲלָקִים וְכָל הַחֲלָקִים שָׁעוֹת וְכָל הַשָּׁעוֹת יָמִים כְּדֶרֶךְ שֶׁתַּחְשֹׁב בַּמּוֹלָדוֹת. וְתִגְרַע מִן הַכּל הַט' שָׁעוֹת וְתרמ''ב חֲלָקִים. וְהַנִּשְׁאָר תַּשְׁלִיךְ חָדְשֵׁי לְבָנָה. וְהַנִּשְׁאָר שֶׁאֵין בּוֹ חֹדֶשׁ לְבָנָה תּוֹסִיף אוֹתוֹ עַל מוֹלַד נִיסָן שֶׁל אוֹתָהּ שָׁנָה. וּבְרֶגַע שֶׁיַּגִּיעַ הַמִּנְיָן בּוֹ תִּהְיֶה תְּקוּפַת נִיסָן שֶׁל אוֹתָהּ שָׁנָה:

6

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.

ו

וְנִרְאִין לִי הַדְּבָרִים שֶׁעַל חֶשְׁבּוֹן תְּקוּפָה זוֹ הָיוּ סוֹמְכִין לְעִנְיַן עִבּוּר הַשָּׁנָה בְּעֵת שֶׁבֵּית דִּין הַגָּדוֹל מָצוּי. שֶׁהָיוּ מְעַבְּרִין מִפְּנֵי הַזְּמַן אוֹ מִפְּנֵי הַצֹּרֶךְ. לְפִי שֶׁחֶשְׁבּוֹן זֶה הוּא הָאֱמֶת יוֹתֵר מִן הָרִאשׁוֹן. וְהוּא קָרוֹב מִן הַדְּבָרִים שֶׁנִּתְבָּאֲרוּ בְּאִצְטַגְנִינוּת יוֹתֵר מִן הַחֶשְׁבּוֹן הָרִאשׁוֹן שֶׁהָיְתָה בּוֹ שְׁנַת הַחַמָּה שס''ה יוֹם וּרְבִיעַ יוֹם:

7

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.4 [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.

ז

וְחֶשְׁבּוֹן שְׁתֵּי תְּקוּפוֹת הָאֵלּוּ שֶׁבֵּאַרְנוּ דַּרְכָּם הַכּל בְּקֵרוּב הוּא וּבְמַהֲלַךְ הַשֶּׁמֶשׁ הָאֶמְצָעִי לֹא בִּמְקוֹמָהּ הָאֲמִתִּי. אֲבָל בִּמְקוֹם הַשֶּׁמֶשׁ הָאֲמִתִּי תִּהְיֶה תְּקוּפַת נִיסָן בִּזְמַנִּים אֵלּוּ בִּכְמוֹ שְׁנֵי יָמִים קֹדֶם שְׁתֵּי הַתְּקוּפוֹת שֶׁיּוֹצְאִין בְּחֶשְׁבּוֹן זֶה. בֵּין בְּחֶשְׁבּוֹן מִי שֶׁחָשַׁב רְבִיעַ יוֹם גָּמוּר בֵּין לְמִי שֶׁמְּחַשֵּׁב לְפָחוֹת מֵרְבִיעַ יוֹם:

Footnotes
1.

This differs from the figure given in Chapter 9, Halachah 3. The reason for this difference is that Rav Ada's calculations (the figures mentioned in this chapter) follow Rabbi Yehoshua's view, which maintains that the world was created in Nisan. In contrast, Shemuel's calculations (those mentioned in Chapter 9) depend more on the view of Rabbi Eliezer, who maintains that the world was created in Tishrei.

2.

So that the calculation will begin from the day of the conjunction of Nisan.

3.

Thus accounting for all the leap years that have passed within the nineteen-year cycle.

4.

As mentioned previously, and as is explained in the subsequent chapters, the mean position or the mean rate of progress of a body in the celestial sphere refers to the average of its monthly or yearly cycle. In actual fact, there are slight inconsistencies between the position of any of these bodies according to these calculations and its actual position as observed in the celestial sphere. Until this point, the Rambam has relied on the mean rate of progress of the celestial bodies for his calculations. In the subsequent chapters, he explains how their exact position in the celestial sphere can be determined.

