Planetary latitudes in medieval Islamic astronomy: an analysis of the non-Ptolemaic latitude parameter values in the Maragha and Samarqand astronomical traditions

Abstract

Some variants in the materials related to the planetary latitudes, including computational procedures, underlying parameters, numerical tables, and so on, may be addressed in the corpus of the astronomical tables preserved from the medieval Islamic period (zīj literature), which have already been classified comprehensively by Van Dalen (Current perspectives in the history of science in East Asia. Seoul National University Press, Seoul, pp 316–329, 1999). Of these, the new values obtained for the planetary inclinations and the longitude of their ascending nodes might have something to do with actual observations in the period in question, which are the main concern of this paper. The paper is in the following sections. In the first section, Ptolemy’s latitude models and their reception in Islamic astronomy are briefly reviewed. In the next section, the medieval non-Ptolemaic values for the inclinations and the longitudes of the nodal lines are introduced. The paper ends with the discussion and some concluding remarks. The derivation of the underlying inclination values from the medieval planetary latitude tables and determining the accuracy of the tables are postponed to “Appendix” in the end of the paper.

1 The Ptolemaic latitude models and their reception in medieval Islamic astronomy

In the Ptolemaic latitude theory for the superior planets in the Almagest (Fig. 1a), the eccentric is inclined to the ecliptic at a fixed angle $i_{0}$ which is equal to $2{;}30^{\circ }$ for Saturn, $1{;}30^{\circ }$ for Jupiter, and $1^{\circ }$ for Mars. The centre of the epicycle revolves on the eccentric in the direction of increasing longitude, reaches the northern and southern limits ¹ at opposite locations on the ecliptic, and crosses it at the nodes. The epicycle itself is inclined to the eccentric with its perigee at the same direction as the eccentric, but at a varying angle $i_{1}$. $i_{1}$ is minimum and equal to $i_{0}$ when the centre of the epicycle is at the nodes and reaches its maximum when the centre of the epicycle is located at the limits: $i_{1\max } = 4{;}30^{\circ }$ for Saturn, $2{;}30^{\circ }$ for Jupiter, and $2{;}15^{\circ }$ for Mars. The direction of these inclinations is fixed with respect to the planets’ apsidal lines. $\angle$ ATN shows the longitudinal difference $\omega _{\mathrm{A}}$ between the apogee A and the northern limit N, which is equal to $+50^{\circ }$ for Saturn, $-20^{\circ }$ for Jupiter, and $0^{\circ }$ for Mars. The model in the Handy Tables (Fig. 1b) is as same as in the Almagest with the exception that $i_{1}$ is fixed at the same values in the Almagest, and that $\omega _{\mathrm{A}} = +40^{\circ }$ for Saturn, a value less precise than that in the Almagest. In the Planetary Hypotheses (Fig. 1c), $i_{1}$ is fixed, but equal to $i_{0}$, that is, the epicycle always remains parallel to the ecliptic. The inclinations are equal to $i_{0}$ in the Almagest/Handy Tables, with an improvement in the case of Mars where $i_{0}=i_{1} = 1{;}50^{\circ }$.

Fig. 1

Ptolemaic latitude model for the superior planets in the a Almagest, b Handy Tables, c Planetary Hypotheses

The Ptolemaic model for latitude of the inferior planets is shown in Fig. 2a for Venus and Fig. 2b for Mercury. Ptolemy observed that the distance $\omega _{\mathrm{A}}$ from the apogee to the northern limit of the eccentric is $0^{\circ }$ for Venus and $+180^{\circ }$ for Mercury while they should be $-1^{\circ }$ and $+196^{\circ }$, respectively, at his time. He also observed that the eccentric of the two planets is always inclined to the ecliptic in one direction, Venus to the north and Mercury to the south. This inclination is varied: when the centre of the epicycle is at either of the apsides, it reaches its maximum, $i_{0\max } = +0{;}10^{\circ }$ for Venus and $-0{;}45^{\circ }$ for Mercury, but when the centre of the epicycle is $\pm 90^{\circ }$ from the apsidal line, $i_{0} = 0^{\circ }$. Consequently, Ptolemy defines a conventional position for the nodal line of the eccentric $\pm 90^{\circ }$ from the apsidal line. Next, when the centre of the epicycle is $\pm 90^{\circ }$ from the apsidal line, the diameter passing through the epicyclic apogee and perigee is inclined to the ecliptic in the line of sight at an angle $i_{1}$. This inclination is also varied, $i_{1\max } = 2{;}30^{\circ }$ for Venus and $6{;}15^{\circ }$ for Mercury. As the epicycle moves towards the apsidal line, it decreases until it vanishes at either of the apsides. Then, when the centre of the epicycle is in the apsidal line, the orthogonal to the diameter passing through the epicyclic apogee and perigee is slanted to the ecliptic across the line of sight at an angle $i_{2}$. The slant is also varied, $i_{2\max } = 3{;}30^{\circ }$ for Venus and $7^{\circ }$ for Mercury when the centre of the epicycle is in the apsidal line. As the epicycle moves away from the apsidal line, the slant decreases until it disappears at $\pm 90^{\circ }$ from the apsidal line. As shown in Fig. 2a, b, the directions of the inclination and slant are reversed for Venus and Mercury. In the Handy Tables and Planetary Hypotheses, all three components of the latitude are fixed, and the inclination and the slant of the epicycle are equal: $i_{1}=i_{2} = 3{;}30^{\circ }$ for Venus and $6{;}30^{\circ }$ for Mercury and $i_{0} = 0{;}10^{\circ }$ for both.²

Fig. 2

Ptolemaic latitude model for a Venus, b Mercury

Although Ptolemy’s latitude models in the Handy Tables and Planetary Hypotheses are simpler and more coherent than those in the Almagest, they nevertheless appear to have been of little influence on medieval Islamic astronomers who adhered to the latitude models in the Almagest. They had, however, adopted a few elements of the latitude models in the Handy Tables and Planetary Hypotheses. For instance, the extremal values for the latitude of all the planets in the Handy Tables have been applied to the Mumtaḥan Zīj (see below, note 18); the maximum values for the latitudes of the inferior planets from the Handy Tables/Planetary Hypotheses are also mentioned in al-Battānī’s Ṣābi’ Zīj and al-Farghānī’s Book on astronomy (although both authors do not identify their sources);³ the table for the latitude of Venus from the Handy Tables is mentioned in Ibn Yūnus’ Ḥākimī Zīj (Cairo, d. 1009) (see below). Ibn al-Shāṭir of Damascus (1304–1375/6) discussed how to adapt the latitude theory in the Almagest and Planetary Hypotheses to his own planetary model, in which two additional small circles are incorporated into the epicycle–eccentric mechanism in order that the motions of heavenly objects be produced solely by uniform circular motions.⁴ The equality of the maximum values of the inclination and slant of each inferior planet, as in the Handy Tables and Planetary Hypotheses, was used in al-Kāshī’s novel method of the computation of the planetary latitudes according to the Almagest models (see below).

