Rambam - 3 Chapters a Day

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

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.)

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).

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.

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.

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

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.

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.

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.

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.

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.)

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.

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

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

See Chapter 13, Halachot 5-6.

See Chapter 13, Halachah 7.

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

See Chapter 13, Halachah 9.

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

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.

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

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.

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.

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°.

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

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

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

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

As stated above, this is the greatest latitude reached.

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

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.

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.

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

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.

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

Our translation is based on authentic manuscript editions of the Mishneh Torah and early printings. There is a printing error in the standard published text.

In this instance as well, our translation is based on authentic manuscript editions of the Mishneh Torah and early printings. There is a printing error in the standard published text, where the words “and the first latitude” were added unnecessarily.

Note the apparent contradiction to the figures the Rambam mentions in Chapter 15, Halachah 2, and the resolution suggested in Note 4 of that chapter.

In these months, the ecliptic (the plane of the sun’s orbit as extended to the celestial sphere) is inclined to the north. After the conjunction, the moon proceeds away from the position of the sun. When the inclination of the ecliptic is northward, this movement places it in a more northerly position. Therefore, the moon will set later than would be foreseen otherwise, resulting in a greater possibility of seeing the new moon. As mentioned in the notes of Chapter 16, as the longitude of the moon increases, seeing the moon also becomes easier. In these months, however, a lesser longitude is required.

In these months, the ecliptic is inclined to the south. As the moon proceeds away from the position of the sun after conjunction, it will be in a more southerly position in these months. Therefore, the moon will set earlier than would be foreseen otherwise, resulting in a lesser possibility of seeing the new moon. To compensate for this difference, a greater longitude is required.

Our translation is based on authentic manuscript editions of the Mishneh Torah and early printings. There is a printing error in the standard published text, and the word “latitude” was added unnecessarily.

To summarize the Rambam’s statements to this point: When the longitude of the moon (the angular distance between the moon and the sun) is minimal, the moon’s crescent will be small and the interval between the time of its setting and that of the sun will be small. Hence, it is unlikely that the moon will be sighted. When the longitude of the moon is greater, the size of the moon’s crescent will increase, as will the interval between the time of its setting and that of the sun. Accordingly, the possibility of sighting the moon will increase.
When the longitude is significantly large, it is obvious that the moon will be seen and no other calculations are necessary. When, however, the longitude is of intermediate size, there is a question whether the moon will be seen. The resolution of this question depends on the inclination of the ecliptic and the latitude of the moon—i.e., the angle—and the direction of that angle—to which the moon is inclined from the plane of the sun.

The sighting adjustment for longitude is based on two different factors: a) whether the constellation is inclined to the north or to the south as it intersects the horizon of Jerusalem, and b) the extent of the southerly position of that constellation. To explain: The constellations intersect the horizon at different angles, reflecting the pattern of their inclination in the heavenly sphere. The constellations from Capricorn until Gemini intersect the horizon at a northerly angle, and the constellations from Cancer to Sagittarius intersect the horizon at a southerly angle. With regard to the second factor, all the constellations are located to the south of Jerusalem. Jerusalem is located 32 degrees north, and the constellation of Cancer, the most northerly of the constellations, is located 23 1/2 degrees north. The more northerly a constellation is located, however, the greater the need for a subtraction from its longitude.

This is the constellation that the moon enters at the vernal (spring) equinox. It is inclined to the north and is not located in an extremely southerly position. Hence, a large sighting adjustment is necessary.

Since the constellation of Taurus intersects the horizon at a northerly inclination and it is located in a relatively northerly position, the largest adjustment is necessary.

Although this constellation is located in a very northerly position, its northerly inclination is less. Hence, a smaller subtraction is made.

Although Cancer is located in the most northerly position of all the constellations, since it has a southerly inclination the sighting adjustment required is less.

This and the four constellations that follow have southerly inclinations. Hence, the figure subtracted from their longitude is less.

This is the smallest sighting adjustment, because this constellation is located in a more southerly position than the others with a southerly inclination.

From this point on, the sighting adjustment increases, because these constellations have a northerly inclination.

The commentaries have noted that although the general thrust of the adjustments suggested by the Rambam conform to the calculations of the astronomers, the exact figures he gives follow neither the classic Greek figures nor those of modern astronomy. It is possible to explain that the Rambam was speaking merely in approximations, giving us a figure useful enough to calculate the position where the moon would be sighted, but not an exact scientific measure. This theory is borne out by the fact that he does not provide different measures for northern and southern latitudes, although according to science these figures vary.

