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Rambam - 1 Chapter a Day

Kiddush HaChodesh - Chapter Fifteen

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Kiddush HaChodesh - Chapter Fifteen

1

If you desire to know the true position of the moon1 on any particular date, first calculate the mean of the moon at the time of the sighting for the desired date. Then calculate the mean of the [moon within its] path and the sun's mean [position] for that date. Subtract the sun's mean from the moon's mean and double the remainder.2 The resulting figure is referred to as the double elongation.

א

אִם תִּרְצֶה לֵידַע מְקוֹם הַיָּרֵח הָאֲמִתִּי בְּכָל יוֹם שֶׁתִּרְצֶה. תּוֹצִיא תְּחִלָּה אֶמְצַע הַיָּרֵחַ לִשְׁעַת הָרְאִיָּה לְאוֹתוֹ הַלַּיְלָה שֶׁתִּרְצֶה. וְכֵן תּוֹצִיא אֶמְצַע הַמַּסְלוּל וְאֶמְצַע הַשֶּׁמֶשׁ לְאוֹתוֹ הָעֵת. וְתִגְרַע אֶמְצַע הַשֶּׁמֶשׁ מֵאֶמְצַע הַיָּרֵחַ. וְהַנִּשְׁאָר תִּכְפּל אוֹתוֹ. וְזֶה הוּא הַנִּקְרָא מֶרְחָק הַכָּפוּל:

2

As mentioned previously, the intent of all the calculations in these chapters is to know how to sight the moon. [In this context, the size of this double elongation is significant.]

[To explain:] It is impossible for this double elongation to be less than five degrees3 or more than 62 degrees4 on the night the moon is to be sighted. Its measure will never exceed or fall short of these numbers.

ב

וּכְבָר הוֹדַעְנוּ שֶׁלֹּא בָּאנוּ בְּכָל אֵלּוּ הַחֶשְׁבּוֹנוֹת שֶׁעָשִׂינוּ בִּפְרָקִים אֵלּוּ אֶלָּא לָדַעַת רְאִיַּת הַיָּרֵחַ. וּלְעוֹלָם אִי אֶפְשָׁר שֶׁיִּהְיֶה מֶרְחָק זֶה הַכָּפוּל בְּלֵיל הָרְאִיָּה שֶׁיֵּרָאֶה בָּהּ הַיָּרֵחַ אֶלָּא מֵחָמֵשׁ מַעֲלוֹת עַד ס''ב מַעֲלוֹת. וְאִי אֶפְשָׁר שֶׁיּוֹסִיף עַל זֶה וְלֹא יִגְרַע מִמֶּנּוּ:

3

Accordingly, it is necessary to contemplate [the length of] this double elongation. If the double elongation is five degrees or near that measure, there is no need to be concerned with an increase, and its measure need not be increased. If [the length of] the double elongation is between six and eleven degrees, one should add one degree to the mean of the [the moon] within its path.5

If [the length of] the double elongation is between twelve and eighteen degrees, one should add two degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between nineteen and 24 degrees, one should add three degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 25 and 31 degrees, one should add four degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 32 and 38 degrees, one should add five degrees to the mean of [the moon] within its path.

If [the length of] the double elongation is between 39 and 45 degrees, one should add six degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 46 and 51 degrees, one should add seven degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 52 and 59 degrees, one should add eight degrees to the mean of [the moon] within its path. If [the length of] the double elongation is between 60 and 63 degrees, one should add nine degrees to the mean of [the moon] within its path.

The mean of [the moon] within its path that results after these additions have been made is referred to as the correct course.

