Rambam - 1 Chapter a Day - Daily Torah Study

1The orbit in which the moon revolves intersects the orbit in which the sun revolves at an angle,1 so that a portion of the moon’s orbit is inclined to the north of the sun’s orbit and a portion is inclined south of the sun’s orbit.2 There are two points, one opposite the other, at which these orbits intersect.3אהָעֲגֻלָּה שֶׁסּוֹבֵב בָּהּ הַיָּרֵחַ תָּמִיד, הִיא נוֹטָה מֵעַל הָעֲגֻלָּה שֶׁסּוֹבֶבֶת בָּהּ הַשֶּׁמֶשׁ תָּמִיד - חֶצְיָהּ נוֹטָה לַצָּפוֹן, וְחֶצְיָהּ נוֹטָה לַדָּרוֹם. וּשְׁתֵּי נְקֻדּוֹת יֵשׁ בָּהּ זוֹ כְּנֶגֶד זוֹ, שֶׁבָּהֶן פּוֹגְעוֹת שְׁתֵּי הָעֲגֻלּוֹת זוֹ בְּזוֹ.

When the moon is at one of these points, it is revolving in the same plane as the sun. As the moon departs from these points, it is proceeding either to the north or to the south of the sun. The point in the moon’s orbit at which it begins to be inclined to the north of the plane of the sun’s orbit is referred to as the head, while the point in its orbit from which it begins to be inclined to the south of the plane of the sun’s orbit is referred to as the tail. This head revolves at a uniform pace,4 proceeding in opposition to the movement of the sphere of the constellations5 without increase or decrease—i.e., it moves from the constellation of Pisces to the constellation of Aquarius. It continuously follows this pattern.לְפִיכָךְ, כְּשֶׁיִּהְיֶה הַיָּרֵחַ בְּאַחַת מִשְׁתֵּיהֶן, נִמְצָא סוֹבֵב בָּעֲגֻלָּה שֶׁל שֶּׁמֶשׁ כְּנֶגֶד הַשֶּׁמֶשׁ בְּשָׁוֶה. וְאִם יֵצֵא הַיָּרֵחַ מֵאַחַת מִשְּׁתֵּי הַנְּקֻדּוֹת, נִמְצָא מְהַלֵּךְ לִצְפוֹן הַשֶּׁמֶשׁ אוֹ לִדְרוֹמָהּ. הַנְּקֻדָּה שֶׁמִּמֶּנָּה יַתְחִיל הַיָּרֵחַ לִנְטוֹת לִצְפוֹן הַשֶּׁמֶשׁ, הִיא הַנִּקְרֵאת 'רֹאשׁ'. וְהַנְּקֻדָּה שֶׁמִּמֶּנָּה יַתְחִיל הַיָּרֵחַ לִנְטוֹת לִדְרוֹם הַשֶּׁמֶשׁ, הִיא הַנִּקְרֵאת 'זָנָב'. וּמַהֲלָךְ שָׁוֶה יֵשׁ לְזֶה הָרֹאשׁ, שֶׁאֵין בּוֹ לֹא תוֹסֶפֶת וְלֹא גֵּרָעוֹן. וְהוּא הוֹלֵךְ בַּמַּזָּלוֹת אֲחוֹרַנִּית מִטָּלֶה לְדָגִים לִדְלִי, וְכֵן הוּא סוֹבֵב תָּמִיד.

2The mean movement of the head in one day is 3 minutes and 11 seconds.6 Thus, its movement in ten days is 31 minutes and 47 seconds; its movement in one hundred days is 5 degrees, 17 minutes, and 43 seconds, in symbols 5° 17’ 43”. In one thousand days, its movement is 52 degrees, 57 minutes and 10 seconds, in symbols 52° 57’ 10”. The remainder of the sum of its progress in ten thousand days is 169 degrees, 31 minutes and 40 seconds, in symbols 169° 31’ 40”.במַהַלַךְ הָרֹאשׁ הָאֶמְצָעִי בְּיוֹם אֶחָד, שְׁלוֹשָׁה חֲלָקִים וְאַחַת עֶשְׂרֵה שְׁנִיּוֹת. נִמְצָא מַהֲלָכוֹ בַּעֲשָׂרָה יָמִים, אֶחָד וּשְׁלוֹשִׁים חֲלָקִים וְשֶׁבַע וְאַרְבָּעִים שְׁנִיּוֹת; וְנִמְצָא מַהֲלָכוֹ בְּמֵאָה יוֹם, חָמֵשׁ מַעֲלוֹת וְשִׁבְעָה עָשָׂר חֲלָקִים וְשָׁלוֹשׁ וְאַרְבָּעִים שְׁנִיּוֹת - סִימָנָם ה' י"ז מ"ג; וְנִמְצָא מַהֲלָכוֹ בְּאֶלֶף יוֹם, שְׁתַּיִם וַחֲמִשִּׁים מַעֲלוֹת וְשִׁבְעָה וַחֲמִשִּׁים חֲלָקִים וְעֶשֶׂר שְׁנִיּוֹת - סִימָנָם נ"ב נ"ז י'; וְנִמְצָא שְׁאֵרִית מַהֲלָכוֹ בַּעֲשֶׂרֶת אֲלָפִים יוֹם, מֵאָה וְתֵשַׁע וְשִׁשִּׁים מַעֲלוֹת וְאֶחָד וּשְׁלוֹשִׁים חֲלָקִים וְאַרְבָּעִים שְׁנִיּוֹת - סִימָנָם קס"ט ל"א מ'.

