Transcription of folios 034r-029v of the Archimedes Palimpsest. (= Archimedes folio Arch52r, Sphere and Cylinder) Archimedes Sponsor The Owner of the Archimedes Palimpsest Responsible for primary transcription (Dublin Core creator) Reviel Netz Responsible for primary transcription (Dublin Core creator) Nigel Wilson Contributor Mike Toth Contributor William Noel Contributor Doug Emery Contributor Alexander Lee Contributor Neel Smith Contributor Christopher Blackwell Contributor Jennifer Adams Contributor Jennifer Curtin Contributor Christopher D'Alessandro Contributor William Dolan Contributor Scott Dubè Contributor Michael Kinney Contributor Stephanie Wheeler Contributor Joshua Whelan Contributor Alana L. Bates Contributor Mary Katherine Benson Contributor Edwin Ranier Brenegar Contributor Harry Briggs Contributor Andrew P. Cannon Contributor Katie Elizabeth Crumpton Contributor Katelyn Marie Ellis Contributor Matthew David Goodson Contributor Bryan Alton Keller Contributor Bethanie V. Kemper Contributor Claire Chamberlyn Kitchens Contributor Adam Charles Race Contributor Peter Eric Soder Contributor Charles David Stolper Contributor Jiayang Wu 5537 Owner of the Archimedes Palimpsest 2008

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 Privately owned parchment codex: "The Archimedes Palimpsest". Multispectral Digital Image Product of the Archimedes Palimpsest (The Owner of the Archimedes Palimpsest, 2008). Heiberg, J. L., Archimedis Opera omnia cum commentariis Eutocii (Leipzig: Teubner, 1910–15; reprinted 1972). Christie’s New York, 29th October 1998 Sale, no. 9058, The Archimedes Palimpsest. A. Papadopoulos-Kerameus, Hierosolymitike Bibliotheke, vol. 4 (St Petersburg, 1899), 329–331, MS 355. 10th Century Manuscript 13th Century Manuscript Archimedes Palimpsest Byzantine Manuscript Content: Against Diondas Content: Against Timandros Content: Archimedes Content: Aristotle Content: Categories Content: Hyperides Content: J. L. Heiberg Content: Method Content: On Floating Bodies Content: On Spiral Lines Content: On the Equilibrium of Planes Content: On the Measurement of the Circle Content: On the Sphere and Cylinder Content: Stomachion Foliation scheme: Undertext foliation, ordered by sequence of undertext Foliation scheme: Undertext foliation, ordered by sequence of columnar undertext Greek Manuscript J. L. Heiberg