new light on the lunar crescent visibility table of yaʿqūb ibn Ṭāriq

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New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq Author(s): Jan P. Hogendijk Source: Journal of Near Eastern Studies, Vol. 47, No. 2 (Apr., 1988), pp. 95-104 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/544381 . Accessed: 07/12/2014 19:19 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of Near Eastern Studies. http://www.jstor.org This content downloaded from 128.235.251.160 on Sun, 7 Dec 2014 19:19:20 PM All use subject to JSTOR Terms and Conditions

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Page 1: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn ṬāriqAuthor(s): Jan P. HogendijkSource: Journal of Near Eastern Studies, Vol. 47, No. 2 (Apr., 1988), pp. 95-104Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/544381 .

Accessed: 07/12/2014 19:19

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journalof Near Eastern Studies.

http://www.jstor.org

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Page 2: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

NEW LIGHT ON THE LUNAR CRESCENT VISIBILITY TABLE OF YACQUB IBN TARIQ

JAN P. HOGENDIJK, State University of Utrecht

I. INTRODUCTION

IN 1968, E. S. Kennedy published in this journal a lunar visibility table attributed to Yacqub ibn Tariq.' This table is of particular interest because it is one of the few extant pieces of evidence from the very beginning of the Islamic mathematical and astronomical tradition. We will present a new interpretation of the table on the basis of a passage in the ZTj (astronomical handbook with tables) of Al-BattdnT (d. A.D. 929). It turns out that the table was computed using a rather sophisticated trigonometrical approximation and a sine table with the base 3438. The table shows that the study of mathematics in the Islamic world had already reached a surprisingly high level by the late eighth century and that complicated Indian mathematical methods were well understood at that time.' We also analyze a related lunar visibility table which has come down to us in the ZTj of AthTr al-DTn al-Abhari (ca. A.D. 1240) but which was probably derived from a late eighth-century archetype analogous to (but not dependent on) Yacqtib's table.

II. THE LUNAR VISIBILITY TABLE OF YACQUJB IBN TARIQ

Since the Muslim new month begins with the sighting of the lunar crescent, the theory of crescent visibility was an important theme in Muslim astronomy. Whether or not the lunar crescent will be visible depends on a large number of factors, such as the position of the sun on the ecliptic, the elongation and the latitude of the moon, the distance between the moon and the earth, the geographical latitude of the locality of the observer, and weather conditions. The Muslim astronomers formulated many visibility conditions;3 some of these involve as one criterion the brightness of the lunar crescent as a function of the distance d between the moon and sun on the celestial sphere. The celestial arc d forms a right spherical triangle with the elongation 11 between sun and moon and the latitude 1 of the moon, so that d itself is determined by 13 and rl.

I Ca. A.D. 780; see F. Sezgin, Geschichte des arabischen Schrifttums, vol. 6, Astronomie his ca. 430 H. (Leiden, 1978), pp. 124-27.

2 Idem, Geschichte des arahischen Schrifttums, vol. 5, Mathematik his ca. 430 H. (Leiden, 1974), p. 12.

3 Cf. D. A. King, "Some Early Islamic Tables for Determining Lunar Crescent Visibility," in D. A. King and G. Saliba, eds., From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval East in Honor of E. S. Kennedy (New York, 1987), pp. 185-225.

[JNES 47 no. 2 (1988)] @ 1988 by The University of Chicago. All rights reserved. 0022-2968/88/ 4702-0002$51.00.

95

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Page 3: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

96 JOURNAL OF NEAR EASTERN STUDIES

Kennedy published and analyzed a double argument lunar visibility table,4 which is attributed to Yacqib ibn Tariq and which I have transcribed here as table 1 (see below). Table 1 is contained in the Zij of Jamal al-Din al-BaghdadT in the Paris manuscript Bibliotheque Nationale, Fonds Arabe 2468.5 The table provides for each integer degree of elongation fl (between 0' and 14') between sun and moon and for each interval of ten minutes of lunar latitude 3 (between 0 10' and 5?) a value L(1, 13), called "arc of light" in the heading.

