Halacha
הלכה א
אִם תִּרְצֶה לֵידַע מְקוֹם הַשְּׁמֶשׁ הָאֲמִתִּי בְּכָל יוֹם שֶׁתִּרְצֶה. תּוֹצִיא תְּחִלָּה מְקוֹמָהּ הָאֶמְצָעִי לְאוֹתוֹ הַיּוֹם עַל הַדֶּרֶךְ שֶׁבֵּאַרְנוּ. וְתוֹצִיא מְקוֹם גֹּבַהּ הַשֶּׁמֶשׁ. וְתִגְרַע מְקוֹם גֹּבַהּ הַשֶּׁמֶשׁ מִמְּקוֹם הַשֶּׁמֶשׁ הָאֶמְצָעִי וְהַנִּשְׁאָר הוּא הַנִּקְרָא מַסְלוּל הַשֶּׁמֶשׁ:
כסף משנה
1.
[The following method should be used] if you wish to know the true position1As stated in Chapter 11, the true position of the sun refers to the position at which it is seen in the heavenly sphere. The difference between the sun's true position and its mean position stems from the fact that the Earth is not located at the exact center of the sun's orbit. of the sun on any particular day you desire: First, it is necessary to calculate the mean position of the sun through the methods of calculation we have explained. Then calculate the position of the apogee of the sun.2The method through which both these figures can be calculated is stated in the previous chapter. Afterwards, subtract the apogee of the sun from the mean position of the sun. The remainder is referred to as the course of the sun.3I.e., the path the sun has traveled in its orbit from the apogee until it reached its present position.הלכה ב
וְתִרְאֶה כַּמָּה מַעֲלוֹת הוּא מַסְלוּל הַשֶּׁמֶשׁ. אִם הָיָה הַמַּסְלוּל פָּחוֹת מִק''פ מַעֲלוֹת. תִּגְרַע מְנַת הַמַּסְלוּל מִמְּקוֹם הַשֶּׁמֶשׁ הָאֶמְצָעִי. וְאִם הָיָה הַמַּסְלוּל יוֹתֵר עַל ק''פ מַעֲלוֹת עַד ש''ס תּוֹסִיף מְנַת הַמַּסְלוּל עַל מְקוֹם הַשֶּׁמֶשׁ הָאֶמְצָעִי. וּמַה שֶּׁיִּהְיֶה אַחַר שֶׁתּוֹסִיף עָלָיו אוֹ תִּגְרַע מִמֶּנּוּ הוּא הַמָּקוֹם הָאֲמִתִּי:
כסף משנה
2.
[The next step is] to calculate the angular distance of the course of the sun.4The course of the sun is an arc extending from the mean position of the sun to its apogee. The angular distance of the course is derived by drawing straight lines from the mean position and the apogee to the center of the sun's orbit.If the apogee of the sun were located at O°, the angular distance of the course and the sun's mean position would be the same. Since, however, the apogee also moves within the heavenly sphere, there is a variance between these two figures. If the angular distance of the course is less than 180 degrees, one should subtract5Before the sun reaches the perigee, the point in its orbit that is closest to the Earth, its true position will always be less than its mean position. Hence, the angle referred to as the angle [determined by the] course must be subtracted from its mean position to arrive at the true position. the angle [determined by the] course6The angle [determined by the] course refers to the extent of the deviation between the position of the sun that can be observed in the sky and its mean position. The manner of determining this figure is described in Halachah 4.
To express these concepts in geometric terms: The sun's true position represents the angle at which it can be found in the sphere of the heavens of which the Earth is the center (c). When the angle that is called the angle [determined by the] course (a) is added to this angle, the sum is equivalent to the angle of the course of the sun (b).
How is this figure derived? Refer to the accompanying diagram: The sum of the angles of the triangle a, c, and d equals 180°, and the angles b and d equal 180°. Hence, b equals c a.
Thus, the true position of the sun is equivalent to its mean position minus the figure referred to as the angle [determined by the] course. Thus, when the course is less than 180 degrees, the sun's true position is always a small amount less than its mean position. from the sun's mean position. If the angular distance of the course is more than 180 degrees, one should add7When the sun passes the perigee, its true position will always be greater than its mean position. Hence, the angle referred to as the angle [determined by the] course must be added to its mean position to arrive at the true position.