The complexity of the Almagest latitude models seems to have been of no concern to medieval astronomers, despite the fact that one of Ptolemy’s most notable remarks on simplicity and complexity in astronomical hypotheses in the Almagest is stated in connection with the latitude theory.⁵ This could give an impression that, since the underlying latitude models in the Handy Tables/Planetary Hypotheses are simpler than those in the Almagest, they might be preferable. But instead they followed the Almagest models with some attempt to remedy them in some aspects, as will be mentioned briefly below.

In the observational aspect, the Middle Eastern Islamic astronomers seem also to have possessed some persuasive observational evidence of problems with the planetary latitudes in some specific zījes; nevertheless, such criticisms had been directed to the adopted parameters, numerical tables, and so on, rather than to the models themselves.⁶

In the philosophical aspect, the following are worthy of consideration: in the latitude models in the Almagest (XIII.2),⁷ Ptolemy embedded small circles that were responsible for rotating the endpoints of the diameters of the epicycles in order to account for their variable inclinations. Some spherical versions of the Ptolemaic models were proposed in Islamic astronomy in order to remove the small circles and give a philosophically justified physical model for the latitudinal tilting of the epicycles: Ibn al-Haytham (965–ca.1040) utilized two contacting concentric spheres with different poles whose distance is equal to the maximum inclination or slant of the epicycle, and rotating in opposite directions with the same velocity as the centre of the epicycle of the planet as seen from the earth.⁸ Naṣīr al-Dīn al-Ṭūsī (1201–1274) deployed three concentric interconnected spheres, two of them producing the geometrical device known as the “Ṭūsī couple”, to solve the difficulties arising from Ptolemy’s small circles.⁹

In the mathematical aspect, it merits mentioning that a geometrical spherical version of the latitude models in the Almagest was proposed by Jamshīd Ghiyāth al-Dīn al-Kāshī (1380–1429); although apparently influenced by al-Ṭūsī, it was essentially intended to improve upon the mathematics of the models.¹⁰ As mentioned above, it uses a distinct feature of the Ptolemaic latitude models in both the Handy Tables and Planetary Hypotheses that the extremal values of the inclination and slant of each inferior planet are equal.¹¹ Al-Kāshī also presented two-argument tables for deriving longitude and latitude; dispensing with lengthy computational procedures, a practitioner only needs enter them with the adjusted eccentric and epicyclic anomalies to derive the ecliptical coordinates.¹² Another example is this: in the Almagest models, the slant component of the latitude of Mercury from the tables should be corrected such that when the planet is in the apogeal half of the eccentric, one-tenth of the slant is subtracted, and inversely, when the planet is in the perigean part of the eccentric, the same amount is added. Muḥyī al-Dīn al-Maghribī rectified the procedure,¹³ and Wābkanawī later commented upon al-Maghribī’s method and improved it in the manner that the amount of the slant itself should be multiplied by the absolute value of the cosine of the planet’s eccentric anomaly;¹⁴ over one century later, Kāshī instead proposed a new interpolation function in order to calculate the slant component of the latitude of Mercury, by means of which there is no need to include this correction.¹⁵

Despite all the above-mentioned refinements, the most important alternatives to Ptolemy’s planetary latitude models during the medieval period were Indian models.¹⁶ They were adopted in some zījes of the early ninth century such as al-Khwārizmī’s Sindhind Z īj (ca. 840)¹⁷ and Yaḥyā b. Abī Manṣūr’s Mumtaḥan Zīj (ca. 832),¹⁸ and were later passed to the Western Islamic astronomical tradition (the Maghrib and Andalus) through al-Khwārizmī’s Zīj; e.g., in the Toledan Tables ¹⁹ and Ibn ‘Azzūz al-Qusanṭīnī’s Muwāfiq Zīj (d. 1354).²⁰ The only medieval astronomer who proposed latitude models different from Ptolemy’s is apparently Levi ben Gerson (1288–1344), who supplied computational procedures and numerical tables as well.²¹

Among the equatoria invented in the late Islamic period for finding the ecliptical positions of the heavenly objects, al-Kāshī’s instrument for computing the planetary latitudes may be worth mentioning.²² However, because of the smallness of the inclinations, such a device could probably be in use only for demonstrational purposes.

2 The values for the inclinations and $\omega _{\mathrm{A}}$ in medieval Islamic astronomy

In the late Islamic period, there was a great deal of systematic planetary observations to determine their underlying parameters while in the early Islamic period the investigation was of solar and lunar motions. The new values for planetary inclinations and longitudes of ascending nodes are found in the zījes written in association with the well-known Islamic observatories established at Maragha (ca. 1260–1320 AD), Beijing (ca. the 1270s), and Samarqand (ca. 1430–1449).

2.1 Inclinations

In the Maragha tradition, new values were observed for the inclinations of the epicycles of the inferior planets. These are mentioned nowhere and can only be derived from the latitude tables. Al-Maghribī explains his own observations and computations made at the Maragha observatory for the purpose of determining the solar, lunar, and planetary parameters in a treatise entitled Talkhīṣ al-majisṭī (Compendium of the Almagest). Nevertheless, since the last two books, IX and X, of this treatise are unfortunately missing from its only surviving manuscript, which is in the author’s handwriting (Leiden: Universiteitsbibliotheek, no. Or. 110), we do not know whether and how he derived his values for the inclinations of the inferior planets and $\omega _{\mathrm{A}}$ of the superior planets (see below) from his own observations. According to the list of its contents, book IX is devoted to the planetary retrograde motions and latitudes and book X to the stereographic projection of the celestial sphere onto a plane surface tangent to the north celestial pole, which is used for the drawing of astrolabe plates.²³

The maximum latitude of an inferior planet is due to the inclination of its epicycle and takes place when the planet is at inferior conjunction with the mean sun, i.e. at the true epicyclic anomaly of $180^{\circ }$. We indicate the inclination component of the latitude as $\beta _{\mathrm{icl}}$ ($\alpha$) where $\alpha$ is the true epicyclic anomaly; $\beta _{\max } = \beta _{\mathrm{icl}}$($180^{\circ }$) in Table 1.