Thus, the distance it travels in twenty-nine days is 1 degree, 32 minutes and 9 seconds, in symbols 1° 32’ 9”. Its progress in a regular year is 18 degrees, 44 minutes and 42 seconds, in symbols 18° 44’ 42”. The mean position of the head on Thursday night of the present year, the starting point for these calculations, is 180 degrees, 57 minutes, and 28 seconds, in symbols 180° 57’ 28”.7וְנִמְצָא מַהֲלָכוֹ לְתִשְׁעָה וְעֶשְׂרִים יוֹם, מַעֲלָה אַחַת וּשְׁנַיִם וּשְׁלוֹשִׁים חֲלָקִים וְתֵשַׁע שְׁנִיּוֹת - סִימָנָם א' ל"ב ט'; וְנִמְצָא מַהֲלָכוֹ לְשָׁנָה סְדוּרָה, שְׁמוֹנֶה עֶשְׂרֵה מַעֲלוֹת וְאַרְבָּעָה וְאַרְבָּעִים חֲלָקִים וּשְׁתַּיִם וְאַרְבָּעִים שְׁנִיּוֹת - סִימָנָם י"ח מ"ד מ"ב. וְאֶמְצַע הָרֹאשׁ בִּתְחִלַּת לֵיל חֲמִישִׁי שֶׁהוּא הָעִיקָר, הָיָה מֵאָה וּשְׁמוֹנִים מַעֲלוֹת וְשִׁבְעָה וַחֲמִשִּׁים חֲלָקִים וּשְׁמוֹנֶה וְעֶשְׂרִים שְׁנִיּוֹת - סִימָנָם ק"פ נ"ז כ"ח.

3If you desire to calculate the position of the head at any given date, you should follow this procedure: First, calculate the mean progress of the head as you calculated the mean of the sun and the mean of the moon. Afterwards, subtract this mean from 360 degrees,8 and the remainder will be the location of the head at that time. The tail’s position will always be the place in the moon’s orbit directly opposite it.גאִם תִּרְצֶה לֵידַע מָקוֹם הָרֹאשׁ בְּכָל עֵת שֶׁתִּרְצֶה - תּוֹצִיא אֶמְצָעוֹ לְאוֹתוֹ הָעֵת כְּדֶרֶךְ שֶׁתּוֹצִיא אֶמְצַע הַשֶּׁמֶשׁ וְאֶמְצַע הַיָּרֵחַ, וְתִגְרַע הָאֶמְצָע מִשְּׁלוֹשׁ מֵאוֹת וְשִׁשִּׁים מַעֲלוֹת. וְהַנִּשְׁאָר, הוּא מָקוֹם הָרֹאשׁ בְּאוֹתָהּ הָעֵת. וּכְנֶגְדּוֹ, לְעוֹלָם יִהְיֶה מָקוֹם הַזָּנָב.

4What is implied? Let us suppose that we desired to know the location of the head on Friday night, the second of Iyar of this year—the starting point for these calculations. There are 29 complete days between the night of the starting point and the date for which we desire to know the location of the head.דכֵּיצַד? הֲרֵי שֶׁרָצִינוּ לֵידַע מָקוֹם הָרֹאשׁ לִתְחִלַּת לֵיל עֶרֶב שַׁבָּת שֶׁיּוֹמוֹ שֵׁנִי לַחֹדֶשׁ אִיָּר מִשָּׁנָה זוֹ שֶׁהִיא שְׁנָת הָעִיקָר, וּמִנְיַן הַיָּמִים הַגְּמוּרִים מִתְּחִלַּת לֵיל הָעִיקָר עַד תְּחִלַּת לֵיל זֶה שֶׁאָנוּ רוֹצִים לֵידַע מָקוֹם הָרֹאשׁ בּוֹ, תִּשְׁעָה וְעֶשְׂרִים יוֹם.

