Ibn al-Zarqalluh’s discovery of the annual equation of the Moon

by S. Mohammad Mozaffari Published on: 9th February 2024

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Ibn al-Zarqālluh (al-Andalus, d. 1100) introduced a new inequality in the longitudinal motion of the Moon into Ptolemy’s lunar model with the amplitude of 24′, which periodically changes in terms of a sine function with the distance in longitude between the mean Moon and the solar apogee as the variable. It can be shown that the discovery had its roots in his examination of the discrepancies between the times of the lunar eclipses he obtained from the data of his eclipse observations over a 37-year period in the latter part of the eleventh century and the predictions made on the basis of the lunar theories in the Mumta-an-zīj (Baghdad, ca. 830) and al-Battānī’s zīj (Raqqa, d. 929), which were available to him at the time. What Ibn al-Zarqālluh found is, in fact, a special case of the annual equation of the Moon, which is applicable in the oppositions and, thus, in the lunar eclipses. The inequality was discovered independently by Tycho Brahe (d. 1601) and Johannes Kepler (d. 1630). As Ibn Yūnus (d. 1009) reports in his ākimī-zīj, Ibn al-Zarqālluh’s medieval Middle Eastern predecessors, the Persian astronomers Māhānī (d. ca. 880) and Nayrīzī (d. 922) as well as ‘Alī b. Amājūr (fl. ca. 920), were already acquainted with the problem of the eclipse timing errors, but it had remained unresolved until Ibn Yūnus provided a provisional, and incorrect, solution by reducing the size of the lunar epicycle. As we argue, the diverse ways to tackle the same problem stem from two different methodologies in astronomical reasoning in the traditions developed separately in the Eastern and Western regions of the medieval Islamic domain.

In nineteenth century, the misinterpretation of a passage in Abu ʾl-Wafāʾ al-Būzjānī’s (10 June 940–after 25 May 997) Majisṭī VII.2.10 by Louis Pierre Eugène Amélie Sédillot (1808–1875, the son of the French orientalist Jean Jacques Emmanuel Sédillot, 1777–1832), gave birth to the blatantly erroneous notion that Abu ʾl-Wafāʾ had discovered a major inequality in the lunar motion in longitude (the so-called “variation”). The controversy over the issue persisted for four decades, before being finally removed by Carra de Vaux (1867–1953). [Sédillot 1845–1849, Vol. 1, p. 42ff; Carra de Vaux 1892. A brief discussion of the controversy is given in Dreyer 1906, p. 252–257 and Neugebauer [1957] 1969, pp. 206–207.]

Over half a century ago now, the late Otto Neugebauer (1899–1990) remarked in his opus magnum that “Muslim astronomers, despite much boasting, restricted themselves by and large to the most elementary parts of Greek astronomy: refinements in the parameters of the solar motion, and increased accuracy in the determination of the obliquity of the ecliptic and the constant of precession.” [Neugebauer 1975, Vol. 1, p. 145.]

Fig 2. Diagram from al-Ṭūsī’s treatise, titled Muʿīniyya fī ʿilm al-hayʾa which shows the trajectory of the center of the Moon’s epicycle in the Ptolemaic model (MS Iran, National Library, no. 21303).
Fig 3. A folio of the Īlkhānī zīj (MS Tehran University, Ḥikmat collection, ff. 34v-35r) ––the formal product of the Maragha observatory under the supervision of Naṣīr al-Dīn al-Tūsī –– which shows the first equation of the Moon.

Although enormous research and studies made by the outstanding figures of the history of medieval Islamic astronomy in the past five decades rendered obsolete Neugebauer’s view, nevertheless, no major breakthrough in observational astronomy, in terms of the discovery of a new anomaly in the complicated motions of the celestial bodies, was claimed to have been achieved by a medieval Muslim astronomer –– maybe, because of the Sédillot case.

In his writings, two eminent figures of the history of astronomy in the western Islamic lands, Juan Vernet (1923–2011) and Julio Samsó [e.g., Samsó 2020, pp. 676–677], already brought into light a modification made by Ibn al-Zarqālluh (al-Andalus, d. 1100) in the Ptolemaic lunar model, consisting of a simple sinusoidal inequality which indicates the lunar motion is somewhat related to the spatial direction of the solar (i.e., Earth’s) orbit. This was the subject of a comprehensive study by R. Puig in 2000. Nevertheless, the nature of Ibn al-Zarqālluh’s refinement and its astronomical connotation was not known to date. Recently, S. M. Mozaffari, an Iranian historian of astronomy, in an article published in the journal Archive for History of Exact Sciences (communicated with Alexander Jones, online on February 2nd, 2024), showed that Ibn al-Zarqālluh in fact discovered the annual equation of the Moon (or, properly speaking, the aggregate effect of the modern lunar inequalities dependent upon the solar anomaly), and reconstructed a way through which the Andalusian astronomer could achieve it.

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Carra de Vaux, 1892, “L’Almageste d’Abû ʾl Wéfâ Albûzdjâni”, Journal asiatique, 8th series, 19, pp. 408–471.
Dreyer, J.L.E., 1906, History of the Planetary Systems from Thales to Kepler, Cambridge: Cambridge University Press.
Mozaffari, Seyyed Mohammad, “Ibn al-Zarqālluh’s discovery of the annual equation of the Moon”, Archive for History of Exact Sciences
Neugebauer, O., [1957] 1969, The Exact Sciences in Antiquity, 2nd edn., New York: Dover.Neugebauer, O., 1975, A History of Ancient Mathematical Astronomy, Berlin-Heidelberg-New York: Springer.
Puig, R., 2000, “The theory of the Moon in the Al-Zīj al-Kāmil fī-l-Taʿālīm of Ibn al-Hāʾim (ca. 1205)”, Suhayl 1, pp. 71–99.
Samsó, J., 2020, On Both Sides of the Strait of Gibraltar: Studies in the history of medieval astronomy in the Iberian Peninsula and the Maghrib, Leiden–Boston: Brill.
Sédillot, Louis Pierre Eugéne Amélie, 1845–1849, Matériaux pour servir à l’histoire comparée des sciences mathématiques chez les Grecs et les Orientaux, 2 vols., Paris: Librairie de Firmin Didot Freres.

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