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Abu al-Wafāʾ al-Būzjānī

Persian Mathematician and Astronomer of the Baghdad Observatory

Abū al-Wafāʾ al-Būzjānī (10 June 940 – July 998 CE) was a distinguished Persian mathematician and astronomer active in Baghdad. He collaborated with the Baghdad observatory team under the Buyid Caliphate and made pioneering contributions across trigonometry, arithmetic, and astronomical instrumentation.:contentReference{index=0}

Mathematics and Trigonometry

Abū al-Wafāʾ was the first to use the tangent function and compiled extremely accurate tables of sines and tangents at 15′ intervals—correct to eight decimal places, far surpassing earlier works.:contentReference{index=1} He also introduced the secant and cosecant functions, and formalized the interrelations of the six trigonometric lines (sine, cosine, tangent, cotangent, secant, cosecant).:contentReference{index=2} Moreover, he proved the law of sines for spherical triangles: \[ \frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c} \]

He also introduced and proved key trigonometric identities, such as:
\[ \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \]
\[ \cos(2a) = 1 - 2\sin^2(a) \]
\[ \sin(2a) = 2\sin(a)\cos(a) \]

Astronomy and Observational Work

Working at the Baghdad observatory established in 988 CE, he constructed the first known wall quadrant for precise celestial measurements.:contentReference He participated in a pioneering experiment in 997 CE, determining the longitude difference between Baghdad and Kath via simultaneous lunar eclipse observations—yielding results very close to modern calculations.:contentReference{index=6} His astronomical writings, especially the first seven treatises of his Almagest, were widely used by later astronomers.:contentReference{index=7}

Practical Works and Arithmetic

He authored accessible mathematical texts, including a guide for scribes and merchants—Kitāb fī mā yaḥtāj ilayhi al-kuttāb wa’l-ʿummāl—notable for containing the first instance of negative numbers in medieval Islamic arithmetic.:contentReference{index=8} He also wrote a geometric construction manual—Kitāb fī mā yaḥtāj ilayhi al-ṣāniʿ min al-aʿmāl al-handasiyya—featuring solutions to classical geometric problems such as constructing the regular heptagon via neusis and fixed-compass methods.:contentReference{index=9}

Legacy and Recognition

• His observational data informed astronomers such as al-Biruni for centuries.:contentReference{index=10}
• The lunar crater Abul Wafāʾ and a Google Doodle (10 June 2015) commemorate his contributions.:contentReference{index=11}
• Modern historians regard him as a pivotal figure in the development of trigonometry, astronomical instrumentation, and applied mathematics during the Islamic Golden Age.:contentReference{index=12}