The Crescent Visibility Table in Al-Khwiirizrni's Zij *

by E. s. KENNEDY and MARDIROS JANJANIAN**

Introduction Muhammad ibn Miis2 al-Khwarizmi (fl. 830, cf. [7l***, p. 463; [9], p. lo), one of the many scientists who were attracted to the Abbasid court at Baghdad, compiled a set of astronomical tables. This zij has not come down to us in its original form, but exists only in several copies of a Latin translation of a revised version. Nevertheless it is of great importance in the history of medieval science because, alone among all the available Islamic Zjes, it is based primarily upon Indian rather than Ptolemaic astronomical theory.

The Muslim calendar is a pure lunar one, and the beginning of each month is determined by the sighting of the new crescent. Hence most zrjes give methods for determining whether or not the new moon will be visible in the western sky after sunset on a clear evening following a sun-moon conjunction. The zij of al-Khw&zmi is no exception in this respect. Chapter 22 explains how to use for this purpose a certain numer- ical table, also given in the text. There is, however, no indication in the zij as to how the table was computed. This paper investigates the question, and without giving a completely satisfactory answer, demonstrates that the table was computed on the basis of Indian visibility theory, but to be used neither in India nor at Baghdad, but at a latitude valid for northern Spain.

The table is found on p. 168 of [6], the published version of the Latin text. There are three entries for each zodiacal sign, i. e., one for each decan (or ten degrees) along the ecliptic. For checking purposes we take the midpoint of each decan, 1 = 5", 15", 25", . . ., 355", as values of the

* A study supported by the National Science Foundation, Washington, D.C. ** American University of Beirut. Lebanon. *** Numbers in square brackets refer to works listed in the bibliography at the end of

the paper.

Centaurus 1965: vol. 1 1 : no. 2: pp. 13-18

6 CENTAURUS. VOL. n

74 E. S. Kennedy and Mardiros Janjanian

argument. The entries are given to degrees and minutes. They are re- produced in the second column of our Table 1, except that the minutes have been converted into decimal fractions, and since the table function is symmetrical with respect to 180°, it suffices to display the entries for the first half of the table only. The data have been plotted in Figure 1 and appear as the set of isolated small circles on the graph.

The text explains that they are to be used as follows: For the evening in question, calculate Am and I,, the true longitudes of the moon and sun respectively. Call T(&) the function shown in the table. Since con- junction has recently occurred, d l = A,,, - 2, > 0. If d l > T(ls) the crescent will be visible, otherwke not. The rule in the text also takes the value of the lunar latitude into consideration, but since it cannot have affected the dculation of the table, we will assume it to be zero.

Fig. 2

The Crescent Visibility Table in AI-Khwcirizmi's Zq 75

The general rationale of the procedure can be understood from Figure 2, which shows the western part of the celestial sphere at a time when the sun has already set and the moon, having passed the sun in longitude, has just arrived at the horizon. If twilight has sufficiently advanced, that is, if it is sufficiently dark before moonset, the crescent will be seen before it goes under. The table sets a limiting value to the Ail, the ecliptic distance between moon and sun. It is natural that the critical elongation should be a function of the solar (or lunar longitude) because the ecliptic passes the horizon at a variable rate, which depends also on the latitude of the locality, cp. The problem is to determine the precise nature of the function T(il), together with the values of any parameters embedded in it.

According to the Almagest, a celestial object will be visible at dawn or twilight at any time when it is above the horizon and when the sun's arc of depression below the horizon equals or exceeds a certain constant which depends on the object (cf. [5], and [3], pp. 136-149). It has previ- ously been remarked (cf. [8], pp. 4244) that the Khw~rizrni visibility table is not based upon this approach of Ptolemy's. Having verified this, it seemed reasonable to us to investigate whether or not the table could be fitted to Indian theory.

The latter prescribes ([l], p. 124; [2], IX,5 and IX,1) that the time elapsed between the setting of the sun and the object in question shall equal or exceed a constant which depends on the particular celestial body involved. In the case of the moon the difference in setting is twelve degrees of daily rotation, where 360" corresponds to twenty-four hours.

Referring again to Figure 2, the critical elongation is found by pro- jecting from the position of the s u n on the ecliptic over to the celestial equator along the horizon trace, thence laying off twelve degrees along the equator as indicated, h a l l y by projecting the end-point back to the ecliptic along the horizon.

Trial computations with plausible values of Q, and E, the inclination of the ecliptic, demonstrated that the resulting curve indeed resembled the plotted data. For Q, = 24" the latitude of Ujjah (the Greenwich of ancient Indian astronomy), however, the resulting m e was much too flat. As cp increases, the peak of the curve rises, but not even for Baghdad (Q, = 33") did the calculated results begin to be comparable with the data. The dotted m e on Figure 1 obtained by putting Q, = 42;36" and E = 24" takes on about the right height, and its general shape is close to the m e plotted from the text.

