1. Since our Sages¹Rosh HaShanah 23b, cited in Chapter 2, Halachah 4. said that among the questions posed to the witnesses [in order to verify their testimony] was "In which direction was [the crescent of] the moon inclined," I feel that it is appropriate to explain how this factor can be calculated. My statements will not be exact, because [this knowledge] is of no consequence regarding the actual sighting of the moon.

The starting point of these calculations is to know the degree of inclination of the constellations' [position].

2. The orbit of the sun that passes through the center [of the sphere] of the constellations does not pass directly through the center of the earth from east to west. Instead, it is inclined above the equator which passes through the center of the earth, [slightly] to the north and to the south. Half of it is northerly in inclination and half is southerly in inclination.

3. There are two points at which the orbit of the sun intersects the equator, which passes through the center of the earth: The first point is the beginning of the constellation of Aries, and the second point, opposite to [the first], is the beginning of the constellation of Libra. Thus, there are six constellations [whose positions] are inclined to the north, those between the beginning of the constellation of Aries and the end of the constellation of Virgo. And there are six [whose positions] are inclined to the south, those between the beginning of the constellation of Libra and the end of the constellation of Pisces.²In our previous notes, we explained that the constellations from Capricorn to Gemini were northerly inclined. There is no contradiction between those statements and the statements above; the difference is in the definition of the word "inclined." In the present context, the Rambam is using the word "inclined" to refer to the position of the ecliptic as it passes through these constellations: Is it located above or below the equator, and what is the extent of its variance from the equator?

Previously, we had used the word "inclined" to refer to the direction of the path of movement of the stars in the sky. In the constellations from Capricorn to Gemini, as the constellations move in their orbit, they will be moving northward. Conversely, in the constellations from Cancer to Sagittarius, as the constellations move in their orbit, they will be moving southward.

The inclination of the constellations can be observed by a careful observer with a field compass. In the spring and in the fall, the sun rises due east and sets due west. In the summer (i.e., when the sun is located in the constellations from Taurus to Virgo), it rises in the northeast and sets in the northwest. In the winter (i.e., when the sun is located in the constellations from Scorpio to Pisces), it rises in the southeast and sets in the southwest. Throughout the year, these constellations rise and set in the place where the sun rises and sets when it is in these constellations.

4. From the beginning of the constellation of Aries, the constellations' [positions] begin to be inclined slightly and diverge from the equator in a northerly direction until the beginning [of the constellation] of Cancer.

The beginning [of the constellation] of Cancer is approximately twenty-three and one half degrees north of the equator. [Afterwards,] the constellations begin to proceed toward the equator slowly until the beginning [of the constellation] of Libra, which is positioned on the equator.

From the beginning of the constellation of Libra, the constellations' [positions] begin to be inclined slightly and diverge from the equator in a southerly direction until the beginning [of the constellation] of Sagittarius.

The beginning [of the constellation] of Sagittarius is approximately twenty-three and one half degrees south of the equator. [Afterwards,] the constellations begin to proceed toward the equator slowly until the beginning [of the constellation] of Aries.

5. Thus, [in their daily orbits,] the beginning [of the constellation] of Aries and the beginning [of the constellation] of Libra revolve on the equator. Therefore, when the sun is positioned in the beginning [of these constellations],³I.e., at the vernal [spring] and autumnal equinoxes. it will not be inclined either to the north or to the south. It will rise due east and set due west, and the daytime and the nighttime [hours] will be equal throughout the world.

6. Thus, it should be clear to you that each of the degrees [in the sphere] of the constellations is inclined to the north or to the south, and that there is a specific measure to their inclination. The greatest inclination will be no more than approximately twenty-three and half degrees.

7. The following represent the extent of inclination from the equator vis-a-vis the number of degrees [in the sphere of the constellations]. Beginning with the constellation of Aries: [A point located at] 10 degrees [in the celestial sphere] will be inclined 4 degrees [from the equator]. [A point located at] 20 degrees will be inclined 8 degrees. [A point located at] 30 degrees will be inclined 11 1/2 degrees.

[A point located at] 40 degrees will be inclined 15 degrees. [A point located at] 50 degrees will be inclined 18 degrees. [A point located at] 60 degrees will be inclined 20 degrees. [A point located at] 70 degrees will be inclined 22 degrees. [A point located at] 80 degrees will be inclined 23 degrees. [A point located at] 90 degrees⁴I.e., the beginning of the constellation of Cancer. In these ninety degrees, we will have passed through three constellations, Aries, Taurus, and Gemini. will be inclined 23 1/2 degrees.