In the accompanying Arabic text published by Kennedy as "Appendix A," the following is stated: if P = 0, the "arc of light" of the moon is i; if 3 # 0, the "arc of light" is L([5/6]fl, 3). The same information is given in a "note on the operation for crescent visibility according to the doctrine of Yacqutb ibn Tariq" in another manu- script also published by Kennedy.6 There it is implied that the numbers in the table are expressed in "digits of light." In Appendix A7 it is stated that the crescent is visible if its digits of light are at least 53/60. The entries in the table must therefore be minutes (i.e., sixtieths) of digits.8

The Arabic texts published by Kennedy do not explain how table I was computed. Because the "arc of light" usually refers to the arc on the celestial sphere between the sun and the moon, Kennedy naturally assumed that the entries in table I represent the distance d on the celestial sphere between the sun and moon, as a function of rl and 13. This interpretation can be supported by means of the following argument.9 The plane through the centers of earth, sun, and moon intersects the moon in a great circle, and the arc of this circle that appears illuminated to an observer on earth is equal to the distance between the moon and the sun on the celestial sphere. Thus d can be used as a measure of the size (and hence the brilliance) of the crescent.

Since both rl and 1 are small, d may be approximated by \/(r12+p2). Kennedy showed that reasonable agreement with the tabulated values is obtained if we assume that Yacqiib used the definition

L(i, p) = c /(12+p2) cd

with c = 60/\(142+52).

The fact that the entries L(r, 13) are "minutes of digits" causes a problem here. In the above-mentioned definition, the term V/(fl2+3p2) approximates the distance d in

4 E. S. Kennedy, "The Lunar Visibility Theory of Yacqub ibn Tariq," JNES 27 (1968): 126-32, re- printed in Studies in the Islamic Exact Sciences by E.S. Kennedy, Colleagues and Former Students (Beirut, 1983).

5 Idem, "A Survey of Islamic Astronomical Ta- bles," Transactions of the American Philosophical Society, vol. 46, pt. 2 (Philadelphia, 1956).

6 Oxford, Bodleian Library, Marsh 663; idem, "Lunar Visibility Theory"; it appears as Appendix B.

7 Ibid. 8 This procedure may be compared with a method

attributed to the ninth-century astronomer Sind ibn CAll. I give a partial translation of the Arabic text quoted in Sezgin, Geschichte des arabischen Schrift-

tums, vol. 6, p. 138:

Knowledge of the crescent and the light in it, by Sind ibn CAll, he said ... multiply the latitude of the moon by itself, and the distance between the two luminaries on the ecliptic (i.e. the elongation) also by itself, then add them and take the square root of the sum. Multiply this by 4 minutes, then one obtains the minutes of light in the moon. If this is between 40' and 50', it will be seen. If this is not so, and the minutes are less than 40, and it has southern latitude, it will not be seen ....

9 Kennedy, "Lunar Visibility Theory," p. 128.

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Page 4: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

YACQUB IBN TARIQ AND LUNAR VISIBILITY 97

TABLE I

A TABLE FOR THE ARC OF LIGHT, ACCORDING TO IBN TARIQ

1= 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0;10 04 08 12 15 19 23 27 31 35 39 43 47 51 55 0;20 04 08 12 15 19 23 27 31 35 39 43 47 51 55

0;30 04 08 12 15 19 23 27 31 35 39 43 47 51 55 0;40 04 08 12 15 19 23 27 31 35 39 43 47 51 55 0;50 04 08 12 15 19 23 27 32 36 40 44 47 51 55

1;00 04 09 13 16 20 24 28 32 36 40 44 47 52 56 1;10 05 09 13 16 20 24 28 32 36 40 44 48 52 56 1;20 05 09 14 16 21 24 28 32 36 40 44 48 52 56

1;30 06 10 14 17 21 24 28 33 37 40 44 48 52 56 1;40 06 10 15 17 22 25 29 33 37 40 44 48 52 56 1;50 07 11 15 17 22 25 29 33 37 41 45 48 53 56

2;00 07 12 16 17 22 25 29 33 37 41 45 48 53 56 2;10 08 12 16 18 23 26 29 33 37 41 45 49 53 56 2;20 09 13 16 18 23 26 30 34 38 41 45 49 53 57 2;30 09 13 16 18 23 26 30 34 38 42 46 49 53 57 2;40 10 14 17 19 24 26 30 34 38 42 46 49 53 57 2;50 11 14 17 19 24 26 30 34 39 42 46 49 53 57

3;00 12 15 17 19 24 27 31 34 39 42 46 49 53 57 3;10 12 15 18 20 25 27 31 35 39 42 46 49 53 57 3;20 13 16 18 20 25 27 31 35 39 43 46 49 53 57 3;30 14 16 18 21 25 27 31 35 39 43 47 49 53 57 3;40 15 17 19 22 25 28 32 35 40 43 47 50 54 57 3;50 16 17 19 22 25 28 32 36 40 43 47 50 54 57

4;00 16 18 20 23 26 28 32 36 40 44 47 50 54 57 4;10 16 18 21 23 26 29 32 36 40 44 47 50 54 57 4;20 176 18 21 243 26 29 3 37 410 44 47 50 55 58 4;30 18 19 21 24 27 29 33 37 41 44 48 51 55 58 4;40 19 20 21 25 27 30 33 38 41 45 48 52 56 59 4;50 19 21 22 25 28 30 34 38 41 45 48 52 57 60 5;00 20 22 23 25 28 30 34 38 41 45 49 52 57 60

degrees, and the factor 60/ /221 serves to convert degrees into "minutes of digits." This means that one digit has to correspond to the improbable value of /221 degrees of arc. As Kennedy remarks, it would be more plausible to let one digit correspond to

1225 = 15 degrees of arc, so that the full moon has the size of 12 digits, as in the eclipse theory in Ptolemy's Almagest.'o If 1 digit is 15?, we have

c-4, so that the

entries in the table must always be at least four times the elongation i1 in degrees.

10 If one digit is 15 degrees, one degree corresponds to 4 minutes (of a digit) as in the procedure of Sind

ibn CAli in n. 7, above.

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Page 5: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

98 JOURNAL OF NEAR EASTERN STUDIES

South

M A

East ,

X E,0D West

North

FIG. 1.

However, for r1>4? and small values of 1, the entries in table I are less than 411. Thus, if we assume that 1 digit is 15', we can no longer maintain that the entries in the table represent the distance d.

III. AL-BATTANI ON THE SIZE AND SHAPE OF THE CRESCENT

Chapter 41 of the ZTj of the famous astronomer Al-Battanil" contains a passage suggesting a curious solution to this problem.12 The passage deals with the determina- tion of the shape of the lunar crescent, its inclination with respect to the ecliptic, and the number of its "digits of light." Al-Battdni first states that the number of "digits of light" is Y15 of the distance between the two luminaries in degrees. He then describes a procedure, which I summarize as follows.

Draw a circle of arbitrary size with center E. The circle is called "circle of the sun." Divide the circle into four quadrants by means of two mutually perpendicular diameters (fig. 1, above)." At the endpoints of the diameters write the four directions "North" (C), "East" (B), "South" (A), and "West" (D). Divide the four quadrants into 90 equal parts. From B and D, mark off arcs BK, DH equal to the latitude 3 of the moon (exaggerated in fig. 1), and in the direction of the latitude (towards the north in fig. 1). Join KH. From A and C, mark off towards the east two arcs AM and CS equal to the elongation 11.

Join MS; let MS and KH intersect at F. Draw the "circle of the moon" with center F and with the same radius as the circle of the sun. Let the two circles intersect at P and Q, and let FE meet the circles at G and T as in figure 1.

It is then stated that the shape of the crescent is PTQG and that GTis the "amount of digits of light" of the moon.14 Note that in general GT is different from d/ 15, the quantity of which was declared to be the "digits of light" in the beginning of the passage. Al-BattdnT says that the inclination of the crescent with respect to the ecliptic

II Sezgin, Geschichte des arabischen Schrifttums, vol. 6, pp. 182-87.

