A New Proof Using Cones for Archimedes’ Volume Propositions on Conoids and Spheroids

This paper presents a new proof of Archimedes’ volume propositions for conoids and spheroids that unifies the treatment of all three conic sections. While Archimedes established that solids generated by rotating conic sections maintain specific volume ratios to cones with the same base and axis, his original arguments—both the exhaustion method in On Conoids and Spheroids and the mechanical approach in Method—require distinct arguments for each type of conic section. Our approach transforms both the conic solid and its reference cone into combinations of cone-related figures through an axis-stretching technique, then establishes volume ratios using only elementary proportional relationships. This unified geometric treatment reveals the underlying structural unity between different conic sections, while employing only tools from Euclid’s Elements and techniques found throughout Archimedes’ works.