## Kiddush HaChodesh - Chapter Eleven

1

As stated in the laws mentioned previously,1 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.2

א

לְפִי שֶׁאָמַרְנוּ בַּהֲלָכוֹת אלּוּ שֶׁבֵּית דִּין הָיוּ מְחַשְּׁבִין בְּדִקְדּוּק וְיוֹדְעִים אִם יֵרָאֶה הַיָּרֵחַ אוֹ לֹא יֵרָאֶה. יָדַעְנוּ שֶׁכָּל מִי שֶׁרוּחוֹ נְכוֹנָה וְלִבּוֹ תָּאֵב לְדִבְרֵי הַחָכְמוֹת וְלַעֲמֹד עַל הַסּוֹדוֹת יִתְאַוֶּה לֵידַע אוֹתָן הַדְּרָכִים שֶׁמְּחַשְּׁבִין בָּהֶם עַד שֶׁיֵּדַע אָדָם אִם יֵרָאֶה הַיָּרֵחַ בְּלַיִל זֶה אוֹ לֹא יֵרָאֶה:

2

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.3 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.4

ב

וְדַרְכֵי הַחֶשְׁבּוֹן יֵשׁ בָּהֶן מַחֲלוֹקוֹת גְּדוֹלוֹת בֵּין חַכְמֵי הַגּוֹיִם הַקַּדְמוֹנִים שֶׁחָקְרוּ עַל חֶשְׁבּוֹן הַתְּקוּפוֹת וְהַגִּימַטְרִיָּאוֹת. וַאֲנָשִׁים חֲכָמִים גְּדוֹלִים נִשְׁתַּבְּשׁוּ בָּהֶן וְנִתְעַלְּמוּ מֵהֶן דְּבָרִים וְנוֹלְדוּ לָהֶן סְפֵקוֹת. וְיֵשׁ מִי שֶׁדִּקְדֵּק הַרְבֵּה וְלֹא פָּגַע בַּדֶּרֶךְ הַנְּכוֹנָה בְּחֶשְׁבּוֹן רְאִיַּת הַיָּרֵחַ אֶלָּא צָלַל בְּמַיִם אַדִּירִים וְהֶעֱלָה חֶרֶס בְּיָדוֹ:

3

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.5

ג

וּלְפִי אֹרֶךְ הַיָּמִים וְרֹב הַבְּדִיקוֹת וְהַחֲקִירוֹת נוֹדְעוּ לְמִקְצָת הַחֲכָמִים דַּרְכֵי חֶשְׁבּוֹן זֶה. וְעוֹד שֶׁיֵּשׁ לָנוּ בְּעִקָּרִים אֵלּוּ קַבָּלוֹת מִפִּי הַחֲכָמִים וּרְאָיוֹת שֶׁלֹּא נִכְתְּבוּ בַּסְּפָרִים הַיְדוּעִים לַכּל. וּמִפְּנֵי כָּל אֵלּוּ הַדְּבָרִים כָּשֵׁר בְּעֵינַי לְבָאֵר דַּרְכֵי חֶשְׁבּוֹן זֶה כְּדֵי שֶׁיִּהְיֶה נָכוֹן לְמִי שֶׁמְּלָאוֹ לִבּוֹ לְקָרְבָה אֶל הַמְּלָאכָה לַעֲשׂוֹת אוֹתָהּ:

4

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].6

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.

ד

וְאַל יִהְיוּ דְּרָכִים אֵלּוּ קַלִּים בְּעֵינֶיךָ מִפְּנֵי שֶׁאֵין אָנוּ צְרִיכִין לָהֶם בַּזְּמַן הַזֶּה. שֶׁאֵלּוּ הַדְּרָכִים דְּרָכִים רְחוֹקִים וַעֲמֻקִּים הֵן. וְהוּא סוֹד הָעִבּוּר שֶׁהָיוּ הַחֲכָמִים הַגְּדוֹלִים יוֹדְעִים אוֹתוֹ וְאֵין מוֹסְרִין אוֹתוֹ לְכָל אָדָם אֶלָּא לִסְמוּכִים נְבוֹנִים. אֲבָל זֶה הַחֶשְׁבּוֹן בַּזְּמַן הַזֶּה שֶׁאֵין בֵּית דִּין לִקְבֹּעַ עַל פִּי הָרְאִיָּה שֶׁאָנוּ מְחַשְּׁבִין בּוֹ הַיּוֹם אֲפִלּוּ תִּינוֹקוֹת שֶׁל בֵּית רַבָּן מַגִּיעִין עַד סוֹפוֹ בִּשְׁלֹשָׁה וְאַרְבָּעָה יָמִים:

5

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.

ה

שֶׁמָּא יִתְבּוֹנֵן חָכָם מֵחַכְמֵי הָאֻמּוֹת אוֹ מֵחַכְמֵי יִשְׂרָאֵל שֶׁלָּמְדוּ חָכְמַת יָוָן בִּדְרָכִים אֵלּוּ שֶׁאֲנִי מְחַשֵּׁב בָּהֶן לִרְאִיַּת הַיָּרֵחַ וְיִרְאֶה קֵרוּב מְעַט בְּמִקְצָת הַדְּרָכִים. וְיַעֲלֶה עַל דַּעְתּוֹ שֶׁנִּתְעַלֵּם מִמֶּנּוּ דָּבָר זֶה וְלֹא יָדַעְנוּ שֶׁיֵּשׁ בְּאוֹתוֹ הַדֶּרֶךְ קֵרוּב. אַל יַעֲלֶה זֶה עַל דַּעְתּוֹ אֶלָּא כָּל דָּבָר שֶׁלֹּא דִּקְדַּקְנוּ בּוֹ מִפְּנֵי שֶׁיָּדַעְנוּ בְּעִקְּרֵי הַגִּימַטְרִיָּאוֹת בִּרְאָיוֹת בְּרוּרוֹת שֶׁאֵין דָּבָר זֶה מַפְסִיד בִּידִיעַת הָרְאִיָּה וְאֵין חוֹשְׁשִׁין לוֹ. לְפִיכָךְ לֹא דִּקְדַּקְנוּ בּוֹ:

6

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.

ו

וְכֵן כְּשֶׁיִּרְאֶה בְּדֶרֶךְ מִן הַדְּרָכִים חֶסְרוֹן מְעַט מֵחֶשְׁבּוֹן הָרָאוּי לְאוֹתָהּ הַדֶּרֶךְ. בְּכַוָּנָה עָשִׂינוּ זֶה לְפִי שֶׁיֵּשׁ כְּנֶגְדּוֹ יִתְרוֹן בְּדֶרֶךְ אַחֶרֶת עַד שֶׁיֵּצֵא הַדָּבָר לַאֲמִתּוֹ בִּדְרָכִים קְרוֹבִים בְּלֹא חֶשְׁבּוֹן אָרֹךְ. כְּדֵי שֶׁלֹּא יִבָּהֵל הָאָדָם שֶׁאֵינוֹ רָגִיל בִּדְבָרִים אֵלּוּ בְּרֹב הַחֶשְׁבּוֹנוֹת שֶׁאֵין מוֹעִילִין בִּרְאִיַּת הַיָּרֵחַ:

7

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 sphere7 is divided into 3608 degrees [and twelve constellations].9 Each constellation includes thirty degrees, beginning with the constellation of Aries the ram.10 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.

ז

הָעִקָּרִים שֶׁצָּרִיךְ אָדָם לֵידַע תְּחִלָּה לְכָל חֶשְׁבּוֹנוֹת הָאִצְטַגְנִינוּת. בֵּין לְדַרְכֵי חֶשְׁבּוֹן הָרְאִיָּה בֵּין לִשְׁאָר דְּבָרִים. אֵלּוּ הֵן. הַגַּלְגַּל מוחלק בְּש''ס מַעֲלוֹת. כָּל מַזָּל וּמַזָּל שְׁלֹשִׁים מַעֲלוֹת. וּמַתְחִיל מִתְּחִלַּת מַזַּל טָלֶה וְכָל מַעֲלָה וּמַעֲלָה ס' חֲלָקִים. וְכָל חֵלֶק וְחֵלֶק ס' שְׁנִיּוֹת. וְכָל שְׁנִיָּה וּשְׁנִיָּה שִׁשִּׁים שְׁלִישִׁיּוֹת. וְכֵן תְּדַקְדֵּק הַחֶשְׁבּוֹן וּתְחַלֵּק כָּל זְמַן שֶׁתִּרְצֶה:

8

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.

