Abu-Abdullah Muhammad ibn Īsa Māhānī (, flourished c. 860 and died c. 880) was a Persian mathematician and astronomer born in Mahan, (in today Kermān, Iran) and active in Baghdad, Abbasid Caliphate. His known mathematical works included his commentaries on Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus' Sphaerica, as well as two independent treatises. He unsuccessfully tried to solve a problem posed by Archimedes of cutting a sphere into two volumes of a given ratio, which was later solved by 10th century mathematician Abū Ja'far al-Khāzin. His only known surviving work on astronomy was on the calculation of azimuths. He was also known to make astronomical observations, and claimed his estimates of the start times of three consecutive lunar eclipses were accurate to within half an hour. Biography Historians know little of Al-Mahani's life due to lack of sources. He was born in Mahan, Persia (hence the Nisba Al-Mahani). He was active in the 9th century CE or 3rd century AH, lived in Baghdad c. 860 and died c. 880. From a reference in Ibn Yunus' Hakimite Tables, he was known to make astronomical observations between 853 and 866, allowing historians to estimate the time of his life and activities. Works Mathematics His works on mathematics covered the topics of geometry, arithmetic, and algebra. Some of his mathematical work might have been motivated by problems he encountered in astronomy. The 10th century catalogue Kitab al-Fihrist mentions al-Mahani's contributions in mathematics but not those in astronomy. He also worked on current mathematical problems at his time. He wrote commentaries on Greek mathematical works: Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus' Sphaerica. In his commentaries he added explanations, updated the language to use "modern" terms of his time, and reworked some of the proofs. He also wrote a standalone treatise Fi al-Nisba ("On Relationship") and another on the squaring of parabola. His commentaries on the Elements covered Books I, V, X and XII; only those on Book V and parts of those on book X and XII survive today. In the Book V commentary, he worked on ratio, proposing a theory on the definition of ratio based on continued fractions that was later discovered independently by Al-Nayrizi. In the Book X commentary, he worked on irrational numbers, including quadratic irrational numbers and cubic ones. He expanded Euclid's definition of magnitudes—which included only geometrical lines—by adding integers