12 C. Nallino, Al-BattanT sive Albatenii opus astronomicum, 3 vols. (Milan, 1899-1907), vol. 3,

pp. 134:16-136:16; vol. I, pp. 89:12-90:26. 13 Notation as in ibid., vol. 1, p. 90. 14 Ibid., vol. 3, p. 136:4.

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Page 6: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

YACQUB IBN TARIQ AND LUNAR VISIBILITY 99

is determined by arcs AP and CQ (it is likely that the average of the two arcs is supposed to measure the inclination).

Al-Battani claims that the procedure is valid for all rl and 3, but it is clear that PTQG resembles the real shape of the moon only if rl is much less than 90O.

IV. ANALYSIS OF AL-BATTANT'S PROCEDURE

In figure 1, BD obviously represents the ecliptic. Points B and D are labeled "East" and "West" for easy reference only.

The crux of the procedure is the determination of segment GT, the width of the crescent. We now determine how GT depends on rl, 1, and d.

Call X the point of intersection of BD and MS. I use the standard notation Sin(x) (with capital) for the sine of an arc of a circle with radius R. Since Sin(x) = R sin(x), we obtain the ordinary sine for R= 1.

Putting ED = R, we have FX = Sin(j), XE = Sin(f), so that

GT= FE = J(Sin2(B)+Sin2(n)). [1]

Note that R is the radius of the full moon.

[1] indicates how the width G T may be computed if 1 and 3 are known, but it is likely that the width was considered to be a function of d. Putting P=0, rl=d in [1], we obtain as a definition of the width in terms of d:

GT = Sin(d).

Thus, Al-Battdni's procedure is based on the relation

sin2(d)= sin2(]) +sin2(G). [2]

Since d, 3P, and rl form a right spherical triangle, we have in fact

cos(d)= cos(P)cos(i). [3]

which is equivalent to

sin2(d) =sin2(11) +sin2( f3) - sin2(f)sin2(f). [4]

If 1 is small (less than 5?) and rl is not too large (say, under 25?), the last term of [4] is small compared to the others, so that [2] is a reasonable approximation.15

The approximation can be explained in geometric terms as follows. Consider a right spherical triangle ABC with sides a = BC, b - CA, and hypotenuse c - AB, and a plane triangle A'B'C' with sides sin(a) = B'C', sin(b) = C'A', and sin(c) = A'B'. If a is small and b is not too large, then we have approximately'6

/ C'~ / C=90, ZA ~ ZA', Z B / ZB'.

In the case of the procedure described by Al-Battani, the principle is applied to the spherical triangle formed by r, 3, and d. The right-angled triangle with sides sin(g),

15 Note, however, that the d2= 2 p-r2 produces in these cases better approximations for d as function of 3 and il.

16 ZC' -- 90 would imply sin(A') = sin(a)/sin(c) and sin(B')= sin(b)/sin(c), so that ZA'= Z A, Z B'= Z B.

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Page 7: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

100 JOURNAL OF NEAR EASTERN STUDIES

sin(1), and sin(d) corresponds to triangle EXF in figure 1. Formula [2] now follows from the Pythagorean theorem in this right triangle. Since the angle between arcs d and 3 on the celestial sphere is approximately equal to angle EFX in figure 1, the inclination of the moon with respect to the ecliptic is approximately equal to the acute angle formed by AC and PQ, which is measured by the average of arcs AP and CQ.

The close relationship between the plane triangle EXF and the spherical triangle formed by the arcs q, 3, and d is confirmed by Al-Battdni's use of the terms "circle of the sun" and "circle of the moon" for the circles with centers E and F respectively. This indicates a special connection between the sun and E and between the moon and F.'7

V. A NEW INTERPRETATION OF YACQUB IBN TARIQ'S TABLE

Since Al-Battani designates GT=Sin(d) as "the amount of digits" of the moon, it is of interest to investigate whether Yacqub ibn Tariq's table 1 could possibly display numbers Sin(d), tabulated in minutes of digits, for a suitable base R.