ח

לְפִיכָךְ אִם יָצָא לְךָ בַּחֶשְׁבּוֹן שֶׁכּוֹכָב פְּלוֹנִי מְקוֹמוֹ בַּגַּלְגַּל בְּע' מַעֲלוֹת וְל' חֲלָקִים וּמ' שְׁנִיּוֹת. תֵּדַע שֶׁכּוֹכָב זֶה הוּא בְּמַזַּל תְּאוֹמִים בַּחֲצִי מַעֲלָה אַחַת עֶשְׂרֵה מִמַּזָּל זֶה. לְפִי שֶׁמַּזַּל טָלֶה ל' מַעֲלוֹת וּמַזַּל שׁוֹר ל' מַעֲלוֹת נִשְׁאַר עֶשֶׂר מַעֲלוֹת וּמֶחֱצָה מִמַּזַּל תְּאוֹמִים וּמ' שְׁנִיּוֹת מֵחֲצִי הַמַּעֲלָה הָאַחֲרוֹן:

9

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.11

ט

וְכֵן אִם יָצָא מְקוֹמוֹ בַּגַּלְגַּל בְּש''כ מַעֲלוֹת. תֵּדַע שֶׁכּוֹכָב זֶה בְּמַזַּל דְּלִי בְּכ' מַעֲלָה בּוֹ. וְעַל דֶּרֶךְ זוֹ בְּכָל הַמִּנְיָנוֹת. וְסֵדֶר הַמַּזָּלוֹת כָּךְ הוּא. טָלֶה שׁוֹר תְּאוֹמִים סַרְטָן אַרְיֵה בְּתוּלָה מֹאזְנַיִם עַקְרָב קֶשֶׁת גְּדִי דְּלִי דָּגִים:

10

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.

י

הַחֶשְׁבּוֹנוֹת כֻּלָּם כְּשֶׁתְּקַבֵּץ שְׁאֵרִית לִשְׁאֵרִית אוֹ כְּשֶׁתּוֹסִיף מִנְיָן עַל מִנְיָן. תְּקַבֵּץ כָּל מִין עִם מִינוֹ. הַשְּׁנִיּוֹת עִם הַשְּׁנִיּוֹת וְהַחֲלָקִים עִם הַחֲלָקִים וְהַמַּעֲלוֹת עִם הַמַּעֲלוֹת. וְכָל זְמַן שֶׁיִּתְקַבֵּץ מִן הַשְּׁנִיּוֹת ס' תָּשִׂים חֵלֶק אֶחָד וְתוֹסִיף עַל הַחֲלָקִים. וְכָל שֶׁיִּתְקַבֵּץ מִן הַחֲלָקִים ס' תָּשִׂים אוֹתוֹ מַעֲלָה וְתוֹסִיף אוֹתָהּ עַל הַמַּעֲלוֹת. וּכְשֶׁתְּקַבֵּץ הַמַּעֲלוֹת תַּשְׁלִיךְ אוֹתָן ש''ס וְהַנִּשְׁאָר מִש''ס וּלְמַטָּה הוּא שֶׁתּוֹפְסִין אוֹתוֹ לְחֶשְׁבּוֹן:

11

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.

יא

בְּכָל הַחֶשְׁבּוֹנוֹת כֻּלָּן כְּשֶׁתִּרְצֶה לִגְרֹעַ מִנְיָן מִמִּנְיָן. אִם יִהְיֶה זֶה שֶׁגּוֹרְעִין אוֹתוֹ יֶתֶר עַל זֶה שֶׁגּוֹרְעִין מִמֶּנּוּ אֲפִלּוּ בְּחֵלֶק אֶחָד תּוֹסִיף עַל זֶה שֶׁגּוֹרְעִין מִמֶּנּוּ ש''ס מַעֲלוֹת כְּדֵי שֶׁיְּהֵא אֶפְשָׁר לִגְרֹעַ זֶה הַמִּנְיָן מִמֶּנּוּ:

12

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.