Why is this so? Refer to the following diagram: d refers to the true position of the sun, b to its mean position and a to the angle referred to as the angle [determined by the] course. E to its true position minus 180° and f refers to its mean position minus 180°. a + f + c equals 180. E + c equals 180. Thus, a + f equals e. Hence, the mean position plus the angle [determined by the] course will be equal to the true position. the angle [determined by the] course to the sun's mean position. The figure remaining after making this addition or subtraction represents [the sun's] true position.
הלכה ג
וְדַע שֶׁאִם יִהְיֶה הַמַּסְלוּל ק''פ בְּשָׁוֶה אוֹ ש''ס בְּשָׁוֶה. אֵין לוֹ מָנָה אֶלָּא יִהְיֶה הַמָּקוֹם הָאֶמְצָעִי הוּא הַמָּקוֹם הָאֲמִתִּי:
כסף משנה
3.
If the course [of the sun] is an even 180 degrees or an even 360 degrees, there will be no angle [determined by the course to add or to subtract]. Instead, the [sun's] mean position is its true position.8I.e., when the sun is at the apogee or perigee, there will be one straight line between the Earth (the center of the heavenly sphere), the center of the sun's orbit, and the actual position of the sun.הלכה ד
וְכַמָּה הִיא מְנַת הַמַּסְלוּל. אִם יִהְיֶה הַמַּסְלוּל עֶשֶׂר מַעֲלוֹת. תִּהְיֶה מְנָתוֹ כ' חֲלָקִים. וְאִם יִהְיֶה כ' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מ' חֲלָקִים. וְאִם יִהְיֶה ל' מַעֲלוֹת תִּהְיֶה מְנָתוֹ נ''ח חֲלָקִים. וְאִם יִהְיֶה מ' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְט''ו חֲלָקִים. וְאִם יִהְיֶה נ' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְכ''ט חֲלָקִים. וְאִם יִהְיֶה ס' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וּמ''א חֲלָקִים. וְאִם יִהְיֶה ע' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''א חֲלָקִים. וְאִם יִהְיֶה פ' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''ז חֲלָקִים. וְאִם יִהְיֶה צ' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''ט חֲלָקִים. וְאִם יִהְיֶה ק' מַעֲלוֹת תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''ח חֲלָקִים. וְאִם יִהְיֶה ק''י תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְנ''ג חֲלָקִים. וְאִם יִהְיֶה ק''כ תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וּמ''ה חֲלָקִים. וְאִם יִהְיֶה ק''ל תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת ל''ג חֲלָקִים. וְאִם יִהְיֶה ק''מ תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְי''ט חֲלָקִים. וְאִם יִהְיֶה ק''נ תִּהְיֶה מְנָתוֹ מַעֲלָה אַחַת וְחֵלֶק אֶחָד. וְאִם יִהְיֶה ק''ס תִּהְיֶה מְנָתוֹ מ''ב חֲלָקִים. וְאִם יִהְיֶה ק''ע תִּהְיֶה מְנָתוֹ כ''א חֲלָקִים. וְאִם יִהְיֶה ק''פ בְּשָׁוֶה אֵין לוֹ מָנָה כְּמוֹ שֶׁבֵּאַרְנוּ אֶלָּא מְקוֹם הַשֶּׁמֶשׁ הָאֶמְצָעִי הוּא מְקוֹמָהּ הָאֲמִתִּי:
כסף משנה
4.
What is the angle [determined by the] course? If the course is ten degrees, the [resulting] angle will be 20 minutes.If the course is twenty degrees, the [resulting] angle will be 40 minutes.
If the course is thirty degrees, the [resulting] angle will be 58 minutes.
If the course is forty degrees, the [resulting] angle will be 1 degree and 15 minutes.
If the course is fifty degrees, the [resulting] angle will be 1 degree and 29 minutes.
If the course is sixty degrees, the [resulting] angle will be 1 degree and 41 minutes.
