Transcription of folios 004r-005v of the Archimedes Palimpsest. (= Archimedes folio Arch47r, 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 5523 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
πυραμὶς βάσιν μὲν ἔχουσα ἰ σόπλευρον τρίγωνον τὸ ΑΒΓ, καὶ ἐ πεζεύχθωσαν αἱ ΔΑ ΔΓ ΔΒ· λέ γω ὅτι τὰ ΑΔΒ ΑΔΓ τρίγωνα ἴσα ἐστὶ τριγώνωι, οὗ ἡ μὲν βάσις ἴση ἐστὶ τῆι περιμέτρωι τοῦ ΑΒΓ τρι γώνου, ἡ δὲ ἀπὸ τῆς κορυφῆς ἐπὶ τὴν βάσιν κάθετος ἴση τῆι καθέτωι τῆι ἀπὸ τοῦ Δ ἐπὶ τὴν ΒΓ ἀγομένην. ἤχθωσαν γὰρ κά θετοι αἱ ΔΚ ΔΛ ΔΜ· αὗται ἄρα ἴσαι ἀλλήλαις εἰσίν . καὶ κείσθω τρί γωνον τὸ ΕΖΗ ἔχον τὴν μὲν ΕΖ βάσιν τῆ περιμέτρωι τοῦ ΑΒΓ τριγώνου ἴσην, τὴν δὲ ΗΘ κάθε τον τῆι ΔΛ ἴσην. ἐπεὶ οὖν τὸ ὑπὸ τῶν ΒΓ ΔΛ διπλάσιόν ἐστιν τοῦ ΔΒΓ τριγώνου, ἔστι δὲ καὶ τὸ μὲν ὑπὸ τῶν ΑΒ ΔΚ διπλάσιον τοῦ ΑΒΔ τριγώνου, τὸ δὲ ὑπὸ ΑΓΔΜ διπλάσιον τοῦ ΑΔΓ τριγώνου, τὸ ἄρα ὑπὸ τῆς περιμέτρου τοῦ ΑΔΓ τριγώνου, τουτ τουτέστι τῆς ΕΖ, καὶ τῆς ΔΛ, τουτ τουτέστι τῆς ΗΘ, διπλάσιόν ἐστι τῶν ΑΔΒ ΒΔΓ ΑΔΓ τριγώνων. ἔστι δὲ καὶ τὸ ὑπὸ ΕΖΗΘ διπλάσι διπλάσιον τοῦ ΕΗΖ τριγώνου· ἴσον ἄρα τὸ ΕΖΗ τρίγωνον τοῖς ΑΛΒ ΒΔΓ ΑΔΓ τρι γώνοις. Θ Ἐὰν περὶ κῶνον ἰσοσκελῆ πυρα μὶς περιγραφῆ ἡ ἐπιφάνεια τῆς πυραμίδος χωρὶς τῆς βάσεως ἴση ἐστὶν τριγώνωι βάσιν μὲν ἔχοντι τὴν ἴσην τῆι περιμέτρωι τῆς βάσεως, ὕψος δὲ τὴν πλευρὰ πλευρὰν τοῦ κώνου. ἔστω κῶνος οὗ βάσις ὁ ΑΒΓ κύκλος καὶ πυραμὶς περι γεγράφθω ὥστε τὴν βάσιν αὐτῆς τουτ τουτέστι τὸ ΔΕΖ πολύγωνον περιγε γραμμένον περὶ τὸν ΑΒΓ κύκλον εἶναι· λέγω ὅτι ἡ ἐπιφάνεια τῆς πυ ραμίδος χωρὶς τῆς βάσεως ἴση ἐστὶ τῶι εἰρημένωι τριγώνωι. ἐπεὶ γὰρ ὁ ἄξων τοῦ κώνου ὀρθός ἐστι πρὸς τὴν βάσιν τουτ τουτέστι πρὸς τὸν ΑΒΓ κύκλον καὶ αἱ ἀπὸ τοῦ κέντρου τοῦ κύκλου ἐπὶ τὰς ἁφὰς ἐπεζευγμέναι εὐθεῖαι κάθετοί εἰσιν ἐπὶ τὰς ἐφαπτομέ νας, ἔσονται ἄρα καὶ αἱ ἀπὸ τῆς κορυφῆς τοῦ κώνου ἐπὶ τὰς ἁφὰς ἐπεζευγμέναι κάθετοι ἐπὶ τὰς ΔΕ ΖΕ ΖΔ. αἱ ΗΑ ΗΒ ΗΓ ἄρα αἱ εἰρημέναι κάθετοι ἴσαι εἰσὶν ἀλλήλαις· πλευραὶ γάρ εἰσιν τοῦ κώνου. κείσθω δὴ τὸ τρί γωνον τοῦ ΘΚΛ ἴσην ἔχων τὴν μὲν ΘΚ τῆι περιμέτρωι τοῦ ΔΕΖ τριγών τριγώνου , τὴν δὲ ΛΜ κάθετον ἴσην τῆι ΗΑ. ἐπεὶ οὖν τὸ μὲν ὑπὸ ΔΕ ΑΗ διπλάσιόν ἐστιν τοῦ ΔΕΗ τριγώνου, τὸ δὲ ὑπὸ ΔΖ ΗΒ διπλάσιόν ἐστιν τοῦ ΔΖΗ τριγώνου, τὸ δὲ ὑπὸ ΕΖ ΓΗ διπλάσιόν ἐστιν τοῦ Ε ΗΖ τριγώνου, ἔστιν ἄρα τὸ ὑπὸ τῆς ΘΚ και τῆς