# Habash al-Hasib al-Marwazi

Ahmad ibn 'Abdallah Habash Hasib Marwazi (766 - d. after 869 in Samarra, Iraq[1] ) was a north-eastern Iranian[2][3][4] astronomer,[5] geographer, and mathematician from Merv in Khorasan who for the first time described the trigonometric ratios: sine, cosine, tangent and cotangent.

He flourished in Baghdad, and died a centenarian after 869. He worked under the Abbasid caliphs al-Ma'mun and al-Mu'tasim.

## Work

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

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
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)