Annular Eclipses in the Ancient and Medieval Periods

The problem of annular eclipsesFootnote ¹ in medieval astronomy is somewhat complicated, involving some historical mechanisms and technical details both on the theoretical and observational levels. First of all, some episodes of annular eclipses, based on definite and undeniable facts, need to be recounted in order to understand better the substantial importance of the problem and its peripheral considerations in the historical and astronomical contexts.

Ptolemy (Almagest V.14) determined the solar angular diameter to be permanently constant (0;31,20°), with “no noticeable difference due to [the variation] in distance,” and equal to the minimum lunar angular diameter; i.e., when the sun and the moon are at their greatest distances from the earth, they subtend the same angle (Toomer Reference Toomer1998, 252, 253). The lunar apparent diameter varies between 0;31,20° and 0;35,20°. Under these conditions, annular solar eclipses are evidently impossible. Based on what Ptolemy himself said, his predecessors (including Hipparchus) assumed that the angular diameters of the sun and the moon are equal, but at their mean distances to the earth; thus annular eclipses are possible (ibid., 252, n. 53). A Pseudo-Eudoxan papyrus from around 190 BCFootnote ² also tells us that solar eclipses can never be total but must remain at most annular.Footnote ³ It is unknown what astronomical implications during the writing of the papyrus led to the belief that the diameter of the moon must always appear smaller than that of the sun.Footnote ⁴ However, since in homocentric models (prevalent in those days) the distances of the celestial bodies to the earth must always remain constant, it seems that finding the angular diameter of the moon to be smaller than that of the sun led to a straightforward conclusion that the moon never covers the sun in a total solar eclipse. It thus appears intriguing that in the time interval of around three centuries, Ptolemy neglected or denied the possibility of annular eclipses, a view diametrically opposed to that of his predecessors. However, one has to acknowledge that the early Greek reports on the possibility of annular eclipses available to Ptolemy were vague and groundless,Footnote ⁵ and he had no opportunity to observe the phenomenon during his career as well.Footnote ⁶

Although the first account of annular eclipses appeared shortly after Ptolemy, by Simplicius in his commentary to Aristotle's De Caelo (Simplicius Reference Heiberg1894, 504, line 16, and 506, line 8),Footnote ⁷ what was passed on from the Greek to the medieval astronomers was instead Ptolemy's denial of annular eclipses. Accordingly, the medieval astronomers had to uncover the variety of solar eclipses and their dependent situations again for themselves.

Turning to the Orient, the Indian hypotheses for determining the solar and lunar angular diameters led to defining the solar and lunar apparent diameters as one-variable functions of their angular velocity (e.g., Súrya Siddhánta IV.1–5; cf. Anon. [1860] 1997, 41). The formulae in their original forms are rather complicated. Nevertheless, after substituting some constants for the solar and lunar diameters and mean distances from the earth, a simple formula in the form of θ = k · v would appear, in which k is a constant (with different values in the case of the sun: 0;33, and the moon: 0;2,26) and v is the diurnal velocity.Footnote ⁸ With a minimum solar velocity of around 0;57° per day and a minimum lunar velocity of around 11;30 °/d, the Indian formula gives 0;31,20° and 0;28° for the minimum solar and lunar angular diameters, respectively.Footnote ⁹ This indicates that annular eclipses are possible in the Indian astronomical context. A minor consequence, however, is of historical importance; that is, the Indian hypotheses produce very nearly the same value for the minimum solar diameter as given by Ptolemy.

In the medieval period, one can numerate four clear direct observations of annular eclipses. Two of them are not in the Ptolemaic context and do not have anything to do with its intrinsic considerations and limitations: 26 October 1147 in Brauweiler, Germany, and 21 January 1292 in Beijing, China (cf. Stephenson Reference Stephenson1997, 62, 258–93, 94),Footnote ¹⁰ but the two others are related to the context of the study and occupy prominent positions within it: the first on 29 July 873 by Abu al-‘Abbās Īrānshahrī in Nīshābūr, Iran (cf. Said and Stephenson Reference Said and Stephenson1997, 45; Goldstein Reference Goldstein1979), and the second on 30 January 1283 by Shams al-Dīn Muḥammad al-Wābkanawī al-Bukhārī in Mughān, Iran.