ג

וְהוֹאִיל וְהַדָּבָר כֵּן. הִתְבּוֹנֵן בְּמֶרְחָק זֶה הַכָּפוּל. אִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל חָמֵשׁ מַעֲלוֹת אוֹ קָרוֹב לְחָמֵשׁ אֵין חוֹשְׁשִׁין לְתוֹסֶפֶת וְלֹא תּוֹסִיף כְּלוּם. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִשֵּׁשׁ מַעֲלוֹת עַד אַחַת עֶשְׂרֵה מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל מַעֲלָה אַחַת. וְאִם יִהְיֶה מֶרְחָק הַכָּפוּל מִשְּׁתֵּים עֶשְׂרֵה מַעֲלוֹת עַד י''ח מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל שְׁתֵּי מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִי''ט מַעֲלוֹת עַד כ''ד מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל שָׁלֹשׁ מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִכ''ה מַעֲלוֹת עַד ל''א מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל ד' מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִל''ב מַעֲלוֹת עַד ל''ח מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל ה' מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִל''ט מַעֲלוֹת עַד מ''ה מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל שֵׁשׁ מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִמ''ו מַעֲלוֹת עַד נ''א מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל שֶׁבַע מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִנ''ב מַעֲלוֹת עַד נ''ט מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל ח' מַעֲלוֹת. וְאִם יִהְיֶה הַמֶּרְחָק הַכָּפוּל מִס' מַעֲלוֹת עַד ס''ג מַעֲלוֹת תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל ט' מַעֲלוֹת. וּמַה שֶּׁיִּהְיֶה אֶמְצַע הַמַּסְלוּל אַחַר שֶׁתּוֹסִיף עָלָיו מַעֲלוֹת אֵלּוּ הוּא הַנִּקְרָא מַסְלוּל הַנָּכוֹן:

4

After the angular distance of the correct course is calculated,6 [the following procedure should be carried out]: If [the result] is less than 180 degrees, the angle of the course7 should be subtracted8 from the mean of the moon at the time of the sighting. If [the result] is more than 180 degrees, the angle of the course should be added9 to the mean of the moon at the time of the sighting.

The figure that remains after this addition or subtraction is made is the true position of the moon at the time of sighting.

ד

וְאַחַר כָּךְ תִּרְאֶה כַּמָּה מַעֲלוֹת הוּא הַמַּסְלוּל הַנָּכוֹן. אִם הָיָה פָּחוֹת מִק''פ מַעֲלוֹת תִּגְרַע מְנַת הַמַּסְלוּל הַזֶּה הַנָּכוֹן מֵאֶמְצַע הַיָּרֵחַ לִשְׁעַת הָרְאִיָּה. וְאִם הָיָה הַמַּסְלוּל הַנָּכוֹן יוֹתֵר עַל ק''פ מַעֲלוֹת עַד ש''ס תּוֹסִיף מְנַת זֶה הַמַּסְלוּל הַנָּכוֹן עַל אֶמְצַע הַיָּרֵחַ לִשְׁעַת הָרְאִיָּה. וּמַה שֶּׁיִּהְיֶה הָאֶמְצַע אַחַר שֶׁתּוֹסִיף עָלָיו אוֹ תִּגְרַע מִמֶּנּוּ הוּא מְקוֹם הַיָּרֵחַ הָאֲמִתִּי לִשְׁעַת הָרְאִיָּה:

5

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

ה

וְדַע שֶׁאִם יִהְיֶה הַמַּסְלוּל הַנָּכוֹן ק''פ בְּשָׁוֶה אוֹ ש''ס בְּשָׁוֶה אֵין לוֹ מָנָה. אֶלָּא יִהְיֶה מְקוֹם הַיָּרֵחַ הָאֶמְצָעִי לִשְׁעַת הָרְאִיָּה הוּא מָקוֹם הָאֲמִתִּי:

6

What is the angle of its course? If the correct course is ten degrees, its angle will be 50 minutes. If the correct course is twenty degrees, its angle will be 1 degree and 38 minutes. If it is 30, its angle will be 2 degrees and 24 minutes. If it is 40, its angle will be 3 degrees and 6 minutes.10 If it is 50, its angle will be 3 degrees and 44 minutes. If it is 60, its angle will be 4 degrees and 16 minutes.

If it is 70, its angle will be 4 degrees and 41 minutes. If it is 80, its angle will be 5 degrees. If it is 90, its angle will be 5 degrees and 5 minutes. If it is 100, its angle will be 5 degrees and 8 minutes.11 If it is 110, its angle will be 4 degrees and 59 minutes. If it is 120, its angle will be 4 degrees and 40 minutes.12

If it is 130, its angle will be 4 degrees and 11 minutes. If it is 140, its angle will be 3 degrees and 33 minutes. If it is 150, its angle will be 213 degrees and 48 minutes. If it is 160, its angle will be 1 degree and 56 minutes. If it is 170, its angle will be 59 minutes. If it is an even 180 degrees, [the course] will not have an angle. Instead, as stated above, the moon's mean position will be identical with its true position.

