Proposition XII

Let the parallelogram μν be perpendicular to the axis [of the circle] ξo [πρ] [Fig. 11]. Draw θμ and θη and erect upon them two planes perpendicular to the plane in which the semicircle oπρ lies and extend these planes on both sides. The result is a prism whose base is a triangle similar to θμη and whose altitude is equal to the axis of the cylinder, and this prism is 1/4 of the entire prism which contains the cylinder. In the semicircle oπρ and in the square μν draw two straight lines κλ and τ υ at equal distances from πξ; these will cut the circumference of the semicircle oπρ at the points κ and τ, the diameter oρ at σ and ζ and the straight lines θη and θμ at φ and χ. Upon κλ and τ υ construct two planes perpendicular to oρ and extend them towards both sides of the plane in which lies the circle ξoπρ; they will intersect the half-cylinder whose base is the semicircle oπρ and whose altitude is that of the cylinder, in a parallelogram one side of which = κσ and the other = the axis of the cylinder; and they will intersect the prism θημ likewise in a parallelogram one side of which is Fig. 11. equal to λχ and the other equal to the axis, and in the same way the half-cylinder in a parallelogram one side of which = τ ζ and the other = the axis of the cylinder, and the prism in a parallelogram one side of which = νφ and the other = the axis of the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .