An Extension of Al-Khal l ’s Qibla Table to the Entire Worldlocomat.loria.fr/khalili/khalili-ext.pdfAn Extension of Al-Khal l ’s Qibla Table to the Entire World Denis Roegel1 4

An Extension ofAl-Khalls Qibla Table

to the Entire World

Denis Roegel

4 August 2008

An Extension of Al-Khalls Qibla Table

to the Entire World

Denis Roegel1

4 August 2008

Abstract

This work provides a modern reconstruction of Al-Khalls qibla table andextends it to the entire world, both with (modern) Arabic numerals, and withthe Abjad numerals used by Al-Khall. The tables themselves are organized ina way very similar to that used by Al-Khall.

1Denis Roegel, LORIA, BP 239, 54506 Vanduvre-les-Nancy cedex, [email protected].

1

Contents

Introduction 51 The qibla and Al-Khall . . . . . . . . . . . . . . . . . . . . . . . 52 The geometrical problem of the qibla . . . . . . . . . . . . . . . . 5

2.1 Spherical trigonometry . . . . . . . . . . . . . . . . . . . . 52.2 Reduction to an acute angle . . . . . . . . . . . . . . . . . 72.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Coordinates of Mecca, now and then . . . . . . . . . . . . 8

3 The recomputed tables . . . . . . . . . . . . . . . . . . . . . . . . 83.1 Abjad numerals . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Al-Khalls qibla table . . . . . . . . . . . . . . . . . . . . 93.3 An extended qibla table . . . . . . . . . . . . . . . . . . . 93.4 Construction and typographic conventions . . . . . . . . . 9

4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Appendix A: Al-Khalls Qibla table 13

Appendix B: Extended table for the Qibla 47

3

Introduction

1 The qibla and Al-Khall

The inh. iraf al-qibla (J.

@

@ Qj

K @

), or in short, the qibla (

J.

@), is the direction

of Mecca and is essential for the prayers in Islam. Indeed, one of the verses ofthe Quran states:

... we will direct you towards a Qibla that you shall like. So, turnyour face in the direction of the sacred mosque.

Wherever you are, turn your face in its direction.

(Surat 2, Ayah 144)2

Since the Hegira, a number of Arab astronomers have provided means to findthe qibla for various locations on Earth, and with various degrees of accuracy.

In particular, in the fourteenth century, the Syrian astronomer Shams al-

Dn Muh.ammad al-Khall (

J

m'@ K

Y@

, ca. 1365 AD) produced a very

accurate qibla table which was certainly computed using a number of tablestabulating auxiliary functions [3, 6, 7].

This qibla table gave the direction of Mecca as a function of latitude andlongitude, for latitudes 10 to 56 degrees North, and for longitudes between 60degrees West and 60 degrees East of Mecca. Copies of this qibla table are foundin Paris and Berlin. Excerpts of the copy from the Bibliotheque Nationale inParis are shown in David A. Kings analyses [4, 5].

2 The geometrical problem of the qibla

The problem of the qibla is purely geometric. It involves angles between short-est lines on a sphere, hence spherical trigonometry. This problem has been achallenge for early Arab astronomers who had not yet fully developed sphericaltrigonometry. But from a modern perspective, it is an almost mundane prob-lem. In this section, we derive the necessary formul, such as may have beenused by Al-Khall.

2.1 Spherical trigonometry

Figure 1 shows the position of Mecca (M) and of some place X on the surface ofthe Earth. N is the North pole. (NX ) and (NM ) are great circles, and thereforemeridians. (XM ) is also a great circle and defines the shortest distance from X

2Our translation, from D. Massons French translation of the Quran (Gallimard, 1967).

5

M

X

N

O

L

q

M

Figure 1: The problem of the Qibla at location X: M is the Mecca, N is theNorth pole and O is the center of the Earth. M and are the latitudes ofMecca and of X. M = 90M and = 90 are the colatitudes of Meccaand X. The difference of longitudes LM and L is L. Then q (measured fromNorth) or q = 180 q (measured from South) is the qibla angle.

to M . The qibla is merely the angle q (measured from South) or q = 180 q(measured from North).

It is towards East if X is less than 180 West of Mecca and towards Westin all other cases. Whether it is North or South is more complex to ascertain(and this shows that North and South are not exactly of the same nature asEast and West).

Two relations of spherical trigonometry are useful in order to compute theqibla:

sin sin q = sinM sin L (1)sin cos q = cosM sin sinM cos cos L (2)

These equations become:

sin sin q = cosM sin L (3)sin cos q = sinM cos cosM sin cos L (4)

If sin sin q 6= 0 and sin cos q 6= 0, one of the equations can be divided bythe other one, giving:

tan q = tan q = sin L cosMsin cos L cosM cos sinM

(5)

=sin L

sin cos L cos tanM(6)

cot q =sin cos L cos tanM

sin L(7)

from which q is easily obtained.

6

Two special cases must be considered, when either sin sin q = 0 or whensin cos q = 0:

1. The first case occurs when sin L = 0. If sin L = 0, q = 0 or q = 180.When L = 0, q = 0 if > M , and q = 180 otherwise. WhenL = 180, q = 0 if < M , and q = 180 otherwise.

2. The second case corresponds to sin cos q = 0. If sin = 0, we are eitherat Mecca or at its antipode, and q cannot be given a definite value. Inany other place than these two, if sin cos q = 0, then cos q = 0 andtherefore the qibla is exactly towards the East or the West, depending onthe respective longitudes.

For the sake of verifying our tables, we give the C code which was used togenerate them. This function takes a latitude and a longitude L in degreesand returns the qibla in degrees from the South direction. It makes use of Mand LM which are the coordinates of Mecca.

#define RAD (3.1415926/180.0)double qibla(double phi,double L) {double DL,q,tanqn,tanqd;DL=fabs(L-L_M);if (DL>180.0)DL=360.0-DL; // smallest longitude difference

tanqn=sin(DL*RAD);tanqd=sin(phi*RAD)*cos(DL*RAD)-cos(phi*RAD)*tan(phi_M*RAD);q=atan2(tanqn,tanqd)/RAD;if (qphi_M?0.0:180.0; // case of Mecca meridian

if (DL==180.0)q=phi

The antipode of Mecca: The qibla is well defined everywhere, except at theexact opposite of Mecca, where Mecca is at equal distance in all directions.

The antipode of Mecca is somewhere in the midst of French Polynesia,and, although it is high sea, it is conceivable that some traveller on acruise ship needs to find the direction of Mecca. In that case, at the exactantipode of Mecca, all directions are equal and the prayers can be givenunconstrained. In all other cases, our tables should give the answer.

The exact case of the antipode is neither present in the original table byAl-Khall (whose world didnt extend beyond the Canary Islands), norin our extended tables, but we come close to it. The closest values arearound 140 W and 21 S. One should for instance peruse the values ofthe qibla for latitude 21 and values of the longitude around 140, andnotice the quick variations of the qibla angle.

The meridian of Mecca: On the meridian of Mecca, the qibla is either to-wards the North (when the location is South of Mecca), or towards theSouth.

The meridian of the antipode of Mecca: This case is symmetrical of theprevious one. In other words, the qibla is either towards the North (whenthe location is North of the antipode of Mecca), or towards the South.