First we determine a value for R. The upper row of table 1 suggests that Sin(d)/d tends to 4 if d (expressed in degrees) tends to zero. Since R sin(d)/d (with d in degrees) tends to R n / 180, it follows that R must be 720/ n minutes of digits, that is to say

R= 12/ n digits.

The last expression for R suggests that we multiply by n to obtain

nRR=12 digits.

Since inR is half the circumference of the circle, it follows that 12 digits correspond to

180?, so that 1 digit corresponds to 15? as above. Thus, the value R=12/n makes sense if we measure arcs and straight lines in the same units of length (digits, degrees, minutes).18

Since R=12/ n digits, we have R= (180/n)o? = (10800/ n)' = 3437.7' 3438'. Sine tables with the base R =3438' are of (pre-Islamic) Indian origin and were known in the early Islamic tradition, although no such tables survive in the Arabic sources. The name of Yacqib ibn T~riq is associated with such a table." Hence there is historical justification for the definition

L(q, P)=Sin(d), with R =3438', and L(q, 3) expressed in "minutes of digits," where 1 digit = 150.

17 It is conceivable that the method somehow originated in the theory of eclipses. For another geometrical construction of the lunar crescent, see the MahdbhaskarTya of Bhaskara I (sixth century A.D.; see K. S. Shukla, Bhdskara I and His Works, pt. 2, Mahdbhdskariya [Lucknow, 1960], pp. 187-88).

18 We note that also in Ptolemy's eclipse theory,

the (linear) digit is the length of a straight segment. However, if we assume R= 12/n digits, the full moon has a width of 24/n digits, not 12 digits as in Ptolemy's theory.

19 Pingree, "The Fragments of the Works of Yacqiib ibn Tariq," JNES 27 (1968): 122.

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Page 8: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

YACQUJB IBN TARIQ AND LUNAR VISIBILITY 101

I have investigated by means of computer calculations whether such a definition is consistent with the values in the table. The following definitions of L(rl, 3) were compared with the table.

1. L(q, P) = (60/ /221)\(1q2 + 32) (Kennedy's approximation); 2. L(r, f) = Sin(d), computed according to the approximation [2]; 3. L(r, 3) = Sin(d), computed according to the correct formula [3].

For each definition, I computed the errors formed by subtracting the computed value L(r, 3) from the entry in the table and then the mean error pt, the (absolute) maximal error m, and the square root o of the mean value of the squares of the errors. The results are as follows:

It m G Def. 1 -0.47 2.0 0.80 Def. 2 -0.02 1.9 0.65 Def. 3 +0.01 2.1 0.67

Thus the second and third definitions are in much better agreement with the table than the first definition. Because these definitions are consistent with a historically plausible concept of "digit," we conclude that Yacqub ibn Tariq's table contained values Sin(d) as indicated, probably obtained by means of [2]. Table 2 displays recomputed values by means of [2]. Over 70 percent of the values in table 1 can be obtained by either rounding or truncating the values in table 2. We can safely assume that both procedures were used indiscriminately because the asymmetry in table 1 (for example, L(3', 4)-= 19, L(4', 3')=20, etc.) suggests that the calculations were made in a sloppy way. The approximation in [2] is a very reasonable one because the maximum difference between the values produced by [2] and [3] is only 0.2 minutes of a digit.

The title "table of the arc of light" for table 1 now appears to be misleading. Someone must have thought that the arc of light and the quantity in the table (the width of the crescent) are the same because they are both expressed in digits. The same confusion is reflected in the instructions in the accompanying text and also in the Zij of Al-BattanT.