To explain: The true position of the moon reflects the line extending from the center of the earth through the center of the moon, as it is projected against the heavenly sphere. Since Jerusalem (or for that matter, any other location on the earth’s surface) is not located at the center of the earth, but rather 4000 miles away, there will be a slight difference between the line described previously and the line extending from a person standing in Jerusalem to the center of the moon, as it is projected against the heavenly sphere. The closer the moon is to the horizon, the larger the sighting adjustment that has to be made.
[The same concept applies with regard to the sun. Nevertheless, since the distance between the earth and the sun is great, the angular difference between these two lines is not of consequence. The moon, by contrast, is located much closer to the earth and, at times, a difference of close to a degree can arise.]
In the evening, the moon will always appear slightly closer to the horizon than it actually is—i.e., it will appear closer to the position of the sun. Therefore, the angular difference between the two lines mentioned above should be subtracted from the moon’s true position. As explained above, the extent of the adjustment to be made depends on the inclination at which constellation intersects the horizon and its latitude in the heavenly sphere.

This principle applies during all the PM hours. During the AM hours, by contrast, the sighting adjustment should be added to the position of the moon (Ralbach).

The sighting adjustment for the moon’s latitude is derived by creating a parallax—i.e., a line directly parallel to the line running from the point of the moon’s first longitude to its first latitude is drawn from the point of its second longitude. A second line is drawn from the position of an onlooker in Jerusalem through the point of the first latitude and intersecting the line of the moon’s second latitude. The point where these two lines intersect is the moon’s second longitude. The adjustment mentioned in the following halachah represents the angle between these two lines.

Because Jerusalem is situated in a more northerly position than all the Zodiac constellations, the moon will always appear more southerly than it actually is. Therefore, if its latitude is northerly, a subtraction is necessary. To use geometric terms: When the moon’s latitude is northerly, its second latitude will always be closer to the point of its longitude than to its first latitude.

Since the moon will always appear more southerly, an addition is required when its original latitude is southerly. In geometric terms: When the moon’s latitude is southerly, its second latitude will always be further removed from the point of its longitude than its first latitude.

This is the point directly after the vernal (spring) equinox, when the sun is inclined northward and enters the northern part of its orbit.

This is the point directly after the autumnal equinox, when the sun is inclined southward and enters the southern part of its orbit.

Here, too, our translation is an emendation of the standard published text, based on authentic manuscript editions of the Mishneh Torah and early printings.

The Rambam is speaking about the second latitude, since it is possible for the sighting adjustment to change a northerly latitude to a southerly one.

The purpose of the calculations that follow (reaching a third longitude and a fourth longitude) is to calculate the time between the setting of the sun and the setting of the moon. The first longitude is sufficient to inform us whether or not the crescent of the moon will be large enough to be visible. The subsequent calculations are necessary to determine whether or not there will be sufficient time for actually sighting the moon. For when the crescent is small, it is difficult to detect unless there is ample time before it sets.
The third longitude reflects the point in the celestial sphere that will set at the same time as the moon does, as seen by a person standing on the equator. This is not the point in the celestial sphere where the moon appears to be located, but rather a point in the celestial sphere that is reached by drawing a line originating at the equator, running parallel to the horizon of the equator, and extending through the center of the moon. The point where this line intersects the celestial sphere is the third longitude.
The reason for associating the moon’s position with the equator is to establish a connection with a standard measure of time. In Jerusalem (and for that matter, anywhere else in the northern or southern hemisphere), the apparent movement of the celestial sphere varies with the seasons. On the equator, by contrast, the movement of the celestial sphere is constant at all times, 15 minutes to the hour.

Cf. Proverbs 2:15.

The angle between each particular constellation in the celestial sphere and the equator varies. The size of the adjustment to be made for the third longitude depends on that angle.

These are the points in the celestial sphere that intersect the horizon of the equator at the greatest angle. Therefore, the largest adjustment is necessary.

These are the points within the celestial sphere that are more or less parallel to the equator.

This means that when the moon’s latitude is northerly, the third longitude will always be closer to the equator.

This means that when the moon’s latitude is southerly, the third longitude will always be further removed from the equator.

These are the constellations that are inclined in a northerly direction.

I.e., the constellations that are inclined in a southerly direction.

This means that when the moon’s latitude is northerly, the third longitude will always be further removed from the equator.

This means that when the moon’s latitude is southerly, the third longitude will always be closer to the equator.

As happens when the moon is located in the beginning of the constellations of Cancer and Capricorn.

The Rambam’s intent in these sets of calculations is to reach a point on the equator that will set at the same time the third longitude sets in Jerusalem. For although the third longitude was able to relate the moon’s position to the equator, it did not take into consideration the difference between the horizon of the equator and the horizon of Jerusalem. This is accomplished by drawing a line from the third longitude to the equator, which is parallel to the horizon of Jerusalem.

Two factors are significant in determining the fourth longitude:
a) The angle of the constellation’s inclination to the horizon of the equator. The greater the inclination of the constellation, the closer the fourth longitude will be located to the equator.
b) whether the constellation is inclined to the north or to the south. If the constellation is inclined to the north, the third longitude, and hence the place on the equator parallel to it, will be located further away from the horizon, resulting in a later setting and thus an extended fourth longitude. Conversely, if the inclination is southerly, the third longitude will be located closer to the horizon, resulting in a shortened fourth longitude.
Of these two factors, the latter is more significant, and causes a larger correction. To explain these factors with regard to the constellations of Pisces and Aries: These constellations are inclined to a great degree, a factor that would reduce the fourth longitude. Since, however, they are northerly inclined, and this is the stronger factor, a modest increase is required.