Parchment Manuscript Private Collection accented ancient Greek in Unicode-C Greek characters English Content: Archimedes Content: On the Sphere and Cylinder Archimedes Palimpsest Greek Manuscript Byzantine Manuscript Parchment Manuscript 13th Century Manuscript 10th Century Manuscript Private Collection Foliation scheme: Undertext foliation, ordered by sequence of columnar undertext
παραλλήλων ἐπιπέδων καὶ τῆς ἴσης ἀμφοτέραις ταῖς ἐκ τῶν κέντρων τῶν παραλλήλων ἐπι πέδων. καὶ τῆς ἴσης ἀμφοτέραις ταῖς ἐκ τῶν κέντρων τῶν κύκλω κύκλων τῶν ἐν τοῖς παραλλήλοις ἐπιπέ δοις ἔστω κῶνος, οὗ τὸ διὰ τοῦ ἄξονος τρίγωνον ἴσον τῶι ΑΒΓ, καὶ τετμήσθω παραλλήλωι ἐπιπέ δωι τῆι βάσει, καὶ ποιείτω τομὴν τὴν ΔΕ, ἄξων δὲ τοῦ κώνου ἔστω ὁ ΒΗ, κύκλος δέ τις ἐκκείσθω, οὗ ἡ ἐκ τοῦ κέντρου μέση ἀνάλογόν ἐστι τῆς τε ΑΔ καὶ συναμφοτέρου τῆς ΔΖ ΗΑ, ἔστω δὲ ὁ κύκλος ὁ Θ· λέγω ὅτι ὁ Θ κύκλος ἴσος ἐστὶ τῆι ἐπιφα νείαι τοῦ κώνου τῆι μεταξὺ τῶν ΔΕ ΑΓ. ἐκκείσθωσαν γὰρ κύκλοι οἱ ΛΚ καὶ τοῦ μὲν Κ κύκλου ἡ ἐκ τοῦ κέντρου δυνάσθω τὸ ὑπὸ τὸ ΒΔΖ, τοῦ δὲ Λ ἡ ἐκ τοῦ κέντρου δυ νάσθω τὸ ὑπὸ ΒΑΗ· ὁ μὲν ἄρα Λ κύκλος ἴσος ἐστὶν τῆι ἐπιφανείαι τ τοῦ ΔΕΒ. καὶ ἐπεὶ τὸ ὑπὸ τῶν ΒΑ ΑΗ ἴσ ἴσον ἐστὶ τῶ τε ὑπὸ τῶν ΒΔ ΔΖ καὶ τῶι ὑπὸ τῆς ΑΔ καὶ συναμφοτέρου τῆς ΔΖ ΑΗ διὰ τὸ παράλληλον εἶναι τὴν ΔΖ τῆι ΑΗ, ἀλλὰ τὸ μὲν ὑπὸ ΑΒ ΑΗ δύναται ἡ ἐκ τοῦ κέντρου τ τοῦ Λ κύκλου, τὸ δὲ ὑπὸ ΒΔ ΔΖ δύναται ἡ ἐκ τοῦ κέντρου τοῦ Κ κύκλου, τὸ δὲ ὑπὸ τῆς ΔΑ καὶ συναμφοτέ ρου τῆς ΔΖ ΑΗ δύναται ἡ ἐκ τοῦ κέντρου τοῦ Θ, τὸ ἄρα ἀπὸ τῆς ἐκ τοῦ κέντρου τοῦ Λ κύκλου ἴσον ἐστὶ τοῖς ἀπὸ τῶν ἐκ τῶν κέντρων τῶν ΚΘ κύκλων· ὥστε καὶ ὁ Λ κύκλος ἴσος ἐστὶ τοῖς ΚΘ κύκλοις. ἀλλ’ ὁ μὲν Λ ἴσος ἐστὶ τῆι ἐπιφανείαι τοῦ ΒΑΓ κώνου, ὁ δὲ Κ τῆι ἐπιφανείαι τοῦ ΔΒΕ κώνου· λοιπὴ ἄρα ἡ ἐπιφάνεια τοῦ κώνου ἡ μεταξὺ τῶν παραλλήλων ἐπιπέδων τῶν ΔΕ ΑΓ ἴση ἐστὶ τῶι Θ κύκλωι. Ἔστω τὸ παραλληλόγραμμον τὸ ΒΑΗ, καὶ διάμετρος αὐτοῦ ἔστω ἡ ΒΗ. καὶ τετμήσθω ἡ ΒΑ πλευρά, ὡς ἔτυχεν, κατὰ τὸ Δ, καὶ διὰ τοῦ Δ ἤχθω πα ράλληλος τῆι ΑΗ ἡ ΔΘ, διὰ δὲ τοῦ Ζ τῆι ΒΑ ἡ ΚΛ· λέγω ὅτι τὸ ὑπὸ ΒΑΗ ἴσ ἴσον ἐστὶ τῶ τε ὑπὸ ΒΔΖ καὶ τῶι ὑπὸ ΔΑ καὶ συναμφοτέρου τῆς ΔΖ ΑΛ. ἐπεὶ γὰρ τὸ μὲν ὑπὸ ΒΑΗ ὅλον ἐστὶ τὸ ΒΗ, τὸ δὲ ὑπὸ ΒΔΖ τὸ ΒΖ, τὸ δὲ ὑπὸ ΔΑ καὶ συ ναμφοτέρου τῆς ΔΖ ΑΛ ὁ ΜΝΞ γνώμων· τῶ μὲν γὰρ ὑπὸ ΔΑΗ ἴ σον ἐστὶν τὸ ΚΗ διὰ τὸ ἴσον εἶναι τὸ ΚΘ παραπλήρωμα τῶ ΔΛ πα ραπληρώματι, τὸ δὲ ὑπὸ ΔΑ ΔΖ τὸ ΔΛ· ὅλον ἄρα τὸ ΒΗ, ὅπερ ἐστὶν τὸ ὑπὸ ΒΑΗ, ἴσον ἐστὶ τῶι τε ὑπὸ ΒΔΖ