ו

וְכַמָּה הִיא מְנַת הַמַּסְלוּל. אִם יִהְיֶה הַמַּסְלוּל הַנָּכוֹן עֶשֶׂר מַעֲלוֹת תִּהְיֶה מְנָתוֹ נ' חֲלָקִים. וְאִם יִהְיֶה הַמַּסְלוּל הַנָּכוֹן כ' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְל''ח חֲלָקִים. וְאִם יִהְיֶה שְׁלֹשִׁים תִּהְיֶה מְנָתוֹ שְׁתֵּי מַעֲלוֹת וְכ''ד חֲלָקִים. וְאִם יִהְיֶה מ' תִּהְיֶה מְנָתוֹ שָׁלֹשׁ מַעֲלוֹת וְשִׁשָּׁה חֲלָקִים. וְאִם יִהְיֶה נ' תִּהְיֶה מְנָתוֹ שָׁלֹשׁ מַעֲלוֹת וּמ''ד חֲלָקִים. וְאִם יִהְיֶה ס' תִּהְיֶה מְנָתוֹ אַרְבַּע מַעֲלוֹת וְט''ז חֲלָקִים. וְאִם יִהְיֶה ע' תִּהְיֶה מְנָתוֹ אַרְבַּע מַעֲלוֹת וּמ''א חֲלָקִים. וְאִם יִהְיֶה פ' תִּהְיֶה מְנָתוֹ חָמֵשׁ מַעֲלוֹת. וְאִם יִהְיֶה צ' תִּהְיֶה מְנָתוֹ חָמֵשׁ מַעֲלוֹת וְה' חֲלָקִים. וְאִם יִהְיֶה ק' תִּהְיֶה מְנָתוֹ ה' מַעֲלוֹת וְח' חֲלָקִים. וְאִם יִהְיֶה ק''י תִּהְיֶה מְנָתוֹ ד' מַעֲלוֹת וְנ''ט חֲלָקִים. וְאִם יִהְיֶה ק''כ תִּהְיֶה מְנָתוֹ ד' מַעֲלוֹת וְכ' חֲלָקִים. וְאִם יִהְיֶה ק''ל תִּהְיֶה מְנָתוֹ ד' מַעֲלוֹת וְי''א חֲלָקִים. וְאִם יִהְיֶה ק''מ תִּהְיֶה מְנָתוֹ ג' מַעֲלוֹת וְל''ג חֲלָקִים. וְאִם יִהְיֶה ק''נ תִּהְיֶה מְנָתוֹ שָׁלֹשׁ מַעֲלוֹת וּמ''ח חֲלָקִים. וְאִם יִהְיֶה ק''ס תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''ו חֲלָקִים. וְאִם יִהְיֶה ק''ע תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''ט חֲלָקִים. וְאִם יִהְיֶה ק''פ בְּשָׁוֶה אֵין לוֹ מָנָה כְּמוֹ שֶׁאָמַרְנוּ אֶלָּא מְקוֹם הַיָּרֵחַ הָאֶמְצָעִי הוּא הַמָּקוֹם הָאֲמִתִּי:

7

If the correct course is more than 180 degrees, you should subtract it from 360 to obtain its angle14, as you did for the course of the sun.15

Similarly, if the correct course includes units as well as tens, you should [calculate the average increase per degree and add the proportionate amount to the lower figure]. The procedure used to calculate the angle of the course for the course of the sun should be used to calculate the angle of the correct course [of the moon].16

ז

וְאִם יִהְיֶה הַמַּסְלוּל הַנָּכוֹן יוֹתֵר עַל ק''פ מַעֲלוֹת. תִּגְרַע אוֹתוֹ מִש''ס וְתֵדַע מְנָתוֹ כְּדֶרֶךְ שֶׁעָשִׂיתָ בְּמַסְלוּל הַשֶּׁמֶשׁ. וְכֵן אִם יִהְיוּ בְּמִנְיַן הַמַּסְלוּל אֲחָדִים עִם הָעֲשָׂרוֹת תִּקַּח מִן הַיּוֹתֵר שֶׁבֵּין שְׁתֵּי הַמָּנוֹת הָאֲחָדִים. כְּדֶרֶךְ שֶׁבֵּאַרְנוּ בְּמַסְלוּל הַשֶּׁמֶשׁ בַּמָּנוֹת שֶׁלּוֹ כָּךְ תַּעֲשֶׂה בַּמַּסְלוּל הַנָּכוֹן בַּמָּנוֹת שֶׁלּוֹ:

8

What is implied? Should we desire to know the true position of the moon on Friday night, the second of Iyar in the present year - the starting point for these calculations - the number of complete days that have passed from the date that is the starting point until the date on which we desire to know the true position of the moon is 29. One should [first] calculate the mean position of the sun for that night; this is 35 degrees, 38 minutes and 33 seconds, in symbols 35° 38' 33".

You should then calculate the mean of the moon at the time of the sighting, which is 53 degrees, 36 minutes and 39 seconds, in symbols 53° 36' 39". Afterwards, calculate the mean of [the moon] within its path for this time, which is 103 degrees, 21 minutes and 46 seconds, in symbols, 103° 21' 46". Then subtract the mean position of the sun from the moon's mean, producing a remainder of 17 degrees, 58 minutes and six seconds. This is the elongation. Doubling this figure produces a double elongation of 35 degrees, 56 minutes and 12 seconds, in symbols 35° 56' 12". Therefore, five degrees should be added to the course, as mentioned. Thus, the correct course will be 10817 degrees and 21 minutes. As mentioned above with regard to the sun,18 the minutes are of no consequence in the calculation of the course.

ח

כֵּיצַד. הֲרֵי שֶׁרָצִינוּ לֵידַע מְקוֹם הַיָּרֵחַ הָאֲמִתִּי בִּתְחִלַּת לֵיל עֶרֶב שַׁבָּת שֶׁיּוֹמוֹ שֵׁנִי לְחֹדֶשׁ אִיָּר מִשָּׁנָה זוֹ שֶׁהִיא שְׁנַת הָעִקָּר. וּמִנְיַן הַיָּמִים הַגְּמוּרִים מִתְּחִלַּת לֵיל הָעִקָּר עַד תְּחִלַּת לַיִל זֶה שֶׁאָנוּ רוֹצִים לֵידַע מְקוֹם הַיָּרֵחַ הָאֲמִתִּי בּוֹ כ''ט יוֹם. תּוֹצִיא אֶמְצַע הַשֶּׁמֶשׁ תְּחִלַּת לַיִל זֶה. יֵצֵא לְךָ אֶמְצָעוֹ ל''ה מַעֲלוֹת וְל''ח חֲלָקִים וְל''ג שְׁנִיּוֹת. סִימָנָם ל''ה ל''ח ל''ג. וְתוֹצִיא אֶמְצַע הַיָּרֵחַ לִשְׁעַת הָרְאִיָּה לְעֵת זוֹ. יֵצֵא לְךָ אֶמְצָעוֹ נ''ג מַעֲלוֹת וְל''ו חֲלָקִים וְל''ט שְׁנִיּוֹת. סִימָנָם נ''ג ל''ו ל''ט. וְתוֹצִיא אֶמְצַע הַמַּסְלוּל לְעֵת זוֹ יֵצֵא לְךָ אֶמְצָעוֹ ק''ג מַעֲלוֹת וְכ''א חֲלָקִים וּמ''ו שְׁנִיּוֹת. סִימָנָם ק''ג כ''א מ''ו. תִּגְרַע אֶמְצַע הַשֶּׁמֶשׁ מֵאֶמְצַע הַיָּרֵחַ יִשָּׁאֵר י''ז מַעֲלוֹת וְנ''ח חֲלָקִים וְשֵׁשׁ שְׁנִיּוֹת. וְזֶה הוּא הַמֶּרְחָק. תִּכְפּל אוֹתוֹ יֵצֵא לְךָ הַמֶּרְחָק הַכָּפוּל ל''ה מַעֲלוֹת וְנ''ו חֲלָקִים וְי''ב שְׁנִיּוֹת. סִימָנָם ל''ה נ''ו י''ב. לְפִיכָךְ תּוֹסִיף עַל אֶמְצַע הַמַּסְלוּל חָמֵשׁ מַעֲלוֹת כְּמוֹ שֶׁהוֹדַעְנוּ וְיֵצֵא לְךָ הַמַּסְלוּל הַנָּכוֹן ק''פ מַעֲלוֹת וְכ''א חֲלָקִים. וְאֵין מַקְפִּידִין עַל הַחֲלָקִים בְּמַסְלוּל כְּדֶרֶךְ שֶׁבֵּאַרְנוּ בַּשֶּׁמֶשׁ:

9

When calculating the angle for a course of 108, the result is 5 degrees and one minute. Since the correct course is less than 180 degrees, this figure should be subtracted from the moon's mean, leaving a remainder of 48 degrees, 3519 minutes and 39 seconds.

The seconds should be rounded off and considered to be a minute. Accordingly, the true position of the moon at this time will be 18 degrees and 36 minutes of the nineteenth degree in the constellation of Taurus, in symbols 18° 36'.

In a similar manner, it is possible for you to calculate the true position of the moon at the time of sighting for any date that you desire from the beginning of this year that was chosen as the starting point until the end of all time.

ט

וּבָאנוּ לַחְקֹר עַל מְנַת זֶה הַמַּסְלוּל הַנָּכוֹן שֶׁהוּא ק''ח נִמְצֵאת מָנָה שֶׁלּוֹ חָמֵשׁ מַעֲלוֹת וְחֵלֶק אֶחָד. וּלְפִי שֶׁהַמַּסְלוּל הַנָּכוֹן הָיָה פָּחוֹת מִק''פ תִּגְרַע הַמָּנָה שֶׁהוּא חָמֵשׁ מַעֲלוֹת וְחֵלֶק אֶחָד מִן אֶמְצַע הַיָּרֵחַ. יִשָּׁאֵר מ''ח מַעֲלוֹת וְל''ג חֲלָקִים וְל''ט שְׁנִיּוֹת. תַּעֲשֶׂה הַשְּׁנִיּוֹת חֵלֶק וְתוֹסִיף עַל הַחֲלָקִים. וְנִמְצָא מְקוֹם הַיָּרֵחַ הָאֲמִתִּי בְּשָׁעָה זוֹ בְּמַזַּל שׁוֹר בְּי''ח מַעֲלוֹת וְל''ו חֲלָקִים מִמַּעֲלַת י''ט. סִימָנָם י''ח ל''ו. וְעַל הַדֶּרֶךְ הַזֶּה תֵּדַע מְקוֹם הַיָּרֵחַ הָאֲמִתִּי בְּכָל עֵת רְאִיָּה שֶׁתִּרְצֶה מִתְּחִלַּת שָׁנָה זוֹ שֶׁהִיא הָעִקָּר עַד סוֹף הָעוֹלָם:

Test Yourself on This Chapter

Footnotes
1.

I.e., the location of the moon as it appears in the sky.

2.

The Rambam is following the notion that the Earth does not lie at the center of the moon's orbit. Hence, like the sun, the moon has an apogee and a perigee.

Also of significance here is the fact that, as mentioned in the notes on the previous chapter, the orbit of the moon is moving from east to west at a relatively fast pace, 11 degrees, 12 minutes and 19 seconds per day.

A conjunction refers to the time the sun, the moon, and the Earth are aligned in a straight line. Accordingly, the sun's rays are reflected back without being observed from the Earth, and the moon is therefore not seen in the heavens. At the time of the conjunction, the moon is always at its apogee.

(This can be explained as follows: When the moon is aligned directly between the sun and the Earth, it is at the point in its orbit that is closest to the sun. Therefore, the gravitational pull of the sun draws the moon away from the Earth.)

As the moon continues in its orbit, after the conjunction, it and its apogee move away from the sun in opposite directions. Significantly, the angular distance traveled by the moon and its apogee from the sun is the same.

To explain: The moon's mean is moving at rate of 13 degrees, 10 minutes and 35 seconds per day from west to east. Since the sun is also moving from west to east at the rate of 59 minutes and eight seconds per day, every day the moon will have traveled 12 degrees, 11 minutes and 27 seconds from the sun. Its apogee is moving from east to west at a speed of 11 degrees, 12 minutes and 19 seconds per day. When the sun's mean motion is added to that figure, the same total, 12 degrees, 11 minutes and 27 seconds, is reached. Thus, the double elongation, which is calculated by doubling the angular distance between the moon's mean and the sun's mean, represents the angular distance between the moon's mean and the apogee of its orbit.