VI. HISTORICAL CONCLUSIONS

The fact that Yacquib tabulated the width of the crescent in a way which cor- responds to the geometrical procedure advocated by Al-Battani, implies that the entire procedure in sec. IV (above) probably originates with Yacquib or that Yacqib and Al-BattanT depend on some common source.20 The most important argument in support of such a dependence is the fundamental implausibility of the value R sin(d)

20 The width of the lunar crescent is considered to be Sin (11) in the Brahmasphutasiddhdnta, so that the general idea that the width is a sine is of Indian origin; see Pingree, "History of Mathematical Astronomy in India," in C. G. Gillispie, ed., Dic-

tionarv of Scientific Biography, vol. 15 (New York, 1980), pp. 533-633. 1 know of no Indian text in which the value Sin(d) is used for the width of the crescent.

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Page 9: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

102 JOURNAL OF NEAR EASTERN STUDIES

TABLE 2

RECOMPUTED VALUES L(T,P)=(3438 / 15) /sin2(T) + sin2(p3)

1=l 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0;10 04.1 08.0 12.0 16.0 20.0 24.0 27.9 31.9 35.9 39.8 43.7 47.7 51.6 55.5 0;20 04.2 08.1 12.1 16.0 20.0 24.0 28.0 31.9 35.9 39.8 43.8 47.7 51.6 55.5 0;30 04.5 08.2 12.2 16.1 20.1 24.0 28.0 32.0 35.9 39.9 43.8 47.7 51.6 55.5 0;40 04.8 08.4 12.3 16.2 20.2 24.1 28.1 32.0 36.0 39.9 43.8 47.7 51.6 55.5 0;50 05.2 08.7 12.4 16.3 20.3 24.2 28.1 32.1 36.0 39.9 43.9 47.8 51.7 55.5

1;00 05.7 08.9 12.6 16.5 20.4 24.3 28.2 32.1 36.1 40.0 43.9 47.8 51.7 55.6 1;10 06.1 09.3 12.9 16.7 20.5 24.4 28.3 32.2 36.2 40.1 44.0 47.9 51.8 55.6 1;20 06.7 09.6 13.1 16.9 20.7 24.5 28.4 32.3 36.2 40.2 44.1 48.0 51.8 55.7 1;30 07.2 10.0 13.4 17.1 20.9 24.7 28.6 32.5 36.4 40.2 44.1 48.0 51.9 55.8 1;40 07.8 10.4 13.7 17.3 21.1 24.9 28.7 32.6 36.5 40.4 44.2 48.1 52.0 55.8 1;50 08.4 10.9 14.1 17.6 21.3 25.1 28.9 32.7 36.6 40.5 44.3 48.2 52.1 55.9

2;00 08.9 11.3 14.4 17.9 21.5 25.3 29.1 32.9 36.7 40.6 44.5 48.3 52.2 56.0 2;10 09.5 11.8 14.8 18.2 21.8 25.5 29.2 33.1 36.9 40.7 44.6 48.4 52.3 56.1 2;20 10.2 12.3 15.2 18.5 22.0 25.7 29.4 33.2 37.0 40.9 44.7 48.6 52.4 56.2 2;30 10.8 12.8 15.6 18.9 22.3 26.0 29.7 33.4 37.2 41.0 44.9 48.7 52.5 56.3 2;40 11.4 13.3 16.0 19.2 22.6 26.2 29.9 33.6 37.4 41.2 45.0 48.8 52.6 56.5 2;50 12.0 13.9 16.5 19.6 23.0 26.5 30.1 33.9 37.6 41.4 45.2 49.0 52.8 56.6

3;00 12.6 14.4 17.0 20.0 23.3 26.8 30.4 34.1 37.8 41.6 45.3 49.1 52.9 56.7 3;10 13.3 15.0 17.4 20.4 23.7 27.1 30.7 34.3 38.0 41.8 45.5 49.3 53.1 56.9 3;20 13.9 15.5 17.9 20.8 24.0 27.4 30.9 34.6 38.3 42.0 45.7 49.5 53.3 57.0 3;30 14.6 16.1 18.4 21.2 24.4 27.7 31.2 34.8 38.5 42.2 45.9 49.7 53.4 57.2 3;40 15.2 16.7 18.9 21.7 24.8 28.1 31.5 35.1 38.7 42.4 46.1 49.9 53.6 57.4 3;50 15.8 17.3 19.5 22.1 25.2 28.4 31.9 35.4 39.0 42.6 46.3 50.1 53.8 57.5

