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TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

Two lunar crescent visibility criteria:

al-Khwarizmi and al-Qallas

Wenceslao Segura González

e-mail: wenceslaoseguragonzalez@yahoo.es

Independent Researcher

Abstract. We show that the criteria of lunar crescent visibility of al-Khwarizmi (9th century) and

al-Qallas (10th century) is not the Indian criterion, according to which the Moon will be visible if

between the moonset and sunset there are more than 48 minutes. Therefore, we distinguished two

new visibility criteria: al-Khwarizmi and al-Qallas, which we analyze and generalize.

1. Introduction

In the past and still today, there were calendars with lunar months, which starting with the

new Moon, then the full Moon is in the middle of the month. In the absence of astronomical

knowledge, the ancient peoples determined the beginning of the month, not by the conjunction with

the Sun, but with the moment on the western horizon, shortly after sunset, the Moon was first seen

after having been new.

Determining the beginning of the month by observation ensures that the calendar does not

deviate from Moon's movement. It has the disadvantage that it is not possible to know in advance

if a lunar month will have 29 or 30 days. We have to wait until sunset on the 29th to know if the

month has 29 or 30 days.

The astronomers of antiquity and the Middle Ages found methods or criteria that allowed

them to find out when the Moon would be seen for the first time after being hidden by the Sun. It

does not appear that these criteria were used to regulate the calendar, but they were useful to

detect errors in observation.

Medieval Muslim astronomers devised numerous techniques to anticipate on what day the

lunar crescent would be visible for the first time (Kennedy 1956; King and Samsó 2001). All these

techniques have not yet been investigated. Still, at least several of these criteria are known (King

1987), and they give us information on how Muslim astronomers dealt with the problem of the

visibility of the lunar crescent.

Among the ancient techniques of interest, in addition to those used by the Babylonians (Fatoohi

et al. 1999; Gautschy 2014; Stern 2008), is the so-called Indian criterion, which says that the lunar

crescent is seen if between the sunset and moonset there are more than 48 minutes (Bruin 1971).

This knowledge passed on to the medieval Muslim astronomers. A matter that has been investigated

by several researchers, which have shown that Muslim astronomers used the Indian criterion

(Fatoohi 1998; Hogendijk 1988; Kennedy and Janjanian 1965).

But the Muslim astronomers did not follow the Indian criteria, but adapted it. They took the

48 minutes mentioned above (or 12º of arc) as an extreme value, but developed criteria that did not

coincide with that of ancient Indian astronomy. This thesis is what we demonstrate in this research.

Firstly, we precisely defined the Indian criterion and found the techniques to determine the

oblique descension and ascension, astronomical concepts, nowadays in disuse, that were used to

develop the criteria of lunar visibility.

We analyze the visibility tables that we associate with al-Khwarizmi and al-Qallas, verifying

that they are different from the Indian criterion. However, they use an arc of 12º as the limit value.

We identified two new visibility criteria, which we have called al-Khwarizmi and al-Qallas, which

we developed and generalized. The knowledge of the techniques developed in the Middle Ages to

determine the vision of the crescent, completes the numerous criteria of lunar visibility that have

been made in recent times (Ilyas 1999).

1

Wenceslao Segura González

2

2. Definitions

The criteria for know when the Moon will be seen for the first time after being in conjunction

with the Sun, we classify as astronomical, physical, and based on experimentation.

The criterion used in India and later adapted by medieval Muslim astronomers is astronomical,

since they only need astronomical data. The physical criteria also use physical laws, such as the

illumination of the sky and the Moon, atmospheric absorption, visual perception of the observer, etc.

Finally, there are the criteria based on observation, which derive rules adjusted to the observations

of the lunar crescent.

We will make the calculations with geocentric magnitudes, and we will not take into account

parallax, refraction, and semi-diameter of the Sun and the Moon. We will assume that the Earth is

a sphere.

Equatorial separation arc

s

a

is the angle measured in the equator in direct sense between

the intersection points of the almucantar of the Sun and almucantar of the Moon with the equator

(arc DE in drawing 1). Ecliptic separation arc

S

a

of the Moon and Sun is the angle measured in the

ecliptic, in direct sense, between the intersection points of the almucantar of both stars with the

ecliptic (arc SP in drawing 1).

The light arc

L

a

is the angle between the centers of the Sun and the Moon. We assume null

the ecliptic latitude of the Sun, but we will take into account the latitude of the Moon, therefore the

light arc will be different from the elongation D or difference between its ecliptic longitudes.

The arc of vision

V

a

is the difference between the heights of the centers of the Moon and

the Sun (arc SB in drawing 1). Lagtime is the difference between the sunset and moonset, regardless

parallax, atmospheric refraction, and semi-diameter.

The mean obliquity of the ecliptic with the equator varies according to

23º 26 21 .45 46 .81 ,

T

T is Julian centuries since 2000. The ecliptic forms the angle i with the horizon, which has a daily

South

North

West horizon

ecliptic

Drawing 1.- We represent the western horizon at the time of the moonset. M is the Moon, and S is the Sun,

which is below the horizon. On the left is the South and on the right the North. The arc S

a SP

is the ecliptic

separation arc; V

a DC SB

is the arc of vision, which in the drawing is equal to the depression d of the

Sun; S

a DE

is the equatorial separation arc;

is the equinox or point of intersection of the equator and the

ecliptic; 90

(

is the geographic latitude) is the angle between the horizon and the equator; i is the

inclination of the ecliptic with the horizon at the time of the moonset;

is the obliquity of the ecliptic or angle

between the ecliptic and the equator; the arc

E is the oblique descension of the Moon; the ecliptic latitude

of the Moon is

and the latitude of the Sun is zero. The almucantar of the Sun is the circle parallel to the

horizon that passes through the Sun, and the verticals are the great circles perpendicular to the horizon. The

drawing corresponds to the beginning of autumn, shortly after the equinox. The latitude of the Moon

corresponds to a positive value and the drawing is the western horizon seen from a position in the northern

hemisphere.

90

i

equator

vertical

almucantar of the Sun

P

B

C

M

S

D

S

a

S

a

V

a

V

a

E

vertical

A

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

and annual variation. The angle that forms the equator with the horizon is always 90

(

is the

geographical latitude).

3 Indian criterion for lunar crescent visibility

The Indian text by unknown author Surya Siddhanta (ca. 600) describes a criterion to find

out the visibility of the lunar crescent, which is called the Indian criterion: «At an arc of separation

of two nadis [1 nadis = 24 minutes of time] she becomes visible in the West, or invisible in the

East». That is to say, that if the time between the sunset and the moonset is greater than 48

minutes, then the crescent will be seen shortly after sunset (Bruin 1971).

The technique to apply this criterion was to determine the moments of the moonset and

sunset, find out their difference, and check if it exceeds 48 minutes.

In drawing 2, we represent the western horizon at the time of sunset. After 48 minutes;

Moon M will reach the horizon and will be at point

M

, moving in a circle parallel to the equator..

During the same time, the Sun will move from S to

S

. Since, on average, the Sun travels 360º in 24

hours, in 48 minutes, it will move 12º along a parallel to the equator. The intersection point between

the almucantar of the Sun and the equator, moves at the same velocity as the Sun, since it also

describes 360º in 24 hours, then in 48 minutes it will move 12º; this arc BA is the arc of separation

s

a

at the time of the moonset. It does not mean that point B "fixed at the equator" reaches point A

in 48 minutes. We consider movement of point of intersection of the almucantar and the equator,

which is not fixed at the equator.

The arc

SS

has 12º measured in a parallel, and the arc BA the same angle but measured at

the equator, therefore the length of the arc

SS

is less than the length of the arc BA (although they

have the same angle) since the radius of the parallel is less than the radius of the equator. We

consider that the celestial sphere has a unit radius, so the length of the arc BA is 12º, the same as its

angle, which does not happen with the arc

SS

since its parallel has a radius smaller than unity..

The apparent speed of the Sun has a small variation with respect to its average value of

3

South

North

West horizon

ecliptic

equator

almucantar of the Sun

S

M

A

B

C

M

S

almucantar of the Moon

parallel

of Sun

parallel

of Moon

Drawing 2.- We draw the western horizon at the time of sunset. The Sun is S, and the Moon is M, which moves

parallel to the equator in the daily movement. We draw the ecliptic and the equator. We assume that the Sun

does not have latitude and therefore is on the ecliptic. According to Indian criteria, in 48 minutes, the Moon

goes from M to

M

, and the Sun goes from S to

S

. The

SS

arc has an angle of 12º since, in 24 hours, it

moves 360º. For 48 minutes the point of intersection of the almucantar of the Sun and the equator goes from

point B to A. This point of intersection has the same speed as the Sun; therefore the arc AB has an angle of

12º (and the arc length is also 12º if we assume that the radius of the celestial sphere is unity). The arc BA is

greater length than the arc

SS

(although they have the same angle) since the length of a parallel is always

less than the equator's length because it has a greater radius than a parallel circumference. BA is the equatorial

separation arc at the time of the moonset, which has the limit value of 12º according to the Indian criterion.

Wenceslao Segura González

15º day

as a consequence of the change of the equation of time

v m

H H E

v

H

and

m

H

are the hour angles of the true Sun, and the mean Sun and E is the equation of time.

The true solar day is the time between two consecutive crosses of the true Sun by the meridian,

that is, the time it takes to increase by 360º the hour angle

v

H

. Since universal time t is the hour

angle of the mean Sun increased by 12 hours

12

m

t H

, then the duration of a true solar day is

24 h

t E

E

is the variation of the equation of time in a day. The extreme values of

E

and the duration of

the true solar day are

max min

24 29 .10 24 0 29 .10; 24 21 .75 23 59 38 .25,

h s h m s h s h m s

t t

then the extreme values of the arc of separation at the moonset are

max min

12º 0 10 .88; 11º59 45 .45,

s s

a a

therefore we consider a mean value of 12º as good. But strictly speaking, the limit value of the

equatorial separation arc at the moonset according to Indian criteria is

360 48 60

12º 1 .

24 24

s

E

aE

The Indian criterion is expressed in three different ways. The first is that the time interval

between sunset and moonset must be greater or equal than 48 minutes. The second that the arc of

separation between both stars at moonset is greater o equal than 12º.

Indian criterion can be put as a function of the depression d of the Sun at moonset, which at

that moment coincides with the arc of vision. Applying the sine theorem in the spherical triangle

DEC of drawing 1

1 1

sin sin cos sin sin12ºcos

V S

d a a

is the geographic latitude.

4 Oblique ascension and oblique descension

The oblique descension

of a point of the ecliptic P located on the western horizon is the

angle measured in the direct sense by the equator from the equinox to the western horizon, that is,

the angle

E of drawing 1 (Chabás 2012).

The oblique ascension

is similar to the oblique descension, with the difference that it takes

the eastern horizon as a reference; therefore

180 180

P P

L L

P

L P

is the ecliptic longitude of point P.

From the spherical triangle

PE of drawing 1, we obtain by the sine theorem

sin

sin sin

cos

P

L

i

is the oblique descension of point P. i is the angle between the ecliptic and the horizon when point

P is on the western horizon.

To calculate the angle i, we use the triangle

PE in drawing 1. First, determine the side

a PE

by the sine theorem.

sin

sin sin

cos

P

L

a

and angle i is calculated using Neper's analogy

cos cos

2 2

180 90 90

tan cot tan tan

2 2 2 2

cos cos

2 2

p p

p p

L a L a

i i

L a L a

with (3) and (4) we calculate the inclination i of the ecliptic at the moment when point P crosses the

horizon, with this result, we determine by (2) the oblique descension of point P.

4

(1)

(2)

(3)

(4)

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

5

5 Adaptation of the Indian criterion

Muslim astronomers adapted the Indian criterion of visibility of the crescent lunar during the

Middle Ages. The method consists of relating the elongation D (or difference in longitude of the

Moon and the Sun) with the equatorial separation arc, that is, finding the function

, , , ,

S

D L L f a L

since D is a monotonically increasing function of

S

a

(as we will see later), then the Indian criterion

12º

S

a

is transformed by (6) into

12º, , ,

S

L f a L

L L L

is the elongation at the time of the moonset.

We make the table of the function

12º, , ,

S

D D a L

. To investigate whether the

crescent will be seen, we compute L,

L

and

at the moonset and determine the corresponding

value of D, if

L D

then the Moon will be visible shortly after sunset.

Frequently the medieval tables to determine the crescent have as argument the longitude of

the Sun L and not the longitude of the Moon

L

. From (6) we find

L

in function of L, and we

change in (6) the argument

L

by L

, , , , , , .

S S

D f a L L g a L

From the ECD triangle of drawing 1 and by application of the sine theorem we find

sin sin cos

V S

a a

of the SPB triangle (drawing 1) and by (8)

sin sin cos

sin .

sin sin

V S

S

a a

a

i i

Applying the cotagent theorem to the triangle MPA

tan

sin

tan

AP

i

as

S s

AP SP AS a L L a D

then by (9) and (10)

1 1

sin cos tan

sin sin ,

sin tan

S

a

D

i i

for small values of

S

a

, D is an monotonically increasing function. The calculation of i is made by

(3) and (4), from which will obtain (12) as a function of

P

L

. To put (12) in function of

L

it is taken

into account that (drawing 1)

1tan

sin .

tan

P

P A AP L L

i

By (12) the Indian criterion is

1 1

sin12ºcos tan

sin sin ,

sin tan

L

i i

the procedure for tabulating (12) is as follows: we give arbitrary values to

P

L

; by (3) and (4), we

determine i; by (12) we calculate D when

12º

S

a. To put D as function of

L

, we make the

change of variable (13). If we want to express D as a function of L, then

L L D

with what we

can do the tabulation with the argument L.

In Table 1 we have obtained, as an example, the value of the function

12º, , , 40º

S

D D a L

for lunar latitudes -5º, 0º, 5º and for average obliquity of the ecliptic of the year 2000, using both the

L and

L

arguments. Drawing 3 is the graphical representation of Table 1, which shows the critical

effect that Moon latitude has on the Indian criterion (14).

6 Depression criterion

Another Indian-like criterion is the depression criterion, which states that the Moon will be

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

Wenceslao Segura González

6

15º

45º

75º

105º

135º

165º

195º

225º

255º

285º

315º

345º

11º 11’

12º 18’

15º 2’

19º 31’

25º 0’

29º 33’

30º 53’

27º 56’

21º 59’

16º 3’

12º 37’

11º 15’

L’

Table 1.- Indian criterion (14) of visibility of lunar crescent for a place of geographical latitude of 40º, for

various values of the ecliptic latitude of the Moon. In the table on the left, the argument is the ecliptic

longitude of the Moon

L

. On the right, the same criterion but with the argument being the ecliptic longitude

of the Sun L.

9º 37’

10º 7’

11º 37’

14º 31’

18º 4’

20º 31’

20º 31’

18º 4’

14º 31’

11º 39’

10º 7’

9º 37’

' 5º

' 0º

5º

D L L

8º 1’

7º 46’

7º 47’

8º 40’

10º 1’

10º 47’

10º 25’

9º 19’

8º 14’

7º 43’

7º 48’

8º 2’

9º 42’

10º 31’

12º 40’

16º 30’

19º 55’

20º 49’

19º 10’

16º 10’

13º 5’

10º 55’

9º 53’

9º 35’

15º

45º

75º

105º

135º

165º

195º

225º

255º

285º

315º

345º

11º 30’

13º 14’

17º 29’

23º 55’

29º 22’

30º 57’

28º 18’

23º 17’

18º 10’

14º 13’

11º 48’

11º 10’

L

' 5º

' 0º

5º

D L L

7º 58’

7º 42’

7º 55’

9º 6’

10º 23’

10º 46’

10º 6’

8º 57’

8º 1’

7º 42’

7º 52’

8º 4’

50 100 150 200 250 300 350

10

15

20

25

30

35

Drawing 3- Effect of lunar latitude on the Indian visibility criterion for a place of 40º of geographical latitude.

The curves correspond to the lunar ecliptic latitudes of +5º, 0º and -5º. We appreciate the remarkable influence

of Moon latitude.

Ecliptic longitude of the Moon

' 5º

' 0º

' 5º

Elongation

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

0

30

60

90

120

150

180

210

240

270

300

330

9º 44’

10º 2’

11º 7’

13º 37’

16º 54’

18º 56’

18º 39’

16º 35’

13º 52’

11º 34’

10º 15’

9º 47’

L

9º 46’

10º 2’

11º 6’

13º 39’

16º 57’

18º 59’

18º 38’

16º 28’

13º 42’

11º 29’

10º 12’

9º 50’

al-

Tabari

Recom-

putado

'

D L L

5º

0º

5º

8º 44’

8º 28’

8º 19’

9º 13’

10º 58’

12º 14’

11º 49’

10º 37’

9º 19’

8º 32’

8º 22’

8º 36’

10º 30’

11º 58’

14º 50’

19º 58’

25º 53’

29º 30’

28º 17’

24º 30’

19º 32’

15º 12’

12º 20’

11º 9’

al-

Tabari

al-

Tabari

Recom-

putado

Recom-

putado

10º 50’

11º 43’

14º 9’

18º 44’

24º 12’

27º 27’

26º 59’

23º 40’

19º 8’

15º 1’

12º 17’

11º 0’

8º 38’

8º 20’

8º 5’

8º 31’

9º 37’

10º 25’

10º 18’

9º 30’

8º 35’

8º 6’

8º 14’

8º 33’

Table 2.- Recomputation of the visibility table of the lunar crescent of al-Tabari (11th century). The argument

of the table is the ecliptic longitude of the Sun. For each of the three lunar latitudes, we give the original value

of al-Tabari's table and the recomputation, using the depression criterion (16).

visible if the Sun's depression at moonset, or arc of vision, is equal to or greater than an extreme

value

0

d

(Hogendijk 1988).

Through the PBS triangle in drawing 1

1sin

sin sin

V

S

a

a

i

by (10) and (11)

1 1

sin tan

sin sin .

sin tan

V

a

D

i i

So the depression criterion requires that if the elongation at moonset fulfill

1 1

0

sin tan

sin sin

sin tan

d

L

i i

the Moon will be visible. The Table 2 is the visibility criterion of the lunar crescent of al-Tabari (11th

century) (King 1987; Fatoohi 1998). The table's argument is the solar ecliptic longitude and calculated

for the lunar latitudes -5º, 0º, and 5º. We see that the table of

' 0

agrees with the recomputation,

calculated for the geographic latitude 35º 50’, the obliqueness of 23º 51’ and the limit depression of

9º 30’, as suggested by Hogendijk (1988). The values tabulated by al-Tabari for the lunar latitudes

of -5º and 5º are significantly different from the recomputed, which means that al-Tabari used other

data different from those in the table of

' 0

.

7

(15)

(16)

Wenceslao Segura González

To calculate Table 2, we give arbitrary values to

P

L

, which is the longitude of a point P on the

ecliptic, that is on the western horizon at the same time as the Moon. By (3) and (4), we determine

the angle i, inclination of the ecliptic with the horizon at the time of moonset. We apply the formula

(15) with

0

V

a d

, and we find a table with argument

P

L

. By (13), we determine the value of L,

and by

L L D

, we find the longitude of the Sun, which allows us tabulate (15) with L as the

argument.

A generalization of this criterion is when the solar depression limit is not constant but depends

on another factor, as in the al-Sanjufini criterion (14th century), in which the depression

0

d

depends

on the Sun's azimuth linearly (Kennedy and Hogendijk 1988). According to these authors, the limit

depression of Sun at moonset depends on the azimuth of the Sun. It does not seem logical criterion

since the observation of the crescent depends on the difference in azimuth between the Sun and

the Moon. In these criteria the formula (16) is still valid, but now,

0

d

is a function dependent on

another parameter, such as the azimuth difference. These criteria include the Fotheringham-Maunder

(Fotheringham 1911, 1921; Maunder 1921).

7 Al-Khwarizmi's criterion

Muslim astronomers of the Middle Ages developed techniques to determine the visibility of

the lunar crescent using the oblique ascension tables. One of these criteria establishes that the

Moon would be visible if it is fulfill

' 180 12º 12º

L L L L

it is calculated at the moonset. We assume that the ecliptic latitude of the Moon is null. Later we

will generalize (17) for non-zero lunar latitudes. (17) has been thought to be another way of expressing

the Indian criterion. We will call inequality (17) the al-Khwarizmi criterion since his lunar crescent

visibility table is the first made with that condition. For (17) to be identical to the Indian criterion, it's

necessary that

,

S

a L L

by (18) and drawing 1 we find

'S

L L a D E ED

which would imply that

L D

But this is not correct, because point D is not on the horizon, since the Sun is below the horizon at

the time of the moonset. Note that the triangle

SD

is not a spherical triangle because the SD side

does not belong to a great circle. Therefore the formulas of spherical trigonometry cannot be

applied to it. The relationship between the ecliptic longitude

S

and the angle

D in drawing 1 is

different from the relationship that would exist if points A and D were on the horizon or in another

position in the sky.

Therefore the al-Khwarizmi criterion is a different criterion from the Indian one, although it

uses 12º as a limit angle. The medieval Muslim astronomers believed that (14) adapted the Indian

criteria, a mistake that has been made again with contemporary researchers.

In drawing 4 we graphically represent the Indian and al-Khwarizmi criteria for the same

observation site and note the difference between the two criteria.

8 Al-Khwarizmi’s criterion for non-zero lunar latitudes

When the Moon has a lunar latitude and therefore is not on the ecliptic, al-Khwarizmi’s

criterion (17) is

12º

P

L L

P is the point of the ecliptic that is on the horizon at the same time as the Moon (see drawing 1), that

is,

P

L

is the oblique descension of the Moon at the time of the moonset. To put

P

L

in

function of

L

we use (13)

1tan

sin 12º

tan

L L

i

8

(17)

(18)

(19)

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

which is al-Khwarizmi's criterion when the Moon has an ecliptic latitude. To find the table of

oblique descensions when there is lunar latitude, we give arbitrary values to

P

L

, and we determine

P

L

by (2) and the angle i by (3) and (4); with (13) we make a change of variables and determine

L

; from (19) we calculate the function

L

. Now the functions

L

and

L

are inverted, and we find L and

L

and the limit value of

D L L

, which we can put as a

function of

L

or L.