(The moon also reaches its apogee when it is full. At this point, the sun, the Earth, and the moon are aligned in a straight line, and the gravitational pull of the sun draws the Earth away from the moon. According to contemporary science, however, these two figures are not alike, and the apogee reached at conjunction is greater than the apogee reached at a full moon.)

3.

At this time, the mean position of the moon will have moved 2 1/2 degrees from the mean position of the sun. Unless it moves that distance, its crescent will be too small to be noticed in the sky. This distance will be covered by the moon in slightly less than five hours. Thus, within five hours of the conjunction of the sun and the moon, it will be possible to sight the new moon.

The commentaries have noted a slight incongruity between the Rambam's statements here and his statements at the beginning of Chapter 17, where he states that the longitude at the night of the sighting of the moon will not be less than nine degrees nor more than twenty-four degrees. There is a difference of approximately seven degrees between these two figures. They attempt to resolve this discrepancy by stating that in this chapter the Rambam is speaking in terms of mean distance, while in Chapter 17 he is speaking in terms of true distance. At times, there can be as great as a seven-degree fluctuation between the two figures (Perush, Ralbach).

4.

At this time, 61 hours will have passed since the time of the conjunction, and the moon will have moved 31 degrees from the sun. Under such circumstances, the crescent of the moon will be large enough to be openly visible to all, and no calculations will be necessary.

5.

The difference between the true position of the moon and its mean position depends on the progress of the moon in its epicycle - i.e., the mean of [the moon] within its path, which is moving at approximately 13 degrees a day from east to west, as stated in the previous chapter. The Rambam is stating that a further adjustment is necessary, depending on the distance between the moon and the sun.

During the first days of the month, as the distance between the sun and the moon increases, the moon's progress in its epicycle will vary from its standard rate of progress. According to the medieval science, this variation depends on the movement of the nekudah hanochachit, the point opposite the center of the moon's orbit. According to modern science, this difference depends on the gravitational pull of the sun and other celestial bodies.

6.

I.e., the progress of the moon in its epicycle.

7.

This refers to the angle between the line extending from the mean position of the moon to the Earth and the line extending from the adjusted position of the moon in its epicycle to the Earth.

8.

When the correct course is less than 180 degrees, the angular distance between the moon's actual position and its mean is less than the mean distance. Hence, a subtraction should be made.

9.

When the correct course is more than 180 degrees, the angular distance between the moon's actual position and its mean is more than the mean distance. Hence, an addition should be made.

10.

The angle of the course of the moon is far larger than the angle of the course of the sun. The rationale for this difference is easily explainable. The moon is far closer to the Earth than the sun. Therefore, the angle between the two lines extending from either end of the course to the Earth will be greater.

11.

The line from the Earth to the moon's mean varies only slightly, while the line from the Earth to the true position of the moon changes to a far greater degree as the moon proceeds along its epicycle, changing the size of the angle of the course. The angle of the course will be largest when the angle between the line extending from the Earth to the true position of the moon and the line extending from the true position of the moon to its mean position is ninety-six degrees - i.e., when these two lines are almost directly perpendicular to each other.

(The reason the largest angle is not at a direct 90-degree angle is that the line to the moon's mean is drawn from the center of the Earth, and the true position takes into consideration the fact that we are looking at the moon from the surface of the Earth, which is removed from its center.)

12.

Our translation is based on the authentic manuscripts and early printings of the Mishneh Torah. The standard printed text reads 4 degrees and 20 minutes.

13.

Here, also, there is a printing error in the standard printed texts of the Mishneh Torah, and those texts read 3 degrees.

14.

For the same angle is produced regardless of whether one makes an increase or a decrease from 180° or 360°.

15.

See Chapter 13, Halachot 5-6.

16.

See Chapter 13, Halachah 7.

17.

In the standard printed texts of the Mishneh Torah, there is a printing error, and those texts read 180 degrees.

18.

See Chapter 13, Halachah 9.

19.

Here also there is a printing error in the standard published text, which reads 33 minutes.

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 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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