A systematic ancient geometric treatise that develops the theory of plane sections of a cone, defining and classifying the parabola, ellipse and hyperbola as distinct curves produced by slicing a cone with a plane and then advancing by a sequence of rigorous propositions to determine their fundamental properties: symmetry and centers, constructions with straightedge and compass, relationships among points, lines and diameters, the determination of tangents and many locus and intersection results; the work provides a synthetic, proposition-by-proposition framework that laid the foundation for later analytic and algebraic approaches to conic sections.