We know the visibility table of the crescent of al-Khwarizmi from four manuscripts analyzed

by King (1987) and that we identify by where they are: El Escorial (9th century), Cairo (11th

century), Paris (11th century), and London (11th century). King has recomputed this criterion for

the latitude of Baghdad 33º 0’, and obliquity of the ecliptic of 23º 51’, using (17) for the moonset.

To reconstruct the table, we give arbitrary values to

L

and by (2), we calculate the oblique

descension of the Moon

L

. From (17), we calculate the function

L

or oblique

descension of the Sun. Now invert this function and calculate L and

D L L

, which is the limit

value of the elongation.

Some authors express the criterion of visibility by a table; others use an adaptation of the

formula (19). Since

’is a small angle we can make the simplifications

9

15

45

75

105

135

165

195

225

255

285

315

345

10º 12’

9º 58’

10º 1’

11º 23’

14º 29’

17º 44’

18º 36’

16º 7’

12º 58’

10º 40’

9º 56’

10º 4’

L

Table 3.- The left is the recomputation of the al-Khwarizmi table for

33º 0'

and ecliptic obliquity 23º 51',

according to King (1987). The first column is the ecliptic longitude of the Moon

L

when it is at the midpoint

of each zodiac sign. The other two columns are those of al-Khwarizmi's text and our recomputation, which

differs slightly from that obtained by King. The second is the table of al-Qallas with the recomputation

proposed by King (1987), calculated for

40º 52

and

23º 51

. To obtain these tables, we give arbitrary

values to

L

, and by (2), we calculate

L

. Then we calculate the function

12

L L

. WeWe

invert the function

L

and determine L and the elongation limit value D for visibility. We can put D

as a function of L or

L

.

10º 10’

9º 56’

10º 7’

11º 38’

14º 32’

17º 25’

18º 14’

16º 19’

12º 37’

10º 31’

9º 50’

10º 15’

al-

Khwarizmi

Recom-

putado

15

45

75

105

135

165

195

225

255

285

315

345

9º 25’

9º 18’

9º 57’

12º 48’

17º 31’

21º 4’

21º 4’

17º 31’

12º 48’

9º 57’

9º 18’

9º 25’

L

9º 25’

9º 14’

9º 49’

11º 48’

15º 55’

19º 32’

21º 2’

18º 28’

13º 13’

10º 26’

9º 22’

9º 45’

al-

Qallas

Recom-

putado

'

D L L

'

D L L

Wenceslao Segura González

1tan tan

sin cot

tan tan tan tan 90i i i

in the last step, angle i has been replaced by its mean value, so the simplified criterion used by some

Muslim astronomers is

cot 12º

L L

knowing

L

, L, and

at the time of the moonset, we found the oblique ascents in the table that

medieval astronomers had at their disposal, and we checked if the previous inequality is fulfilled.

9 Al-Qallas lunar visibility criterion

The name of al-Qallas (10th century) is associated with other criterion of visibility (Suter

1914; Newgebauer 1962; Kennedy and Janjanian 1965; King 1987; Hogendijk 1988). Table 3

shows King's computation, calculated by (17) at the time of the moonset. But the fit is not good and

there are excessive differences, which indicates that the table of al-Qallas was not calculated by

(17).

Kennedy and Janjanian suggest that this table was calculated when the intermediate point P,

equidistant between the positions of the Sun and the Moon (both assumed in the ecliptic), is on the

western horizon (see drawing 5). Although the criterion for reproducing the table of al-Qallas

fulfills the relation (17), nevertheless it is not the criterion of al-Khwarizmi.

According to Kennedy and Janjanian, to make the table of al-Qallas, it is assumed that the

equatorial separation arc of Sun and the Moon when P is on the horizon has a limit of 12º. But this

is not the Indian criterion, which requires that the equatorial separation arc be equal to or greater

than 12º at the moonset. The reasoning continues assuming that the oblique descension of the

Moon and the Sun are

L C

and

L A

, which is not correct, since the oblique

descensions are defined when, the Moon or the Sun, are on the western horizon, which in our

reasoning does not occur (drawing 5). Finally, the reasoning of Kennedy and Janjanian establishes

that there is visibility when

6º 6º ,

P P

L L L L

we find (17) when the two inequalities (20) are subtracted. Note that the inverse reasoning is not

possible, that is, to deduce (20) from (17), which means that we have another different criterion,

10

20 60 100 140 180 220 260 300 320

11º

13º

15º

17º

19º

Ecliptic longitude of the Moon

Drawing 4.- Comparison between the Indian criterion and the al-Khwarizmi criterion when

0

and for the

same geographical latitude. The thick line is the limit value of the elongation so that there is visibility

according to the Indian criterion. The discontinuous line is the criterion of al-Khwarizmi. The argument is the

Moon's longitude. The Indian criterion curve is symmetric respect to 180º.

