al-Battani

Al-Battānī, whose full name was Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī al-Ṣābiʾ al-Battānī, and whose Latinized name was Albategnius, was born before 858 in Harran in Bilād ash-Shām (Islamic Syria), 44 kilometres (27 mi) southeast of the modern Turkish city of Urfa. He was the son of Jabir ibn Sinan al-Harrani, a maker of astronomical instruments.[3] The epithet al-Ṣabi’ suggests that his family belonged to the pagan Sabian sect of Harran,[4][5] whose religion featured star worship, and who had inherited the Mesopotamian legacy of an interest in mathematics and astronomy.[2][n 3] His contemporary, the polymath Thābit ibn Qurra, was also an adherent of Sabianism, which died out during the 11th century.[7]

Although his ancestors were likely Sabians, al-Battānī was a Muslim, as shown by his first name.[5] Between 877 and 918/19 he lived in Raqqa, now in north central Syria, which was an ancient Roman settlement beside the Euphrates, near Harran. During this period he also lived in Antioch,[3] where he observed a solar and a lunar eclipse in 901. According to the Arab biographer Ibn al-Nadīm, the financial problems encountered by al-Battānī in old age forced him to move from Raqqa to Baghdad.[8]

Al-Battānī died in 929 at Qasr al-Jiss,[2] near Samarra, after returning from Baghdad where he had resolved an unfair taxation grievance on behalf of a clan from Raqqa.[9]

Al-Battānī is considered to be the greatest[10][11][12] and most famous of the known astronomers of the medieval Islamic world. He made more accurate observations of the night sky than any of his contemporaries,[3] and was the first of a generation of new Islamic astronomers that followed the founding of the House of Wisdom in the 8th century.[13] His meticulously described methods allowed others to assess his results, but some of his explanations about the movements of the planets were poorly written, and have mistakes.[14]

Sometimes referred to as the "Ptolemy of the Arabs",[15] al-Battānī's works reveal him to have been a devout believer in Ptolemy's geocentric model of the cosmos. He refined the observations found in Ptolemy's Almagest,[3] and compiled new tables of the Sun and the Moon, previously long accepted as authoritative.[5] Al-Battānī established his own observatory at Raqqa. He recommended that the astronomical instruments there were greater than one metre (3 ft 3 in) in size.[8] Such instruments, being larger—and so having scales capable of measuring smaller values—were capable of greater precision than had previously been achieved.[16] Some of his measurements were more accurate than those taken by the Polish astronomer and mathematician Nicolaus Copernicus during the Renaissance. One reason for this is thought to be that al-Battānī's location for his observations at Raqqa was closer to the Earth's equator, so that the ecliptic and the Sun, being higher in the sky, were less susceptible to atmospheric refraction.[5] The careful construction and alignment of his astronomical instruments enabled him to achieve an accuracy of observations of equinoxes and solstices that had previously been unknown.[8]

An annular solar eclipse. Al-Battānī was one of the first astronomers to understand why such phenomena can occur.

A representation of the celestial equator and Earth's ecliptic

Al-Battānī was one of the first astronomers to observe that the distance between the Earth and the Sun varies during the year, which led him to understand the reason why annular solar eclipses occur.[3][17][18] He saw that the position in the sky at which the angular diameter of the Sun appeared smallest was no longer located where Ptolemy had stated it should be,[3] and that since Ptolemy's time, the longitudinal position of the apogee had increased by 16°47'.[12]

Al-Battānī was an excellent observer.[19] He improved Ptolemy's measurement of the obliquity of the ecliptic (the angle between the planes of the equator and the ecliptic),[9] producing a value of 23° 35';[5][n 4] the accepted value is around 23°.44.[20] Al-Battānī obtained the criterion for observation of the lunar crescent—i.e., if the longitude difference between the Moon and the Sun is greater than 13° 66˝ and the Moon's delay after sunset is more than 43.2 minutes, the crescent will be visible.[2] His value for the solar year of 365 days, 5 hours, 46 minutes and 24 seconds, is 2 minutes and 22 seconds from the accepted value.[5]