Table 1 New values for the inclination of the inferior planets in the Maragha tradition

In the Īlkhānī Zīj, the maximum tabular value for the latitude of Venus is $8{;}40^{\circ }$,²⁴ corresponding to $i_{1\max } \approx 3{;}25^{\circ }$. This is close to the extremal value $8{;}52^{\circ }$ as derived from the Handy Tables and Planetary Hypotheses in which $i_{1\max } = 3{;}30^{\circ }$.²⁵ Nevertheless, the table itself suffers from some difficulties as explained in the “Appendix”, so that it is actually impossible to assign one single value to $i_{1\max }$ from which the table has been computed. It is noteworthy that Ibn Yūnūs in his Ḥākimī Zīj has an extra table for the latitude of Venus where $\beta _{\mathrm{icl}}(0^{\circ }) = 1{;}29^{\circ }$ and $\beta _{\mathrm{icl}}(180^{\circ }) = 8{;}52^{\circ }$.²⁶ With Ibn Yūnus’ non-Ptolemaic $e \approx 1{;}3$ for the eccentricity of Venus (corresponding to the maximum equation of centre $=\,2{;}0{,}30^{\circ }$)²⁷ and $r \approx 43{;}28$ for the radius of the epicycle (corresponding to the maximum epicyclic equation $=\,46{;}25^{\circ }$ at mean distance),²⁸ it can be found that both latitude values result from the rounded $i_{1\max } = 3{;}30^{\circ }$. It seems that Ibn Yūnus obtained this table either directly or via a medium from the Handy Tables. Note that a mixed tradition of the Almagest-Handy Tables for the latitudes of the inferior planets already existed in earlier Islamic works (see note 3 above). The values $3{;}25^{\circ }$ or $3{;}30^{\circ }$ are in better agreement with the correct inclination of the planet than $2{;}30^{\circ }$ in the Almagest and other Islamic zījes.

In al-Maghribī’s Adwār al-anwār,²⁹ the maximum latitude of Venus is $6{;}40^{\circ }$, which corresponds to $i_{1\max } \approx 2{;}37^{\circ }$. In his first zīj, the Tāj al-azyāj, written at Damascus in 1258 before his joining Maragha, he has the value $8{;}30^{\circ }$ for the maximum latitude of Venus,³⁰ which corresponds to $i_{1\max } \approx 3{;}21^{\circ }$. His value for the maximum latitude of Mercury in the Adwār is $4{;}35^{\circ }$, corresponding to $i_{1\max }\approx 7{;}2^{\circ }$.³¹ This is approximately equal to the Ptolemaic maximum value for the slant of Mercury, i.e., $i_{2\max } = 7{;}0^{\circ }$. Since al-Maghribī’s table of the slant is identical to the table in Almagest XIII.5,³² it is straightforwardly deduced that al-Maghribī took the maximum inclination of Mercury equal to its maximum slant, as in the Handy Tables/Planetary Hypotheses, although in the Handy Tables and Planetary Hypotheses Ptolemy takes in the inclination as $6{;}30^{\circ }$, not $7{;}0^{\circ }$. It should be noted that the values near $7{;}0^{\circ }$ are in excellent agreement with the true value of the inclination of the planet.³³ These values are summarized in Table 1.

It appears to have been generally accepted after the Maragha observatory that the value $3{;}30^{\circ }$ for the inclination of Venus was a recent improvement upon the parameters of the Almagest. For instance, Ibn al-Shāṭir refers to it as “what the moderns have amended [in the Almagest]”.³⁴ Also, one may conclude that in each of the two independent observational programs carried out at the Maragha observatory, i.e., by al-Maghribī and by the main staff of the observatory, who were engaged in preparing the Īlkhānī Zīj, it was found that the maximum values of the inclination and slant of one of the inferior planets are equal, although neither maintained this for the other inferior planet. This can be considered a partial rediscovery of Ptolemy’s latitude model in the Handy Tables and Planetary Hypotheses. Over a century after the Maragha observatory, al-Kāshī appears to have achieved the same result that in the new observations made by recent predecessors and modern scholars both the inclination and the slant of Venus and Mercury are, respectively, equal to $3{;}30^{\circ }$ and $7{;}0^{\circ }$.³⁵ He was familiar with both the zījes of the Maragha tradition, so that he was involved in the task of revising the Īlkhānī Zīj, and hence named his own work Zīj-i Khāqānī dar takmīl-i Zīj-i Īlkhānī (Khāqānī Zīj; the improvement of the Īlkhānī Zīj); and in his discussion of the latitude theory, he explicitly refers to al-Maghribī and his Zīj al-kabīr (Great Zīj),³⁶ an alternative name for al-Maghribī’s Adwār al-anwār.³⁷

The tables of the latitude of Venus in Ibn al-Shāṭir’s Jadīd Zīj,³⁸ Ulugh Beg’s Sulṭānī Zīj ³⁹ and al-Kāshī’s Khāqānī Zīj ⁴⁰ are identical to the corresponding table in the Īlkhānī Zīj, as are the tables of the latitude of Mercury aside from the correction for distance in computing the slant.⁴¹ This illustrates the influence that the formal work of the Maragha astronomy exerted on later Middle Eastern astronomers and that the inclinations of the inferior planets from the Maragha observatory appeared trustworthy to them. Perhaps for this reason, the astronomers at the Samarqand observatory turned to measuring the inclinations of the superior planets. The values derived for the inclinations of the superior planets from the critical entries in the northern and southern latitude tables in Ulugh Beg’s Sulṭānī Zīj ⁴² are as listed in Table 2. The critical entries are the latitudes for the true anomaly of $90^{\circ }$ and of $180^{\circ }$; the first is used to determine the inclination $i_{0}$ of the eccentric and the latter to derive the maximum inclination $i_{1\max }$ of the epicycle. The subscripts ‘N’ and ‘S’ denote, respectively, the northern and southern latitudes. Note that our derivation of the inclinations from the extremal latitudes as explained in “Appendix” are based on the new values observed for the eccentricities and the radii of the epicycles of the superior planets at the Samarqand observatory, which are summarized in the “Appendix” (Tables 8, 9). This explains why the values of $i_{0}$ for Saturn and Jupiter are Ptolemy’s while the corresponding latitudes are non-Ptolemaic.

Table 2 New values obtained for the inclinations of the superior planets at the Samarqand observatory

2.2 The values of $\omega _{\mathrm{A}}$

In Ptolemaic astronomy, the apsidal and nodal lines of the planets are sidereally fixed. Indian models have completely different features: the apsidal lines of the planets move in the direction of increasing longitude, but their nodal lines in the retrograde direction, like that of the moon, at unequal rates.⁴³ The majority of (especially, early) Islamic zījes have Ptolemy’s values for $\omega _{\mathrm{A}}$.⁴⁴ The non-Ptolemaic values for $\omega _{\mathrm{A}}$ from the Maragha and Samarqand traditions are listed in Table 3 (the true modern values at the times are indicated in the columns headed Mod.). These values are either explicitly mentioned in the canons to the tables or can be extracted from the tables or columns for “the minutes of proportion”. The minutes of proportion $c_{5}$ is a simple cosine function of the argument of latitude $\omega _{\mathrm{p}} = \kappa + \omega _{\mathrm{A}}$ where $\kappa$ is the true eccentric anomaly of the planet. In the Almagest, the user has to add the values of $\omega _{\mathrm{A}}$ to $\kappa$ and then enter the table with the result. This is also the case with both al-Maghribī’s and Wābkanawī’s zījes, but in the Sulṭānī Zīj the entries are shifted by $- \omega _{\mathrm{A}}$ with respect to the corresponding entries in the Almagest tables. As a result, the procedure is shortened by one step.

Table 3 Difference $\omega _{\mathrm{A}}$ between the longitudes of the apogees and the northern limits of the superior planets in the zījes of the Maragha and Samarqand observatories

3 Discussion and conclusion

Unlike the detailed examples in the Almagest for determining the parameters of the planets’ motions in longitude, Ptolemy did not mention specific observations for planetary latitudes, only what appear to be approximate extreme values, which gave rise to some medieval objections.⁴⁵ Consequently, the medieval astronomers had no access to specialized methods and techniques required for making observations to determine the latitude parameters.⁴⁶ The required observations for the derivation of the inclinations and the values of $\omega _{\mathrm{A}}$ should essentially be made at some specific conditions which are very rarely possible to satisfy: in order to measure the inclinations of the eccentric and the epicycle of a superior planet, it should be observed when it is both close to a limit and at opposition to (or as near as possible to conjunction with) the mean sun. This can take place once in about 30 years for Saturn, 12 years for Jupiter, and 15 or 17 years for Mars. The longitude of the ascending node can be determined when the planet crosses the ecliptic, in this case, when the centre of the epicycle is at a node, whose longitude is thus equal to the mean longitude of the planet.⁴⁷ As mentioned briefly in Sect. 1, early Islamic astronomers employed Indian methods or the ungrounded (if not meaningless) mixed versions of the Hindu and Ptolemaic latitude theories, as well as having difficulties with understanding the latitude models in the Almagest. Among the surviving early Islamic zījes, presumably al-Battānī’s is one of the first works in which the Almagest latitude theory appeared in an understandable and correct abridged form (though, having a minor fault; see above, note 6), i.e., at the turn of the tenth century. These perhaps explain the reason it took a relatively long period until the first attempts were made to measure the inclinations of the planets (at the Maragha observatory in the mid-thirteenth century). Also, for the same reason, the derivation of the latitude parameters from the observational data can be considered the acme of medieval observational astronomy.

Concerning the Eastern branch of Islamic astronomy, a partial improvement upon Ptolemy’s values for the inclinations of the inferior planets in the Almagest was made at the Maragha observatory. This can, however, be considered a rediscovery of Ptolemy’s latitude models of the inferior planets both in the Handy Tables and Planetary Hypotheses, of which the Maragha astronomers were quite probably aware. Of course, al-Maghribī’s value $7{;}2^{\circ }$ for the inclination of Mercury is in excellent agreement with the modern data, and thus clearly better than Ptolemy’s values in the Almagest ($6{;}15^{\circ }$) and in the Handy Tables/Planetary Hypotheses ($6{;}30^{\circ }$). As the contents of al-Maghribī’s Talkhīṣ al-majisṭī reveal, there is no doubt that he had made the actual observations at the Maragha observatory, as well as having the technical and mathematical skills to derive the parameters from the observational data. In the beginning of 1262, the observations were begun in the Maragha observatory (al-Maghribī’s first documented observation in the Talkhīṣ is the lunar eclipse on 7 March 1262) and in April–May 1276, he completed the canons of his last zīj, Adwār al-anwār, and then engaged in preparing its tables.⁴⁸

Ulugh Beg’s planetary parameters (Table 4; “Appendix”, Tables 8, 9) can explicitly indicate systematic observations at Samarqand, although nothing about them appears to have been documented. Unlike the improvement made at the Maragha observatory in the case of the inferior planets, the values the Samarqand observers determined for the inclinations of the superior planets are either equal to or larger than the corresponding values in the Almagest while they should be very close (if not essentially equal) to each other. Accordingly, it may be that the attempts made in the Samarqand observatory resulted in a deterioration in the Almagest ’s underlying values for the inclinations of the superior planets, while these had already been improved significantly in Ptolemy’s other two works. We are not told how and when the Samarqand observers reached their own values for the extremal latitudes of the superior planets; in his Commentary on Zīj of Ulugh Beg, Qūshčī only mentions the maximum and minimum latitudes at the limits, expressing “we found” them, but without giving any more information.⁴⁹ The true values of the maximum latitudes of the three planets at oppositions, the zodiacal signs where they occurred in the period of 1430–49 when the Samarqand observatory was active, and their differences from the extremal values in the Sulṭānī Zīj (Table 2) are given below. It is noteworthy that these errors are comparable with those found in the stellar latitudes in Ulugh Beg’s star catalogue incorporated into his zīj.⁵⁰

About the values of $\omega _{\mathrm{A}}$, it should be noted that observing no new value in the case of the inferior planets cannot be so strange because unlike the superior planets, the directions of the nodal lines of the eccentrics of the inferior planets in Ptolemaic astronomy are an intrinsic convention of the model that they are $\pm 90^{\circ }$ from the apsidal lines. It is not imaginable that an observer could recognize that the two are not perpendicular to each other when the centre of the epicycle is at either of the apsides, but make angles as small as about $1^{\circ }$ and $16^{\circ }$ to the perpendicular. In particular for Mercury, it should be considered that Ptolemy’s longitude of the apogee of the planet, $191^{\circ }$ in 135 AD, is over $-30^{\circ }$ from the true value of about $223^{\circ }$. The same error can also be found in the majority of the Islamic zījes with the exception of those based upon Indian parameters (e.g., al-Khwārizmī’s).⁵¹

Table 4 Longitude of the apogee of the superior planets in the zījes of the Maragha and Samarqand traditions

The two interrelated problems in the values of $\omega _{\mathrm{A}}$ for the superior planets in Table 3 are as follows:

(a) These might initially be interpreted as indicating that the medieval astronomers were aware of the change in the relative directions of the apsidal and nodal lines. Then, an inevitable consequence of the continuous increase in $\omega _{\mathrm{A}}$ with respect to Ptolemy’s values would be that the apsidal and nodal lines of the superior planets do not share the same motions, in contrast to the fundamental assumption in Ptolemaic astronomy that they are sidereally fixed. Nevertheless, such a conclusion was never posed in the Eastern (Middle Eastern) Islamic astronomy, where permitting any secular change in the Ptolemaic constants or fundamental assumptions and hypotheses was considered very unlikely. Accordingly, any new value, variation, or change found in a parameter was undoubtedly attributed to faulty observations, defects and discrepancies in instruments, and/or untrustworthy ancient data. However, a diametrically opposed point of view was adopted in the Western Islamic astronomy, where it was allowed to consider proper motions and/or models to account for the observationally found secular changes.⁵² For instance, about 1075, Ibn al-Zarqālluh found that the solar apogee had the proper motion of $1^{\circ }$ in 279 Julian years in the direction of increasing longitude, which is then added to the constant rate of precession $1^{\circ }/\hbox {66y}$. Several Andalusian astronomers applied this proper motion to the apogees of the planets as well.⁵³ Nevertheless, whether this proper motion was thought to be the case for the nodal lines of the planets, or whether other such proper motions were proposed for them, must await a thorough inspection of the materials related to the planetary latitudes in the Western Islamic zījes.

(b) How were they obtained? These medieval values perhaps were the result of actual observations. Needless to say, the values of $\omega _{\mathrm{A}}$ are dependent upon the values adopted for the longitudes of the apogees. By means of the two, one can also examine to what extent the values obtained for the longitudes of the nodes/limits were accurate.

