# Muḥammad ibn Jābir al-Ḥarrānī al-Battānī : biography

c. 858 - 929

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 Muslim astronomer, astrologer, and mathematician. He introduced a number of trigonometric relations, and his Kitāb az-Zīj was frequently quoted by many medieval astronomers, including Copernicus.

## 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. His epithet aṣ-Ṣabi’ suggests that among his ancestry were members of the Sabian sect; however, his full name indicates that he was Muslim. Some western historians state that he is of noble origin, like an Arab prince, but traditional Arabic biographers make no mention of this. He lived and worked in ar-Raqqah, a city in north central Syria.

## Honors

• The crater Albategnius on the Moon is named after him.
• In the fictional Star Trek universe, the Excelsior-class starship USS Al-Batani [sic] NCC-42995, mentioned on Star Trek: Voyager as Kathryn Janeway's first deep space assignment, is named after him.
• The Doctor Who novel Night of the Humans, features a solar system called Battani 045.

## 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:

b \sin (A) = a \sin (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°.

## 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.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. In his zij, he provided descriptions of a quadrant instrument.

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. 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.

## Astronomy

One of al-Battānī's best-known achievements in astronomy was the determination of the solar year as being 365 days, 5 hours, 46 minutes and 24 seconds.

He was able to correct some of Ptolemy's results and compiled new tables of the Sun and Moon, long accepted as authoritative. Some of his measurements were even more accurate than ones taken by Copernicus many centuries later. Researchers have ascribed this phenomenon to al-Battānī being in a geographical location that is closer to the southern latitude, which might have been more favorable for such observations.

Al-Battānī discovered that the direction of the Sun's apogee, as recorded by Ptolemy, was changing. (In modern heliocentric terms this is due to the changing direction of the eccentricity vector of the Earth's orbit). He also introduced, probably independently of the 5th century Indian astronomer Aryabhata, the use of sines in calculation, and partially that of tangents. He also calculated 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'). 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 development of science and astronomy. Copernicus mentioned his indebtedness to al-Battānī and 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, and his data continues to be used in geophysics.Dalmau, W. (1997) ', Surveys in Geophysics 18: 213-223.

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