2.4 Coordinates of Mecca, now and then

The (modern) coordinates of Mecca are 3949 E and 2125 N. Al-Khall, how-ever, uses 2130 N, which is a very good approximation of the latitude of Mecca.For the longitude, Al-Khall used 67. This is the longitude obtained when con-sidering the prime meridian of the Fortunate Islands (now the Canaries), whichwas the reference adopted by Ptolemy (120 AD) in his Geography.

Abjad @ ' H k X P h

Value 1 2 3 4 5 6 7 8 9

Abjad

Value 10 20 30 40 50 60 70 80 90

Abjad P H H p X

Value 100 200 300 400 500 600 700 800 900 1000

Figure 2: The Abjad numerals.

3 The recomputed tables

Al-Khall original table makes use of the Abjad (or alphabetic) numerals, whichwere in use before the Hindu-Arabic numerals 0, ..., 9. Our recomputed tablesuse the Abjad and Hindu-Arabic numerals, on two facing pages.

3.1 Abjad numerals

Figure 2 shows the correspondence between Abjad numerals and their values. Anumerical value is obtained by writing the Arabic letters, in decreasing weight

8

from right to left. For example: 327 is written 7 + 20 + 300, hence P + + ,

hence finally Q .

For more details on the Abjad numerals, see Iranis article [2] and Ifrahsbook on the history of numbers [1] (in spite of the shortcomings of this book).

We provide two sets of tables: a reproduction of Al-Khalls table, and anextended table.

3.2 Al-Khalls qibla table

The first set of tables is a reproduction of Al-Khalls qibla table, rendered withArabic and Abjad numerals. These tables give the qibla for latitudes 10 to56, as well as for 3330 (the latitude of Damascus). The longitudes run from7 to 127 (East of the Canaries), hence approximately from 20 W to 100 E(Greenwich-based longitudes), hence the total number of entries is 5760. Thecalculations use the coordinates of Mecca known to Al-Khall. Some of thevalues of Al-Khalls original table do slightly differ from ours, either becausehe made mistakes, or because he used interpolation schemes.

3.3 An extended qibla table

The second set of tables is an extended qibla table, along the lines of Al-Khallstable. It extends from latitudes 90 S to 90 N, and for all degrees of longitude,counted from the Greenwich meridian (positively eastwards). There are a totalof 65160 entries. These tables are given both with Arabic and with Abjadnumerals, although the latter will probably be of little use. These tables usethe modern coordinates of Mecca.

3.4 Construction and typographic conventions

In Al-Khalls qibla table, as well as in our extension of the table, the qibla is anacute angle and degrees are given first (hence to the right), before the minutes(hence to the left). This also applies to the modern representation, so that 4776 really represents 7647. This should be no problem once one is accustomedto read from right to left.

The computations were done assuming a spherical Earth. Ignoring the flat-tening of the Earth should have only neglectible consequences. The originaltables by Al-Khall slightly differ from our tables, and the differences werepinpointed in Kings article [4].

Negative values of the longitude are shown in red in the extended table, bothwith Arabic and with Abjad numerals. Southern latitudes are also shown in redin the extended table.

The values 0 were rendered by 0, although this is not what Al-Khall used(for instance in his original table entry 820 for latitude 23 and longitude 18,although the correct value is 8159). We hope to improve this representationin a future version of the tables.

9

Also, like in Al-Khalls qibla table, when the qibla is measured towards theNorth, its value is shown in black. Otherwise it is in red. Northern and southerndirections are also shown by a subscript N or S, so that the tables can be printedon black and white printers and yet remain useable.

4 Examples

We consider several examples and show how the tables can be used to find theqibla.

Example 1 :

Find the qibla for latitude 24 and longitude 43 in Al-Khalls originaltable.

Solution: Locate the two tables for latitude 24, the left-hand onebeing in Arabic, and the right-hand one in English. Find the lon-gitude 43 in the first four columns on the right. 43 appears underthe head Second number, which means that the column Secondunder latitude 24 is to be used. Finally, the value shown is 21 88Swhich means that the qibla is 8821 from the South. As shown byKing, Al-Khall had the value 8822, hence only 1 of error. More-over, the qibla is directed eastward, because the longitude 43 is Westof that of Mecca, taken to be 67. Alternately, the first and thirdcolumns from the right correspond to an eastward direction, whereasthe second and fourth column correspond to a westward direction.The table on the left-hand page shows the corresponding value withAbjad numerals.

Example 2 :

Find the qibla for latitude 13 and longitude 72 in Al-Khalls originaltable.

Solution: Locate the two tables for latitude 13, the left-hand one be-ing in Arabic, and the right-hand one in English. Find the longitude72 in the first four columns on the right. 72 appears under the headSecond number, which means that the column Second under lat-itude 13 is to be used. Finally, the value shown is 37 28N which meansthat the qibla is 2837 from the North. Again, Al-Khall was off byonly 1 with 2838. Moreover, it is directed westward, because thelongitude 72 is East of that of Mecca.

Example 3 :

Find the qibla for latitude 13 N and longitude 72 E (= +72) in theextended qibla table.

Solution: Locate the tables for latitude 13 N (= +13), the left-handones being in Arabic, and the right-hand ones in English. Thereare six such tables. Only one contains the value 72 in its first twocolumns on the right. 72 appears in the column with the headingFirst number. On the same line and under the heading First oflatitude 13 N, we find the value 12 70N, that is 7012, North. Andsince 72 (counted from the Greenwich meridian) is East of Mecca,the qibla is 7012 from North, and westward.

10

Example 4 :

Find the qibla for latitude 49 N and longitude 40 W (= 40) inthe extended qibla table.

Solution: Locate the tables for latitude 49 N (= +49), the left-handones being in Arabic, and the right-hand ones in English. There aresix such tables. Only one contains the value 40 in its first twocolumns on the right. 40 appears in the column with the headingSecond number. On the same line and under the heading Secondof latitude 49, we find the value 50 82N, that is 8250, North. Andsince 40 (counted from the Greenwich meridian) is West of Mecca,the qibla is 8250 from North, and eastward.

Example 5 :

Find the qibla for latitude 4835 N and longitude 745 E in theextended qibla table.

Solution: This value is not present in the tables, but we can use asimple interpolation scheme. For 48 N and 8 E, we find 551 (fromSouth) and for 49 N and 8 E, we find 5356 (from South). Thisleads to a qibla of 551 + 3560 (53

56 551) = 551 3560 15 =

5423 for the location (4835 N, 8 E).Then, for 48 N and 7 E, we find 5615 (from South) and for 49 Nand 7 E, we find 5511 (from South). This leads to a qibla of5615 + 3560 (55

11 5615) = 5615 3560 14 = 5538 for the

location (4835 N, 7 E).Finally, an approximation of the qibla for the location (4835 N,745 E) is 5538 + 4560 (54

23 5538) = 5538 4560 115 =

5442. The real value is indeed 5442.

5 Acknowledgements

First, I want to thank Klaus Lagally whose ArabTEX package was heavily usedin the tables as well as in this introduction. Among other things, this packageprovides facilities for typesetting Abjad numerals.

The Arabic headings of the tables as well as the various Arabic translationswere obtained through the kind help of Nazim Benassa who also suggestedseveral improvements for the Arabic typographic correctness of the tables.

Finally, Robert Marche suggested improvements in the derivation of for-mula (6).

Bibliography

[1] Georges Ifrah. The Universal History of Numbers: From Prehistory to theInvention of the Computer. New York: John Wiley & Sons, 1999. [This bookshould be supplemented by Joseph Daubens critical review which appearedin the Notices of the AMS, volume 49, number 1, January 2002.].