The longitudes of the apogees of the superior planets in the Maragha and Samarqand astronomical traditions are listed in Table 4.⁵⁴ The values in brackets are the true values computed for the times. We round both the historical and modern values to the nearest $0.5^{\circ }$ which is sufficient for our purpose.

As seen in Table 4, al-Maghribī has more precise values for the longitudes of the apogees of the superior planets than those applied to the Sulṭānī Zīj. As regards Tables 3 and 4, one can deduce that the values obtained by al-Maghribī for the longitudes of the nodes/limits of the three planets are about $+1^{\circ }$, $+5^{\circ }$, $-2.5^{\circ }$ in error, respectively. Wābkanawī’s value for $\omega _{\mathrm{A}}$ of Mars increases the error in the longitude of the nodes/limits of the planet to $+15.5^{\circ }$. The corresponding errors in the Sulṭānī Zīj are $-1.5^{\circ }$, $+3^{\circ }$, and $+3.5^{\circ }$.

As we shall see below, the possibility exists that al-Maghribī derived his values of $\omega _{\mathrm{A}}$ from actual observations as accurate as the others documented in his treatise Talkhīṣ al-majisṭī. Following him, Wābkanawī mentioned in his zīj some observations performed during his 40-year career, which were directed to reconciling theory and observation.⁵⁵ Let us first consider whether and how the astronomers at Maragha and Samarqand were able to obtain their own values for $\omega _{\mathrm{A}}$ from observations in the periods of the activities in the two observatories.

In the period of al-Maghribī’s observations at Maragha (1262–76), he had one opportunity in the case of Jupiter and Saturn to find that Ptolemy’s value of $\omega _{\mathrm{A}}$ for each planet is invalid at the time and another to determine a correct value. In this period, the mean longitude of Saturn changed from about $21^{\circ }$ to about $197^{\circ }$, the longitude of its perihelion was about $79^{\circ }$, and its ascending node was at the longitude of about $107^{\circ }$. From the three observations of Saturn made on 25 October 1263, 9 December 1266, and 27 February 1273 when the planet crossed the meridian of Maragha, by means of Ptolemy’s iterative algorithmic procedure, al-Maghribī obtained the longitude of the eccentric apogee of the planet as $258.5^{\circ }$ (Table 4), which, of course, could be taken for the whole period of his observations at Maragha. Then, with Ptolemy’s value $\omega _{\mathrm{A}} = +50^{\circ }$, the latitude of the planet should have been zero when its mean longitude was about $\lambda _{\Omega }= \lambda _{\mathrm{A}} - \omega _{\mathrm{A}} - 90^{\circ } = 258.5^{\circ } - 50^{\circ }- 90^{\circ } = 118.5^{\circ }$; this took place, according to modern theories, about mid-December 1269, and according to al-Maghribī’s Adwār, about mid-November 1269. About this time, the planet culminated at Maragha during nights and its latitude was about $+0{;}43^{\circ }$,⁵⁶ which is in the range of the accuracy of the instruments of the Maragha observatory.⁵⁷ It is thus probable that if al-Maghribī had made systematic observations in this time interval, he could achieve the result that the planet had a sizable latitude while, adopting Ptolemy’s $\omega _{\mathrm{A}} = +50^{\circ }$, its epicycle centre should have been located very near the ascending node, and thus its latitude should have been about zero.

About mid-May to mid-December 1268, for some period before the last visibility of Saturn on 11 June and after its first visibility on 18 July, the longitude of the planet changed from about $99^{\circ }$ to $113^{\circ }$ and its latitude from about $-0{;}10^{\circ }$ to $+0{;}10^{\circ }$ and reached zero on 14 September. From observations of the planet near the horizon as accurate as those recorded in his Talkhīṣ al-majisṭī, al-Maghribī might then conclude that the latitude of the planet had reached zero somewhere in this period. His tables give about $100^{\circ }$ to $107^{\circ }$ for the mean longitude of Saturn in this time interval.⁵⁸ This could lead to the result that $\omega _{\mathrm{A}}$ for Saturn should be about $61.5^{\circ }$–$68.5^{\circ }$. Therefore, it is probable that al-Maghribī’s denial of Ptolemy’s $\omega _{\mathrm{A}} = +50^{\circ }$ and his own rounded $+60^{\circ }$ were found by observation.

In the similar situation for Jupiter, with the Ptolemaic $\omega _{\mathrm{A}} = -20^{\circ }$, the planet should have crossed the ecliptic when its mean longitude was equal to $\lambda _{\Omega }= \lambda _{\mathrm{A}} - \omega _{\mathrm{A}} - 90^{\circ } = 178^{\circ } + 20^{\circ }- 90^{\circ } = 108^{\circ }$. This took place according to both the modern ephemeris and al-Maghribī’s Adwār about 9–21 April 1267 while the latitude of the planet was about $+0{;}26^{\circ }$. Unlike Saturn, the culmination of Jupiter in this interval occurred during daylight, but an observer had time enough to observe the planet after sunset and find that it had a slight latitude, while according to Ptolemy its latitude should be about zero.

In the period of the activities at the Samarqand observatory (1430–49), the longitude of Saturn changed from about $277^{\circ }$ to $174^{\circ }$, its ascending node was at about $109^{\circ }$, and its perihelion/perigee at about $82^{\circ }$. The Samarqand observers could thus observe the planet when crossing its nodes, southern limit, and perigee, and this 20-year period was sufficient for Jupiter and Mars to be observed at all the critical points.

To sum up, this paper has established that new values for the latitude parameters, inclinations, and directions of the lines of the limits/nodes were observed at the Maragha and Samarqand observatories in late medieval Islamic astronomy, and has analysed their accuracy. However, in the absence of further documented information, it has not settled the question of how these values were determined, although some ways for deriving them from observations were proposed.

Appendix: Planetary inclinations and latitude tables

In order to derive the inclinations from the planetary extremal latitudes and parameter values adopted in the zījes of the Maragha and Samarqand traditions, we make use of both modern and Ptolemy’s methods. The first consists in solving trigonometrical equations resulted from the configuration of the ecliptic, eccentric, and epicycle at the northern or southern limit (in the case of the superior planets) and at the nodes (for the inferior planets). The latter is to extract the inclinations from the maximum and minimum latitudes by interpolation in the epicyclic equation tables, an approximate method explained in Almagest XIII.3.⁵⁹