8. If the number [of degrees] also includes units, you should calculate [their degree of inclination] by taking an average between the two figures, as was explained with regard to [the determination of the position of] the sun and the moon.⁵See Chapter 13, Halachah 7, and Chapter 15, Halachah 7.

What is implied? Five degrees will be inclined 2 degrees. If the number of degrees [in the celestial sphere] is twenty-three, the inclination [from the equator] will be nine degrees. This same pattern should be followed whenever calculating a number that has both units and tens.

9. Since you know the extent of the inclination of all the degrees from one until ninety, you will be able to calculate the degree of inclination [of the entire celestial sphere] according to the method of calculation explained with regard to the moon's latitude.⁶See Chapter 16, Halachah 13. For if the number is between 90 and 180, the number should be subtracted from 180.⁷I.e., 110° has the same degree of inclination as 70°. For the angle of inclination proceeds in an even-balanced arc with 90 degrees as its center.

If the number is between 180 and 270, 180 should be subtracted from the number.⁸I.e., the southerly inclination beginning from the constellation of Libra parallels exactly the northerly inclination beginning from Aries. If the number is between 270 and 360, the number should be subtracted from 360.⁹Producing the inverse of the figures mentioned in the previous quadrant. [After these subtractions have been made], you will know the degree of inclination of the remainder, for it is the degree of inclination of the numbers mentioned previously without any addition or subtraction.

10. If you desire to know the number of degrees by which the moon's position deviates from the equator to the north or to the south, [you should follow this procedure]: Calculate the inclination of the degree [in the celestial sphere] that will be the true position, and see if its inclination is northerly or southerly. Afterwards, calculate the moon's first latitude and see whether it is northerly or southerly.

If the moon's latitude and the inclination of the degree [in the celestial sphere where it is located] have the same direction, then they should be added together.¹⁰E.g., if they are both northerly, the moon's latitude will cause it to appear further north than the constellation in which its true position is located. If they are in different directions - i.e., one is southerly and one is northerly - the smaller figure should be subtracted from the larger one.¹¹I.e., if the moon's true position is located in a northerly position and its latitude is southerly, the latitude should be subtracted from the angular distance of its true position. The remainder is the [angular] distance of the moon from the equator to the direction in which the larger figure was inclined.

11. What is implied? Let us say that we are trying to calculate the degree of inclination of the moon from the equator on the night of sighting, the second day of the month of Iyar of this year. You have already established that the angle [in the celestial sphere] at which the moon was located is the nineteenth degree of the constellation of Taurus. Thus, its inclination to the north will be approximately 18 degrees.¹²Actually, the true figure is a few minutes larger.

The latitude of the moon is approximately four degrees to the south.¹³Actually, the true figure is a few minutes less. After you subtract the lesser figure from the larger one, the position of the moon will be fourteen degrees to the north of the equator, for the largest figure is the eighteen degrees north [at which the moon's true position was located]. All [of the phases] of this calculation are approximations and are not exact, because they are of no consequence regarding the sighting.

12. If you desire to know the direction to which the moon will appear to be inclined [when sighted, it is governed by the following principles]: Calculate [the moon's angular] distance from the equator. If it is located on the equator or within two or three degrees to the north or to the south,¹⁴The variance of two or three degrees will not produce a difference that is of consequence. it will appear due west and its crescent will appear to be pointed due east.

13. If [the moon] is inclined to the north of the equator, it will appear in the northwest, and its crescent will appear to be pointed southeast.

14. If [the moon] is inclined to the south of the equator, it will appear in the southwest, and its crescent will appear to be pointed northeast. To the extent of [the moon's] distance [from the equator], its inclination will increase.

15. Among the other dimensions of the examination of the witnesses is the question: "How high [in the sky did the moon] appear?" This factor can be determined by the arc of sighting. If the arc is short, the moon will appear close to the earth, and when it is long, it will appear high above the earth. The length of the arc of sighting [will determine] the height at which [the moon] will appear above the earth to the witnesses.

16. Thus, we have explained all the calculations necessary for the sighting [of the moon] and the examination of the witnesses, so that everything will be comprehensible to men of understanding, and they will not lack awareness of any of the Torah's paths. [Therefore,] they will not venture forth in search of it in other texts. "Seek out of the book of God, read it. None of these will be lacking" [Isaiah 34:16].¹⁵With this halachah, the Rambam explains one of his motives for the inclusion of this section in the Mishneh Torah: so that a serious student would not have to resort to other texts, particularly those of the gentile scientific community, to obtain this knowledge. In this manner, he affirms one of the purposes for the Mishneh Torah stated in its introduction, to provide a text that includes every dimension of Jewish law.