If the course is seventy degrees, the [resulting] angle will be 1 degree and 51 minutes.
If the course is eighty degrees, the [resulting] angle will be 1 degree and 57 minutes.
If the course is ninety degrees, the [resulting] angle will be 1 degree and 59 minutes.
If the course is one hundred degrees, the [resulting] angle will be 1 degree and 58 minutes.9The largest angle determined by the course is when the angular distance of the course itself is 96 degrees - i.e., shortly after the mean position of the sun passes directly above the center of its orbit. After this point is reached, the angle begins to decrease.
Significantly, the rate of the decrease does not correspond exactly to the rate of increase as the angles approach 96 degrees. The reason for this difference is that, as stated above, the position of the Earth is not at the center of the sun's orbit. Hence, at 100 and 80 degrees, although the mean position of the sun has moved an equal distance from the center of its orbit, it has moved different distances from the Earth.
If the course is one hundred ten degrees, the [resulting] angle will be 1 degree and 53 minutes.
If the course is one hundred twenty degrees, the [resulting] angle will be 1 degree and 45 minutes.
If the course is one hundred thirty degrees, the [resulting] angle will be 1 degree and 33 minutes.
If the course is one hundred forty degrees, the [resulting] angle will be 1 degree and 19 minutes.
If the course is one hundred fifty degrees, the [resulting] angle will be 1 degree and 1 minute.
If the course is one hundred sixty degrees, the [resulting] angle will be 42 minutes.
If the course is one hundred seventy degrees, the [resulting] angle will be 21 minutes.
If the course is an even one hundred eighty degrees, it has no measure. Instead, its mean position is its true position, as we explained.
הלכה ה
הָיָה הַמַּסְלוּל יֶתֶר עַל ק''פ מַעֲלוֹת. תִּגְרַע אוֹתוֹ מִש''ס מַעֲלוֹת וְתֵדַע מְנָתוֹ. כֵּיצַד. הֲרֵי שֶׁהָיָה הַמַּסְלוּל ר' מַעֲלוֹת. תִּגְרַע אוֹתוֹ מִש''ס תִּשָּׁאֵר ק''ס מַעֲלוֹת. וּכְבָר הוֹדַעְנוּ שֶׁמְּנַת ק''ס מַעֲלוֹת מ''ב חֲלָקִים. וְכֵן מְנַת הַמָּאתַיִם מ''ב חֲלָקִים:
כסף משנה
5.
[The following procedure should be used] if the course [of the sun] is greater than one hundred eighty degrees: One should subtract the course from three hundred sixty degrees and [calculate the resulting] angle accordingly.10Thus, our computations will be based on the negative of the angle measured previously. To put the Rambam's statements in layman's terms: The angle formed will be the same regardless of whether the mean position of the sun is measured in an increase from O° or a decrease from 360°.What is implied? If the course is 200 degrees, that figure should be subtracted from 360 degrees, leaving a remainder of 160 degrees. Since you already know that the [resulting] angle of a course of 160 degrees is 42 minutes, that same figure will be the [resulting] angle of a course of 200 degrees.
הלכה ו
וְכֵן אִם הָיָה הַמַּסְלוּל ש' מַעֲלוֹת. תִּגְרַע אוֹתוֹ מִש''ס יִשָּׁאֵר ס'. וּכְבָר יָדַעְתָּ שֶׁמְּנַת ס' מַעֲלוֹת מַעֲלָה אַחַת וּמ''א חֲלָקִים. וְכֵן הִיא מְנַת הַש' מַעֲלוֹת. וְעַל דֶּרֶךְ זוֹ בְּכָל מִנְיָן וּמִנְיָן:
כסף משנה
6.
