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# Muhyi l'din al-Maghribi

(About 1220 in Spain- about 1283 in Maragha, Iran)

Muhyi al-din al-Maghribi was an eminent astronomer who was born in Spain, but who first worked in Damascus in Syria. His life seems to have been greatly affected by the wars of the period and he seems to have found favour with the winning side eventually working withal-Tusi at the Mongol observatory at Maragha, Iran.

In 1256 the castle of Alamut was attacked by the forces of the Mongol leader Hulegu, a grandson of Genghis Khan, who was at that time set on extending Mongol power in Islamic areas. Some claim thatal-Tusi, who was in the castle at this time, betrayed the defenses of Alamut to the invading Mongols. Certainly Hulegu's forces destroyed Alamut and since Hulegu was himself interested in science, he treatedal-Tusi with great respect. Hulegu attacked Baghdad in 1258, laid siege to the city, and entered it in February 1258. Hulegu, however, had made Maragha, in the Azerbaijan region of northwestern Iran, his capital.

Muhyi l'din went to Maragha in 1258 as a guest of Hulegu.Al-Tusi and Muhyi l'din were involved in the construction of an Observatory. Work began in 1259 west of Maragha, and traces of it can still be seen there today. The observatory at Maragha became operational in 1262. There is a unique manuscript by Muhyi l'din in which he lists precise observations made at the Maragha Observatory between 1262 and 1274. The author of [1] discusses the three observations of the sun and the mathematical methods which Muhyi l'din used to find the solar eccentricity andapogee [2].

Perhaps Muhyi l'din is most famous for his work on trigonometry. He wrote Book on the theorem ofMenelaus and Treatise on the calculation of sines. In this second work he used interpolation to calculate an approximate value for the sine of one degree. He did this by two different methods, then compared the values he obtained achieving an accuracy of 4 places. A more accurate value was not obtained until the work ofQadi Zada andal-Kashi. In doing this work Muhyi l'din also found an approximate value for which he compared with the bounds obtained byArchimedes using 96inscribed [3]andcircumscribed[4]polygons.

Muhyi l'din also considered the classical problem ofdoubling the cube which he approached byHippocrates' method of finding two mean proportionals between two given lines.

Another important aspect of Muhyi l'din's work was the critical commentaries which he produced on some of the classic Greek works such asEuclid's Elements,Apollonius'sConics,Theodosius's Spherics, andMenelaus's Spherics. A particularly important commentary by Muhyi l'din is that on Book XV of the Elements (which was not written byEuclid).Hypsicles added a Book XIV to the Elements which dealt with the mensuration of the regulardodecahedron[5] andicosahedron[6]. Later Book XV was written in Arabic by an unknown author, perhaps using Greek works which are now lost. Book XV has common features with Book XIV byHypsicles but contains considerably more.

The original Arabic version of Book XV is lost but there are four surviving manuscripts containing Muhyi l'din's commentary on it. We know that there was more than one version of the Arabic Book XV, for recently a Hebrew translation of Book XV has been discovered which has been translated from a different version to that which Muhyi l'din used for his commentary. Muhyi l'din's Book XV contains [7]:-

... the ratios of (1) the edges, (2) the faces, (3) the surface areas, (4) the perpendicular distances from the centre to a face and (5) the volumes of the five regular polyhedra inscribed in one sphere.

## Article by:

J J O'Connor and E F Robertson Note:

1-G Saliba, Solar observations at the Maraghah observatory before 1275 : a new set of parameters, J. Hist. Astronom. 16 (2) (1985), 113-122.

2-The apogee is the point where a heavenly body is furthest away from the centre of its orbit.The nearest point is called the perigee.

3-A circle is said to be inscribed to a triangle or other polygon if the edges are tangents to the circle.

The polygon is than said to becircumscribed to the circle.

The (unique) circle inscribed to a triangle is called the incircle and its centre is the incentre.

4-A circle is said to be circumscribed to a triangle or other polygon if the vertices of the polygon lie on the circle.

The polygon is than said to beinscribed in the circle.

The (unique) circle circumscribed to a triangle is called the circumcircle and its centre is the circumcentre.

5-A dodecahedron is aregular polyhedron with 12 faces each of which is a regular pentagon.

6-An icosahedron is aregular polyhedron with 20 faces each of which is an equilateral triangle.

7-J P Hogendijk, an Arabic text on the comparison of the five regular polyhedra: "Book XV" of the "Revision of the Elements" by Muhyi al-Din al-Maghribi, Z. Gesch. Arab.-Islam. Wiss. 8 (1993), 133-233.