The three Banu Musa brothers are almost indistinguishable and most of the information is at this link. However, there is some information specific to the eldest: Jafar Muhammad ibn Musa ibn Shakir.
It certainly appears that of the three brothers, he was the best mathematician. In addition to making perhaps the major contribution to the geometry text described above, Jafar Muhammad also wrote Premises of the book of conics which was a critical revision of Apollonius's Conics.
However, Jafar Muhammad was also the most politically active of the brothers, particularly in the last part of his life when the Turks were gaining control of the empire. It appears that his dispute with al-Kindi coloured Jafar Muhammad actions and he opposed anyone who had been a friend of al-Kindi's. For this reason he campaigned successfully to have al-Musta'in become Caliph. However, there were internal struggles and al-Musta'in's brother besieged Baghdad with his army. In fact Jafar Muhammad was sent by al-Musta'in to negotiate the terms of his surrender with the besieging forces.
Article by: J J O'Connor and E F Robertson