The two latter observations, however, do not fall under the same conditions in their context. The first took place in the early Islamic period and was preserved incidentally as a simple observation, which only Bīrūnī (d. 1048) reported without giving further details and used as the evidence for rejecting the homocentric solar model by an Islamic astronomer named Abū Ja‘far al-Khāzīn (d. ca. 961–971) (Bīrūnī Reference al-Bīrūnī1954, v.2, 632; Samsó [Reference Samsó1977] 1994, 274). The latter is not a simple record. Rather, Wābkanawī calculated its parameters (timing, magnitude, . . .) based on Muḥyī al-Dīn al-Maghribī's Adwār al-anwār, a zīj written in 1275 on the basis of the extensive systematic observations performed at the Maragha Observatory from 1261 through 1274.Footnote ¹¹ He evidentially determined the eclipse to be annular.Footnote ¹² In moving from the first to the latter, some historical mechanisms need to be considered, which appear to have shaped the annular eclipse as a justified phenomenon strictly grounded in the astronomical tradition of the late Islamic period.

We commence with the first report in Bīrūnī's language. It is necessary to first mention that since the maximum lunar diameter (0;35,20°) is sufficiently larger than the solar one (0;31,20°) within Ptolemy's considerations, the solar eclipse occurring under this condition must show a duration of darkness (Ar. makth, “staying”). As a result, in the table of solar eclipses in the Almagest (VI.8), a fifth column with only one entry is needed to show the motion of the moon from the instant of complete immersion through the instant of maximum eclipse. In fact, such an entry had already been included in the table of solar eclipses in the Almagest, but most Arabic manuscripts of Almagest do not have that one-entry column (Toomer Reference Toomer1998, 305, n. 63. Tables on 306–7). Bīrūnī started with the observation by Muḥammad b. 'Isḥāq al-Sarakhsī of the total solar eclipse of 27 May 876 AD whose duration, Bīrūnī said, was fully perceptible:

Bīrūnī then reported Īrānshahrī's observation and added that:

Then, Bīrūnī returned to the problem of the duration of darkness (“staying”) in solar eclipses and posed the possibility that:

He finally concluded on Abū Ja‘far al-Khāzīn's concentric solar model that:

The Islamic medieval astronomers’ acquaintance with the Indian hypotheses and above-mentioned simple relation for determining the angular diameters of the sun and the moon precedes their familiarity with the Ptolemaic considerations. However, even after Greek astronomy was adopted completely and became the dominant astronomy in the Islamic period, the Indian hypotheses and formulae concerning the solar and lunar diameters not only remained but also gained superiority over the Ptolemaic considerations, becoming the substitutes for them. Consequently, one may find them everywhere in the Islamic zīj literature, as a reflection of the mathematical astronomy of that tradition.Footnote ¹⁴ Bīrūnī's clear mention (the second quotation above) of the Indian astronomical tradition and the report on the observation of the annular eclipse go side by side. It thus appears convenient to argue that this leaves no room to doubt that Īrānshahrī's report served as the observational evidence that led to the use of the Indian hypotheses and the already-mentioned simple formula as the standards for determining the solar and lunar angular diameters in Islamic medieval astronomy. In fact, applying the Indian hypotheses seems to have been the only accessible way that could possibly provide a justified ground for annular eclipses.

Nevertheless, this does not give a sufficient justification for the variety in the range of the solar and lunar apparent diameters that existed in earlier Islamic astronomical sources. Applying the Indian hypotheses, some early Islamic astronomers such as Khwārizmī held the Ptolemaic constant value for the solar apparent diameter as the minimum value but accepted the range 0;31,20° to 0;33,48° for it as well as changed the range of variation in the lunar apparent diameter as 0;29,16° to 0;34,34°. Khwārizmī flourished around half a century before both Īrānshahrī and Sarakhsī made their observations. A simple check with the aid of the NASA's Five Millennium Catalogue of Solar and Lunar Eclipses also shows that in the period of 800–950, no total or annular solar eclipse occurred at Baghdad and Raqqa, where the early Islamic astronomers worked. This means the annular solar eclipse or the solar eclipse showing a perceptible darkness does not serve as observational evidence to explain why the early Islamic astronomers used the Indian hypotheses and relation to determine the solar and lunar apparent diameters.Footnote ¹⁵ However, there is enough evidence that the same observations by Īrānshahrī and Sarakhsī established the annular eclipse as a certain type of solar eclipse in the late Islamic period (see below).