4;00 16.5 17.9 20.0 22.6 25.6 28.8 32.2 35.7 39.3 42.9 46.6 50.3 54.0 57.7 4;10 17.1 18.5 20.5 23.1 26.0 29.2 32.5 36.0 39.5 43.1 46.8 50.5 54.2 57.9 4;20 17.8 19.1 21.1 23.6 26.4 29.6 32.9 36.3 39.8 43.4 47.0 50.7 54.4 58.1 4;30 18.4 19.7 21.6 24.1 26.9 30.0 33.2 36.6 40.1 43.7 47.3 50.9 54.6 58.3 4;40 19.1 20.3 22.2 24.6 27.3 30.4 33.6 36.9 40.4 44.0 47.5 51.2 54.8 58.5 4;50 19.7 20.9 22.7 25.1 27.8 30.8 34.0 37.3 40.7 44.2 47.8 51.4 55.1 58.7 5;00 20.4 21.5 23.3 25.6 28.3 31.2 34.3 37.6 41.0 44.5 48.1 51.7 55.3 58.9

for the width of the crescent. The actual width of the crescent is the orthogonal projection R(1-cos(d)) of the illuminated arc d mentioned in the introduction.21 For small d, the value R sin(d) of Yacqilb and Al-Battdni is much too large.

Thus table 1 is a rather sophisticated piece of very early Islamic mathematics with little relation to physical reality. It is clear that table 1 was computed first and that the accompanying visibility criteria were formulated afterwards and possibly modified in the light of experience.22

21 The width is stated to be R(1-Cos (11)) in the Mahdbhdskarfya of Bhdskara I; see Shukla, Bhdskara land His Works, pp. 187-88.

22 This may explain the curious factor 5/6 in the texts in Appendixes A and B in Kennedy, "Lunar Visibility Theory."

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Page 10: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

YACQUB IBN TARIQ AND LUNAR VISIBILITY 103

Since the geometrical procedure described by Al-Battani can now be dated back to the very beginning of the Islamic scientific tradition in the eighth century, the following two aspects of the procedure must have been known at that time:

1. If the three arcs a, b, and c form a right spherical triangle, the plane triangle with sides sin(a), sin(b), and sin(c) has approximately the same angles as the spherical triangle, provided that a is small and b is not too large. Traces of this method can also be found in (pre-Islamic) Indian astronomy;23 thus, it is likely that the method was transmitted from India to the Islamic world.

2. Constructions involving sines by means of a circle divided into four quadrants by points designated by the cardinal directions and with a circumference divided into 4 times 90 degrees. Here it should be noted that another construction in Al-Battani's Z-j, namely the well-known approximate construction of the direction of Mecca24 is of a similar kind. Hence that construction may also date back to the eighth century.

VII. ANOTHER TABLE OF THE "LIGHT OF THE MOON"

The Dublin manuscript (Chester Beatty 4076) of the Zij of Athir al-Din al-Abhari, a thirteenth-century astronomer and mathematician,25 contains on its last leaf a table of the "light of the moon," similar to the table of Yacquib ibn Tariq. This table, which is transcribed below as table 3, is said to be computed by "Muhammad al- . . ."; unfortunately, the second part of the name is not legible. Table 3 contains for all integer degrees of elongation q from 1 to 30' and for all multiples of 10' of latitude 3 (from 10' to 2'30') a value L(q, P3). It is likely that the continuation of table 3 for latitudes from 2?40' to 5' was once contained in a part of the manuscript which is now lost.

It is immediately evident that table 3 consists of two essentially different parts. 1. The part for elongations from 1 to 14', which is very similar to the correspond-

ing part of table 1; computer calculations show that the agreement with the formula

L(q, 3) = Sin(d) is almost as good as in the case of table 1

(t=-0.07, c-=0.62 for the

second definition). It is likely that this part of the table also dates back to the eighth century. However, the italicized values in table 3, which are different from those in table 1, show that table 1 and the first part of table 3 are independent.