76 E. S. Kennedy and Mardiros Janjanian

However, the dotted m e is skewed slightly to the right, whereas the text function is strictly symmetrical. Indeed, a function calculated on the basis of the Indian theory must be asymmetrical. This follows from the fact that if a pair of solar longitudes is chosen at equal distances from a solstitial point, the pair of spherical quadrilaterals employed to calculate the corresponding A ' s will not be congruent.

Fig. 3

In order to obtain a symmetrical function, yet retain the 12" difference in setting time basic to the Indian scheme, the somewhat artificial ex- pedient illustrated in Figure 3 was adopted. For any chosen A, a pair of equatorial points was chosen 6" above and 6" below its horizontal pro- jection onto the equator. This pair of points was projected back on the ecliptic and calculations made for the resulting A t .

The work involved is a routine application of spherical trigonometry. When computed by hand, however, the determination of each d l is a lengthy and laborious operation. With the assistance of Mi. John Golds- berry of the IBM, the recomputation of the entire table was programmed in the FORTRAN symbolic language for the IBM 1401 electronic com- puter (and later the 1620) at the American University of Beirut. Further- more, the computer was programmed to sweep through a lattice of values for and E, calculating for each ordered couple (9, E ) the sum of the squared deviations between text and computation. In principle, the minimUm value of the resulting matrix is the computed curve which best fits the data.

The results are given in Table I, and the continuous Curve on Figure 1 was plotted by using the numbers in the third column of this table.

The Crescent Visibility Table in AI-Khwcirizmi's Zij

Table I

Recalculation of Khwarizmi's Lunar Ripeness Table

77

for

Lambda 5.0

15.0 25.0 35.0 45.0 55.0 65.0 75.0 85.0 95.0

105.0 115.0 125.0 135.0 145.0 155.0 165.0 175.0

Epsilon = 23.50

Text Calculation 9.433 9.335 9.417 9.306 9.350 9.258 9.317 9.21 8 9.300 9.219 9.350 9.307 9.550 9.531 9.950 9.945

10.617 10.596 11.483 11.518 12.800 12.715 14.250 14.151 15.967 15.747 17.517 17.385 19.183 18.927 20.333 20.232 21.067 21.176 21.283 21.672

Phi = 42.67

Difference -.097 -. 110 -.09 1 -.098 -.080 -.042 -.018 -. 004 -.020

.035 -.084 -.098 -.219 -.131 -.255 -.100

.109

.389

SS = .371

Text and calculations are seen to be quite close, sufficiently so to permit the conclusion that the theory underlying the text is very near that used by the Sanskrit sources. The table cannot have been computed for use in Baghdad or India; it is probably the work of the Spanish Arab, Maslama ibn 4 m a d al-Majriti ([9], p. 76) the redactor of Khw&bni's zij from whose Arabic version the Latin translation was made.

While the above inferences are reasonably safe, it is nevertheless a fact that the function we have exhibited is an approximation only, and that there is a systematic divergence between text and computations which cannot be attributed to random computational errors in the text. Because of this there seems no point yet in trying to deduce whether the E embedded in the text is the standard Indian 24" or one of the more accurate Greek and Muslim values in the vicinity of 23;30". For the same reason, it seems fruitless to speculate extensively about the precise value of g~ involved. Al-Majriti is supposed to have worked at Cordoba,

78 E. S. Kennedy and Mardiros Janjanian

for which the medieval texts report latitudes ranging from 35;O" to 38;50" (cf. [4]). Its actual value is near 38". Toledo, another Spanish medieval center of scientific activity has a latitude close to 40;0", the value reported by most texts.

BIB LIO G RAPH Y 1. Brahmagupta, The KhaOQakhgdyaka, transl. by P. C. Sengupta. Calcutta, 1934. 2. Burgess, Ebenezer, (transl.) The SBrya-Siddhanta, Juurn. Am. Or. SOC., vol. 6 (1860),

pp. 141-498; reprinted Calcutta, 1935. 3. Gandz, S., Obermann. J., and Neugebauer, 0.. The Code of Maimonides, Book 3.

Treatise 8. Sanctification of the New Moon, New Haven. 1956. 4. Haddad, F. I., and Kennedy, E. S., Place Names of Medieval Islam, Geographical

Review, 54 (1964), pp. 439-440. 5. Kennedy, E. S., and Agha, Muhammad, Planetary Visibility Tables in Islamic Astron-

omy, Centaurus, 7 (1960). pp. 134-140. 6. KhwBrizmi. Die astronomischen Tafeln des Muhammad ibn MhsB al-Khwarizmi . . .

herausgeg. von H. Suter, Kopenhagcn, 1914 (= Kgl. Danske Vidensk. Selsk. Skrifter 7.R hist.-6101. Af'd. 111. 1).

7. Nallino, C. A., Raccolta di scritti &ti e incditi. vol. 5. Rome, 1944. 8. Neugebauer, 0. (transl.), The Astronomical Tables of al-KhwilrizmI, Hist. Filos. Skr.

Dan. Vid. Selsk. 4, no. 2. Copenhagen. 1961. 9. Suter. H.. Die Mathematiker und Astronomen der Araber . . ., Abhandlungen zur

Grschichte der mathematischrn Wissenschaften . . . , X Heft, Leipdg. 1900.