Al-Mahani's Sphere and Cylinder commentary

Al-Mahani's Sphere and Cylinder commentary is associated with Archimedes' book II, proposition4: dividing a sphere into pieces with a prescribed volume ratio. The short label is descriptive, based on TDV's work list, not a manuscript title page inspected by the museum ¹.

The sources connect the work with a cubic reduction involving a number and a square. Al-Mahani did not solve it; al-Khazin later solved the equation using conics. The commentary does not become a general cubic-solution manual, and the importance assigned to formulation in Khayyam's reported account does not supply a missing final success ^{2 3}.

The invention article supplies a modern volume derivation and independent examples, not the original text or a complete recovery of its proof order. We have not examined al-Mahani's reasoning line by line or discovered our5/32 example in a copyist's manuscript ⁴.

TDV reports Leiden remains and a commentary by al-Quhi; Dold-Samplonius qualifies the commentator as probably al-Quhi. That qualification remains, and a work fragment is not equated with a complete later commentary. Multiple reference descriptions are not a museum manuscript catalogue based on fresh inspection ^{1 3}.

Dating remains the ninth-century lifetime horizon, not an eclipse observation or a newly found completion signature. The accounts read here establish no precise composition day. The biographies, work list and DSB reproduction were read, not the Leiden fragment, al-Mahani's original or all the editions listed in their bibliographies ^{5 6}.