Abu al-Wafa' al-Buzjani

Astronomy

Abū al-Wafāʾ was the first to build a wall quadrant to observe the sky. It has been suggested that he was influenced by the works of al-Battani as the latter described a quadrant instrument in his Kitāb az-Zīj. His use of the concept of the tangent helped solve problems involving right-angled spherical triangles. He developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.

In 997, he participated in an experiment to determine the difference in local time between his location, Baghdad, and that of al-Biruni (who was living in Kath, now a part of Uzbekistan). The result was very close to present-day calculations, showing a difference of approximately 1 hour between the two longitudes. Abū al-Wafāʾ is also known to have worked with Abū Sahl al-Qūhī, who was a famous maker of astronomical instruments. While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.

Almagest

Among his works on astronomy, only the first seven treatises of his Almagest (Kitāb al-Majisṭī) are now extant. The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.

Mathematics

He defined the tangent function, and he established several trigonometric identities in their modern form, where the ancient Greek mathematicians had expressed the equivalent identities in terms of chords. The trigonometric identities he introduced were:

${\displaystyle \sin (a\pm b)=\sin (a)\cos (b)\pm \cos (a)\sin (b)}$

${\displaystyle \cos (2a)=1-2{\sin }^2(a)}$

${\displaystyle \sin (2a)=2\sin (a)\cos (a)}$

He has discovered the law of tangents and may have discovered the law of sines for spherical triangles, however, other scholars like Abu-Mahmud Khojandi have been credited with the latter achievement:

${\displaystyle \frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}}$

where ${\displaystyle A,B,C}$ are the sides of the triangle (measured in radians on the unit sphere) and ${\displaystyle a,b,c}$ are the opposing angles.

Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.

Legacy

• The crater Abul Wáfa on the Moon is named after him.
• On 10 June 2015, Google changed its logo in memory of Abū al-Wafāʾ al-Būzjānī.