Matemáticos y Astronomos PErsas de la Era de Oro de los Islamicos en Iraq Persia - Trigonometria Esferica

 Creadores de la tangente y de algunas identidades trigonométricas y otras cosas de astronomia

https://en.wikipedia.org/wiki/Habash_al-Hasib_al-Marwazi

Habash al-Hasib al-Marwazi

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Work[edit]

He made observations from 825 to 835, and compiled three astronomical tables: the first were still in the Hindu manner; the second, called the 'tested" tables, were the most important; they are likely identical with the "Ma'munic" or "Arabic" tables and may be a collective work of al-Ma'mun's astronomers; the third, called tables of the Shah, were smaller.

Apropos of the solar eclipse of 829, Habash gives us the first instance of a determination of time by an altitude (in this case, of the sun); a method which was generally adopted by Muslim astronomers.

In 830, he seems to have introduced the notion of "shadow", umbra (versa), equivalent to our tangent in trigonometry, and he compiled a table of such shadows which seems to be the earliest of its kind. He also introduced the cotangent, and produced the first tables of for it.^[6][7]

The Book of Bodies and Distances[edit]

Al-Hasib conducted various observations at the Al-Shammisiyyah observatory in Baghdad and estimated a number of geographic and astronomical values. He compiled his results in The Book of Bodies and Distances, in which some of his results included the following:^[8]

Earth

• Earth's circumference: 20,160 miles (32,444 km)
• Earth's diameter: 6414.54 miles (10323.201 km)
• Earth radius: 3207.275 miles (5161.609 km)

Moon

• Moon's diameter: 1886.8 miles (3036.5 km)
• Moon's circumference: 5927.025 miles (9538.622 km)
• Radius of closest distance of Moon: 215,208;9,9 (sexagesimal) miles
• Half-circumference of closest distance of Moon: 676,368;28,45,25,43 (sexagesimal) miles
• Radius of furthest distance of Moon: 205,800;8,45 (sexagesimal) miles
• Diameter of furthest distance of Moon: 411,600.216 miles (662,406.338 km)
• Circumference of furthest distance of Moon: 1,293,600.916 miles (2,081,848.873 km)

Sun

• Sun's diameter: 35,280;1,30 miles (56,777.6966 km)
• Sun's circumference: 110,880;4,43 miles (178,444.189 km)
• Diameter of orbit of Sun: 7,761,605.5 miles (12,491,093.2 km)
• Circumference of orbit of Sun: 24,392,571.38 miles (39,256,038 km)
• One degree along orbit of Sun: 67,700.05 miles (108,952.67 km)
• One minute along orbit of Sun: 1129.283 miles (1817.405 km)

https://en.wikipedia.org/wiki/Abu_al-Wafa%27_Buzjani

Abu al-Wafa' Buzjani

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Astronomy[edit]

Abu Al-Wafa' was the first to build a wall quadrant to observe the sky.^[5] It has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kitāb az-Zīj.^[5] His use of tangent helped to solve problems involving right-angled spherical triangles, and developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.^[6]

In 997, he participated in an experiment to determine the difference in local time between his location 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. Abu al-Wafa is also known to have worked with Abū Sahl al-Qūhī, who was a famous maker of astronomical instruments.^[6] While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.^[6]

Almagest[edit]

Among his works on astronomy, only the first seven treatises of his Almagest (Kitāb al-Majisṭī) are now extant.^[7] The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.^[5][6]

Mathematics[edit]

He established several trigonometric identities such as sin(a ± b) in their modern form, where the Ancient Greek mathematicians had expressed the equivalent identities in terms of chords.^[8]

${\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$

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

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

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

He also discovered the law of sines for spherical triangles:

${\displaystyle {\frac {\sin A}{sina}}={\frac {\sin B}{sinb}}={\frac {\sin C}{sinc}}}$

where A, B, C are the sides (measured in radians on the unit sphere) and a, b, c are the opposing angles.^[8]

Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.^[8]

Works[edit]

• Almagest (كتاب المجسطي Kitāb al-Majisṭī).
• A book of zij called Zīj al‐wāḍiḥ (زيج الواضح), no longer extant.^[6]
• "A Book on Those Geometric Constructions Which Are Necessary for a Craftsman", (كتاب في ما یحتاج إليه الصانع من الأعمال الهندسية Kitāb fī mā yaḥtāj ilayh al-ṣāniʿ min al-aʿmāl al-handasiyya).^[9] This text contains over one hundred geometric constructions, including for a regular heptagon, which have been reviewed and compared with other mathematical treatises. The legacy of this text in Latin Europe is still debated.^[10][11]
• "A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen", (كتاب في ما يحتاج إليه الكتاب والعمال من علم الحساب Kitāb fī mā yaḥtāj ilayh al-kuttāb wa’l-ʿummāl min ʾilm al-ḥisāb).^[9] This is the first book where negative numbers have been used in the medieval Islamic texts.^[6]

He also wrote translations and commentaries on the algebraic works of Diophantus, al-Khwārizmī, and Euclid's Elements.^[6]