יב

כֵּיצַד. הֲרֵי שֶׁהִצְרִיכְךָ הַחֶשְׁבּוֹן לִגְרֹעַ מָאתַיִם מַעֲלוֹת וְנ' חֲלָקִים וּמ' שְׁנִיּוֹת. סִימָן רנ''ם. מִק' מַעֲלוֹת וְכ' חֲלָקִים וְל' שְׁנִיּוֹת. סִימָנָן קכ''ל. תּוֹסִיף עַל הַק' ש''ס יִהְיוּ הַמַּעֲלוֹת ת''ס וְתַתְחִיל לִגְרֹעַ הַשְּׁנִיּוֹת מִן הַשְּׁנִיּוֹת תָּבוֹא לִגְרֹעַ אַרְבָּעִים מִשְּׁלֹשִׁים אִי אֶפְשָׁר. תָּרִים חֵלֶק אֶחָד מִן הָעֶשְׂרִים חֲלָקִים וְתַעֲשֶׂה אוֹתוֹ שִׁשִּׁים שְׁנִיּוֹת וְתוֹסִיף עַל הַשְּׁלֹשִׁים וְנִמְצְאוּ הַשְּׁנִיּוֹת תִּשְׁעִים. תִּגְרַע מֵהֶם הַמ' יִשָּׁאֵר חֲמִשִּׁים שְׁנִיּוֹת. וְתַחֲזֹר לִגְרֹעַ חֲמִשִּׁים חֲלָקִים מִי''ט חֲלָקִים שֶׁכְּבָר הֲרִימוֹתָ מֵהֶם חֵלֶק אֶחָד וַעֲשִׂיתוֹ שְׁנִיּוֹת. וְאִי אֶפְשָׁר לִגְרֹעַ חֲמִשִּׁים מִתִּשְׁעָה עָשָׂר. לְפִיכָךְ תָּרִים מַעֲלָה אַחַת מִן הַמַּעֲלוֹת וְתַעֲשֶׂה אוֹתָהּ שִׁשִּׁים חֲלָקִים וְתוֹסִיף עַל הַתִּשְׁעָה עָשָׂר וְנִמְצְאוּ הַחֲלָקִים ע''ט. תִּגְרַע מֵהֶן הַחֲמִשִּׁים יִשָּׁאֵר תִּשְׁעָה וְעֶשְׂרִים חֲלָקִים. וְתַחֲזֹר לִגְרֹעַ הַמָּאתַיִם מַעֲלוֹת מִן אַרְבַּע מֵאוֹת וְנ''ט מַעֲלוֹת שֶׁכְּבָר הֲרִימוֹתָ מַעֲלָה אַחַת וְעָשִׂיתָ חֲלָקִים. יִשָּׁאֵר מָאתַיִם וְתֵשַׁע וַחֲמִשִּׁים מַעֲלוֹת וְנִמְצָא הַשְּׁאָר סִימָנוֹ רנ''ט כ''ט נ'. וְעַל דֶּרֶךְ זוֹ בְּכָל גֵּרָעוֹן וְגֵרָעוֹן הַשֶּׁמֶשׁ וְהַיָּרֵחַ:

13

The sun, the moon, and the remainder of the seven stars,12 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,13 the earth is not at the center of [their orbits].

יג

וְכֵן שְׁאָר הַשִּׁבְעָה כּוֹכָבִים מַהֲלַךְ כָּל אֶחָד וְאֶחָד מֵהֶן בַּגַּלְגַּל שֶׁלּוֹ מַהֲלָךְ שָׁוֶה. אֵין בּוֹ לֹא קַלּוּת וְלֹא כְּבֵדוּת אֶלָּא כְּמוֹ מַהֲלָכוֹ הַיּוֹם כְּמוֹ מַהֲלָכוֹ אֶמֶשׁ כְּמוֹ מַהֲלָכוֹ לְמָחָר כְּמוֹ מַהֲלָכוֹ בְּכָל יוֹם וָיוֹם. וְגַלְגַּל שֶׁל כָּל אֶחָד מֵהֶם אַף עַל פִּי שֶׁהוּא מַקִּיף אֶת הָעוֹלָם אֵין הָאָרֶץ בְּאֶמְצָעוֹ:

14

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.14 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.15

יד

לְפִיכָךְ אִם תַּעֲרֹךְ מַהֲלַךְ כָּל אֶחָד מֵהֶן לַגַּלְגַּל הַמַּקִּיף אֶת הָעוֹלָם שֶׁהָאָרֶץ בְּאֶמְצָעוֹ שֶׁהוּא גַּלְגַּל הַמַּזָּלוֹת. יִשְׁתַּנֶּה הִלּוּכוֹ וְנִמְצָא מַהֲלָכוֹ בְּיוֹם זֶה בְּגַלְגַּל הַמַּזָּלוֹת פָּחוֹת אוֹ יוֹתֵר עַל מַהֲלָכוֹ אֶמֶשׁ אוֹ עַל מַהֲלָכוֹ לְמָחָר:

15

The uniform speed at which a planet, the sun, or the moon progresses is referred to as its mean motion.16 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 sun17 or the true position of the moon.18

טו

הַמַּהֲלָךְ הַשָּׁוֶה שֶׁמְּהַלֵּךְ הַכּוֹכָב אוֹ הַשֶּׁמֶשׁ אוֹ הַיָּרֵחַ בְּגַלְגַּלּוֹ הוּא הַנִּקְרָא אֶמְצַע הַמַּהֲלָךְ. וְהַמַּהֲלָךְ שֶׁיִּהְיֶה בְּגַלְגַּל הַמַּזָּלוֹת שֶׁהוּא פְּעָמִים יוֹתֵר וּפְעָמִים חָסֵר הוּא הַמַּהֲלָךְ הָאֲמִתִּי. וּבוֹ יִהְיֶה מְקוֹם הַשֶּׁמֶשׁ אוֹ מְקוֹם הַיָּרֵחַ הָאֲמִתִּי:

16

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,19 the third of Nisan, of the present year, the seventeenth year of the 260th [nineteen-year] cycle - i.e., the year 4938 since creation20 - which is the year 1489 with regard to contracts,21 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.

טז

כְּבָר אָמַרְנוּ שֶׁאֵלּוּ הַדְּרָכִים שֶׁאָנוּ מְבָאֲרִים בַּהֲלָכוֹת אֵלּוּ אֵינָן אֶלָּא לְחֶשְׁבּוֹן רְאִיַּת הַיָּרֵחַ בִּלְבַד. לְפִיכָךְ עָשִׂינוּ הָעִקָּר שֶׁמִּמֶּנּוּ מַתְחִילִין לְעוֹלָם לְחֶשְׁבּוֹן זֶה מִתְּחִלַּת לֵיל חֲמִישִׁי שֶׁיּוֹמוֹ יוֹם שְׁלִישִׁי לְחֹדֶשׁ נִיסָן מִשָּׁנָה זוֹ שֶׁהִיא שְׁנַת י''ז מִמַּחֲזוֹר ר''ס. שֶׁהִיא שְׁנַת תתקל''ח וְאַרְבַּעַת אַלְפַּיִם לַיְצִירָה. שֶׁהִיא שְׁנַת תפ''ט וְאֶלֶף לַשְּׁטָרוֹת. שֶׁהִיא שְׁנַת ק''ט וְאֶלֶף לְחֻרְבַּן בַּיִת שֵׁנִי. וְזוֹ הִיא שֶׁאָנוּ קוֹרְאִים אוֹתָהּ שְׁנַת הָעִקָּר בְּחֶשְׁבּוֹן זֶה:

17

Since the sighting of the moon is significant only in Eretz Yisrael as explained,22 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.23

This location is situated approximately 32 degrees north of the equator,24 [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,25 [and the surrounding areas extend] from 21° to 27° [west].

יז

וּלְפִי שֶׁהָרְאִיָּה לֹא תִּהְיֶה אֶלָּא בְּאֶרֶץ יִשְׂרָאֵל כְּמוֹ שֶׁבֵּאַרְנוּ. עָשִׂינוּ כָּל דַּרְכֵי חֶשְׁבּוֹן הַזֶּה בְּנוּיִים עַל עִיר יְרוּשָׁלַיִם וְלִשְׁאָר הַמְּקוֹמוֹת הַסּוֹבְבִין אוֹתָהּ בִּכְמוֹ שִׁשָּׁה אוֹ שִׁבְעָה יָמִים שֶׁבָּהֶן רוֹאִין אֶת הַיָּרֵחַ תָּמִיד וּבָאִים וּמְעִידִים בְּבֵית דִּין. וּמָקוֹם זֶה הוּא נוֹטֶה מִתַּחַת הַקַּו הַשָּׁוֶה הַמְסַבֵּב בְּאֶמְצַע הָעוֹלָם כְּנֶגֶד רוּחַ צְפוֹנִית בִּכְמוֹ ל''ב מַעֲלוֹת עַד ל''ה וְעַד כ''ט. וְכֵן הוּא נוֹטֶה מֵאֶמְצַע הַיִּשּׁוּב כְּנֶגֶד רוּחַ מַעֲרָב בִּכְמוֹ כ''ד מַעֲלוֹת עַד כ''ז וְעַד כ''א:

Footnotes
1.