5We should then calculate the mean of the head according to the familiar manner, adding its distance traveled in 29 days to the starting point. Thus, the mean of the head is 182 degrees, 29 minutes and 37 seconds, in symbols 182° 29’ 37”. This mean should be subtracted from 360, leaving a remainder of 177 degrees, 30 minutes and 23 seconds, in symbols 177° 30’ 23”. This is the location of the head. The seconds are of no consequence. Thus, the position of the head will be 27 degrees and 30 minutes within the constellation of Virgo. The position of the tail will be directly opposite it: 27 degrees and 30 minutes within the constellation of Pisces.התּוֹצִיא אֶמְצַע הָרֹאשׁ לָעֵת הַזֹּאת עַל הַדֶּרֶךְ שֶׁיָּדַעְתָּ, וְהוּא שֶׁתּוֹסִיף מַהֲלָכוֹ לְתִשְׁעָה וְעֶשְׂרִים יוֹם עַל הָעִיקָר, יֵצֵא לָךְ אֶמְצַע הָרֹאשׁ מֵאָה וּשְׁתַּיִם וּשְׁמוֹנִים מַעֲלוֹת וְתִשְׁעָה וְעֶשְׂרִים חֲלָקִים וְשֶׁבַע וּשְׁלוֹשִׁים שְׁנִיּוֹת - סִימָנָם קפ"ב כ"ט ל"ז; תִּגְרַע אֶמְצָע זֶה מִשְּׁלוֹשׁ מֵאוֹת וְשִׁשִּׁים, יִשָּׁאֵר לָךְ מֵאָה וְשֶׁבַע וְשִׁבְעִים מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים וְשָׁלוֹשׁ וְעֶשְׂרִים שְׁנִיּוֹת - סִימָנָם קע"ז ל' כ"ג; וְזֶה הוּא מָקוֹם הָרֹאשׁ, וְאַל תִּפְנֶה אֶל הַשְּׁנִיּוֹת. נִמְצָא מָקוֹם הָרֹאשׁ - בְּמַזַּל בְּתוּלָה בְּשֶׁבַע וְעֶשְׂרִים מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים, וּמָקוֹם הַזָּנָב כְּנֶגְדּוֹ - בְּמַזַּל דָּגִים בְּשֶׁבַע וְעֶשְׂרִים מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים.

6There will always be an even half of the celestial sphere between the position of the head and the position of the tail. Therefore, whenever the head is in a particular constellation, the tail will be seven constellations further in the order of constellations, at the same position with regard to degrees and minutes. For example, if the head is ten degrees within a particular constellation, the tail will be ten degrees within the seventh constellation from it.ולְעוֹלָם יִהְיֶה בֵּין הָרֹאשׁ וּבֵין הַזָּנָב חֲצִי הַגַּלְגַּל בְּשָׁוֶה. לְפִיכָךְ כָּל מַזָּל שֶׁתִּמְצָא בּוֹ מָקוֹם הָרֹאשׁ, יִהְיֶה הַזָּנָב בְּמַזַּל שְׁבִיעִי מִמֶּנּוּ בִּכְמוֹ מִנְיַן הַמַּעֲלוֹת וְהַחֲלָקִים בְּשָׁוֶה - אִם יִהְיֶה הָרֹאשׁ בְּעֶשֶׂר מַעֲלוֹת בְּמַזַּל פְּלוֹנִי, יִהְיֶה הַזָּנָב בְּעֶשֶׂר מַעֲלוֹת מִמַּזַּל שְׁבִיעִי מִמֶּנּוּ.

7After having established the position of the head, the position of the tail, and the true position of the moon, consider these three figures: If the position of the moon is the same, both in degrees and in minutes, as its head or tail, then the moon will not be inclined to the north or the south.9 If the position of the moon has passed10 the head11 and it is proceeding in the direction of the tail, know that the moon will be inclined to the north of the plane of the sun’s orbit. If the position of the moon is before the tail12 and it is proceeding in the direction of the head, know that the moon will be inclined to the south of the plane of the sun’s orbit.זוּמֵאַחַר שֶׁתֵּדַע מָקוֹם הָרֹאשׁ וּמָקוֹם הַזָּנָב וּמָקוֹם הַיָּרֵחַ הָאֲמִתִּי, הִתְבּוֹנֵן בִּשְׁלָשְׁתָּן: אִם מָצָאתָ הַיָּרֵחַ עִם הָרֹאשׁ אוֹ עִם הַזָּנָב בְּמַעֲלָה אַחַת וְחֵלֶק אֶחָד, תֵּדַע שֶׁאֵין הַיָּרֵחַ נוֹטֶה לֹא לִצְפוֹן הַשֶּׁמֶשׁ וְלֹא לִדְרוֹמָהּ; וְאִם רָאִיתָ מָקוֹם הַיָּרֵחַ לִפְנֵי מָקוֹם הָרֹאשׁ וְהוּא הוֹלֵךְ כְּנֶגֶד הַזָּנָב, תֵּדַע שֶׁהַיָּרֵחַ נוֹטֶה לִצְפוֹן הַשֶּׁמֶשׁ; וְאִם הָיָה הַיָּרֵחַ לִפְנֵי מָקוֹם הַזָּנָב וַהֲרֵי הוּא הוֹלֵךְ כְּנֶגֶד הָרֹאשׁ, תֵּדַע שֶׁהַיָּרֵחַ נוֹטֶה לִדְרוֹם הַשֶּׁמֶשׁ.

8The inclination of the moon to the north or to the south is referred to as the moon’s latitude.13 If the moon’s incline is northerly, it is referred to as a northerly latitude. If the moon’s incline is southerly, it is referred to as a southerly latitude. If the moon is positioned at either the head or the tail, it has no latitude, as explained above.חהַנְּטִיָּה שֶׁנּוֹטֶה הַיָּרֵחַ לִצְפוֹן הַשֶּׁמֶשׁ אוֹ לִדְרוֹמָהּ, הִיא הַנִּקְרֵאת 'רֹחַב הַיָּרֵחַ'. וְאִם הָיָה נוֹטֶה לַצָּפוֹן - נִקְרָא 'רֹחַב צְפוֹנִי', וְאִם הָיָה נוֹטֶה לַדָּרוֹם - נִקְרָא 'רֹחַב דְּרוֹמִי'. וְאִם הָיָה הַיָּרֵחַ בְּאַחַת מִשְּׁתֵּי הַנְּקֻדּוֹת - לֹא יִהְיֶה לוֹ רֹחַב, כְּמוֹ שֶׁבֵּאַרְנוּ.

9The moon’s latitude14 will never exceed five degrees, whether to the north or to the south. This is the pattern it follows. The moon begins at the head and diverges slightly from the sun’s orbit, as it proceeds on its own orbit. The size of this divergence continues to increase until it reaches five degrees.15 At this point, the moon begins to come slightly closer to the sun’s orbit, until it has no latitude at all when it reaches its tail.טלְעוֹלָם לֹא יִהְיֶה רֹחַב הַיָּרֵחַ יָתֵר עַל חָמֵשׁ מַעֲלוֹת, בֵּין בַּצָּפוֹן בֵּין בַּדָּרוֹם. אֶלָא כָּךְ הוּא דַּרְכּוֹ: יַתְחִיל מִן הָרֹאשׁ וְיִתְרַחֵק מִמֶּנּוּ מְעַט מְעַט, וְהַמֶּרְחָק הוֹלֵךְ וְנוֹסָף עַד שֶׁיַּגִּיעַ לְחָמֵשׁ מַעֲלוֹת, וְיַחְזֹר וְיִתְקָרֵב מְעַט מְעַט, עַד שֶׁלֹּא יִהְיֶה לוֹ רֹחַב כְּשֶׁיַּגִּיעַ לַזָּנָב.

After it reaches the tail, the moon will again begin to diverge slightly from the sun’s orbit, until this divergence reaches five degrees. It will then begin to approach the sun’s orbit, until ultimately it has no latitude at all.וְיַחְזֹר וְיִתְרַחֵק מְעַט מְעַט, וְהַמֶּרְחָק נוֹסָף עַד שֶׁיַּגִּיעַ לְחָמֵשׁ מַעֲלוֹת; וְיַחְזֹר וְיִתְקָרֵב, עַד שֶׁלֹּא יִהְיֶה לוֹ רֹחַב.

10The following procedure should be applied if you desire to determine the latitude of the moon, and to know whether it is northerly or southerly: First, calculate the position of the head and the true position of the moon at the desired date. Then subtract the position of the head from the true position of the moon. The remainder is referred to as “the course of the latitude.”16יאִם תִּרְצֶה לֵידַע רֹחַב הַיָּרֵחַ כַּמָּה הוּא בְּכָל עֵת שֶׁתִּרְצֶה, וְאִם צְפוֹנִי הוּא אוֹ דְּרוֹמִי - תּוֹצִיא מָקוֹם הָרֹאשׁ וּמָקוֹם הַיָּרֵחַ הָאֲמִתִּי לְאוֹתָהּ הָעֵת, וְתִגְרַע מָקוֹם הָרֹאשׁ מִמָקוֹם הַיָּרֵחַ הָאֲמִתִּי, וְהַנִּשְׁאָר הוּא הַנִּקְרָא מַסְלוּל הָרֹחַב.

