# Muḥammad ibn Jābir al-Ḥarrānī al-Battānī

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"Albategnius" redirects here. For the lunar crater, see Albategnius (crater).
Born A modern artist's impression of al-Battānī holding an astrolabe c. 858 CE Harran 929 CE Qasr al-Jiss (near Samarra) Islamic Golden Age Caliphate Mathematics, Astronomy, Astrology Trigonometrical relationships Kitāb az-Zīj Influenced by Influenced

Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī al-Ṣābiʾ al-Battānī (Arabic: محمد بن جابر بن سنان البتاني) (Latinized as Albategnius, Albategni or Albatenius) (c. 858, Harran – 929, Qasr al-Jiss, near Samarra) was an Arab astronomer, astrologer, and mathematician. He elaborated to a specified degree a number of trigonometric relations which were transmitted from India and Greco-Rome, and his Kitāb az-Zīj was frequently quoted by many medieval European astronomers, including by Copernicus.[1]

## Life

Little is known about al-Battānī's life beside that he was born in Harran near Urfa, in Upper Mesopotamia, which is now in Turkey, and his father was a famous maker of scientific instruments.[1] His family were members of the Sabian sect; however, his full name indicates that he might have been a Muslim.[2] Some Western historians state that he is of poor origin, like an Arab slave,[3] but traditional Arabic biographers make no mention of this.[1] He lived and worked in Ar-Raqqah, a city in north central Syria.

## Astronomy

One of Al-Battani's best-known achievements in astronomy was the refinement of existing values for the length of the year.[4] Ptolemy calculated the solar year values for the length of the year as being 365 days, 5 hours, 55 minutes and 12 seconds.[5] Al-Battani recalculated the solar year values for the length of the year as being 365 days, 5 hours, 46 minutes and 24 seconds.[2] Researchers have ascribed this phenomenon to Al-Battani as being in a geographical location that is closer to the southern latitude (although not as southernly as Ptolemy) than later observers such as Copernicus, which might have been more favorable for such observations.[2]

He was able to correct some of Ptolemy's results and compiled new tables of the Sun and Moon, long accepted as authoritative.[3]

Al-Battānī rediscovered that the direction of the Sun's apogee, as recorded by Ptolemy, was changing.[6] (In modern heliocentric terms this is due to the changing direction eccentricity vector of the Earth's orbit). He also elaborated to a specified degree a number of trigonometric relations, the use of sines in calculation, and partially that of tangents.[3] He elaborated to a specified degree the work of an Indian astronomer Aryabhata(476–550 CE) and a Greek astronomer Pythagoras (570 BC – c. 495 BC). He also recalculated the values for the precession of the equinoxes (54.5" per year, or 1° in 66 years) and the obliquity of the ecliptic (23° 35'), which was an elaboration of Hipparchus' work.[2][5] 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.

Al-Battānī's work is considered instrumental in the elaboration to a specified degree of science and astronomy.[2] Copernicus also quoted him in the book that initiated the Copernican Revolution, the De Revolutionibus Orbium Coelestium. Al-Battānī was frequently quoted by Tycho Brahe, Riccioli, among others. Kepler and Galileo showed interest in some of his observations.[1] And his elaborations to a specified degree continue to be used in geophysics.[7]

## Mathematics

In mathematics, al-Battānī 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:

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

He gives other trigonometric formulae, such as:[2]

$b \sin (A) = a \cos (90^\circ - A)$

Al-Battānī used al-Marwazi's idea of tangents ("shadows") to develop equations for calculating tangents and cotangents, compiling tables of them. He also discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants, which he referred to as a "table of shadows" (in reference to the shadow of a gnomon), for each degree from 1° to 90°.[8]

## Works

Al-Battānī's major work is Kitāb az-Zīj, or the book of astronomical tables, also known as az-Zīj aṣ-Ṣābi’. It was largely based on Ptolemy's theory, and other Greco-Syriac sources, while showing little Indian or Persian influence.[1][9] In his zij, he provided descriptions of a quadrant instrument.[10]

This book went through many translations to Latin and Spanish, including a Latin translation as De Motu Stellarum by Plato of Tivoli in 1116, which was later reprinted with annotations by Regiomontanus.[3] A reprint appeared at Bologna in 1645. The original MS. is preserved at the Vatican; and the Escorial library possesses in MS. a treatise of some value by him on astronomical chronology.[3]

## Notes

1. Hartner, Willy (1970–80). "Al-Battānī, Abū ʿAbd Allāh Muḥammad Ibn Jābir Ibn Sinān al-Raqqī al-Ḥarrānī al–Ṣābi". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 0-684-10114-9.
2.  Chisholm, Hugh, ed. (1911). "Albategnius". Encyclopædia Britannica (11th ed.). Cambridge University Press.
3. ^ Missing or empty |title= (help)
4. ^ a b Missing or empty |title= (help)
5. ^ Singer, Charles Joseph (1997). A short history of science to the nineteenth century. Courier Dover Publications. p. 135. ISBN 978-0-486-29887-0.
6. ^ Dalmau, W. (1997) CRITICAL REMARKS ON THE USE OF MEDIEVAL ECLIPSE RECORDS FOR THE DETERMINATION OF LONG-TERM CHANGES IN THE EARTH'S ROTATION', Surveys in Geophysics 18: 213-223.
7. ^ "trigonometry". Encyclopædia Britannica. Retrieved 2008-07-21.
8. ^ E. S. Kennedy, A Survey of Islamic Astronomical Tables, (Transactions of the American Philosophical Society, New Series, 46, 2), Philadelphia, 1956, pp. 10–11, 32–34.
9. ^ Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy (Cambridge University Press) 21 (1). doi:10.1017/S095742391000007X.