and fractions as rational magnitudes as well as square and cubic roots as irrational magnitudes. He called square roots "plane irrationalities" and cubic roots "solid irrationalities", and classified the sums or differences of these roots, as well as the results of the roots' additions or subtractions from rational magnitudes, also as irrational magnitudes. He then explained Book X using those rational and irrational magnitudes instead of geometric magnitudes like in the original. His commentaries of the Sphaerica covered book I and parts of book II, none of which survive today. His edition was later updated by Ahmad ibn Abi Said al-Harawi (10th century). Later, Nasir al-Din al-Tusi (1201–1274) dismissed Al-Mahani and Al-Harawi's edition and wrote his own treatment of the Sphaerica, based on the works on Abu Nasr Mansur. Al-Tusi's edition became the most widely known edition of the Sphaerica in the Arabic-speaking world. Al-Mahani also attempted to solve a problem posed by Archimedes in On the Sphere and Cylinder, book II, chapter 4: how to divide a sphere by a plane into two volumes of a given ratio. His work led him to an equation, known as "Al-Mahani's equation" in the Muslim world: . However, as documented later by Omar Khayyam, "after giving it lengthy meditation" he eventually failed to solve the problem. The problem was then considered unsolvable until 10th century Persian mathematician Abu Ja'far al-Khazin solved it using conic sections. Astronomy His astronomical observations of conjunctions as well as solar and lunar eclipses was cited in the zij (astronomical tables) of Ibn Yunus (c. 950 – 1009). Ibn Yunus quoted Al-Mahani as saying that he calculated their timings with an astrolabe. He claimed his estimates of the start times of three consecutive lunar eclipses were accurate to within half an hour. He also wrote a treatise, Maqala fi ma'rifat as-samt li-aiy sa'a aradta wa fi aiy maudi aradta ("On the Determination of the Azimuth for an Arbitrary Time and an Arbitrary Place"), his only known surviving work on astronomy. In it, he provided two graphical methods and an arithmetic one of calculating the azimuth—the angular measurement of a heavenly object's location. The arithmetic method corresponds to the cosine rule in spherical trigonometry, and was later used by Al-Battani (c. 858 – 929). He wrote another treatise, whose title, On the Latitude of the Stars, is known but its content is entirely lost. According to later astronomer Ibrahim