Al-Battani

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Abū ʿAbdullāh Muḥammad ibn Jābir ibn Sinān ar-Raqqī al-Ḥarrani aṣ-Ṣabiʾ al-Battānī (Arabic: أبو عبد الله محمد بن جابر بن سنان الحراني الصابي البتاني. Latinized: Albategnius, Albategni, Albatenius. c. 850 - c. 923) was an Arab[1] astronomer/astrologer and mathematician, born in Harran near Urfa, which is now in Turkey. His epithet as-Sabi suggests that among his ancestry were members of the Sabian sect who worshiped the stars, however, his full name affirms that he was Muslim[citation needed]. His best-known achievement was the determination of the solar year as being 365 days, 5 hours, 46 minutes and 24 seconds.

He produced a number of trigonometrical relationships:

\tan a = \frac{\sin a}{\cos a}
\sec a = \sqrt{1 + \tan^2 a }

He also solved the equation sin x = a cos x discovering the formula:

\sin a = \frac{a}{\sqrt{1 + a^2}}

which approximates the sine around the point 0. He also used al-Marwazi's idea of Tangents ("shadows") to develop equations for calculating tangents and cotangents, compiling tables of them.

Al Battani worked in Syria, at ar-Raqqah and at Damascus, where he died. He was able to correct some of Ptolemy's results and compiled new tables of the Sun and Moon, long accepted as authoritative, discovered the movement of the Sun's apogee, treats the division of the celestial sphere, and introduces, probably independently of the 5th century Indian astronomer Aryabhata, the use of sines in calculation, and partially that of tangents, forming the basis of modern trigonometry. He also calculated the values for the precession of the equinoxes (54.5" per year, or 1° in 66 years) and the inclination of Earth's axis (23° 35'). He used a uniform rate for precession in his tables, choosing not to adopt the theory of trepidation attributed to his colleague Thabit ibn Qurra.

His most important work is the Kitāb al-zīğ (Arabic: كتاب الزيج "the book of tables") with 57 chapters, which by way of Latin translation as De Motu Stellarum by Plato Tiburtinus (Plato of Tivoli) in 1116 (printed 1537 by Melanchthon, annotated by Regiomontanus), had great influence on European astronomy. A reprint appeared at Bologna in 1645. Plato's original manuscript is preserved at the Vatican; and the Escorial Library possesses in manuscript a treatise by Al Battani on astronomical chronology.

Copernicus mentioned his indebtedness to Al-Battani and quoted him, in the book that gave new meanings to revolution, De Revolutionibus Orbium Coelestium.

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