If the course of the latitude is between one degree and 180 degrees, the latitude of the moon is northerly. If course of the latitude is more than 180 degrees, the latitude of the moon is southerly. If the course is an even 180 degrees or an even 360 degrees, the moon does not have any latitude at all.אִם יִהְיֶה מַסְלוּל הָרֹחַב מִמַּעֲלָה אַחַת עַד מֵאָה וּשְׁמוֹנִים, תֵּדַע שֶׁרֹחַב הַיָּרֵחַ צְפוֹנִי; וְאִם הָיָה הַמַּסְלוּל יָתֵר עַל מֵאָה וּשְׁמוֹנִים, תֵּדַע שֶׁרֹחַב הַיָּרֵחַ דְּרוֹמִי; וְאִם הָיָה מֵאָה וּשְׁמוֹנִים בְּשָׁוֶה אוֹ שְׁלוֹשׁ מֵאוֹת וְשִׁשִּׁים בְּשָׁוֶה, אֵין לַיָּרֵחַ רֹחַב כְּלָל.

Afterwards, determine the size of the angle of the course of the latitude17 —i.e., the extent to which the moon is inclined to the north or to the south. This figure is referred to as the moon’s southerly latitude or northerly latitude, as we explained.וְתַחְזֹר וְתִרְאֶה מְנַת מַסְלוּל הָרֹחַב כַּמָּה הִיא, וְהוּא שִׁעוּר נְטִיָּתוֹ לַצָּפוֹן אוֹ לַדָּרוֹם, וְהוּא הַנִּקְרָא רֹחַב הַיָּרֵחַ הַדְּרוֹמִי אוֹ הַצְּפוֹנִי, כְּמוֹ שֶׁבֵּאַרְנוּ.

11How large is the angle of the course of the latitude? If the course of the latitude is ten degrees, its angle will be 52 minutes. If the course is twenty degrees, its angle will be one degree and 43 minutes. If the course is thirty degrees, its angle will be 2 degrees and 30 minutes. If the course is forty degrees, its angle will be 3 degrees and 13 minutes.יאוְכַמָּה הִיא מְנַת מַסְלוּל הָרֹחַב? אִם יִהְיֶה מַסְלוּל הָרֹחַב עֶשֶׂר מַעֲלוֹת, תִּהְיֶה מְנָתוֹ שְׁנַיִם וַחֲמִשִּׁים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל הַזֶּה עֶשְׂרִים מַעֲלוֹת, תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וּשְׁלוֹשָׁה וְאַרְבָּעִים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל שְׁלוֹשִׁים, תִּהְיֶה מְנָתוֹ שְׁתֵּי מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל אַרְבָּעִים, תִּהְיֶה מְנָתוֹ שָׁלוֹשׁ מַעֲלוֹת וּשְׁלוֹשָׁה עָשָׂר חֲלָקִים.

If the course is fifty degrees, its angle will be 3 degrees and 50 minutes. If the course is sixty degrees, its angle will be 4 degrees and 20 minutes. If the course is seventy degrees, its angle will be 4 degrees and 42 minutes. If the course is eighty degrees, its angle will be 4 degrees and 55 minutes. If the course is ninety degrees, its angle will be 5 degrees.18וְאִם יִהְיֶה הַמַּסְלוּל חֲמִשִּׁים מַעֲלוֹת, תִּהְיֶה מְנָתוֹ שָׁלוֹשׁ מַעֲלוֹת וַחֲמִשִּׁים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל שִׁשִּׁים, תִּהְיֶה מְנָתוֹ אַרְבַּע מַעֲלוֹת וְעֶשְׂרִים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל שִׁבְעִים, תִּהְיֶה מְנָתוֹ אַרְבַּע מַעֲלוֹת וּשְׁנַיִם וְאַרְבָּעִים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל שְׁמוֹנִים, תִּהְיֶה מְנָתוֹ אַרְבַּע מַעֲלוֹת וַחֲמִשָּׁה וַחֲמִשִּׁים חֲלָקִים; וְאִם יִהְיֶה הַמַּסְלוּל תִּשְׁעִים, תִּהְיֶה מְנָתוֹ חָמֵשׁ מַעֲלוֹת.

