Habash al-Hasib al-Marwazi

Habash al-Hasib al-Marwazi (766 – d. after 869 in Samarra, modern Iraq) was a Persian astronomer, geographer, and mathematician from Merv in Khorasan, who was the first to describe the trigonometric ratios tangent, and cotangent.

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

Work
Ahmad ibn 'Abdallah Habash Hasib Marwazi made astronomical observations from 825 to 835, and compiled three zijes (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.

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:


 * 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)