Similarly, if the course was three hundred degrees, one should subtract that figure from three hundred sixty, leaving a remainder of sixty. Since you already know that the [resulting] angle of a course of 60 degrees is 1 degree and 41 minutes, that same figure will be the [resulting] angle of a course of 300 degrees. Similar procedures should be followed in calculating other figures.הלכה ז
הֲרֵי שֶׁהָיָה הַמַּסְלוּל ס''ה מַעֲלוֹת. וּכְבָר יָדַעְנוּ שֶׁמְּנַת הַשִּׁשִּׁים הִיא מַעֲלָה אַחַת וּמ''א חֲלָקִים. וּמְנַת הָע' הִיא מַעֲלָה אַחַת וְנ''א חֲלָקִים. נִמְצָא בֵּין שְׁתֵּי הַמָּנוֹת י' חֲלָקִים. וּלְפִי חֶשְׁבּוֹן הַמַּעֲלוֹת יִהְיֶה לְכָל מַעְלָה חֵלֶק אֶחָד. וְיִהְיֶה מְנַת הַמַּסְלוּל שֶׁהוּא ס''ה מַעֲלָה אַחַת וּמ''ו חֲלָקִים:
כסף משנה
7.
[How is the angle determined by the course calculated] when the course is [an intermediate figure - e.g.,] 65 degrees? You already know that the [resulting] angle of 60 degrees is 1 degree and 41 minutes. And you know that the [resulting] angle of 70 degrees is 1 degree and 51 minutes. Thus, there are ten minutes between these [two] measures. Thus, [an increase of] a degree [of the course] will bring an increase of a minute [in the resulting angle]. Thus, the [resulting] angle of a course of 65 degrees will be 1 degree and 46 minutes.11Although the correspondence between the angular length of the course and the angle [determined by the] course is not uniform over a large span, within a span of ten degrees the difference between the actual figure and the approximation arrived at by the Rambam is not of consequence.הלכה ח
וְכֵן אִלּוּ הָיָה הַמַּסְלוּל ס''ז הָיְתָה מְנָתוֹ מַעֲלָה אַחַת וּמ''ח חֲלָקִים. וְעַל דֶּרֶךְ זוֹ תַּעֲשֶׂה בְּכָל מַסְלוּל שֶׁיִּהְיֶה בְּמִנְיָנוֹ אֲחָדִים עִם הָעֲשָׂרוֹת. בֵּין בְּחֶשְׁבּוֹן הַשֶּׁמֶשׁ בֵּין בְּחֶשְׁבּוֹן הַיָּרֵחַ:
כסף משנה
8.
Similarly, if the course was 67 degrees, the [resulting] angle would be 1 degree and 48 minutes. A similar procedure should be followed regarding any course that has both units and tens, both for calculations regarding the sun and for calculations regarding the moon.הלכה ט
כֵּיצַד. הֲרֵי שֶׁרָצִינוּ לֵידַע מְקוֹם הַשֶּׁמֶשׁ הָאֲמִתִּי בִּתְחִלַּת לֵיל הַשַּׁבָּת י''ד יוֹם לְחֹדֶשׁ תַּמּוּז מִשָּׁנָה זוֹ. תּוֹצִיא אֶמְצַע הַשֶּׁמֶשׁ תְּחִלָּה לָעֵת הַזֹּאת. וְסִימָנוֹ ק''ה ל''ז כ''ח כְּמוֹ שֶׁבֵּאַרְנוּ. וְתוֹצִיא מְקוֹם גֹּבַהּ הַשֶּׁמֶשׁ לָעֵת הַזֹּאת. יֵצֵא לְךָ סִימָנוֹ פ''ו מ''ה כ''ג. וְתִגְרַע מְקוֹם הַגֹּבַהּ מִן הָאֶמְצָעִי. יֵצֵא לְךָ הַמַּסְלוּל י''ח מַעֲלוֹת וְנ''ב חֲלָקִים וּשְׁתֵּי שְׁנִיּוֹת. סִימָנָם י''ח נ''ב ב'. וְאַל תַּקְפִּיד בְּכָל מַסְלוּל עַל הַחֲלָקִים אֶלָּא אִם יִהְיוּ פָּחוֹת מִשְּׁלֹשִׁים אַל תִּפְנֶה אֲלֵיהֶם. וְאִם הָיוּ שְׁלֹשִׁים אוֹ יוֹתֵר תַּחְשֹׁב אוֹתָם מַעֲלָה אַחַת וְתוֹסִיף אוֹתָהּ עַל מִנְיַן מַעֲלוֹת הַמַּסְלוּל. לְפִיכָךְ יִהְיֶה מַסְלוּל זֶה י''ט מַעֲלוֹת וְתִהְיֶה מְנָתוֹ עַל הַדֶּרֶךְ שֶׁבֵּאַרְנוּ ל''ח חֲלָקִים:
כסף משנה
9.