1.1 Maragha: inferior planets

Figure 3 shows the configuration of the eccentric and epicycle of Venus when the centre of the epicycle is at either of the nodes of the eccentric, i.e., $\pm 90^{\circ }$ from the apsidal line; in this position, the inclination of the epicycle is in the line of sight. The radius of the eccentric $\textit{OC} = R = 60$, the eccentricity $\textit{TO} = e$, the distance from the earth to the centre of the epicycle $\textit{TC} = (R^{2}- e^{2})^{1/2}$, and the radius of the epicycle $\textit{CP} = r$. We let the true epicyclic anomaly $\alpha = \angle \textit{ACP}$, so that $\textit{PP}^{\prime } = \textit{QQ}^{\prime }= r\hbox { sin }\alpha$, $\textit{PQ} = P^{\prime }Q^{\prime } = r\hbox { cos }\alpha \hbox { sin }i_{1\max }$, and $\textit{CQ}^{\prime }= r\hbox { cos }\alpha \hbox { cos }i_{1\max }$. For Mercury, the direction of the inclination of the epicycle is reversed, as the epicyclic apogee A is inclined to the north; also, $\textit{TO} = 2e$. The latitude $\beta = \angle \textit{PTQ}$ is found from the below formula (negative sign for Venus and positive sign for Mercury):

$$\begin{aligned} \sin \beta _{{\mathrm{icl}}} (\alpha )=\pm \frac{\textit{PQ}}{\sqrt{(\textit{TC}+C{Q}^{\prime })^{2}+Q{Q}^{\prime 2}+\textit{PQ}^{2}}}. \end{aligned}$$

(1)

If $\alpha = 180^{\circ }$, for which the latitude of the inferior planets reaches its maximum value, all that is necessary is to solve triangle TBC in which $\angle BTC = \beta _{\max }$:

$$\begin{aligned} \frac{\textit{TC}}{\sin (\angle \textit{TBC})}=\frac{\textit{CB}}{\sin (\angle \textit{CTB})}\hbox { or } \frac{\textit{TC}}{\sin (\beta _{\max } +i_{1\max } )}=\frac{r}{\sin (\beta _{\max })} \end{aligned}$$

(2)

and so, $i_{1\max } = (\beta _{\max }+i_{1\max }) - \beta _{\max }$. Therefore, $i_{1\max }$ can be conveniently found from TC, r, and $\beta _{\max }$.

Fig. 3

Configuration of the epicycle and the eccentric of Venus when the centre of the epicycle is at the ascending node of the eccentric, $\pm 90^{\circ }$ from the apsidal line

The underlying planetary parameters in the zījes of the Maragha tradition are as follows: the eccentricity of Venus in the Īlkhānī Zīj is $e\approx 1{;}2$ where $R = 60$, computed from the maximum tabular value $q_{\max } = 1{;}59^{\circ }$ for the equation of centre of the planet,⁶⁰ in al-Maghribī’s Adwār al-anwār e $\approx$ 1;3, from $q_{\max } = 2{;}0^{\circ }$.⁶¹ The value adopted for the eccentricity of Mercury, $e = 3;0$, and the radii of the epicycles of both the inferior planets in these two zījes are Ptolemaic, Venus: 43; 10, Mercury: 22;30.

Due to Ptolemy’s complicated model for Mercury’s motion in longitude of a movable eccentric, in Fig. 3: $\textit{TC} \ne (R^{2} -e^{2})^{1/2}$, but with Ptolemy’s value $e = 3{;}0, \textit{TC} \approx 56{;}40$.⁶² Then, with al-Maghribī’s value $\beta _{\max } = \beta _{\mathrm{icl}}(180^{\circ }) = 4{;}35^{\circ }$ for the maximum latitude of Mercury, one can compute $i_{1\max } \approx 7{;}2^{\circ }$ from (2).

Our historical method, i.e., Ptolemy’s solution, of which al-Maghribī quite probably made use, is an analogy between the extremal latitudes and inclination and the epicyclic equations and true epicyclic anomalies, respectively, in the sense that the extremal latitudes $\beta _{\mathrm{icl}}$(0) ($=\angle \textit{ATC}$ in Fig. 3; line AT not drawn) and $\beta _{\mathrm{icl}}(180^{\circ })\,(=\angle \textit{BTC})$ can be taken as the epicyclic equations in the correction tables for longitude corresponding, respectively, to the true epicyclic anomalies corresponding to the anomalies $\pm i_{1\max }$ and $180^{\circ }\pm i_{1\max }$ (in Fig. 3, assume that the epicycle is perpendicular to the ecliptic). When the centre of the epicycle of an inferior planet is at either of the nodes, i.e., $\pm 90^{\circ }$ from the apsidal line, its true eccentric anomaly, $\angle \textit{OTC}$ in Fig. 3, is $\kappa = \pm 90^{\circ }$, which corresponds to the mean eccentric anomaly $\bar{{\kappa }}\approx \pm 93^{\circ }$. At this position, from al-Maghribī’s table of the equations of Mercury, the following values for the epicyclic equation p can be derived:⁶³

$$\begin{aligned} 5{;}12^{\circ }\hbox { for } \alpha= & {} 180^{\circ } - i_{1\max }= 172^{\circ }\hbox { and}\\ 4{;}34^{\circ }\hbox { for } \alpha= & {} 180^{\circ }- i_{1\max } = 173^{\circ }. \end{aligned}$$

Al-Maghribī’s $\beta _{\max } = 4{;}35^{\circ }$ falls between these two values. So, with taking it as the epicyclic equation, the linear interpolation between $p = 5{;}12^{\circ }$ and $p = 4{;}34^{\circ }$ results in $\alpha = 180^{\circ }- i_{1\max } \approx 172{;}58^{\circ }$, and thus $i_{1\max }\approx 7{;}2^{\circ }$, which is in best agreement with our earlier derivation.

Table 5 shows that a table computed from this value is in good agreement with al-Maghribī’s tabular entries.⁶⁴

Table 5 Al-Maghribī’s table of the inclination of Mercury $i_{1\max } = 7;2^{\circ }$

In the case of Venus, using formula (2) with al-Maghribī’s value $\beta _{\max } = \beta _{\mathrm{icl}}(180^{\circ }) = 6{;}40^{\circ }$ results in $i_{1\max } = 2{;}37^{\circ }$ (note that because the eccentricity of Venus is small, $\textit{TC} \approx 60$). But this value cannot be derived from al-Maghribī’s table of the epicyclic equation of Venus according to Ptolemy’s procedure. As mentioned earlier, both in the Īlkhānī Zīj and in al-Maghribī’s Adwār, the radius of the epicycle of Venus is Ptolemaic, and as shown in Table 6, the tables of the epicyclic equation of the planet at mean distance in both zījes are equivalent to the corresponding one in the Almagest. Nevertheless, al-Maghribī has different values for the epicyclic equation for arguments $171^{\circ }$–$179^{\circ }$ (enclosed by the dashed lines in Table 6). I cannot explain why these entries are different, larger than the Almagest entries. As mentioned earlier, Ibn Yūnus has a greater value for the radius of the epicycle of Venus than Ptolemy ($r \approx 43{;}28$; see Table 6).⁶⁵ But, al-Maghribī’s equations for arguments $171^{\circ }$–$179^{\circ }$ can in no way be related to Ibn Yūnus’s.