2. The part of table 3 for elongations of 15' or more, which was computed by means of a different method (if by any method at all). Note that for q1 15' We have L(q, 10')= 4r in table 3, whereas we should have L(q, 10')<4rq according to Yacqfib's method. Many values in this part of table 3 satisfy L(, 13) 4 + 3P. It seems that the compiler of table 3 took the part for ql 14' from an earlier source, in which the method of computation was not explained. He probably guessed the values for rq> 14 .

23 Pingree, "History of Mathematical Astronomy in India," p. 562, formulas (a) and (V.32).

24 In chap. 56, Nallino, Al-Battdnr, vol. 3, pp. 206- 7; and translation, ibid., vol. 1, pp. 136-37. See also King, "The Earliest Islamic Mathematical Methods

and Tables for Finding the Direction of Mecca," Zeitschrift fiir Geschichte der arabisch-islamischen Wissenschaften 3 (1986): 103-7.

25 Kennedy, "Survey of Islamic Astronomical Tables," p. 133, no. 56.

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Page 11: New Light on the Lunar Crescent Visibility Table of Yaʿqūb ibn Ṭāriq

104 JOURNAL OF NEAR EASTERN STUDIES

TABLE 3

A TABLE BY MEANS OF WHICH THE LIGHT OF THE MOON CAN BE FOUND FROM

THE ABSOLUTE DISTANCE UNDER THE LATITUDE OF THE

MOON, COMPUTED BY MUHAMMAD...

1= 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0;10 04 08 12 16 20 24 28 31 35 39 43 47 51 55 0;20 04 08 12 16 20 24 28 31 36 39 43 47 51 55 0;30 04 08 12 16 20 24 28 31 36 39 43 47 51 55 0;40 04 08 12 16 20 24 28 32 36 40 43 47 51 55 0;50 05 09 13 16 21 25 28 32 36 40 43 47 51 55

1;00 05 09 13 16 21 25 29 32 36 40 44 47 52 56 1;10 05 09 13 16 21 25 29 32 36 40 44 48 52 56 1;20 06 10 14 16 21 25 29 33 36 40 44 48 52 56 1;30 06 10 14 17 21 25 29 33 36 40 44 48 52 56 1;40 07 11 15 17 22 25 29 34 37 41 44 48 52 56 1;50 07 11 15 17 22 25 29 34 37 41 44 48 52 57

2;00 08 12 16 17 22 26 30 34 38 41 45 49 53 57 2;10 08 12 16 18 23 26 30 34 38 41 45 49 53 57 2;20 09 13 16 18 23 26 30 34 38 41 45 49 53 57 2;30 10 13 17 18 24 26 30 34 38 41 45 49 53 57

l= 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0;I0 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 0;20 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 0;30 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 0;40 60 64 69 72 76 80 84 88 92 96 100 104 108 112 116 120 0;50 60 64 69 72 76 80 84 89 92 96 100 104 108 112 116 120

1;00 60 64 69 73 77 81 84 89 93 96 101 104 108 113 116 120 1;10 61 65 69 73 77 81 85 89 93 97 101 104 109 113 117 121 1;20 61 65 69 73 77 81 85 89 93 97 101 104 109 113 117 121 1;30 61 65 69 73 77 81 85 89 93 97 101 104 109 113 117 121 1;40 61 65 69 74 78 81 85 90 93 98 102 104 109 113 117 121 1;50 61 65 69 74 78 82 86 90 94 98 102 104 110 114 118 121

2;00 61 65 69 74 78 82 86 90 94 98 102 104 110 114 118 122 2;10 61 65 70 74 78 82 86 90 94 98 102 105 110 114 118 122 2;20 61 66 70 75 79 83 87 91 95 99 103 105 110 115 119 122 2;30 61 66 70 75 79 83 87 91 95 99 103 105 110 115 119 122

NOTE: the entries for equal latitudes are in columns in the manuscript; I have rearranged table 3 in order to facilitate comparisons with tables I and 2.

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