These constellations are inclined to the north, and the degree of their inclination is less than that of Pisces and Aries. Hence, a greater increase is required.

Here, the constellations begin a southerly inclination. Hence, although they are more parallel to the horizon of the equator, no addition is made.

In this instance, the degree of inclination of these constellations is great and their inclination is southerly. Both of these factors lead to a reduction in the fourth longitude. Hence, the greatest subtraction is required.

The Ralbach questions why the Rambam refers to the first latitude. Seemingly, it would be appropriate to make this correction based on the second latitude, for there is a significant difference between it and the first latitude. According to trigonometry, it also would appear that the calculations should be based on the second latitude.

Although the fourth longitude established a relationship between the equator and Jerusalem, it is still dependent on the third longitude, which relates to the moon and the celestial sphere as they set on the horizon of the equator. Through the correction mentioned here, we find a place on the extension of the equator that will set at the same as the moon sets in Jerusalem. Having reached this point, we can calculate the difference in time (15 degrees to the hour) between the setting of the sun and this point (which will set at the same time as the setting of the moon). Accordingly, we will be able to determine whether or not this interval will allow for the sighting of the moon.
The correction for geographic longitude is reached by drawing a line from the position of the moon parallel to the horizon of Jerusalem. One might ask: If this was the Rambam’s intent, why were so many intermediate steps—the definition of the second, third, and fourth latitudes—necessary? Why didn’t he suggest drawing the above- mentioned line at the very beginning of his calculations?
The explanation is that the Rambam allowed an individual to follow his own steps in arriving at this final figure. I.e., these lines and distances are all artificial and can be determined only by calculations. Through trigonometry, if one knows the length of one side of a triangle and two angles, or the length of two sides and one angle, it is possible to calculate the size of all three angles and all three sides. To find the line extending from the moon to the equator parallel to the horizon of Jerusalem, the Rambam had to build sets of triangles, and calculate angles based on the relationship of one triangle to another. The process he followed is reflected in the series of corrections he offers.

A northerly latitude means that the actual position of the moon is further removed from the horizon than the third longitude. This will result in a later setting of the moon. Accordingly, the correction based on geographic latitude will require addition to the fourth longitude. This applies regardless of whether the inclination of the constellation in which the moon is located is northerly or southerly.

A southerly latitude means that the actual position of the moon is closer to the horizon than the third longitude. This will result in an earlier setting of the moon. Accordingly, the correction based on geographic latitude will require subtraction from the fourth longitude. This applies regardless of whether the inclination of the constellation in which the moon is located is northerly or southerly.

Our translation represents a correction of the standard printed text of the Mishneh Torah.

It is possible that the Rambam’s wording alludes to a concept mentioned previously, that the calculations he suggests are applicable only at the beginning of the month, when the new moon might be sighted.

I.e., barring clouds, as explained at the beginning of the following chapter.

As mentioned at the beginning of this chapter, the first longitude gives us information regarding the size of the moon’s crescent and the difference between the moon’s setting and that of the sun. When the first longitude is sufficiently large or when it is sufficiently small, it is possible to determine whether or not the moon will be sighted without considering extenuating factors—e.g., its longitude, the inclination of the constellation in which it is located, and the extent of that inclination. When, however, the first longitude is of intermediate length, these extenuating factors must be considered. The establishment of a systematic method of considering these factors is the purpose of all the computations mentioned in this chapter.

See Halachot 13 and 14.

As the Rambam mentioned at the very beginning of this discussion (Chapter 11, Halachah 6), the figures that he gives are not exact. They do, however, give us sufficient information to determine when and where the moon will be sighted.

Rosh HaShanah 25a, commenting on Psalms 104:19.

Loc. cit.

Commenting on I Chronicles 12:32, “From the descendants of Yissachar, men who had understanding of the times...,” Bereshit Rabbah 72:5 explains that the sages of the tribe of Yissachar were those responsible for the determination of the calendar. (See also the commentary of the Radak on this verse.)

The context of this commentary is not a proper place for a full discussion of the Rambam’s perspective on the supposed conflicts between science and the Torah. It must be noted, however, that the statements made here, emphasizing the importance of the empirical evidence of science, should not be interpreted as indicating that the perspective science adopts at any given time should be accepted in place of the Torah’s teachings. In this context, it is worthy to quote the Rambam’s statements in Hilchot Shechitah 10:13: Similarly, with regard to the conditions that we have enumerated as causing an animal to be trefah (unable to live for an extended period): Even though it appears from the medical knowledge available to us at present that some of these conditions are not fatal... all that is significant to us is what our Sages said, as [implied by Deuteronomy 17:11]: “[You shall act] according to the instructions that they will give you.”

Our translation is based on authoritative manuscripts and early printings of the Mishneh Torah; it differs slightly from the standard printed text.