[2] Rida A. K. Irani. Arabic Numeral Forms. Centaurus, 4:112, 1955.

[3] David A. King. Al-Khalls Auxiliary Tables for Solving Problems of Spher-ical Astronomy. Journal for the History of Astronomy, 4:99110, 1973.

11

[4] David A. King. Al-Khalls Qibla Table. Journal of Near Eastern Studies,34(2):81122, April 1975.

[5] David A. King. World-maps for finding the direction and distance to Mecca:innovation and tradition in Islamic science. Leiden: Brill, 1999.

[6] Glen van Brummelen. The numerical structure of al-Khalls auxiliary ta-bles. Physis, 28:667698, 1991.

[7] Glen van Brummelen and Kenneth Butler. Determining the Interdependenceof Historical Astronomical Tables. Journal of the American Statistical As-sociation, 92(437):4148, March 1997.

12

Appendix A

Al-Khalls Qibla table

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41 67N 0 72N 7 66N 25 71N 35 64N 50 70N 97 37 127 7

14 67N 1 72N 37 65N 24 71N 2 64N 48 70N 96 38 126 8

45 66N 2 72N 4 65N 23 71N 27 63N 46 70N 95 39 125 9

13 66N 2 72N 29 64N 22 71N 49 62N 43 70N 94 40 124 10

39 65N 1 72N 51 63N 20 71N 8 62N 40 70N 93 41 123 11

1 65N 0 72N 10 63N 18 71N 23 61N 37 70N 92 42 122 12

21 64N 59 71N 26 62N 15 71N 35 60N 32 70N 91 43 121 13

36 63N 57 71N 38 61N 12 71N 44 59N 28 70N 90 44 120 14

48 62N 55 71N 45 60N 8 71N 48 58N 22 70N 89 45 119 15

55 61N 52 71N 48 59N 4 71N 47 57N 16 70N 88 46 118 16

57 60N 49 71N 46 58N 59 70N 41 56N 10 70N 87 47 117 17

54 59N 45 71N 39 57N 53 70N 30 55N 3 70N 86 48 116 18

45 58N 40 71N 25 56N 47 70N 13 54N 55 69N 85 49 115 19

28 57N 35 71N 4 55N 40 70N 48 52N 46 69N 84 50 114 20

4 56N 29 71N 36 53N 33 70N 17 51N 37 69N 83 51 113 21

31 54N 23 71N 59 51N 24 70N 37 49N 27 69N 82 52 112 22

49 52N 15 71N 12 50N 15 70N 47 47N 16 69N 81 53 111 23

55 50N 7 71N 15 48N 5 70N 48 45N 4 69N 80 54 110 24

49 48N 58 70N 6 46N 54 69N 38 43N 51 68N 79 55 109 25

29 46N 48 70N 45 43N 42 69N 16 41N 37 68N 78 56 108 26

53 43N 38 70N 9 41N 29 69N 41 38N 22 68N 77 57 107 27

0 41N 26 70N 17 38N 15 69N 52 35N 6 68N 76 58 106 28

48 37N 13 70N 8 35N 0 69N 48 32N 49 67N 75 59 105 29

15 34N 59 69N 42 31N 44 68N 29 29N 30 67N 74 60 104 30

21 30N 44 69N 58 27N 26 68N 54 25N 10 67N 73 61 103 31

4 26N 27 69N 55 23N 7 68N 4 22N 49 66N 72 62 102 32

25 21N 9 69N 34 19N 47 67N 0 18N 26 66N 71 63 101 33

25 16N 50 68N 56 14N 24 67N 43 13N 1 66N 70 64 100 34

7 11N 29 68N 6 10N 0 67N 15 9N 34 65N 69 65 99 35

37 5N 6 68N 5 5N 35 66N 39 4N 6 65N 68 66 98 36

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38 72N 50 73N 57 70N 13 73N 18 69N 36 72N 97 37 127 7

22 72N 55 73N 37 70N 16 73N 54 68N 38 72N 96 38 126 8

4 72N 59 73N 15 70N 20 73N 28 68N 40 72N 95 39 125 9

44 71N 4 74N 51 69N 22 73N 0 68N 42 72N 94 40 124 10

22 71N 8 74N 24 69N 25 73N 30 67N 43 72N 93 41 123 11

57 70N 11 74N 55 68N 27 73N 56 66N 43 72N 92 42 122 12

30 70N 14 74N 23 68N 29 73N 20 66N 44 72N 91 43 121 13

0 70N 17 74N 48 67N 30 73N 40 65N 43 72N 90 44 120 14

26 69N 19 74N 9 67N 31 73N 56 64N 42 72N 89 45 119 15

49 68N 21 74N 26 66N 31 73N 8 64N 41 72N 88 46 118 16

8 68N 23 74N 38 65N 31 73N 15 63N 39 72N 87 47 117 17

22 67N 24 74N 46 64N 30 73N 16 62N 37 72N 86 48 116 18

30 66N 25 74N 47 63N 29 73N 12 61N 34 72N 85 49 115 19

33 65N 25 74N 42 62N 27 73N 1 60N 31 72N 84 50 114 20

28 64N 25 74N 30 61N 25 73N 42 58N 27 72N 83 51 113 21

15 63N 24 74N 9 60N 22 73N 15 57N 22 72N 82 52 112 22

52 61N 22 74N 38 58N 19 73N 37 55N 17 72N 81 53 111 23

18 60N 20 74N 56 56N 15 73N 48 53N 10 72N 80 54 110 24

31 58N 18 74N 0 55N 10 73N 46 51N 4 72N 79 55 109 25

28 56N 14 74N 49 52N 5 73N 30 49N 56 71N 78 56 108 26

7 54N 10 74N 20 50N 58 72N 56 46N 47 71N 77 57 107 27

24 51N 6 74N 31 47N 51 72N 4 44N 38 71N 76 58 106 28

15 48N 0 74N 18 44N 43 72N 50 40N 27 71N 75 59 105 29

36 44N 54 73N 37 40N 34 72N 12 37N 16 71N 74 60 104 30

20 40N 46 73N 26 36N 24 72N 8 33N 3 71N 73 61 103 31

24 35N 38 73N 41 31N 13 72N 37 28N 49 70N 72 62 102 32

42 29N 28 73N 21 26N 0 72N 38 23N 34 70N 71 63 101 33

13 23N 18 73N 25 20N 47 71N 12 18N 17 70N 70 64 100 34

59 15N 6 73N 57 13N 32 71N 23 12N 59 69N 69 65 99 35

10 8N 53 72N 5 7N 15 71N 16 6N 39 69N 68 66 98 36

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55 77N 44 75N 8 76N 6 75N 22 74N 28 74N 97 37 127 7

51 77N 53 75N 59 75N 13 75N 9 74N 34 74N 96 38 126 8

46 77N 2 76N 50 75N 21 75N 56 73N 40 74N 95 39 125 9

40 77N 11 76N 39 75N 28 75N 40 73N 45 74N 94 40 124 10

32 77N 19 76N 26 75N 35 75N 22 73N 51 74N 93 41 123 11

23 77N 27 76N 12 75N 41 75N 3 73N 56 74N 92 42 122 12

13 77N 35 76N 55 74N 47 75N 41 72N 1 75N 91 43 121 13

0 77N 43 76N 36 74N 53 75N 16 72N 5 75N 90 44 120 14

46 76N 50 76N 15 74N 59 75N 49 71N 9 75N 89 45 119 15

29 76N 57 76N 51 73N 4 76N 18 71N 13 75N 88 46 118 16

10 76N 4 77N 24 73N 9 76N 43 70N 16 75N 87 47 117 17

47 75N 10 77N 53 72N 14 76N 4 70N 19 75N 86 48 116 18

21 75N 16 77N 18 72N 18 76N 21 69N 21 75N 85 49 115 19

51 74N 22 77N 38 71N 22 76N 31 68N 23 75N 84 50 114 20

17 74N 28 77N 52 70N 26 76N 35 67N 25 75N 83 51 113 21

36 73N 33 77N 59 69N 29 76N 31 66N 26 75N 82 52 112 22

49 72N 38 77N 58 68N 32 76N 19 65N 27 75N 81 53 111 23

54 71N 42 77N 48 67N 34 76N 56 63N 27 75N 80 54 110 24

49 70N 46 77N 26 66N 36 76N 20 62N 26 75N 79 55 109 25

31 69N 50 77N 49 64N 37 76N 28 60N 25 75N 78 56 108 26

57 67N 53 77N 55 62N 38 76N 19 58N 24 75N 77 57 107 27

4 66N 56 77N 40 60N 38 76N 47 55N 21 75N 76 58 106 28

45 63N 59 77N 56 57N 38 76N 47 52N 18 75N 75 59 105 29

52 60N 0 78N 38 54N 37 76N 14 49N 15 75N 74 60 104 30

13 57N 2 78N 35 50N 35 76N 0 45N 10 75N 73 61 103 31

32 52N 2 78N 34 45N 33 76N 57 39N 5 75N 72 62 102 32

26 46N 2 78N 22 39N 30 76N 56 33N 58 74N 71 63 101 33

24 38N 2 78N 42 31N 26 76N 51 26N 51 74N 70 64 100 34

56 27N 0 78N 26 22N 21 76N 41 18N 42 74N 69 65 99 35

53 14N 58 77N 41 11N 15 76N 37 9N 33 74N 68 66 98 36

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27 83N 43 77N 35 81N 3 77N 44 79N 23 76N 97 37 127 7

36 83N 56 77N 40 81N 15 77N 45 79N 34 76N 96 38 126 8

46 83N 9 78N 44 81N 27 77N 44 79N 44 76N 95 39 125 9

55 83N 22 78N 48 81N 38 77N 43 79N 54 76N 94 40 124 10

3 84N 35 78N 51 81N 49 77N 41 79N 4 77N 93 41 123 11

12 84N 48 78N 54 81N 1 78N 37 79N 14 77N 92 42 122 12

20 84N 1 79N 56 81N 12 78N 33 79N 23 77N 91 43 121 13

28 84N 13 79N 56 81N 22 78N 27 79N 32 77N 90 44 120 14

35 84N 26 79N 56 81N 33 78N 20 79N 41 77N 89 45 119 15

43 84N 38 79N 55 81N 44 78N 11 79N 50 77N 88 46 118 16

49 84N 50 79N 53 81N 54 78N 0 79N 58 77N 87 47 117 17

56 84N 2 80N 50 81N 4 79N 46 78N 7 78N 86 48 116 18

1 85N 14 80N 44 81N 14 79N 31 78N 15 78N 85 49 115 19

6 85N 26 80N 37 81N 24 79N 12 78N 23 78N 84 50 114 20

11 85N 37 80N 28 81N 33 79N 49 77N 30 78N 83 51 113 21

14 85N 49 80N 16 81N 43 79N 22 77N 38 78N 82 52 112 22

16 85N 0 81N 0 81N 52 79N 51 76N 45 78N 81 53 111 23

17 85N 12 81N 41 80N 1 80N 12 76N 51 78N 80 54 110 24

16 85N 23 81N 17 80N 10 80N 26 75N 58 78N 79 55 109 25

13 85N 34 81N 46 79N 19 80N 31 74N 4 79N 78 56 108 26

7 85N 45 81N 8 79N 27 80N 23 73N 10 79N 77 57 107 27

58 84N 56 81N 19 78N 35 80N 59 71N 15 79N 76 58 106 28

43 84N 6 82N 16 77N 43 80N 13 70N 20 79N 75 59 105 29

21 84N 17 82N 52 75N 51 80N 58 67N 25 79N 74 60 104 30

48 83N 27 82N 59 73N 58 80N 2 65N 29 79N 73 61 103 31

58 82N 38 82N 20 71N 5 81N 6 61N 33 79N 72 62 102 32

38 81N 48 82N 27 67N 12 81N 39 55N 37 79N 71 63 101 33

19 79N 58 82N 21 61N 18 81N 53 47N 39 79N 70 64 100 34

37 74N 8 83N 55 50N 24 81N 33 36N 42 79N 69 65 99 35

36 61N 18 83N 46 31N 30 81N 24 20N 43 79N 68 66 98 36

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8 89N 45 79N 14 87N 4 79N 20 85N 23 78N 97 37 127 7

31 89N 3 80N 32 87N 20 79N 34 85N 38 78N 96 38 126 8

55 89N 21 80N 51 87N 36 79N 48 85N 53 78N 95 39 125 9

41 89S 38 80N 11 88N 52 79N 2 86N 7 79N 94 40 124 10

15 89S 56 80N 31 88N 8 80N 17 86N 22 79N 93 41 123 11

49 88S 13 81N 51 88N 24 80N 31 86N 36 79N 92 42 122 12

21 88S 31 81N 12 89N 40 80N 46 86N 50 79N 91 43 121 13

52 87S 48 81N 34 89N 56 80N 1 87N 4 80N 90 44 120 14

22 87S 6 82N 57 89N 12 81N 16 87N 18 80N 89 45 119 15

50 86S 23 82N 39 89S 28 81N 31 87N 32 80N 88 46 118 16

16 86S 41 82N 14 89S 43 81N 47 87N 46 80N 87 47 117 17

39 85S 58 82N 48 88S 59 81N 4 88N 0 81N 86 48 116 18

1 85S 16 83N 20 88S 15 82N 20 88N 14 81N 85 49 115 19

19 84S 34 83N 50 87S 31 82N 38 88N 28 81N 84 50 114 20

33 83S 52 83N 18 87S 47 82N 56 88N 42 81N 83 51 113 21

44 82S 10 84N 43 86S 2 83N 15 89N 55 81N 82 52 112 22

49 81S 28 84N 5 86S 18 83N 35 89N 9 82N 81 53 111 23

48 80S 46 84N 24 85S 34 83N 57 89N 23 82N 80 54 110 24

39 79S 5 85N 37 84S 50 83N 40 89S 36 82N 79 55 109 25

20 78S 23 85N 44 83S 6 84N 15 89S 50 82N 78 56 108 26

48 76S 42 85N 43 82S 23 84N 47 88S 3 83N 77 57 107 27

1 75S 1 86N 31 81S 39 84N 15 88S 17 83N 76 58 106 28

52 72S 21 86N 5 80S 55 84N 38 87S 31 83N 75 59 105 29

13 70S 40 86N 18 78S 12 85N 54 86S 44 83N 74 60 104 30

52 66S 0 87N 2 76S 29 85N 59 85S 58 83N 73 61 103 31

31 62S 20 87N 0 73S 46 85N 47 84S 11 84N 72 62 102 32

40 56S 41 87N 43 68S 3 86N 5 83S 25 84N 71 63 101 33

29 48S 2 88N 12 62S 20 86N 23 80S 39 84N 70 64 100 34

49 36S 23 88N 22 51S 38 86N 18 75S 52 84N 69 65 99 35

27 20S 45 88N 52 31S 55 86N 53 61S 6 85N 68 66 98 36

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11 85S 51 81N 4 87S 9 81N 58 88S 27 80N 97 37 127 7

34 84S 13 82N 32 86S 29 81N 30 88S 46 80N 96 38 126 8

56 83S 35 82N 59 85S 50 81N 2 88S 5 81N 95 39 125 9

17 83S 57 82N 24 85S 10 82N 32 87S 24 81N 94 40 124 10

35 82S 19 83N 47 84S 31 82N 1 87S 43 81N 93 41 123 11

51 81S 41 83N 9 84S 52 82N 29 86S 2 82N 92 42 122 12

5 81S 4 84N 29 83S 12 83N 55 85S 21 82N 91 43 121 13

17 80S 26 84N 47 82S 33 83N 19 85S 40 82N 90 44 120 14

25 79S 49 84N 2 82S 54 83N 41 84S 0 83N 89 45 119 15

30 78S 12 85N 15 81S 15 84N 1 84S 19 83N 88 46 118 16

32 77S 35 85N 24 80S 37 84N 19 83S 39 83N 87 47 117 17

29 76S 58 85N 29 79S 58 84N 33 82S 58 83N 86 48 116 18

21 75S 22 86N 30 78S 20 85N 44 81S 18 84N 85 49 115 19

7 74S 46 86N 26 77S 42 85N 50 80S 38 84N 84 50 114 20

47 72S 10 87N 16 76S 4 86N 52 79S 58 84N 83 51 113 21

19 71S 34 87N 0 75S 26 86N 48 78S 18 85N 82 52 112 22

42 69S 59 87N 35 73S 49 86N 38 77S 38 85N 81 53 111 23

54 67S 25 88N 0 72S 12 87N 19 76S 59 85N 80 54 110 24

53 65S 50 88N 14 70S 35 87N 50 74S 20 86N 79 55 109 25

37 63S 16 89N 13 68S 58 87N 8 73S 41 86N 78 56 108 26

2 61S 43 89N 54 65S 23 88N 11 71S 2 87N 77 57 107 27

5 58S 50 89S 14 63S 47 88N 53 68S 24 87N 76 58 106 28

41 54S 22 89S 5 60S 12 89N 10 66S 46 87N 75 59 105 29

44 50S 54 88S 22 56S 37 89N 52 62S 9 88N 74 60 104 30

7 46S 25 88S 55 51S 56 89S 48 58S 32 88N 73 61 103 31

43 40S 55 87S 31 46S 30 89S 42 53S 55 88N 72 62 102 32

24 34S 24 87S 57 39S 2 89S 12 47S 19 89N 71 63 101 33

4 27S 52 86S 1 32S 34 88S 49 38S 44 89N 70 64 100 34

46 18S 20 86S 33 22S 5 88S 6 28S 51 89S 69 65 99 35

37 9S 46 85S 42 11S 35 87S 55 14S 25 89S 68 66 98 36

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35 79S 0 84N 26 81S 17 83N 18 83S 34 82N 97 37 127 7

47 78S 26 84N 41 80S 41 83N 37 82S 57 82N 96 38 126 8

56 77S 52 84N 55 79S 6 84N 55 81S 20 83N 95 39 125 9

3 77S 19 85N 6 79S 31 84N 10 81S 44 83N 94 40 124 10

8 76S 45 85N 15 78S 56 84N 24 80S 8 84N 93 41 123 11

9 75S 12 86N 21 77S 22 85N 35 79S 31 84N 92 42 122 12

8 74S 39 86N 24 76S 47 85N 44 78S 55 84N 91 43 121 13

3 73S 7 87N 24 75S 13 86N 49 77S 20 85N 90 44 120 14

54 71S 35 87N 21 74S 39 86N 51 76S 44 85N 89 45 119 15

41 70S 3 88N 13 73S 6 87N 50 75S 9 86N 88 46 118 16

23 69S 31 88N 1 72S 32 87N 44 74S 34 86N 87 47 117 17

59 67S 0 89N 43 70S 0 88N 33 73S 59 86N 86 48 116 18

30 66S 30 89N 20 69S 27 88N 17 72S 24 87N 85 49 115 19

53 64S 59 89N 50 67S 55 88N 55 70S 50 87N 84 50 114 20

9 63S 30 89S 13 66S 23 89N 25 69S 16 88N 83 51 113 21

17 61S 0 89S 27 64S 52 89N 47 67S 43 88N 82 52 112 22

14 59S 28 88S 31 62S 39 89S 0 66S 10 89N 81 53 111 23

0 57S 56 87S 23 60S 9 89S 1 64S 38 89N 80 54 110 24

34 54S 24 87S 3 58S 39 88S 49 61S 54 89S 79 55 109 25

53 51S 50 86S 27 55S 8 88S 21 59S 26 89S 78 56 108 26

56 48S 16 86S 34 52S 36 87S 35 56S 57 88S 77 57 107 27

41 45S 41 85S 21 49S 4 87S 28 53S 27 88S 76 58 106 28

6 42S 6 85S 45 45S 31 86S 55 49S 56 87S 75 59 105 29

9 38S 29 84S 43 41S 57 85S 53 45S 25 87S 74 60 104 30

48 33S 51 83S 13 37S 22 85S 16 41S 53 86S 73 61 103 31

3 29S 12 83S 12 32S 46 84S 1 36S 20 86S 72 62 102 32

52 23S 31 82S 38 26S 8 84S 4 30S 46 85S 71 63 101 33

18 18S 50 81S 33 20S 30 83S 23 23S 11 85S 70 64 100 34

25 12S 7 81S 0 14S 50 82S 2 16S 34 84S 69 65 99 35

16 6S 22 80S 6 7S 9 82S 10 8S 57 83S 68 66 98 36

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14 73S 40 86N 3 75S 55 85N 54 76S 10 85N 96 38 126 8

13 72S 11 87N 5 74S 24 86N 0 76S 38 85N 95 39 125 9

10 71S 42 87N 5 73S 54 86N 3 75S 6 86N 94 40 124 10

3 70S 13 88N 1 72S 24 87N 3 74S 34 86N 93 41 123 11

53 68S 45 88N 55 70S 54 87N 1 73S 3 87N 92 42 122 12

39 67S 17 89N 45 69S 24 88N 55 71S 32 87N 91 43 121 13

22 66S 49 89N 31 68S 55 88N 45 70S 1 88N 90 44 120 14

0 65S 38 89S 14 67S 26 89N 31 69S 30 88N 89 45 119 15

34 63S 5 89S 51 65S 57 89N 13 68S 0 89N 88 46 118 16

3 62S 31 88S 24 64S 31 89S 50 66S 30 89N 87 47 117 17

27 60S 57 87S 51 62S 58 88S 22 65S 59 89S 86 48 116 18

44 58S 22 87S 12 61S 25 88S 47 63S 28 89S 85 49 115 19

55 56S 47 86S 26 59S 52 87S 5 62S 56 88S 84 50 114 20

59 54S 11 86S 32 57S 17 87S 16 60S 24 88S 83 51 113 21

54 52S 34 85S 31 55S 43 86S 18 58S 51 87S 82 52 112 22

41 50S 57 84S 20 53S 7 86S 11 56S 18 87S 81 53 111 23

19 48S 19 84S 59 50S 31 85S 52 53S 43 86S 80 54 110 24

46 45S 39 83S 27 48S 54 84S 22 51S 9 86S 79 55 109 25

3 43S 59 82S 43 45S 16 84S 39 48S 33 85S 78 56 108 26

8 40S 18 82S 45 42S 37 83S 41 45S 57 84S 77 57 107 27

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8 18S 46 77S 43 19S 20 79S 36 21S 55 80S 71 63 101 33

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44 69S 11 88N 28 71S 25 87N 21 72S 3 87N 96 38 126 8

37 68S 44 88N 24 70S 57 87N 18 71S 34 87N 95 39 125 9

28 67S 18 89N 17 69S 30 88N 13 70S 6 88N 94 40 124 10

15 66S 52 89N 7 68S 2 89N 5 69S 38 88N 93 41 123 11

59 64S 33 89S 54 66S 36 89N 53 67S 10 89N 92 42 122 12

40 63S 59 88S 38 65S 51 89S 38 66S 43 89N 91 43 121 13

17 62S 23 88S 17 64S 17 89S 19 65S 44 89S 90 44 120 14

50 60S 47 87S 52 62S 43 88S 56 63S 11 89S 89 45 119 15

18 59S 11 87S 23 61S 8 88S 28 62S 37 88S 88 46 118 16

41 57S 34 86S 49 59S 33 87S 55 60S 2 88S 87 47 117 17

0 56S 56 85S 10 58S 57 86S 17 59S 27 87S 86 48 116 18

12 54S 18 85S 24 56S 20 86S 33 57S 51 86S 85 49 115 19

19 52S 39 84S 33 54S 43 85S 43 55S 15 86S 84 50 114 20

19 50S 59 83S 34 52S 5 85S 45 53S 38 85S 83 51 113 21

12 48S 19 83S 28 50S 26 84S 40 51S 0 85S 82 52 112 22

58 45S 37 82S 14 48S 47 83S 26 49S 22 84S 81 53 111 23

36 43S 55 81S 52 45S 7 83S 4 47S 43 83S 80 54 110 24

6 41S 12 81S 20 43S 25 82S 31 44S 2 83S 79 55 109 25

27 38S 28 80S 38 40S 43 81S 49 41S 21 82S 78 56 108 26

38 35S 43 79S 46 37S 0 81S 55 38S 39 81S 77 57 107 27

41 32S 56 78S 43 34S 16 80S 50 35S 56 80S 76 58 106 28

34 29S 8 78S 29 31S 30 79S 33 32S 11 80S 75 59 105 29

18 26S 19 77S 5 28S 43 78S 3 29S 26 79S 74 60 104 30

52 22S 29 76S 29 24S 55 77S 22 25S 39 78S 73 61 103 31

19 19S 37 75S 43 20S 5 77S 29 21S 50 77S 72 62 102 32

37 15S 43 74S 47 16S 14 76S 26 17S 0 77S 71 63 101 33

49 11S 47 73S 43 12S 21 75S 13 13S 8 76S 70 64 100 34

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48 64S 33 89S 24 66S 41 89N 3 68S 56 88N 96 38 126 8

35 63S 56 88S 13 65S 42 89S 54 66S 31 89N 95 39 125 9

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38 59S 1 87S 21 61S 52 87S 8 63S 43 88S 92 42 122 12

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44 56S 42 85S 30 58S 35 86S 21 60S 29 87S 90 44 120 14

12 55S 1 85S 59 56S 56 85S 52 58S 52 86S 89 45 119 15

35 53S 20 84S 24 55S 17 85S 18 57S 14 86S 88 46 118 16

54 51S 39 83S 44 53S 37 84S 40 55S 35 85S 87 47 117 17

9 50S 56 82S 0 52S 56 83S 56 53S 56 84S 86 48 116 18

19 48S 13 82S 10 50S 14 83S 8 52S 16 84S 85 49 115 19

24 46S 29 81S 15 48S 32 82S 13 50S 35 83S 84 50 114 20

24 44S 44 80S 15 46S 49 81S 13 48S 54 82S 83 51 113 21

19 42S 59 79S 8 44S 5 81S 6 46S 12 82S 82 52 112 22

8 40S 12 79S 56 41S 20 80S 52 43S 28 81S 81 53 111 23

51 37S 25 78S 37 39S 34 79S 31 41S 44 80S 80 54 110 24

28 35S 36 77S 11 37S 47 78S 3 39S 59 79S 79 55 109 25

59 32S 46 76S 38 34S 59 77S 27 36S 13 79S 78 56 108 26

24 30S 55 75S 59 31S 10 77S 43 33S 26 78S 77 57 107 27

43 27S 3 75S 12 29S 20 76S 51 30S 37 77S 76 58 106 28

56 24S 9 74S 19 26S 28 75S 51 27S 47 76S 75 59 105 29

4 22S 14 73S 19 23S 35 74S 43 24S 56 75S 74 60 104 30

6 19S 18 72S 13 20S 40 73S 28 21S 4 75S 73 61 103 31

3 16S 19 71S 0 17S 43 72S 5 18S 9 74S 72 62 102 32

56 12S 19 70S 43 13S 45 71S 36 14S 13 73S 71 63 101 33

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25 43S 12 79S 57 44S 12 80S 35 46S 12 81S 85 49 115 19

32 41S 24 78S 3 43S 25 79S 41 44S 27 80S 84 50 114 20

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45 29S 46 71S 52 30S 50 72S 3 32S 55 73S 80 54 110 24

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38 50S 28 81S 41 51S 11 82S 46 52S 54 82S 97 37 127 7

19 49S 38 80S 22 50S 21 81S 27 51S 5 82S 96 38 126 8

58 47S 47 79S 1 49S 32 80S 6 50S 16 81S 95 39 125 9

36 46S 56 78S 38 47S 41 79S 43 48S 27 80S 94 40 124 10

12 45S 4 78S 14 46S 50 78S 19 47S 37 79S 93 41 123 11

47 43S 12 77S 48 44S 59 77S 52 45S 46 78S 92 42 122 12

19 42S 19 76S 20 43S 6 77S 24 44S 55 77S 91 43 121 13

51 40S 25 75S 51 41S 14 76S 54 42S 3 77S 90 44 120 14

20 39S 30 74S 20 40S 20 75S 22 41S 10 76S 89 45 119 15

48 37S 35 73S 46 38S 26 74S 48 39S 17 75S 88 46 118 16

14 36S 40 72S 11 37S 31 73S 12 38S 23 74S 87 47 117 17

39 34S 43 71S 35 35S 35 72S 33 36S 28 73S 86 48 116 18

2 33S 46 70S 56 33S 39 71S 53 34S 33 72S 85 49 115 19

23 31S 47 69S 15 32S 41 70S 11 33S 36 71S 84 50 114 20

42 29S 48 68S 33 30S 43 69S 27 31S 39 70S 83 51 113 21

0 28S 48 67S 49 28S 44 68S 40 29S 41 69S 82 52 112 22

16 26S 48 66S 3 27S 44 67S 52 27S 41 68S 81 53 111 23

31 24S 46 65S 15 25S 43 66S 2 26S 41 67S 80 54 110 24

44 22S 43 64S 26 23S 41 65S 10 24S 40 66S 79 55 109 25

56 20S 39 63S 35 21S 38 64S 16 22S 37 65S 78 56 108 26

7 19S 34 62S 42 19S 33 63S 21 20S 34 64S 77 57 107 27

16 17S 28 61S 49 17S 28 62S 24 18S 29 63S 76 58 106 28

24 15S 21 60S 53 15S 21 61S 25 16S 23 62S 75 59 105 29

31 13S 13 59S 57 13S 14 60S 25 14S 16 61S 74 60 104 30

37 11S 3 58S 59 11S 4 59S 24 12S 8 60S 73 61 103 31

42 9S 52 56S 1 10S 54 57S 22 10S 58 58S 72 62 102 32

46 7S 40 55S 2 8S 42 56S 18 8S 46 57S 71 63 101 33

50 5S 27 54S 2 6S 29 55S 14 6S 34 56S 70 64 100 34

54 3S 12 53S 2 4S 15 54S 10 4S 19 55S 69 65 99 35

57 1S 56 51S 1 2S 58 52S 5 2S 3 54S 68 66 98 36

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43 47S 23 79S 39 48S 5 80S 38 49S 46 80S 97 37 127 7

25 46S 30 78S 21 47S 13 79S 19 48S 55 79S 96 38 126 8

5 45S 37 77S 0 46S 20 78S 58 46S 4 79S 95 39 125 9

44 43S 43 76S 39 44S 27 77S 36 45S 11 78S 94 40 124 10

22 42S 49 75S 16 43S 34 76S 13 44S 19 77S 93 41 123 11

58 40S 54 74S 52 41S 39 75S 48 42S 25 76S 92 42 122 12

33 39S 58 73S 26 40S 45 74S 21 41S 31 75S 91 43 121 13

7 38S 2 73S 59 38S 49 73S 53 39S 37 74S 90 44 120 14

39 36S 6 72S 30 37S 53 72S 24 38S 41 73S 89 45 119 15

10 35S 8 71S 0 36S 56 71S 53 36S 46 72S 88 46 118 16

40 33S 10 70S 29 34S 59 70S 20 35S 49 71S 87 47 117 17

9 32S 11 69S 56 32S 1 70S 46 33S 51 70S 86 48 116 18

36 30S 12 68S 22 31S 2 69S 10 32S 53 69S 85 49 115 19

2 29S 11 67S 46 29S 2 68S 33 30S 54 68S 84 50 114 20

27 27S 10 66S 9 28S 2 67S 54 28S 55 67S 83 51 113 21

51 25S 8 65S 31 26S 1 66S 14 27S 54 66S 82 52 112 22

13 24S 6 64S 52 24S 58 64S 33 25S 52 65S 81 53 111 23

34 22S 2 63S 11 23S 55 63S 50 23S 50 64S 80 54 110 24

55 20S 57 61S 29 21S 51 62S 5 22S 46 63S 79 55 109 25

14 19S 52 60S 46 19S 46 61S 20 20S 42 62S 78 56 108 26

32 17S 46 59S 2 18S 40 60S 33 18S 37 61S 77 57 107 27

50 15S 38 58S 17 16S 33 59S 45 16S 30 60S 76 58 106 28

6 14S 30 57S 30 14S 25 58S 56 14S 22 59S 75 59 105 29

22 12S 20 56S 44 12S 16 57S 6 13S 14 58S 74 60 104 30

37 10S 10 55S 56 10S 6 56S 16 11S 4 57S 73 61 103 31

52 8S 58 53S 8 9S 55 54S 24 9S 53 55S 72 62 102 32

6 7S 46 52S 19 7S 42 53S 32 7S 40 54S 71 63 101 33

20 5S 32 51S 29 5S 28 52S 40 5S 26 53S 70 64 100 34

34 3S 17 50S 40 3S 13 51S 47 3S 12 52S 69 65 99 35

47 1S 1 49S 50 1S 57 49S 53 1S 55 50S 68 66 98 36

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8 45S 21 77S 58 45S 2 78S 50 46S 42 78S 97 37 127 7

50 43S 26 76S 40 44S 7 77S 31 45S 49 77S 96 38 126 8

32 42S 31 75S 21 43S 13 76S 12 44S 55 76S 95 39 125 9

13 41S 35 74S 1 42S 17 75S 51 42S 0 76S 94 40 124 10

52 39S 39 73S 40 40S 21 74S 30 41S 5 75S 93 41 123 11

31 38S 42 72S 18 39S 25 73S 7 40S 9 74S 92 42 122 12

8 37S 44 71S 54 37S 28 72S 43 38S 13 73S 91 43 121 13

45 35S 46 70S 30 36S 31 71S 17 37S 16 72S 90 44 120 14

20 34S 47 69S 4 35S 33 70S 51 35S 19 71S 89 45 119 15

55 32S 48 68S 38 33S 34 69S 23 34S 21 70S 88 46 118 16

28 31S 48 67S 10 32S 34 68S 54 32S 22 69S 87 47 117 17

1 30S 47 66S 41 30S 34 67S 24 31S 22 68S 86 48 116 18

32 28S 46 65S 11 29S 34 66S 53 29S 22 67S 85 49 115 19

3 27S 44 64S 41 27S 32 65S 20 28S 21 66S 84 50 114 20

33 25S 42 63S 9 26S 30 64S 47 26S 20 65S 83 51 113 21

2 24S 38 62S 36 24S 27 63S 12 25S 17 64S 82 52 112 22

30 22S 34 61S 2 23S 23 62S 37 23S 14 63S 81 53 111 23

57 20S 29 60S 28 21S 19 61S 0 22S 10 62S 80 54 110 24

24 19S 24 59S 52 19S 14 60S 23 20S 5 61S 79 55 109 25

49 17S 17 58S 16 18S 7 59S 44 18S 59 59S 78 56 108 26

14 16S 10 57S 39 16S 0 58S 5 17S 52 58S 77 57 107 27

39 14S 2 56S 1 15S 52 56S 25 15S 45 57S 76 58 106 28

3 13S 53 54S 23 13S 44 55S 44 13S 36 56S 75 59 105 29

26 11S 43 53S 44 11S 34 54S 2 12S 26 55S 74 60 104 30

49 9S 32 52S 4 10S 23 53S 20 10S 16 54S 73 61 103 31

11 8S 21 51S 24 8S 11 52S 38 8S 4 53S 72 62 102 32

34 6S 8 50S 44 6S 59 50S 55 6S 51 51S 71 63 101 33

55 4S 54 48S 3 5S 45 49S 11 5S 38 50S 70 64 100 34

17 3S 40 47S 22 3S 30 48S 28 3S 23 49S 69 65 99 35

39 1S 24 46S 41 1S 15 47S 44 1S 7 48S 68 66 98 36

45

Appendix B

Extended tablefor the Qibla

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42 8N 21 38N 45 8N 35 38N 49 8N 49 38N 31 1

