Sphere and Cylinder, On Spirals & Conoids — Analytical Summary

Scope: Heiberg Vol. I, pp. 1–255, 272–499; Vol. II, pp. 1–141 · Heath (1897), pp. 1–90, 99–188

Master Synthesis of Curved Surfaces, Solids & Kinematic Loci

Across these three treatises addressed to Dositheus, Archimedes builds the complete pre-Newtonian integral and differential calculus of circles, spheres, quadrics of revolution, and polar spirals:

1. Convexity & Arc/Surface Length Foundations (*Sphere & Cylinder* I, Assump. 1–5): Because neither Euclid nor Eudoxus possessed an axiom comparing curved arc lengths or curved surface areas against rectilinear bounds, Archimedes formulates ἐπὶ τὰ αὐτὰ κοῖλα (convexity) so that nested convex perimeters and surfaces obey strict monotonicity.

2. Discrete Power Sums as Definite Integrals: - Linear sum Paraboloid segment cylinder cone (Conoids Props. 19–22). - Quadratic sum Spiral sector circle (On Spirals Props. 10, 24), Hyperboloid (Conoids Props. 25–26), and Spheroid (Conoids Props. 27–32).

3. Differential Tangent Construction (*On Spirals* Props. 18–20): First rigorous tangent to a non-conic mechanical curve in mathematical history, yielding .

Summary Table of Exact Area & Volume Ratios

Geometric Figure	Treatise Locus	Exact Canonical Theorem
Sphere surface	Sphere & Cyl. I.33	great circle ()
Sphere & Circumscribed Cylinder	Sphere & Cyl. I.34 Cor.	and
Spherical cap surface	Sphere & Cyl. I.42–43	Circle of radius = slant chord from pole to base rim
Spherical segment volume	Sphere & Cyl. II.2	Cone of same base and height
Cubic sphere division	Sphere & Cyl. II.4 (Eutocius)	via parabola hyperbola; max at
Spiral polar subtangent	On Spirals Props. 18–20	(kinematic rectification of circle)
Spiral first-turn area	On Spirals Prop. 24	of first circle ()
Ellipse area	Conoids Prop. 4	()
Paraboloid segment	Conoids Props. 21–22	inscribed cone ( enveloping cylinder)
Spheroid hemisphere & segment	Conoids Props. 27–32	Half-spheroid cone; segment ratio