Elongation

Indian criterion

criterion of

al-Khwarizmi

(20)

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

which we will call al-Qallas. Although the above reasoning to deduce (20) is not satisfactory, (20)

represents a new lunar visibility criterion, more general than al-Khwarizmi's criterion.

If

is an angle between 0 and 12, then the generalization of (20) is

and 12 12,

P P

L L L L L L

we can make a new generalization.

and

are two angles that define the criterion, so the Moon

will be visible if both relations are simultaneously fulfilled

' and ' .

P P

L L L L L L

When

0

and

12

, we find al-Khwarizmi criterion (14). If

6

and

6

we have

criterion (20) of al-Qallas. And if 12

, then we have the generalization (21).

Table 4 shows the re-computation of the al-Qallas criterion, according to (20). The steps to

make table 4 are:

* We give an arbitrary value to

P

L

and find the table

P

L

by (2), (3) and (4).

* We calculate the limits of oblique descensions of the Moon and the Sun:

6

P

L L

and

6

P

L L

.

* We invert tables

'

L

and

L

and find L and

'

L

.

* We calculate

D L L

.

* The argument of the table is

P

L

, being P a point of the ecliptic that is at the same distance from

the points M and S (drawing 5), that is to say, that

P

L

is the half-sum of the longitude of the Sun and

the Moon

2

P

L L L

.

* To put the table in function of

'

L

or L we take into account that

2

P

L L D

and

2

P

L L D

.

10 Al-Qallas criterion when there is lunar latitude

When the Moon has an ecliptic latitude, it is necessary to correct criteria (20), (21) and (22),

which now take the form

11

South

North

West horizon

ecliptic

Drawing 5.- Al-Qallas criterion of visibility of the crescent, according to Kennedy and Janjanian. M is the

Moon, and S is the Sun, both in the ecliptic, that is, we assume that there is no ecliptic latitude of the Moon.

The ecliptic longitude of the Sun is

L S

, and the Moon is

L M

. Since we assume that P is the

midpoint between Sun and Moon, then

PS PM

and, therefore,

BA BC

. The hypothesis assumes that

the extreme value of the equatorial separation arc between the Moon and the Sun s

a

AC

is 12 degrees,

therefore

6º

AB BC

. The oblique descension of point P is

P

B

L

, therefore

6º

P

AL

and

6º

P

C L

.

L

and

L

are the oblique descension of the Moon and the Sun (not

represented in the drawing). Kennedy and Janjanian mistakenly identify

L C

and

L A

. So

the suggested criterion is that the extreme value for visibility is

6

P

L L

, which implies

6

P

LL

. Since

P

L

is known, we determine

L

and

L

, and by inverting both functions,

we calculate L and

L

and therefore

D L L

.

90

i

equator

almucantar of the Sun

P

S

M

almucantar

of the Moon

A

B

C

(21)

(22)

Wenceslao Segura González

' and ,

P P

M L L L

in drawing 6, we see that

'

M N NM L NM

therefore it is necessary to calculate

NM

. From drawing 6, we see that the triangle

MNM

is not

spherical because the arc

MM

does not belong to a great circle. Then the procedure to follow is

to determine the arc

MM

in the case that it belongs to a great circle, then it will be possible to

apply the theorems of spherical trigonometry. Let us observe that given the points

M

and M, it is

always possible to find a great circle (whose center is the center of the celestial sphere) that passes

through both points.

The procedure is as follows:

a) Find the height

h

of the Moon. From triangle ABC of drawing 6, we determine by the sine

theorem the height

h

of the Moon

sin cos sin

h

and

(drawing 6) are the two parameters that characterize the generalized criterion of al-Qallas.

b) We determine the azimuth of the Moon

A

by the cosine theorem applied to the triangle PnZM

(drawing 7)

12

5º

15º

25º

35º

45º

55º

65º

75º

85º

95º

105º

115º

125º

135º

145º

155º

165º

175º

al-

Qallas

Recom-

putado

2

L L

9º 20’

9º 18’

9º 16’

9º 13’

9º 13’

9º 18’

9º 32’

9º 57’

10º 36’

11º 35’

12º 43’

14º 9’

15º 45’

17º 23’

18º 56’

20º 14’

21º 11’

21º 40’

9º 26’

9º 25’

9º 21’

9º 19’

9º 18’

9º 21’

9º 33’

9º 57’

10º 37’

11º 29’

12º 48’

14º 15’

15º 58’

17º 31’

19º 11’

20º 20’

21º 4’

21º 17’

Table 4- Recomputation of the al-Qallas visibility criterion according to the Kennedy and Janjanian proposal.