[To apply these principles]: Should we desire to know the true position of the sun at the beginning of Friday night, the fourteenth of Tammuz for this present year: First, we should calculate the mean position of the sun for this time, which is, as explained,12Chapter 12, Halachah 2. 105° 37' 25". We should then calculate the apogee of the sun at this time, which is 86° 45' 23". When the apogee is subtracted from the mean position, the remainder, the course [of the sun], will be 18 degrees, 52 minutes and 2 seconds, in symbols 18° 52' 2 ".With regard to the course [of the sun], the minutes are of no consequence. If they are less than thirty, they should be disregarded entirely. If they are more than thirty, they should be considered an additional degree and added to the sum of the degrees. Accordingly, it should be considered as if there are 19 degrees in this course. The [resulting] angle of such a course can be calculated to be 38 minutes in the manner that we explained.
הלכה י
וּלְפִי שֶׁהַמַּסְלוּל הַזֶּה הָיָה פָּחוֹת מִק''פ. תִּגְרַע הַמָּנָה שֶׁהִיא ל''ח חֲלָקִים מֵאֶמְצַע הַשֶּׁמֶשׁ יִשָּׁאֵר ק''ד מַעֲלוֹת וְנ''ט חֲלָקִים וְכ''ה שְׁנִיּוֹת. סִימָנָם ק''ד נ''ט כ''ה. וְנִמְצָא מְקוֹם הַשֶּׁמֶשׁ הָאֲמִתִּי בִּתְחִלַּת לֵיל זֶה בְּמַזַּל סַרְטָן בְּט''ו מַעֲלוֹת בּוֹ פָּחוֹת ל''ה שְׁנִיּוֹת. וְאַל תִּפְנֶה אֶל הַשְּׁנִיּוֹת כְּלָל לֹא בִּמְקוֹם הַשֶּׁמֶשׁ וְלֹא בִּמְקוֹם הַיָּרֵחַ וְלֹא בִּשְׁאָר חֶשְׁבּוֹנוֹת הָרְאִיָּה. אֶלָּא חֲקֹר עַל הַחֲלָקִים בִּלְבַד. וְאִם יִהְיוּ הַשְּׁנִיּוֹת קָרוֹב לִשְׁלֹשִׁים עֲשֵׂה אוֹתָם חֵלֶק אֶחָד וְהוֹסִיפוֹ עַל הַחֲלָקִים:
כסף משנה
10.
Since the course is less than 180 degrees, the [resulting] angle [of the course], 38 minutes, should be subtracted from the mean position of the sun, leaving a remainder of 104 degrees, 59 minutes and 25 seconds, in figures 104° 59' 25". Thus, the true position of the sun at the beginning of this night will be fifteen degrees less 35 seconds in the constellation of Cancer.One need not pay attention to the seconds at all, neither with regard to the position of the sun, nor with regard to the position of the moon, nor in any other calculations regarding the sighting [of the moon]. Instead, if the number of seconds is approximately13The commentaries have questioned the Rambam's use of the word "approximately." Our bracketed additions are made in that light. thirty [or more], they should be considered a minute, and added to the sum of the minutes.
הלכה יא
וּמֵאַחַר שֶׁתֵּדַע מְקוֹם הַשֶּׁמֶשׁ בְּכָל עֵת שֶׁתִּרְצֶה. תֵּדַע יוֹם הַתְּקוּפָה הָאֲמִתִּי כָּל תְּקוּפָה שֶׁתִּרְצֶה. בֵּין תְּקוּפוֹת הַבָּאוֹת אַחַר עִקָּר זֶה שֶׁמִּמֶּנּוּ הִתְחַלְנוּ. בֵּין תְּקוּפוֹת שֶׁעָבְרוּ מִשָּׁנִים קַדְמוֹנִיּוֹת:
כסף משנה