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 that al-Tusi, who was in the castle at this time, betrayed the defences of Alamut to the invading Mongols. Certainly Hulegu's forces destroyed Alamut and since Hulegu was himself interested in science, he treated al-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  discusses the three observations of the sun and the mathematical methods which Muhyi l'din used to find the solar eccentricity and apogee.
Perhaps Muhyi l'din is most famous for his work on trigonometry. He wrote Book on the theorem of Menelaus 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 of Qadi Zada and al-Kashi. In doing this work Muhyi l'din also found an approximate value for π which he compared with the bounds obtained by Archimedes using 96 inscribed and circumscribed polygons.
Muhyi l'din also considered the classical problem of doubling the cube which he approached by Hippocrates' 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 as Euclid's Elements, Apollonius's Conics, Theodosius's Spherics, and Menelaus's Spherics. A particularly important commentary by Muhyi l'din is that on Book XV of the Elements (which was not written by Euclid). Hypsicles added a Book XIV to the Elements which dealt with the mensuration of the regular dodecahedron and icosahedron. 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 by Hypsicles 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 :-
... 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