Table 6 Tables of the epicyclic equation of Venus in the Īlkhānī zīj, al-Maghribī’s Adwār, and Ibn Yūnus’ Ḥākimī zīj

With Ptolemy’s procedure, the maximum latitude $\beta _{\max } = \beta _{\mathrm{icl}}(180^{\circ }) = 6{;}40^{\circ }$ taken as the epicyclic equation at mean distance corresponds to an epicyclic anomaly $\alpha = 180^{\circ }- i_{1\max } = 177{;}28{,}38^{\circ }$, and thus $i_{1\max } = 2{;}31{,}22^{\circ }$ which is nearly equal to the Ptolemaic $2{;}30^{\circ }$. It can be assumed that $6{;}40^{\circ }$ is estimated from the observed latitudes of Venus near inferior conjunction with the sun, although estimating latitude in this location is extremely difficult and al-Maghribī’s error, like Ptolemy’s is close to $-2^{\circ }$, but it cannot be determined which of the two values for $i_{1\max }$ al-Maghribī actually derived from it. However, since the tabular entries for the inclination of Venus in his Adwār are in better agreement with computation from $i_{1\max } = 2{;}37^{\circ }$ than from $i_{1\max } = 2{;}30^{\circ }$ (see below), it seems to be safe to conclude that it is the first value that, in reality, underlies al-Maghribī’s table in the Adwār.

In Table 7, Cols. 2 and 3, respectively, indicate the Almagest ⁶⁶ and al-Maghribī’s table of the inclination of Venus in the Adwār. Cols. 4 and 5 give the recomputed values from the two values for $i_{1\max }$. Neither of the two sets of the re-computed latitudes is as consistent with the tabular entries as we have seen earlier in the case of al-Maghribī’s table of the inclination of Mercury (Table 5). The table was probably computed according to the following procedure: the entries for the arguments $0^{\circ }$–$96^{\circ }$) are nearly identical to the corresponding entries in the Almagest, except for the slightly improved increments of $0;1^{\circ }$ for the first five entries; this seems reasonable, since a small change in $i_{1\max }$ from $2{;}30^{\circ }$ to $2{;}37^{\circ }$ causes no critical change in the latitudes in this range (at most, $0{;}3^{\circ }$). But most of the entries in the latter part of the table appear to have been derived from multiplying the corresponding entries in the Almagest table by 6;40/6;22 (see Col. 6). It is noteworthy that applying similar procedures to computing the planetary equation tables can be found in Islamic astronomy, more remarkably, in al-Maghribī’s Tāj al-azyāj ⁶⁷ as well as in the Īlkhānī Zīj.⁶⁸

Table 7 Tables of the inclination $\beta _{\mathrm{icl}}$ of Venus in the Almagest, al-Maghribī’s Tāj, and Adwār the Īlkhānī Zīj

In the Tāj al-azyāj, al-Maghribī has for Venus $\beta _{\max } =\beta _{\mathrm{icl}}(180^{\circ }) = 8{;}30^{\circ }$. The table of the planet’s epicyclic equation in this zīj does not have those strange entries found in the Adwār, but is identical to the Almagest/Handy Tables, and so to the Īlkhānī Zīj (Table 6). Both formula (2) and interpolation in the anomalistic equation table of Venus result in $i_{1\max } \approx 3{;}21^{\circ }$. Al-Maghribī’s values for the inclination and the recomputed values are shown in Table 7 (Cols. 7 and 8).

The entries in the latitude tables of the inferior planets in the Īlkhānī Zīj are given for each integer degree of the argument. Col. 9 in Table 7 shows the entries in the table of the inclination of Venus in the Īlkhānī Zīj. Both solving the trigonometrical equation (2) with $\beta _{\max } = \beta _{\mathrm{icl}}(180^{\circ }) = 8{;}40^{\circ }$ and interpolation in the table of the epicyclic equation of Venus results in $i_{1\max } \approx 3{;}25^{\circ }$. But the first half of the table, in the region of the arguments $0^{\circ }$–$90^{\circ }$, is identical to the corresponding table in the Almagest computed from $i_{1\max } = 2{;}30^{\circ }$. Moreover, re-computation of the table from $i_{1\max } = 3{;}25^{\circ }$ (Col. 10) shows that the results are by no means in agreement with the entries in the latter part of the table. In MS. P (fol. 44r) of the Īlkhānī Zīj, an unknown commentator has restored the latter half of the table to the Ptolemaic numbers and thus reduced the maximum latitude of the planet to $6{;}22^{\circ }$. We are told that this modification was done on the basis of the corresponding table in al-Maghribī’s Zīj, while, as we have seen above, his Tāj and Adwār have, respectively, the values $8{;}30^{\circ }$ and $6{;}40^{\circ }$ for the maximum latitude of Venus. It is worth noting that in his commentary on the Īlkhānī Zīj, al-Nīshābūrī briefly refers to this problem in the table of the latitude of Venus as a subtle point, but he offers no solution.⁶⁹ Since the table is intrinsically contradictory, I cannot speculate about the reasoning behind it. However, it was apparently reasonable enough for Ibn al-Shāṭir and the Samarqand astronomers to quote it in their own zījes.

Table 8 Equations of centre and eccentricities in Ulugh Beg’s Sulṭānī Zīj

1.2 Samarqand: superior planets

The new planetary parameters obtained at the Samarqand observatory, as deduced from Ulugh Beg’s Sulṭānī zīj, are summarized in Tables 8 for the equation of centre and 5 for the equation of anomaly. In both, Cols. 2 and 4 show, respectively, the maximum and minimum equations tabulated in this zīj and Cols. 3 and 5 show the arguments, mean eccentric or true epicyclic anomalies, for which the extremal values are given.

The tables of the equation of centre in the Almagest are symmetrical and additive for half the table and subtractive for the other half, but in the Sulṭānī Zīj, they are displaced and always additive; the values $k_{1} = 1/2(\hbox {Max} + \hbox {Min})$ of the displacements and $q_{\max } = 1/2(\hbox {Max} - \hbox {Min})$ of the maximum equations of centre (note that $k_{1} > q_{\max })$, and the corresponding eccentricities $e=R \cdot \hbox {tan }1/2\,q_{\max }$, where the radius of the eccentric $R = 60$, are given in the last three columns of Table 8.