Al-Battānī observed changes in the direction of the Sun's apogee, as recorded by Ptolemy,[21] and that as a result, the equation of time was subject to a slow cyclical variation.[22] His careful measurements of when the March and September equinoxes took place allowed him to obtain a value for the precession of the equinoxes of 54.5" per year, or 1 degree in 66 years,[5][9] a phenomenon that he realised was altering the Sun's annual apparent motion through the zodiac constellations.[23]

It was impossible for al-Battānī, who adhered to the ideas of a stationary Earth and geocentricism, to understand the underlying scientific reasons for his observations or the importance of his discoveries.[23]

The fundamental trigonometric functions defined from a right-angled triangle: sine, cosine, and tangent

A spherical triangle with sides a, b, and c

One of al-Battani's greatest contributions was his introduction of the use of sines and tangents in geometrical calculations, especially spherical trigonometric functions, to replace Ptolemy's geometrical methods. Al-Battānī's methods involved some of the most complex mathematics developed up to that time.[23] He was aware of the superiority of trigonometry over geometrical chords, and demonstrated awareness of a relation between the sides and angles of a spherical triangle, now given by the expression:[12]

${\displaystyle \cos a=\cos b\cos c+\sin b\sin c\cos A}$

Al-Battānī produced a number of trigonometrical relationships:[24]

${\displaystyle \tan \alpha ={\frac {\sin \alpha }{\cos \alpha }}}$

${\displaystyle \sec \alpha ={\sqrt {1+\tan ^{2}\alpha }}}$, where ${\displaystyle \sec \alpha ={\frac {1}{\cos \alpha }}}$.

He also solved the equation

${\displaystyle \sin x=y\cos x}$,

discovering the formula

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

Al-Battānī used the Iranian astronomer Habash al-Hasib al-Marwazi's idea of tangents to develop equations for calculating and compiling tables of both tangents and cotangents. He discovered their reciprocal functions, the secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°, which he referred to as a "table of shadows", in reference to the shadow produced on a sundial.[24]

A geometrical representation of the method used by al-Battānī to determine the qibla, shown as q from O (the observer) to M (Mecca)[25][n 5]

Using these trigonometrical relationships, al-Battānī created an equation for finding the qibla, which Muslims face in each of the five prayers they practice every day.[26] The equation he created did not give accurate directions, as it did not take into account the fact that Earth is a sphere. The relationship he used was precise enough only for a person located in (or close to) Mecca, but was still a widely used method at the time. Al-Battānī's equation for ${\displaystyle q}$, the angle of the direction of a place towards Mecca is given by:[25]

${\displaystyle \tan q={\frac {\sin \Delta \lambda }{\sin \Delta \phi }}}$

where ${\displaystyle \Delta \lambda }$ is the difference between the longitude of the place and Mecca, and ${\displaystyle \Delta \phi }$ is the difference between the latitude of the place and Mecca.

Al-Battānī's equation was superseded a century after it was first used, when the polymath al-Biruni summarized several other methods to produce results that were more accurate than those that could be obtained using al-Battānī's equation.[27]

A small work on trigonometry, Tajrīd uṣūl tarkīb al‐juyūb ("Summary of the principles for establishing sines") is known. Once attributed to the Iranian astronomer Kushyar Gilani by the German orientalist Carl Brockelmann, it is a fragment of al‐Battānī's zīj. The manuscript is extant in Istanbul as MS Carullah 1499/3. [2] The authenticity of this work has been questioned, as scholars believe al-Battānī would have not have included al‐juyūb for "sines" in the title.[8]

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• Sarton, George (1960). A History of Science: Ancient Science Through the Golden Age of Greece. Vol. 1. Cambridge, Massachusetts: Harvard University Press. OCLC 179869164.