12If the course of the latitude has both units and tens, you should calculate the average increase per degree and add the proportionate amount to the lower figure, as was done with regard to the course of the sun and the course of the moon.19 What is implied? When the course of the latitude is 53 degrees, the size of the angle should be determined as follows. It has already been established that when the course is 50 degrees, its angle is 3 degrees and 50 minutes. When the course is 60 degrees, its angle is 4 degrees and 20 minutes. Therefore, there is a difference of 30 minutes between them, 3 minutes for each degree. Accordingly, when calculating the angle for a course of 53 degrees, the result will be 3 degrees and 59 minutes. A similar process should be followed with regard to all other figures.יבוְאִם יִהְיוּ אֲחָדִים עִם הָעֲשָׂרוֹת - תִּקַּח הָרָאוּי לָהֶם לְפִי הַיָּתֵר שֶׁבֵּין שְׁתֵּי הַמָּנוֹת, כְּמוֹ שֶׁעָשִׂיתָ בְּמַסְלוּל הַשֶּׁמֶשׁ וּבְמַסְלוּל הַיָּרֵחַ. כֵּיצַד? הֲרֵי שֶׁהָיָה מַסְלוּל הָרֹחַב שָׁלוֹשׁ וַחֲמִשִּׁים מַעֲלוֹת - וּכְבָר יָדַעְתָּ שֶׁאִלּוּ הָיָה הַמַּסְלוּל חֲמִשִּׁים, הָיְתָה מְנָתוֹ שָׁלוֹשׁ מַעֲלוֹת וַחֲמִשִּׁים חֲלָקִים, וְאִלּוּ הָיָה הַמַּסְלוּל שִׁשִּׁים, הָיְתָה מְנָתוֹ אַרְבַּע מַעֲלוֹת וְעֶשְׂרִים חֲלָקִים; נִמְצָא הַיָּתֵר בֵּין שְׁתֵּי הַמָּנוֹת, שְׁלוֹשִׁים חֲלָקִים, שְׁלוֹשָׁה חֲלָקִים לְכָל מַעֲלָה; וְנִמְצָא לְפִי חֶשְׁבּוֹן מַסְלוּל זֶה שֶׁהוּא שָׁלוֹשׁ וַחֲמִשִּׁים, מְנָתוֹ שָׁלוֹשׁ מַעֲלוֹת וְתִשְׁעָה וַחֲמִשִּׁים חֲלָקִים. וְעַל דֶּרֶךְ זוֹ תַּעֲשֶׂה בְּכָל מִנְיָן וּמִנְיָן.

13Since you know the angles for all the values of the course of the latitude until ninety degrees, as was mentioned, you will be able to know the angle for all possible values of the course. For if the course is between 90 and 180 degrees, subtract the course from 180 and find the angle for the remainder.20יגמֵאַחַר שֶׁתֵּדַע מְנָתוֹ שֶׁל מַסְלוּל הָרֹחַב עַד תִּשְׁעִים, כְּמוֹ שֶׁהוֹדַעְנוּךָ, תֵּדַע מָנוֹת שֶׁל כָל מִנְיָנוֹת הַמַּסְלוּל; שֶׁאִם יִהְיֶה הַמַּסְלוּל יָתֵר עַל תִּשְׁעִים עַד מֵאָה וּשְׁמוֹנִים - תִּגְרַע הַמַּסְלוּל מִמֵּאָה וּשְׁמוֹנִים, וְהַנִּשְׁאָר תֵּדַע בּוֹ הַמָּנָה.

14Similarly,21 if the course is between 180 and 270 degrees,22 subtract 180 from the course and find the angle for the remainder.23ידוְכֵן אִם הָיָה הַמַּסְלוּל יָתֵר מִמֵּאָה וּשְׁמוֹנִים עַד מָאתַיִם וְשִׁבְעִים - תִּגְרַע מִמֶּנּוּ מֵאָה וּשְׁמוֹנִים, וְהַנִּשְׁאָר תֵּדַע בּוֹ הַמָּנָה.

15Similarly, if the course is between 270 and 360 degrees, subtract the course from 360 and find the angle for the remainder.24טווְאִם הָיָה הַמַּסְלוּל יָתֵר עַל מָאתַיִם וְשִׁבְעִים עַד שְׁלוֹשׁ מֵאוֹת וְשִׁשִּׁים - תִּגְרַע אוֹתוֹ מִשְּׁלוֹשׁ מֵאוֹת וְשִׁשִּׁים, וְהַנִּשְׁאָר תֵּדַע בּוֹ הַמָּנָה.

16What is implied? If the course is 150°, subtract 150 from 180, leaving 30. As mentioned previously, the angle of a course of 30 degrees will be 2 degrees and 30 minutes. Thus, the angle of a course of 150 degrees also will be 2 degrees and 30 minutes.טזכֵּיצַד? הֲרֵי שֶׁהָיָה הַמַּסְלוּל מֵאָה וַחֲמִשִּׁים - תִּגְרַע אוֹתוֹ מִמֵּאָה וּשְׁמוֹנִים, נִשְׁאָר שְׁלוֹשִׁים. וּכְבָר יָדַעְתָּ שֶׁמְּנַת שְׁלוֹשִׁים, שְׁתֵּי מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים; וְכָּךְ תִּהְיֶה מְנַת מֵאָה וַחֲמִשִּׁים, שְׁתֵּי מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים.