Furthermore, it also does not explain why the use of the Indian simple relation persisted in Islamic astronomy; for example, al-Battānī (who flourished a generation after Īrānshahrī, d. 929) held that the solar apparent diameter varied between 0;31,20° and 0;33,40°, and the lunar apparent diameter from 0;29,30° through 0;35,20°, but he did not hint at the possibility of annular eclipses. This is the case, too, with a great number of the Islamic astronomers. There is no evidence showing that the astronomers who preceded Bīrūnī were aware of Īrānshahrī's observation. In addition, there is no evidence to ascertain whether Īrānshahrī wrote in Arabic or Persian. That only Bīrūnī referred to Īrānshahrī in his works and that he alluded to him as a philosopher who followed his own constructed religion rather than any other opinion (Bīrūnī [Reference al-Bīrūnī and Sachau1910] 1964, 1: 6), neither Islam as the new religion acknowledged by Iranians nor the ancient Iranian religions, gives the impression that Īrānshahrī was likely not writing in Arabic.

The non-dependency of the Indian hypotheses and the observational evidence is not exclusive to early Islamic astronomy but is the case with Indian astronomy itself. Neither the possibility nor the report of the occurrence of annular eclipses is given in the Indian texts (e.g., Súrya Siddhánta; cf. Anon. [1860] 1997, 174).

After all, since the Indian hypotheses and relations were already available in Islamic astronomy, one cannot estimate to what degree the fact that the Islamic astronomers considered the lunar angular diameter to be less than the solar apparent diameter might have been due to Īrānshahrī's observation. In particular, since there is no allusion to the annular eclipse as one of three possible types of solar eclipses in the early Islamic treatises or zījes prior to Bīrūnī (i.e., before the eleventh century), any hope of establishing a justified linkage between the two fades in the absence of further historical facts.Footnote ¹⁶

Bīrūnī proposed (the third quotation above) three possible ways that the solar and lunar apparent diameters can change only in order to explain solar eclipses that showed a duration of darkness. The third way appears to have been adopted in Islamic astronomy two hundred years before Bīrūnī. Note that he did not speak of the relation between the apparent diameters and annular eclipses simply because such a consideration is not necessary. Indeed, in order to include annular eclipses in the Ptolemaic context, one need only to deny that the angular diameter of the sun is permanently constant or, in Bīrūnī's language, to accept the Indian hypotheses that the solar apparent diameter varies. As soon as this is done, the possibility of annular eclipses can be held. In other words, the equality of the solar and lunar apparent diameters at their maximum distances from the earth presents no obstacle to the possibility of annular eclipses. For example, al-Maghribī, who gave a comprehensive account of the possible conditions involved in annular eclipses (see below), accepted the equality of the solar and lunar apparent diameters at their greatest distances from the earth but determined them to be equal to 0;31,8°.Footnote ¹⁷ Although this leaves a season-based time limitation in the occurrence of annular eclipses (namely, solar eclipses can no longer show annularity when they occur in the months when the sun is more distant from the earth),Footnote ¹⁸ nevertheless, it does not make annular eclipses impossible.

After Bīrūnī's time, we encounter either incidental references to, or complete accounts of, annular eclipses, kusūf ḥalqat al-nūr (“bright ring eclipse”). The first known incidental hint of annular eclipses without mentioning a real example appeared in the Zīj al-mu‘tabar al-sanjarī (written ca. 1115) (al-Khāzinī, Wajīz, fol. 28r).Footnote ¹⁹ After the founding of the Maragha Observatory, numerous references to annular eclipses could be found. The astronomical treatises and the zijes written in this period present a clear account of the annular eclipse and numerate it as a type of solar eclipses; e.g., the Īlkhānī zīj II.9 (C: p. 50; T: fol. 24r; P: fol. 24r);Footnote ²⁰ al-Ṭūsī's Memoir II.13 (ed. Ragep 1993, 238–239);Footnote ²¹ Shīrāzī, Nihāya (fol. 134r), Ikhtiyārāt II.12.3 (fol. 102v; written in Persian around 1285–1305), and Tuḥfa II.15 (fol. 96r; written in Arabic around 1285);Footnote ²² Nīshābūrī, Kashf (fol. 177v) and Sharḥ (fol. 69v);Footnote ²³ al-Maghribī's Talkhīṣ al-majisṭī (see below); Wābkanawī's Muḥaqqaq zīj III.13.22 (fols. 126v–127r); Kamālī's Ashrafī zīj V.18 (fols. 152v–153r); Kāhsī's Khāqānī zīj III.14 (fol. 89r).

