Histories & Culture
The Solar Equation in the Zij of Yahya bin abi Mansur

E.S. Kennedy, Cairo


The astronomer named in the title is a figure of considerable interest in the history of medieval science. A convert to Islam from the Caspian region of Iran, his work may be expected to exhibit the mixture of Sasanian, Indian, and Hellen¬istic techniques characteristic of his time, the first half of the ninth century.

Unfortunately, only two of his writings are extant (see [8] ,p. 8), one a short astrological treatise, MS 177 of the Kandilli Observatory library, Istan¬bul. The second is Escorial MS Ar. 927, a collection of astronomical tables of the category known as zijes. Parts of this manuscript are undoubtedly the work of Yahya, but some sections are specifically attributed to other scientists in the zij itself. For previous work on this document the reader may consult items [2], [3], [5], and [9] in the numbered bibliography at the end of the paper.

Our present concern is with the mode of computing the sun's position employed in the original zij. The matter would seem to be settled by a table of the solar equa¬tion on ff. 14r and 15r of the Escorial manuscript. But this evidence is contradicted by a table on f. 236r of Paris (Bibl. Nat.) MS Suppl. Pers. 1488. This book, called the Zij-i Ashrafi was written in Persian by one Muhammad b. Abdallah Sanjar al-Kamili, Sayf-i Munajjim (fl. 1300) in Shiraz, Iran. It is a standard zij, compiled with great care and precision, but in addition to the usual explanations and tables the author states that he will make it possible for the user to calculate planetary positions according to the methods given in all the zijes current in his time. With this in mind he gives planetary mean motions and epoch positions for each of twelve zijes, six of these being no longer extant. Where necessary he also has tables to be used in converting from mean to true longitudes. Hence the Ashrafi zij is a rich mine for the extraction of otherwise unavailable medieval astronomy. What follows is a first essay in exploiting it, but it will be necessary first to set up some apparatus.

Ancient Models for the Solar Motion

In Hellenistic astronomy it was customary to regard the earth as fixed in space, with the sun moving at constant speed about it in a circular orbit, one revolution per annum. The earth was assumed to be slightly displaced from the orbit center, the point on the circle at greatest distance from the earth being called the apogee. The apsidal line joins earth to apogee. This eccentricity imparts a slight annual variation to the angular velocity of the sun as seen from the earth; the sun moves at its slowest when passing through the apogee, conversely its angular velocity is maximum when it reaches the opposite point, the perigee. The upshot of this is that the sun's longitude, λ, defined here for simplicity as the angle between the earth-sun vector and the apsidal line, is

λ =¯λ – e (¯λ)

where ¯λ is the mean longitude. Further, e , the solar 'equation', has a period of 360°, is positive in the first and second quadrants, negative in the third and fourth, and

e( 180°●n) = 0 , n = 0, 1,2,3, ... .

For the eccentric model, the calculation of e is an exercise in plane trigono¬metry, see, e.g. [4] ,

p. 115. The curve of the function resembles a sine wave, except that its maximum occurs somewhat beyond 90°, and hence the, line ¯λ = 90 ° is not an axis of symmetry. Solar positions calculated by this method con¬form closely to the actual situation of the sun.

Indian astronomers used the loss accurate 'method of sines', putting

(1) e(¯λ) = max e • sin ¯λ

A modification of this was the 'method of declinations, which made

(2) e(¯λ) = ( max e • δ(¯λ) ) ∕ ε

where δ (x) is the declination of an ecliptic point having longitude x; and ε = max δ ,the inclination of the ecliptic. This technique has been found described only once, in a treatise of Biruni [4 ], p. 118. It was apparently unknown to the Indians, and is conjectured to be either of Sasanian or early Islamic origin. The solar and lunar equation tables in al-Khwarizmi's zij ([6], p. 95; C7], pp. 132 -¬137) have been computed by the method of declinations. Note that for both ex¬pressions (1) and (2) e is symmetrical about A = 90°.

The Table in the Ashrafi Zij

This bears the title 'Table of the Solar Equation for the Zij of Yahya ibn Mansur'. Entries have been computed to seconds of arc for every degree of the argument from 1° to 360°. They are symmetrical with respect to 90°, hence cannot be based on the eccentric model. Moreover since 2e(30°) ≠ e(90°) = max e, the method of sines is also ruled out.

A computer program using (2) was written and run, recomputing the table under the assumption that the parameter ε = 24°, the standard Indian value, and incorpo¬rating the max e of the text,

1; 59,56°. (As customary, we use commas to sep¬arate sexagesimal digits, the semicolon being the sexagesimal point).