17If the course is 200°, subtract 180 from 200, leaving 20. As mentioned previously, the angle of a course of 20 degrees will be 1 degree and 43 minutes. Thus, the angle of a course of 200 degrees also will be 1 degree and 43 minutes.יזהֲרֵי שֶׁהָיָה הַמַּסְלוּל מָאתַיִם - תִּגְרַע מִמֶּנּוּ מֵאָה וּשְׁמוֹנִים, יִשָּׁאֵר עֶשְׂרִים. וּכְבָר יָדַעְתָּ שֶׁמְּנַת עֶשְׂרִים, הִיא מַעֲלָה אַחַת וּשְׁלוֹשָׁה וְאַרְבָּעִים חֲלָקִים; וְכָּךְ תִּהְיֶה מְנַת מָאתַיִם, תִּהְיֶה מַעֲלָה אַחַת וּשְׁלוֹשָׁה וְאַרְבָּעִים חֲלָקִים.

18If the course is 300°, subtract 300 from 360, leaving 60. As mentioned previously, the angle of a course of 60 degrees will be 4 degrees and 20 minutes. Thus, the angle of a course of 300 degrees also will be 4 degrees and 20 minutes. A similar process should be followed with regard to all other values.יחהֲרֵי שֶׁהָיָה הַמַּסְלוּל שְׁלוֹשׁ מֵאוֹת - תִּגְרַע אוֹתוֹ מִשְּׁלוֹשׁ מֵאוֹת וְשִׁשִּׁים, נִשְׁאָר שִׁשִּׁים. וּכְבָר יָדַעְתָּ שֶׁמְּנַת שִׁשִּׁים, אַרְבַּע מַעֲלוֹת וְעֶשְׂרִים חֲלָקִים; וְכָּךְ הִיא מְנַת שְׁלוֹשׁ מֵאוֹת, אַרְבַּע מַעֲלוֹת וְעֶשְׂרִים חֲלָקִים. וְעַל דֶּרֶךְ זוֹ, בְּכָל הַמִּנְיָנוֹת.

19The following procedure should be applied if one desires to know the latitude of the moon and whether it is either northerly or southerly at the beginning of Friday night, the second of Iyar of this year: It has already been established that the true position of the moon on this night is 18 degrees and 36 minutes within the constellation of Taurus, in symbols 18° 36”. Similarly, it has been established that the position of the head at that time is 27 degrees and 30 minutes within the constellation of Virgo, in symbols 27° 30’. To arrive at the latitude, you must subtract the position of the head from the position of the moon, leaving a course of the latitude of 231 degrees and 6 minutes, in symbols 231° 6’. As mentioned, the minutes are of no consequence with regard to the course. Therefore, according to the principles explained in this chapter, the angle of this course will be 3 degrees and 53 minutes. This is the latitude of the moon at the beginning of this night. It is southerly, for the course is larger than 180 degrees.יטהֲרֵי שֶׁרָצִינוּ לֵידַע רֹחַב הַיָּרֵחַ כַּמָּה הוּא, וּבְאֵיזֶה רוּחַ הוּא אִם צְפוֹנִי וְאִם דְּרוֹמִי, בִּתְחִלַּת לֵיל עֶרֶב שַׁבָּת שֵׁנִי לְחֹדֶשׁ אִיָּר מִשָּׁנָה זוֹ. וּכְבָר יָדַעְתָּ שֶׁמָקוֹם הַיָּרֵחַ הָאֲמִתִּי הָיָה בְּלֵיל זֶה בִּשְׁמוֹנֶה עֶשְׂרֵה מַעֲלוֹת וְאַרְבָּעִים חֲלָקִים מִמַּזַּל שׁוֹר - סִימָנוֹ י"ח מ'; וּמָקוֹם הָרֹאשׁ הָיָה בְּאוֹתָהּ הָעֵת, בְּשֶׁבַע וְעֶשְׂרִים מַעֲלוֹת וּשְׁלוֹשִׁים חֲלָקִים מִמַּזַּל בְּתוּלָה - סִימָנוֹ כ"ז ל'. תִּגְרַע מָקוֹם הָרֹאשׁ מִמָקוֹם הַיָּרֵחַ, יֵצֵא לָךְ מַסְלוּל הָרֹחַב מָאתַיִם אַחַת וּשְׁלוֹשִׁים מַעֲלוֹת וַעֲשָׂרָה חֲלָקִים - סִימָנוֹ רל"א י' - לְפִי שֶׁאֵין מַשְׁגִּיחִין עַל הַחֲלָקִים, בְּכָל מַסְלוּל. וְנִמְצֵאת הַמָּנָה שֶׁל מַסְלוּל זֶה בַּדְּרָכִים שֶׁבֵּאַרְנוּ בְּפֵרֶק זֶה, שָׁלוֹשׁ מַעֲלוֹת וּשְׁלוֹשָׁה וַחֲמִשִּׁים חֲלָקִים. וְזֶה הוּא רֹחַב הַיָּרֵחַ בִּתְחִלַּת לֵיל זֶה, וְהוּא דְּרוֹמִי - שֶׁהֲרֵי הַמַּסְלוּל יָתֵר עַל מֵאָה וּשְׁמוֹנִים.

