Al-Battānī

Al-Battānī was born around 858 in or near Ḥarrān, in Mesopotamia (modern Turkey), and died in 929 near Sāmarrāʾ in Iraq, on a return journey to ar-Raqqah after travelling to Baghdad to protest an unjust tax against a group from his adopted city. His family belonged to the Sabian community of Ḥarrān, a religious group of star observers; his own name, Abū ʿAbd Allāh Muḥammad, indicates that he himself was Muslim. MacTutor identifies his father, with a hedge, as Jābir ibn Sinān al-Ḥarrānī, a well known instrument maker.

From about 877 he made astronomical observations at Antioch and, principally, at ar-Raqqah in Syria, a working record MacTutor dates as continuing until 918. His star catalogue is calibrated to the epoch year 880. Britannica places the career in the same shape: From 877 he carried out many years of remarkably accurate observations at ar-Raqqah in Syria.

Principal works

• Kitāb al-Zīj, his principal astronomical treatise, in 57 chapters. MacTutor summarises its structure: an opening on the division of the celestial sphere and the signs of the zodiac; sexagesimal arithmetic and the trigonometric functions introduced together in the early chapters; observational data of his own in chapter 4; astronomical problems in the manner of the Almagest across chapters 5 to 26; solar, lunar and planetary motion in chapters 27 to 31; astrological problems in chapters 49 to 55; and a closing pair on the construction of a sundial and other astronomical instruments.
• A commentary on Ptolemy's Tetrabiblos.
• On ascensions of the signs of the zodiac, MacTutor notes that the Fihrist lists this as a separate work, but a chapter of the same title appears inside Kitāb al-Zīj, so the identification is uncertain.
• On the quantities of the astrological applications.
• Latin translation of the Zīj, made by Plato of Tivoli around 1116 as De motu stellarum; a Spanish translation followed in the thirteenth century, and printed editions appeared in 1537 and 1645.

The shift from Ptolemy's chord to sine

Al-Battānī's central methodological move, the one that matters to a modern mathematics classroom, is the replacement of Ptolemy's geometrical chord methods with a systematic trigonometric approach. MacTutor puts it plainly: Rather than using geometrical methods, as Ptolemy had done, al-Battānī used trigonometrical methods which were an important advance. Britannica echoes the same shift: he improved Ptolemy's astronomical calculations by replacing geometrical methods with trigonometry.

The concrete form this takes in his own text is a set of formulae for right angled triangles that a modern student recognises immediately, for example, in the relation \( b \sin A = a \sin(90^\circ - A) \) that MacTutor quotes directly from the Zīj. The text puts the trigonometric identities to practical use.

Refined astronomical constants

Working from years of his own careful observation, al-Battānī refined the numerical values that underpin observational astronomy:

• The length of the solar year: 365 days, 5 hours, 46 minutes, 24 seconds.
• The precession of the equinoxes: 54.5 arcseconds per year.
• The obliquity of the ecliptic: 23° 35′.
• A catalogue of 489 stars, calibrated to the year 880.
• The demonstration that the Sun's farthest distance from the Earth varies, establishing the possibility of annular solar eclipses alongside total eclipses.

Hartner, quoted at length by MacTutor, characterises al-Battānī as taking no critical attitude towards the Ptolemaic kinematics in general while showing a very sound scepticism in regard to Ptolemy's practical results. Relying on his own observations, he corrects Ptolemy's numerical errors, sometimes tacitly, sometimes openly, on the main parameters of planetary motion. The distinction matters: al-Battānī's strongest contribution is his careful observational practice and the values that superseded Ptolemy's.

An honest characterisation

The Fihrist calls al-Battānī one of the famous observers and a leader in geometry, theoretical and practical astronomy, and astrology, and adds: Nobody is known in Islam who reached similar perfection in observing the stars and scrutinising their motions. This tenth century bibliographer's assessment is preserved in MacTutor. His work offers a MOTA classroom a case study in what a systematic trigonometric approach recovers over Ptolemaic geometry. It also shows the lasting value of corrected measurements reported with care.

Reception in Europe

Plato of Tivoli's Latin translation of the Zīj, De motu stellarum, was in circulation in Europe from 1116 through printed editions in 1537 and 1645. MacTutor records the influence on Tycho Brahe, Kepler, Galileo and Copernicus, specifically on the reception of the trigonometric method and the refined constants.

Related curriculum links in the curriculum

Al-Battānī's shift to trigonometric method is the direct precursor to Abū al-Wafāʾ al-Būzjānī's tangent, secant and cosecant tables a century later, and to the trigonometric geodesy of Al-Bīrūnī. The Continuum places him in his short form chronological arc alongside those figures. The short Heritage narrative on the About page gives the broader institutional context.

Sources

1. MacTutor History of Mathematics, University of St Andrews, Al-Battānī biography (birth c. 858 in or near Ḥarrān, death 929 in Qaṣr al-Jiṣṣ, full Arabic name, Sabian family origin with the hedge on his father Jābir ibn Sinān al-Ḥarrānī, the 57 chapter structure of Kitāb al-Zīj, the trigonometric method quotation, the star count of 489, the length of year and precession values, the obliquity of 23° 35′, the annular eclipse result, the Hartner and Swerdlow assessments, and the account of his final journey and death). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Battani/
2. Encyclopædia Britannica, al-Battānī biography (secondary verification of dates c. 858 to 929, birth in or near Ḥarrān, death near Sāmarrāʾ, ar-Raqqah as the observation site from 877, the trigonometry over geometry shift, the annular eclipse result, the refined constants for the year and the seasons, and the Latin transmission as De motu stellarum). https://www.britannica.com/biography/al-Battani