The results came out very close to those of the Ashrafi table; only once did the difference between text and computation exceed a second of arc. The differ¬ences were consistent, however, occurring in the middle of the table, with the text entries, where differing, always greater than the corresponding machine results. This prompted the thought that perhaps a variation in the parameter would produce closer correspondence. To this end the program was rerun with ε = 23;35° a better value than the Indian one and adopted by Battani and Biruni. But now the divergence was greater, except that where differences exist, frequently as much as two seconds, the text is less than the computation. An ε seemed in¬dicated located between the other two trials. The value 23; 51° is such. It is the parameter used by Ptolemy, and has cropped up in a different part of Yahya's zij ([3], p. 24). A third run, with the Ptolemaic parameter inserted, gave about as close correspondence with the source as can be expected.

There were only twenty-four divergences between text and machine, the num¬ber of negative differences equalling the positives. An excerpt appears in the se¬cond column of the table below, chosen at intervals of ten degrees. Wherever the text differs from the computer, the value from the former is shown within paren¬theses above the corresponding machine result.

The solar equation


Excerpts from the solar equation tables attributed to Yahya:













































 1;43, 5





 1; 52 ,18


1;51,24 j









 1;58, 0





 1;59, 56





The solar equation table in the Escorial manuscript has been described in[5] and its entries have been excerpted at intervals of ten degrees in the third and fourth columns of our table above. It is clearly based on the eccentric model. Since the Ashrafi and Escorial tables are so manifestly different, which one is Yahya's?

Speaking for the former is the categorical attribution to him by the Ashrafi au¬thor and the use of the archaic and artificial method of declinations, typical of the astronomers of his generation and nation.

Militating against it is the: maximum equation of 1;59,56°, found nowhere else in the literature. In the zij of Ibn Yunus ([1], p. 57), two observational determinations of the maximum solar equation are reported. These were made by the 'Ashab al-Mumtahan', a group of astronomers with whom Yahya is associated. Observing at Baghdad they obtained 1;59°, the parameter of the Escorial table, and Yahya is named as a participant. At Damascus the result was 1;59,51°, close to the Ashrafi value, but Yahya is not named. So the question remains open.

In any case, the mere existence of the Ashrafi table is worth noting as a cu¬riosity. For whoever calculated it in the first place unwittingly combined two bad choices: from among several available values he took the inaccurate ε of Ptole¬my; at the same time he abandoned Ptolemy's eccentric model in favor of the in¬ferior method of declinations.



[1] Caussin de Perceval, 'Le livre de la grande Table Hakémite . . . ' par Ebn Iounis.. . ., Notices et extraits des mss. de la bibl. nationale .... tome septiéme, Paris, an XII (= 1803/4) de la république.
[2] Kennedy, E.S., `A Survey of Islamic Astronomical Tables' , Transactions of the American Philosophical Society, N.S. 46, Pt. 2, Philadelphia 1956.
[3] Kennedy, E.S., and Faris, Nazim, 'The Solar Eclipse Technique of Yahva b. Abi Mansur, Journal for the History of Astronomy, 1 (1970) , pp.28-30.
[4] Kennedy, E.S., and Muruwwa, Ahmadm Biruni on the Solar Equation' , Journal of Near Eastern Studies, 17 ( 1958) .pp. 112-121.
[5] Salam, Hala, and Kennedy, E.S. , 'Solar and Lunar Tables in Early Is¬lamic Astronomy' . Journal of the American Oriental Society, 8 (1967), pp. 492-497.
[6] Neugebauer, O. , 'The Astronomical Tables of al-Khwarizmi' , Hist. Filos. Skr. Dans. Vid. Selsk. 4, no. 2, Copenhagen, 1962.
[7] Suter, H., 'Die astronomischen Tafeln des Muhamrnad ibn Musa al-¬Khwarizmi ... , Kg1. Danske Vidensk. Skrifter, 7. R. , Hist. og filos. Afd. 3, 1, Copenhagen, 1914.
[8] Suter. H. , 'Die Mathematiker und Astronomen der Araber und ihre Wer¬ke' , Abhandlungen zur Geschichte der mathematischen Wisscnschaften ... X Heft, Leipzig, 1900.
[9] Vernet, J. . "Las 'Tabulae probatae"', Homenaje a Millas-Vallicrosa, vol. 2, Consejo Superior de Investigaciones Cientificas, Barcelona, 1956. p. 510-522.

Published by Franz Steiner Verlag GmbH Wiesbaden, 1977