Sphere and Cylinder, On Spirals & Conoids — Philological & Mathematical Notes

1. Cicero and the Tomb of Archimedes at Syracuse (75 BC)

In Tusculan Disputations V.23, Cicero records how, serving as quaestor in Sicily in 75 BC, he cleared away brambles outside the Agrigentine Gate of Syracuse and rediscovered the forgotten tomb of Archimedes by recognizing on a small column the sphere and circumscribed cylinder of Sphere and Cylinder I.34 Corollary, accompanied by worn Greek iambic verses declaring that the cylinder’s volume and surface stood in the sesquialter ratio to the sphere. As The Method, Prop. 2 reveals in Archimedes’ own words, he discovered the volume first by weighing slices on a virtual balance, and then conjectured by analogy with Measurement of a Circle Prop. 1 ()!

2. Eutocius’s Recovery of Archimedes’ Doric Cubic Manuscript (II.4)

In Sphere and Cylinder II.4, Archimedes reduces cutting a sphere in a given ratio to dividing a line at so that , adding “an analysis and synthesis of both problems will be given at the end.” For centuries the supplement seemed lost, inspiring Dionysodorus and Diocles to invent independent conic solutions. As Eutocius (eutocius:in-archimedem, Heiberg Vol. III, pp. 130–132; heath:1897, p. xxxvi) recounts, he stumbled upon a worn old book (ἔν τινι παλαιῷ βιβλίῳ) written in Syracusan Doric dialect and using Archimedes’ archaic pre-Apollonian conic names (ὀρθογωνίου κώνου τομή for parabola, ἀμβλυγωνίου κώνου τομή for hyperbola), proving beyond doubt that it was Archimedes’ own missing diorismos!

3. GEOMETOR Cross-Corpus Concordance

• Euclid (`geometor/euclid`) (geometor:euclid): Covers four required cross-corpus citations in Heath’s Euclid: - archimedes:sphere-and-cylinder (Euclid I.Def.4 straight line, X.1 exhaustion, XII.18 spheres); - archimedes:sphere-and-cylinder:ii-2 (Euclid XII commentary on spherical segments); - archimedes:conoids-and-spheroids (Euclid XII.2 affine scaling of circles to ellipses & XII.10 cones); - eutocius:in-archimedem (Eutocius’s catalogue of 12 cube-duplication constructions & II.4 cubic).

• Proclus (`geometor/proclus`) (proclus:in-euclidem): At Friedlein p. 110, Proclus contrasts Euclid’s definition of the straight line (ἥτις ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται) with Archimedes’ metric postulate (ἐλαχίστη τῶν τὰ αὐτὰ πέρατα ἐχουσῶν, Assump. 1), arguing that minimum length is an accidental quantitative symptom rather than the ontological essence of straightness.

• Pappus (`geometor/pappus`) (pappus:synagoge:iv): In Synagoge IV.21–25, Pappus simplifies Archimedes’ spiral quadrature and discusses why Archimedes’ neusis tangents in On Spirals Props. 5–9 employ solid/marked-straightedge means where conic/planar reductions exist.