In the last three works, a separate chapter is devoted to an account of the annular eclipse and its variants with regard to the apparent (topocentric) latitude of the moon:

Similar accounts appear in Wābkanawī's and Kamālī's zījes. The first gives more explanations about the variations in the annularity, such that his account covers the situation in which the lunar disk is tangent to the solar one in a direction.Footnote ²⁴

A common characteristic shared by the above-mentioned works is that they were written in the “eastern” Islamic realm, and their authors were well acquainted with Bīrūnī and his reports on the two famous observations (previously quoted).Footnote ²⁵ This makes a clear difference in the contextual situation of annular eclipses between this period and the earlier one. Here, one may not hesitate to argue that both the observational evidence and the hypothetical grounds were sufficient to accept, justify, and predict annular eclipses. It was in such a context that Wābkanawī had the opportunity to predict and observe the solar eclipse of 1283 at Mughān as annular.

Finally, there are two more remarks left to say about the Indian hypotheses and their interchange with Islamic astronomy. First, in the simple relation θ = k · v, the above-mentioned values for k are derived from the magnitudes of the diameters of the sun and the moon and their mean distances from the earth, as accepted in Indian astronomy. The simple relation appears to be the standard formula for determining the angular diameters of the sun and the moon in Islamic astronomy; nevertheless, the Islamic astronomers were not aware of the original formulae and the simple relation being derived from them because they used different values for the planetary magnitudes and distances. The Indian astronomical texts available to them, e.g., Brahmagupta's Khandakhadyaka (Brahmagupta, 62 and 118–120), did not supply them with any further information about the procedure but only mentioned the simple relation. In addition, in his India, Bīrūnī only mentioned the simple relation without explaining how it was derived (Bīrūnī [Reference al-Bīrūnī and Sachau1910] 1964, 79–80). Second, determining the θ values depends on the instantaneous velocity. Although the medieval formulae for computing the angular diameters originated from Indian astronomy, the velocities of the sun and the moon were calculated on the basis of the Ptolemaic solar and lunar models, according to the instructions in Almagest VI.4 implemented with a minor refinement made during the medieval period. As a result, it appears that the medieval astronomers had to include the Indian simple relation into the framework of the Ptolemaic models. In other words, here we see a synthesis of Indian astronomy with Greek astronomy or, strictly speaking, a successful setting of a relatively primitive formula with a carefully calculated variable.Footnote ²⁶ The adjusted Indian formulae found their way into the European astronomy of the late medieval period (Swerdlow Reference Swerdlow1973, 98 and 105, n. 9).

Discussion and Conclusion

Medieval science is by-and-large tradition-based. The medieval astronomers then tended to experience or speak of what they were looking for, that is, what was “defined” for them. In such an atmosphere, a well-“defined” phenomenon, “observational evidence” for supporting it, and a “ready-to-use tradition” in order to justify it worked together. No not-already-known phenomenon was permitted to penetrate its solid working context unless all three conditions (and, maybe, some others) were satisfied.

In order to consider the observation of the annular eclipse in the pre-telescopic era, both the optical essentials and the historical considerations related to it should be taken into account. For the first, the optical essential: In the annular eclipse, “the ring phase is less marked since a portion of the sun stays unobscured at all times. Hence, the loss of daylight is usually relatively small” (Stephenson Reference Stephenson1997, 62). It does not thus absorb much attention. Also, it is almost impossible to figure out the depth of the eclipse by the unaided eye. Nevertheless, the annulus could be seen when the sun is close to the horizon (a while after sunrise or before sunset) during the ring phase (as was the case with Irānshahrī's observation of 873) or through mist or light cloud (as was the case with the Chinese observation of 1292), due to the atmospheric extinction of sunlight. The ring is also clearly visible in the pinhole image devices, a model of which has been described in Wābkanawi's zīj, and it had probably been used in his observation of 1283.Footnote ²⁷ For the latter, the historical considerations: assuming all the necessary optical conditions to have been provided, the fact should be seriously considered that the paths of annular eclipses on the Globe's surface are usually narrow. In order to observe an annular eclipse, its path should have passed on the specific locations and the important cities where astronomers were working and observing. And at the time of the occurrence of the annular eclipse, a good number of the astronomers or the observers with sufficient astronomical knowledge and keen eyesight (neither local astrologers nor timekeepers) should have been present to observe the phenomenon and distinguish it from a total eclipse. It therefore seems less likely that all the above-mentioned conditions would be present together at the same time,Footnote ²⁸ as the rarity of the trustable historical reports show. From 873 when Īrānshahī observed the annular eclipse to the time when Bīrūnī referred to his observation, we have a period of around one century and a half. This interval saw the extensive observations of the eclipses in the Middle East (e.g., some group of astronomers in Baghdad, al-Battānī in Raqqa and in Antakiya, Ibn Yūnus in Egypt, Bīrūnī in the northeast of Iran, and so on), which were collected and studied by some scholars, notably, by F. R. Stephenson and his colleagues. But an annular eclipse neither occurred nor was observable in the locations of the Middle East where the astronomers were working and at the times when they were active in making observations. Moreover, they had not likely access to Īrānshahrī's report. (Because, as previously mentioned, Īrānshahrī did not accept Islam and so there is no evidence to show that he wrote in Arabic; his works are all lost, and among the early Islamic astronomers only Bīrūnī refers to him.)

