A lacuna in Book I of Archimedes' \textit{Sphere and Cylinder} (Q1238790)

Almost all commentators on Archimedes' \textit{Sphere and Cylinder: Book I} observe that there is a gap in the proof of Proposition 9 of that work, namely the unproved assertion that any two of the triangles formed by three generators of a finite isosceles cone are larger than the third triangle formed by these generators. In the present paper the author reviews the proofs -- both faulty and valid -- that have been presented, and then supplies a new one that shows the result is an easy consequence of Proposition 3 of the same work.