Footnotes

More specifically, as mentioned in Halachah 9, the maximum angular distance between the two orbits is 5 degrees.

The importance of the concept the Rambam introduces here, the difference in latitude between the planes of the orbits of the sun and the moon, becomes significant in the following chapter. To explain briefly: In the previous chapters, it was explained that the visibility of the moon depends upon the distance in longitude between it and the sun. At the time of conjunction, the sun and the moon are at the same longitudinal point. Therefore, they set at the same time. As the difference in longitude between them increases, the crescent of the moon grows and the time of its setting becomes later, increasing the chances of its visibility. Nevertheless, the moon’s latitude also affects its visibility. The greater the latitude of the moon [i.e., its inclination from the orbit of the sun] the larger its crescent will appear. Also, a northerly latitude causes the moon to set later and thus makes it easier to be sighted. A southerly latitude, by contrast, causes the moon to set earlier and thus makes sighting it more difficult.

The difference in latitude between the orbits of the sun and the moon explains why there is not a lunar eclipse at every full moon, and why there is not a solar eclipse at every conjunction—although at the time of the full moon, the sun, the earth and the moon are aligned in a single line, and at the time of conjunction, the sun, the moon and the earth are aligned in a single line. Although the longitude of the sun and the moon is the same at these times, since their latitudes are different, the moon’s shadow does not interfere with the light of the sun at a conjunction, and the earth’s shadow does not prevent the light of the sun from reaching the moon at a full moon. Only when a conjunction or a full moon takes place at (or near) the point where the orbits of the moon and the sun intersect does an eclipse take place.

Because of the revolution of the head, the determination of the moon’s longitude will require several stages of computation.

I.e., the head revolves from east to west.

As evident from the later figures given by the Rambam, this number is an approximation, and the actual figure is several thirds less.

The Rambam is giving a negative figure here, his intent being 360°—180° 57’ 28. In positive terms, it would be a position of 179° 2’ 32.

This subtraction is necessary, since, as mentioned above, the head revolves from east to west, opposite to the direction of the heavenly sphere as a whole. Thus, we begin with a negative value as a starting point and add to it the distance traveled by the head. When that total is subtracted from 360, we have a positive figure that is the true position of the head. The reason the Rambam uses a negative figure for his starting point is that as the numbers increase, it is easier to add the mean distance traveled by the head to the starting point of 180° 57’ 28 and subtract the total from 360, than to define the starting point in positive terms and subtract the mean progress from it.

As mentioned previously, the head and the tail are the positions where the moon’s orbit intersects with that of the sun. Thus, if the moon is at the head or the tail, it is not at all inclined.

10.

We have used a non-literal translation of the word לִפְנֵי in this and the following sentence based on the context in this halachah.

11.

I.e., the difference between the position of the moon and the position of the head is less than 180 degrees, as stated in Halachah 10.

12.

I.e., the difference between the position of the moon and the position of the head is more than 180 degrees, as stated in Halachah 10.

13.

The Hebrew term רֹחַב הַיָּרֵחַ literally means “the width of the moon.” It was given this name because its range from 0° to 5° is far less than that of the longitude of the moon, אֹרֶךְ הַיָּרֵחַ, the angular distance between the moon and the sun, which ranges from 0° to 360°.

14.

According to contemporary science, the Rambam is making an approximation, for the latitude of the moon can reach 5 degrees and 9 minutes.

15.

As indicated from Halachah 11, this is the mid-point between the head and the tail, 90° and 270°.

16.

I.e., the distance the moon has traveled in its orbit from the head to its present position.

17.

This refers to the angle between the plane of the sun’s orbit and the position of the moon in its orbit.

18.

As stated above, this is the greatest latitude reached.

19.

See Chapter 13, Halachah 7, and Chapter 15, Halachah 7.

20.

For, as mentioned above, 90 degrees is the even mid-point of the course, and its angle increases and decreases in the same proportions as one approaches or leaves that point.

21.

In the standard printed texts of the Mishneh Torah, there is a printing error, and the concluding phrase from the chapter was added here by mistake.

22.

At 270°, as at 90°, the course reaches its maximum latitude, 5 degrees.

23.

For, as mentioned above, the course begins to increase as it progresses after reaching its tail in the same proportions as it increases as it progresses from its head.

24.

For the rate of the angular decrease from 270 to 180 is equivalent to the rate of decrease from 90 to 180.

Rambam - 1 Chapter a Day

Kiddush HaChodesh - Chapter 16

Kiddush HaChodesh - Chapter 16