They had denied Ptolemy's hypothesis of the unvarying solar apparent diameter and had also accepted the Indian simple formula admitting variation in the solar diameter and had utilized it in the framework of the Ptolemaic models since the early stages of flourishing astronomy in the Islamic period. They did so simply because Ptolemy's hypothesis appeared to them to be unreasonable and restrictive (the last quotation from Bīrūnī above) and giving permission to the ill-reasoned homocentric solar models (e.g., Abū Ja‘far al-Khāzin's) to be posed as well. The line of reasoning was that the solar distance to the Earth is varied, thus its angular diameter should change. From this, it follows that the annular eclipse might be possible. Nevertheless, they had not mentioned or become aware of the incidental consequence of the denial of the Ptolemy's hypothesis, i.e., the possibility of annular eclipses. (As already-mentioned, it was also the case with the Indian astronomy itself.) This may be due to the fact that neither was the phenomenon defined for the early Islamic astronomers (none of their preceding oriental-occidental astronomical traditions had provided them with the knowledge of it), nor was there enough observational evidence for it.

But, the situation drastically changed during the late Islamic period (from ~1000 AD onwards). The observational evidence confirmed by an outstanding figure, i.e., Bīrūnī, was now available. By Bīrūnī, it had also played a role, as empirical evidence, in dealing with theoretical astronomy. The matters concerning the justification of the phenomenon, i.e., the Indian hypotheses of the solar and lunar angular diameters, and the mathematical aspect of the problem, i.e., the Indian simple formula for computing the angular diameter, not only were already known, but also well-established. No difficulty would therefore be expected to arise if the phenomenon, for which the term “Bright Ring” was coined, was defined and identified as a further type of solar eclipses. It was in such an atmosphere that in nearly all the astronomical treatises (even in the most elementary of them) written in the Middle East during this period, a separate chapter has been devoted to annular eclipses while no account of the phenomenon or even a tentative allusion to its possibility can be found in the astronomical treatises or in the zījes written in the early Islamic period of (before ~1000 AD). (This situation is also the case with the “western” Islamic astronomy, i.e., the south of Spain and Morocco where a distinguished separate tradition of Islamic astronomy flourished: no allusion to annular eclipses may be addressed in it.)

To sum up, the early astronomers possessed the preliminaries to “justify” the phenomenon, but neither was the phenomenon “defined” to them, nor do they appear to have any “observational evidence” for it. The phenomenon became readily known when all the mentioned three factors were present. The situation of the annular eclipse in medieval astronomy is one of the rare instances that the process from defining and establishing the phenomenon to justifying and later predicting it had happened inside its context by the aid of the amazing interconnections of the available traditions working into it, without the need for any further concept to be created or any further computational technique to be invented. (Otherwise, it was less probable that such a remarkable result could have been achieved.)

In the end, it also deserves noting that, other than the medieval astronomy in the Middle East in the period of the eleventh century onwards when the annular eclipse was widely known, it was usually associated with the theoretical astronomy in the ancient and medieval periods. The discussion on the annular eclipse was put back on the astronomical agenda three times when the idea of the homocentricity in the planetary configurations arose: the sixth century: Simplicius vs. the homocentric models prevalent in his day; the eleventh century: Bīrūnī vs. Abū Jafar al-Khāzin's solar model; and the fifteenth century: a Pseudo-Regiomontani text entitled Refutatio errorum Alpetragii de motibus celestibus vs. the homocentric models by al-Biṭrūjī (d. ca. 1204).Footnote ²⁹ It was used as the exclusive evidence for showing their glaring deficiency and thus for disproving and rejecting them.