43 7N 21 37N 46 7N 35 37N 49 7N 49 37N 32 2

44 6N 22 36N 46 6N 35 36N 49 6N 49 36N 33 3

45 5N 23 35N 47 5N 36 35N 49 5N 49 35N 34 4

45 4N 23 34N 47 4N 36 34N 49 4N 49 34N 35 5

46 3N 24 33N 47 3N 36 33N 49 3N 49 33N 36 6

47 2N 25 32N 48 2N 37 32N 49 2N 49 32N 37 7

48 1N 25 31N 48 1N 37 31N 49 1N 49 31N 38 8

48 0N 26 30N 49 0N 37 30N 49 0N 49 30N 39 9

11 0N 27 29N 11 0N 38 29N 11 0N 49 29N 40 10

10 1N 27 28N 11 1N 38 28N 11 1N 49 28N 41 11

9 2N 28 27N 10 2N 38 27N 11 2N 49 27N 42 12

9 3N 29 26N 10 3N 39 26N 11 3N 49 26N 43 13

8 4N 30 25N 9 4N 39 25N 11 4N 49 25N 44 14

7 5N 30 24N 9 5N 39 24N 11 5N 49 24N 45 15

6 6N 31 23N 9 6N 40 23N 11 6N 49 23N 46 16

5 7N 32 22N 8 7N 40 22N 11 7N 49 22N 47 17

5 8N 32 21N 8 8N 40 21N 11 8N 49 21N 48 18

4 9N 33 20N 7 9N 41 20N 11 9N 49 20N 49 19

3 10N 34 19N 7 10N 41 19N 11 10N 49 19N 50 20

2 11N 35 18N 7 11N 42 18N 11 11N 49 18N 51 21

2 12N 35 17N 6 12N 42 17N 11 12N 49 17N 52 22

1 13N 36 16N 6 13N 42 16N 11 13N 49 16N 53 23

0 14N 37 15N 5 14N 43 15N 11 14N 49 15N 54 24

59 14N 38 14N 5 15N 43 14N 11 15N 49 14N 55 25

59 15N 38 13N 5 16N 44 13N 11 16N 49 13N 56 26

58 16N 39 12N 4 17N 44 12N 11 17N 49 12N 57 27

57 17N 40 11N 4 18N 44 11N 11 18N 49 11N 58 28

56 18N 41 10N 3 19N 45 10N 11 19N 49 10N 59 29

56 19N 41 9N 3 20N 45 9N 11 20N 49 9N 60 30

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36 50N 55 20N 53 50N 3 21N 11 51N 11 21N 91 61

35 51N 54 21N 53 51N 2 22N 11 52N 11 22N 92 62

35 52N 53 22N 52 52N 2 23N 11 53N 11 23N 93 63

34 53N 53 23N 52 53N 2 24N 11 54N 11 24N 94 64

34 54N 52 24N 52 54N 1 25N 11 55N 11 25N 95 65

33 55N 51 25N 52 55N 1 26N 11 56N 11 26N 96 66

33 56N 51 26N 52 56N 1 27N 11 57N 11 27N 97 67

32 57N 50 27N 51 57N 0 28N 11 58N 11 28N 98 68

32 58N 49 28N 51 58N 0 29N 11 59N 11 29N 99 69

31 59N 49 29N 51 59N 59 29N 11 60N 11 30N 100 70

31 60N 48 30N 51 60N 59 30N 11 61N 11 31N 101 71

30 61N 47 31N 50 61N 59 31N 11 62N 11 32N 102 72

30 62N 46 32N 50 62N 58 32N 11 63N 11 33N 103 73

30 63N 46 33N 50 63N 58 33N 11 64N 11 34N 104 74

29 64N 45 34N 50 64N 58 34N 11 65N 11 35N 105 75

29 65N 44 35N 50 65N 57 35N 11 66N 11 36N 106 76

29 66N 44 36N 50 66N 57 36N 11 67N 11 37N 107 77

28 67N 43 37N 49 67N 57 37N 11 68N 11 38N 108 78

28 68N 43 38N 49 68N 56 38N 11 69N 11 39N 109 79

28 69N 42 39N 49 69N 56 39N 11 70N 11 40N 110 80

27 70N 41 40N 49 70N 56 40N 11 71N 11 41N 111 81

27 71N 41 41N 49 71N 56 41N 11 72N 11 42N 112 82

27 72N 40 42N 49 72N 55 42N 11 73N 11 43N 113 83

26 73N 40 43N 49 73N 55 43N 11 74N 11 44N 114 84

26 74N 39 44N 48 74N 55 44N 11 75N 11 45N 115 85

26 75N 38 45N 48 75N 54 45N 11 76N 11 46N 116 86

26 76N 38 46N 48 76N 54 46N 11 77N 11 47N 117 87

25 77N 37 47N 48 77N 54 47N 11 78N 11 48N 118 88

25 78N 37 48N 48 78N 54 48N 11 79N 11 49N 119 89

25 79N 36 49N 48 79N 53 49N 11 80N 11 50N 120 90

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34 69S 25 80N 11 69S 48 80N 49 68S 11 81N 151 121

34 68S 25 81N 11 68S 48 81N 49 67S 11 82N 152 122

33 67S 25 82N 11 67S 48 82N 49 66S 11 83N 153 123

33 66S 24 83N 11 66S 48 83N 49 65S 11 84N 154 124

33 65S 24 84N 11 65S 48 84N 49 64S 11 85N 155 125

32 64S 24 85N 10 64S 48 85N 49 63S 11 86N 156 126

32 63S 24 86N 10 63S 48 86N 49 62S 11 87N 157 127

32 62S 24 87N 10 62S 48 87N 49 61S 11 88N 158 128

31 61S 24 88N 10 61S 47 88N 49 60S 11 89N 159 129

31 60S 24 89N 10 60S 47 89N 49 59S 49 89S 160 130

30 59S 36 89S 9 59S 13 89S 49 58S 49 88S 161 131

30 58S 36 88S 9 58S 13 88S 49 57S 49 87S 162 132

30 57S 36 87S 9 57S 13 87S 49 56S 49 86S 163 133

29 56S 36 86S 9 56S 13 86S 49 55S 49 85S 164 134

29 55S 36 85S 9 55S 12 85S 49 54S 49 84S 165 135

28 54S 36 84S 8 54S 12 84S 49 53S 49 83S 166 136

28 53S 36 83S 8 53S 12 83S 49 52S 49 82S 167 137

27 52S 36 82S 8 52S 12 82S 49 51S 49 81S 168 138

27 51S 36 81S 8 51S 12 81S 49 50S 49 80S 169 139

26 50S 36 80S 7 50S 12 80S 49 49S 49 79S 170 140

26 49S 36 79S 7 49S 12 79S 49 48S 49 78S 171 141

25 48S 36 78S 7 48S 12 78S 49 47S 49 77S 172 142

25 47S 35 77S 7 47S 12 77S 49 46S 49 76S 173 143

24 46S 35 76S 6 46S 12 76S 49 45S 49 75S 174 144

24 45S 35 75S 6 45S 12 75S 49 44S 49 74S 175 145

23 44S 35 74S 6 44S 12 74S 49 43S 49 73S 176 146

22 43S 35 73S 5 43S 12 73S 49 42S 49 72S 177 147

22 42S 35 72S 5 42S 12 72S 49 41S 49 71S 178 148

21 41S 34 71S 5 41S 11 71S 49 40S 49 70S 179 149

21 40S 34 70S 4 40S 11 70S 49 39S 49 69S 180 150

53

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54

Documents

An Extension of Al-Khal l ’s Qibla Table to the Entire Worldlocomat.loria.fr/khalili/khalili-ext.pdfAn Extension of Al-Khal l ’s Qibla Table to the Entire World Denis Roegel1 4