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 youngest brother:
al-Hasan ibn Musa ibn Shakir.
He wrote The elongated circular figure which is a work
on the ellipse. This book is lost
except for a fragment in Hebrew of a compilation by Ibn al-Samh. From this
fragment Rashed in [3] deduces that al-Hasan had two objectives. One objective
was to measure a curved area while the other was to study the geometric
properties of curves. Rashed claims, as we have suggested above, that while Archimedes'
texts were being translated into Arabic for the first time, the Banu Musa
(perhaps al-Hasan in particular) was trying to give new proofs of the Greek
results as well as trying to prove results going beyond what the Greeks had
achieved.