The tables of the equation of the epicyclic anomaly are also always additive, but they use a different type of the displacement explained as follows: for all the planets, the first half of the table, for arguments $0^{\circ }$–$180^{\circ }$, corresponds to the equation $p_\mathrm{A}$ of the epicyclic anomaly at greatest distance ($R+e$; for Mercury: $R +3e$), when the centre of the epicycle is located at the apogee, but the latter half of the table, for arguments $180^{\circ }$–$360^{\circ }$, to the equation $p_{\Pi }$ of the epicyclic anomaly at the least distance ($R-e$, excepting Mercury), when the centre of the epicycle is located at the perigee. In the case of Saturn and Jupiter, all the entries are increased by an amount $k_{2}$. For Mars and the two inferior planets, the first half of the table gives $p_\mathrm{A}$ while the entries in the latter part of the table were increased by $360^{\circ }$, i.e., $360^{\circ } -p_{\Pi }$.⁷⁰ The values $k_{2}$ are shown in Col. 6. Cols. 7 and 8 display the maximum values of the epicyclic equation, respectively, at apogee and perigee (for Saturn and Jupiter: $p_{\mathrm{A}, \max } = \hbox {Max} - k_{2}$ and $p_{\Pi , \max }=k_{2} - \hbox {Min}$). The last column gives the corresponding epicyclic radii (for all the planets, except Mercury, $r = \hbox {sin}(p_{\mathrm{A}, \max }) \cdot (R+e) = \hbox {sin}(p_{\Pi , \max }) \cdot (R - e))$. In the Sulṭānī zīj, there is a table for the differences $p_{\Pi }- p_{\mathrm{A}}$ and two interpolation tables for each planet, one for the epicyclic anomaly $0^{\circ }$–$180^{\circ }$ and another for the epicyclic anomaly $180^{\circ }$–$360^{\circ }$. The entries of the equation tables are carried to two sexagesimal places, except Mercury to one place, and are given for each integer degree of the argument (for Mars, from $150^{\circ }$ to $210^{\circ }$, the arguments are in steps of $0{;}20^{\circ }$). Their reconstruction on the basis of the values derived above is in good agreement with the tabular entries.

It merits noting that in his Commentary on Zīj of Ulugh Beg, Qūshčī explains the layout of the equation and mean motion tables in this zīj. The values he mentions for the maximum equations agree with what can be extracted from the tables, with the exception of very slight differences in $q_{\max }$ of Saturn ($6{;}37,4{{\mathbf {6}}}^{\circ }$) and Jupiter ($5{;}18, {{\mathbf {42}}}^{\circ }$). The parameters we deduced from the tables are in agreement with those he gives. As Qūshčī remarks, for all the planets, the mean eccentric anomalies are reduced and, inversely, the mean epicyclic anomalies are increased by the above-mentioned values for $k_{1}$ and are then tabulated; and for Saturn and Jupiter, the longitudes of the apogees are decreased by the above-mentioned values for $k_{2}$.⁷¹

Figure 4 shows the configuration of the epicycle and eccentric, inclined to the ecliptic, of a superior planet when the centre of the epicycle is at the northern limit. In order to derive the inclinations from the extremal latitudes in the Sulṭānī zīj, we should first know the distance ${ TC} = \rho ^{\prime }$ between the earth T and the centre C of the epicycle of a superior planet at the northern limit, which can be computed from the eccentricity (Table 8) and the value adopted for $\omega _{\mathrm{A}}$ as mentioned earlier in Table 3. Where $R = 60$:

Fig. 4

Configuration of the epicycle and the eccentric of the superior planets with respect to the ecliptic when the centre of the epicycle is at the northern limit

In Fig. 4, the distance $\textit{TC} = \rho ^{\prime }$ and the radius of the epicycle $\textit{CP} = r$. We again let the true epicyclic anomaly $\alpha = \angle \textit{ACP}$, and $\angle \textit{CTN}^{\prime } = i_{0}$, so that $\textit{PP}^{\prime } = \textit{QQ}^{\prime } = r\hbox { sin } \alpha , \textit{PQ} = P^{\prime }Q^{\prime } = r\hbox { cos }\alpha \hbox { sin }i_{1 \mathrm{max}}$, and $\textit{CQ}^{\prime } = r\hbox { cos } \alpha \hbox { cos }i_{1 \mathrm{max}}$, and now $QN = Q^{\prime }N^{\prime } = (\textit{TC} + \textit{CQ}^{\prime })$ sin $i_{0}$. The latitude $\beta = \angle PTN$ at either limit can be computed from:

$$\begin{aligned} \sin \left| {\beta (\alpha )} \right|= & {} \frac{(\textit{TC}+C{Q}^{\prime })\sin i_0 -\textit{PQ}\cos i_0 }{\sqrt{(\textit{TC}+C{Q}^{\prime })^{2}+Q{Q}^{\prime 2}+PQ^{2}}}\nonumber \\= & {} \frac{r\cos \alpha \sin (i_0 -i_{1\max } )+{\rho }^{\prime }\sin i_0 }{\sqrt{({\rho }^{\prime }+C{Q})^{\prime 2}+Q{Q}^{\prime 2}+PQ^{2}}}. \end{aligned}$$

(3)

First, in order to derive $i_{0}$, we let $\alpha = 90^{\circ }$ in (3); thus, we have:

$$\begin{aligned} \sin i_0 =\sqrt{1+\frac{r^{2}}{{\rho }^{\prime 2}}}\cdot \sin \left( \left| {\beta _{{\mathrm{icl}}} (90^{\circ })} \right| \right) \end{aligned}$$

(4)

Then, in order to derive $i_{1}$, we can solve a long trigonometric equation resulting from (3) for $\alpha = 180^{\circ }$, after substituting $i_{0}$ computed from (4), a given value for the northern or southern latitude for this epicyclic anomaly ($\beta _{\max }=\angle BTN^{\prime }$), and r and $\rho ^{\prime }$ in (3). Instead, similar to what we have done earlier in the case of the inferior planets, a simpler solution is to solve triangle TCB in which $\angle BTC = \beta _{\max }- i_{0}$:

$$\begin{aligned} \frac{\textit{TC}}{\sin (\angle \textit{TBC})}=\frac{CB}{\sin (\angle \textit{CTB})}\hbox { or} \frac{{\rho }^{\prime }}{\sin (\beta _{\max } -i_0 +i_{1\max } )}=\frac{r}{\sin (\beta _{\max } -i_0 )} \end{aligned}$$

(5)

and thus, $i_{1\max } = (\beta _{\max }- i_{0 }+i_{1\max })- ( \beta _{\max }- i_{0})$.

Our formulae (4) and (5), applying the planetary parameters and the tabular latitudes in Ulugh Beg’s zīj (as mentioned in Tables 2, 8, 9), result in the inclinations in Table 2. Using Ptolemy’s solution, in which the extremal latitudes at the limits are taken as the epicyclic equations and the inclinations as the true epicyclic anomalies, as we have explained for the derivation of $i_{1\max }$ of the inferior planets, results in the same inclinations.

The tables in the Sulṭānī Zīj were generally calculated carefully, showing only minor deviations in the cases that have been examined,⁷² which is also true of the latitude tables at the northern and southern limits. Re-computation of the latitudes from the values derived for the inclinations is in good agreement with the tables in the Sulṭānī Zīj with no difference exceeding $0{;}1^{\circ }$ in the case of Saturn, $0{;}3^{\circ }$ for Jupiter, and $0{;}4^{\circ }$ for Mars. Table 10 is a specimen for Mars at the southern limit, which shows the greatest differences.

Table 9 Equations of the epicyclic anomaly and radii of the epicycles in Ulugh Beg’s Sulṭānī Zīj

Table 10 Ulugh Beg’s table of the latitude of Mars at the southern limit $i_{0} = 1{;}20^{\circ }$, $i_{1\max } = 2{;}7^{\circ }$