One-Page Learn · The Halachos at a glance
קִדּוּשׁ הַחֹדֶשׁ
Sanctification of the New Month 19
Sefer Zemanim · The tilt of the constellations against the equator, how it combines with the moon's own latitude to fix which way the crescent leans, and the closing declaration that ends the treatise
23° 30'
The greatest inclination the constellations ever reach
6
Constellations inclined north, and six inclined south
14
Degrees north of the equator, the moon on the second of Iyar
2
Questions to the witnesses this whole chapter exists to answer
Part 1The tilt of the constellations, and the table that measures it
- A chapter written for the court, and openly approximate. Since our Sages listed among the questions put to the witnesses the direction in which the crescent was inclined, the Rambam explains how that can be calculated, stating at the outset that his figures will not be exact because the knowledge is of no consequence for the actual sighting; the starting point is the inclination of the constellations. (19:1)
- The sun's orbit crosses the equator at exactly two points. The orbit does not run through the center of the earth from east to west but is inclined, half northerly and half southerly, meeting the equator at the beginning of Aries and the beginning of Libra, so six constellations from Aries to the end of Virgo incline north and six from Libra to the end of Pisces incline south. (19:2, 19:3)
- The inclination peaks at Cancer and Sagittarius and vanishes at the crossings. From Aries it grows to roughly twenty three and a half degrees north at the beginning of Cancer, returns to the equator at Libra, grows to roughly twenty three and a half degrees south at the beginning of Sagittarius, and returns again; when the sun stands at the beginning of Aries or Libra it rises due east and sets due west and day and night are equal throughout the world. (19:4, 19:5, 19:6)
- The table, and how fast it flattens. Ten degrees along the band inclines four degrees, twenty inclines eight, thirty inclines eleven and a half, forty inclines fifteen, fifty inclines eighteen, sixty inclines twenty, seventy inclines twenty two, eighty inclines twenty three, and ninety inclines twenty three and a half, the increments shrinking from four degrees to half a degree across the run. (19:7)
- Units are averaged, and the circle is folded into the first quarter. A figure carrying units as well as tens is handled by taking an average between the two table entries, so five degrees inclines two and twenty three inclines nine; and a number between 90 and 180 is subtracted from 180, one between 180 and 270 has 180 subtracted from it, and one between 270 and 360 is subtracted from 360, the remainder carrying the inclination already tabulated. (19:8, 19:9)
RememberFrom eighty degrees to ninety the sun travels ten degrees and gains half a degree of latitude. That is the solstice: not a halt in motion but a halt in direction.
Part 2The moon's distance from the equator
- Two figures, combined by direction rather than by size. To find how far the moon stands from the equator, take the inclination of the degree at which its true position falls and note whether it is northerly or southerly, then take the moon's first latitude and note the same; matching directions are added, opposing directions call for subtracting the smaller from the larger. (19:10)
- The remainder leans in the direction of the larger figure. Whatever is left after the addition or subtraction is the angular distance of the moon from the equator, and it lies toward whichever of the two figures was the greater, which is what makes the sign of the result depend on a comparison rather than on the moon alone. (19:10)
- The worked example, on the night of the second of Iyar. The moon's true position having already been fixed at the nineteenth degree of Taurus, that degree inclines approximately eighteen degrees north, while the moon's latitude is approximately four degrees south; subtracting the lesser from the greater places the moon fourteen degrees north of the equator, north because the eighteen was the larger. (19:11)
- He states the disclaimer a second time inside the example. All the phases of this calculation, the Rambam writes, are approximations and are not exact, because they are of no consequence regarding the sighting, which is the same caution he opened the chapter with rather than an admission of a difficulty he ran into. (19:11)
RememberThe moon's distance from the equator is never read off the moon. It is the difference between where its degree leans and where the moon leans.
Part 3Which way it points, how high it stood, and the last word
- On the equator, the crescent points due east. If the moon is located on the equator, or within two or three degrees north or south of it, it will appear due west and the horns of its crescent will appear to point due east. (19:12)
- North and south reverse the horns, and distance exaggerates them. A moon inclined north of the equator appears in the northwest with its crescent pointing southeast, and a moon inclined south appears in the southwest with its crescent pointing northeast, the inclination increasing in proportion to the moon's distance from the equator. (19:13, 19:14)
- The second examination question is answered by the arc of sighting. The witnesses were also asked how high the moon appeared, and this is determined by the arc of sighting: a short arc means the moon appeared close to the earth, and a long arc means it appeared high above it. (19:15)
- The treatise closes by telling the student to look nowhere else. Everything needed for the sighting of the moon and the examination of the witnesses has now been explained, so that it will be comprehensible to men of understanding and they will lack awareness of none of the Torah's paths and will not venture out in search of it in other texts, sealed with the verse from Isaiah, seek out of the book of God and read, none of these shall be lacking. (19:16)
RememberThe horns never point at the moon. They point back toward a sun already below the horizon, which is exactly why the question tests a witness.