The argument is the half-sum of the ecliptic longitudes of the Moon and the Sun. The table has been

calculated for criterion (20) of al-Qallas, which is different from criterion (17) of al-Khwartizmi. We have used

interpolation to do the computation, calculated for a geographic latitude of 42º.67 and an oblique of 23º.50.

(23)

(24)

(25)

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

sin cos sin

cos .

sin cos

i

Ai

c) To determine the azimuth of point

M

(drawing 7) we apply (26), but now the ecliptic latitude is

zero

cos sin

cos .

sin cos

i

Ai

d) The next step is to determine the straight distance between points

M

and M, that is, the chord

that connects

M

and M. If

r

is the distance from points

M

or M to the axis of the sphere

perpendicular to the horizon plane and r is the radius of the celestial sphere, then

' cos .

r r h

e) The radii that join the axis of the celestial sphere perpendicular to the horizon with points

M

and

M form the angle

A A A

or the azimuth difference of points

M

and M. Therefore the chord

of the arc

MM

is

' 2 sin 2 cos sin .

2 2

A A

MM r r h

f) Segment

MM

is the chord of the minor circle that joins

M

and M, and also the chord of the arc

of the maximum circle that joins both points. So this last arc has the length

1 1

'

2 sin 2 sin cos sin .

2 2

MM A

s r r h

r

g) We go back to triangle

MNM

in drawing 6 and determine the arc

NM

. First, apply the sine

theorem and calculate the angle

sin

sin

sin

s

and then we apply the cotagent theorem to the triangle

MNM

13

South

North

West horizon

ecliptic

i

equator

almucantar of the Sun

P

S

M

almucantar of the Moon

M

N

Drawing 6.- Adaptation of the al-Qallas criterion when there is lunar ecliptic latitude. In this drawing, the

Moon M has latitude

and therefore is not on the ecliptic. It is necessary to express the longitude

M

as

a function of

L N

. Thus we must solve the triangle

MNM

, but it is not a spherical triangle because the

MM

side is an arc of a minor circle. Therefore we cannot apply the formulas of spherical trigonometry. Too

solve the problem, we build a new spherical triangle with the same vertices, but with the arc

MM

belonging

to a great circle. The text indicates how the arc

MM

is calculated; after finding it, we solve the triangle

MNM

and calculate the

MN

side, and finally, we find

M L NM

.

and

are the parameters that

characterize the generalized criterion of al-Qallas. Note that angle

is very close to angle i, but they are not

equal.

h

A

B

C

D

(26)

(27)

(28)

(29)

(30)

(31)

Wenceslao Segura González

tan

sin ,

tan

NM

then the criterion of al-Qallas (23) is

' and .

P P

L NM L L L

The procedure to apply (33) is as follows:

* We give arbitrary values to

P

L

, and by (2), (3), and (4), we calculate the angle i and the oblique

descension

P

L

. By (33), we calculate

M

and

L

. Both functions are inverted,

and we determine

M

and L. By (32), we calculate

NM

and determine

L

by (24). Finally, we

find the limit elongation D for visibility. The argument can be

P

L

, L, or

L

.

11 Conclusions

We have deduced formula (3) to find the table of oblique descension, which medieval

astronomers used for the visibility criteria of the lunar crescent. The Indian criterion has been

precisely defined, and we have shown the importance of lunar ecliptic latitude.

We examined the depression criterion for its similarity to the Indian criterion and verified that

Muslim astronomers applied this criterion. We have defined the visibility criterion of al-Khwarizmi

(17), and we have confirmed that it is different from the Indian criterion. The al-Khwarizmi criterion

has been generalized for non-zero lunar latitude.

When examining the table of al-Qallas, we verify that it corresponds to a new criterion of

lunar visibility. We have generalized it for when the Moon has lunar latitude.

In summary, we verified that the visibility tables of the lunar crescent of al-Khwarizmi and

al-Qallas are different criteria from the Indian criterion, although undoubtedly inspired by the latter.

12 Bibliography

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Doctoral thesis. Durham University.

14

Horizon

Ecliptic

M

i

90

N

P

Z

90 '

h

Drawing 7- Celestial sphere. M is the Moon with ecliptic latitude

and height above the horizon

h

.

n

P

is

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A

is the azimuth of the Moon measured from the south in the

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P

ZM

and determine the azimuth of the Moon.

(32)

(33)

TWO LUNAR CRESCENT VISIBILITY CRITERIA: AL-KHWARIZMI AND AL-QALLAS

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15