E.g., Chapter 1, Halachah 6; Chapter 2, Halachah 4; Chapter 6, Halachah 1.

2.

This concept, the calculation of the place and position of the new moon, and the determination of when it will be visible, is the subject of this and the following eight chapters. In the present chapter, the Rambam outlines the general principles and ground rules governing his calculations.

3.

See Chapter 17, Halachah 24, where the Rambam states that in this text he refers to the works of Greek scientists, because the books written by the Sages of Israel on the subject were not available to him. In the following halachah, however, he mentions having accepted traditions from the Rabbis.

4.

The poetic wording is borrowed, out of context, from Bava Kama 91a.

5.

The latter phrase is borrowed, also out of context, from Exodus 36:2.

6.

I.e., to receive this knowledge, one had to have received semichah as described in Hilchot Sanhedrin, Chapter 4. Nevertheless, not all the Sages who received semichah were privileged to this knowledge.

7.

This term is used, because from man's perspective, the earth appears flat and the heavens appear as a sphere that revolves around him, only half of which is visible at any given time.

8.

The number 360 was chosen because it can be divided by all the cardinal integers except for seven (Perush).

9.

One corresponding roughly to each of the months of the year.

10.

For the year begins in spring, and during the spring the sun is located in the constellation of Aries.

11.

See Hilchot Yesodei HaTorah 3:7, which states that these constellations appeared in these forms at the time of the flood, and then they were given these names. At present, the stars have changed position somewhat, and some creativity is required to perceive how the images suggested by these names are appropriate for these constellations.

12.

The Rambam appears to be referring to his statements in Hilchot Yesodei HaTorah 3:1, which relate that there are nine spheres in which the stars revolve: The moon revolves in the first, then Mercury, Venus, the Sun, Mars, Jupiter, and then Saturn. In the eighth sphere revolve all the stars that are visible, and the ninth sphere includes and encircles all existence.

13.

The Rambam is following the theory of an earth-centered universe. The term "center" must, however, be understood loosely, because the earth does not lie at the exact center of all these spheres.

14.

The early astronomers realized that at some given times, the sun appears to travel faster or slower than at others - i.e., the pace at which it appears to proceed in the heavens varies between approximately 1 1/2 degrees per day and 58 1/2 minutes per day. Similarly, they saw that at different times of the year, the sun appears larger or smaller. By postulating that the earth was not the center of the sun's orbit, they were able to resolve these anomalies.

15.

Were the earth to lie at the center of all the planets' orbits, the speed at which the planets progress would not only be uniform, it would appear uniform. Since the earth is not in the center, although the planets are proceeding at a uniform pace, this does not always appear to be the case.

16.

See the notes on Chapter 6, Halachah 1.

17.

I.e., the angular location in the heavenly sphere at which the sun can be found. The stars cannot be seen during the daytime. Hence, we cannot actually see the constellations in which the sun is located. Throughout this text, the term "the position of the sun" generally refers to the angular position of the celestial sphere that is just below the horizon when the sun sets.

18.

The place of the moon in the heavenly sphere can be seen at night. It is possible for us to determine its angular position in comparison to the constellations of the Zodiac.

19.

I.e., Wednesday night.

20.

This year corresponds to 1178 C.E. There are several other dates cited within the Mishneh Torah with regard to the composition of that text.

21.

In Talmudic times, legal contracts were dated from the year when Alexander the Great ascended to the throne.

22.

For the New Moon can be sanctified only in Eretz Yisrael, as stated in Chapter 1, Halachah 8.

23.

The positive value of testimony from locales of more than a day's journey is mentioned in Chapter 3, Halachot 15-18.

24.

More precisely, Jerusalem is 31° 47' north of the equator.

25.

The populated area refers to the land mass of Europe and Asia, for at the time the Rambam wrote his text, America had not been discovered. The center of the populated area refers to a line approximately 90° east of "western edge of civilization". Thus, Jerusalem, which is 66° east of "western edge of civilization," is 24° west of this line.

The significance of the latitude and longitude of Jerusalem with regard to these calculations is mentioned in Chapter 17.