Transcription of folios 109r-106v of the Archimedes Palimpsest. (= Archimedes folio Arch45r, Spiral Lines, 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 5603 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 Spiral Lines 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
καὶ τὸ τρίτον μέρος τοῦ ἀπὸ τ τᾶς ΗΑ, τὸ δὲ ΝΠ χωρίον ποτὶ τὸ Π τ τοῦ τον ἔχει τὸν λόγον, ὃν τὰ συναμ φότερα τό τε ὑπὸ τᾶν ΗΘΑ καὶ τὸ τρίτον μέρος τοῦ ἀπὸ τᾶς ΗΑ τετραγώνου ποτὶ συναμφότερ συναμφότερον τό τε ὑπὸ τᾶν ΗΑΘ καὶ τὸ τρίτο τρίτον μέρος τοῦ ἀπὸ τᾶς ΗΑ τετρα γώνου, ἕξει καὶ τὸ Ξ ποτὶ τὸ Π τοῦτον τὸν λόγον, ὃν ἔχει συναμ φότερα τό τε ὑπὸ τᾶν ΘΗ ΗΑ καὶ δύο τριταμόρια τοῦ ἀπὸ τᾶς ΗΑ ποτὶ συναμφότερον τό τε ὑπὸ τᾶν ΘΗ ΗΑ καὶ δύο τρι ταμόρια τοῦ ἀπὸ τᾶς ΗΑ ποτὶ συναμφότερον τό τε ὑπὸ τᾶν ΘΗ ΗΑ καὶ τὸ τρίτον μέρος τοῦ ἀπὸ τᾶς ΗΑ. τὰ δὲ συναμφό τερα τό τε ὑπὸ τᾶν ΘΗ ΗΑ καὶ δύο τριταμόρια τοῦ ἀπὸ τᾶς ΗΑ ποτὶ συναμφότερα τό τε ὑπὸ τᾶν ΘΗΑ καὶ τὸ τρίτον μέ ρος τοῦ ἀπὸ τᾶς ΗΑ τετραγώ νου τοῦτον ἔχει τὸν λόγον, ὃν ἔχει συναμφότερα ἅ τε ΘΗ καὶ δύο τριταμόρια τᾶς ΗΑ ποτὶ συναμφοτέραν τὰν ΘΗ καὶ τὸ τρίτον μέρος τᾶς ΗΑ· δῆλ δῆλον οὖν, ὅτι καὶ τὸ Ξ χωρίον ποτὶ τὸ Ν χωρίον τοῦτον ἔχει τὸν λόγο λόγον , ὃν συναμφότερα ἅ τε ΘΗ καὶ δύο τριταμόρια τᾶς ΗΑ ποτὶ συναμφότερον τὰν ΘΗ καὶ τὸ τρίτον μέρος τᾶς ΗΑ. ΑΡΧΙΜΗΔ ΑΡΧΙΜΗΔους ΠΕΡΙ ΕΛΙΚΩΝ ΑΡΧΙΜΗΔΟΥΣ ΠΕΡΙ Τ ΤΗΣ ΣΦΑΙΡΑΣ ΚΑΙ ΚΥΛΙ ΚΥΛΙΝ ΔΡΟΥ Ἀρχιμήδης Δοσιθέωι χαίρειν πρότερον μὲν ἀπέσταλκά σοι τῶν ὑφ’ ἡμῶν τεθεωρημέ νων γράψας μετὰ ἀποδείξε ἀποδείξεως , ὅτι πᾶν τμῆμα τὸ περιεχόμε νον ὑπό τε εὐθείας καὶ ὀρθογω νίου κώνου τομῆς ἐπίτριτόν ἐστι τριγώνου τοῦ τὴν αὐτὴν βάσιν ἔχοντος τῶι τμήματι καὶ ὕψ ὕψος ἴσον· ὕστερον δὲ ἡμῖν ὑποπε σόντων θεωρημάτων ἀξίων λόγου πεπραγματεύμεθα περὶ τὰς ἀποδείξεις αὐτῶν. ἔστι δὲ τάδε· πρῶτον μέν, ὅτι πάσης σφαίρας ἡ ἐπιφάνεια τετραπλα σία ἐστὶν τοῦ μεγίστου κύκλου τῶν ἐν αὐτῆι· ἔπειτα δέ, ὅτι παντὸς τμήματος σφαίρας τῆι ἐπιφα νείαι ἴσος ἐστὶ κύκλος, οὗ ἡ ἐκ τοῦ κέντρου ἴση ἐστὶ τῆι εὐθείαι τῆι ἀ πὸ τῆς κορυφῆς τοῦ τμήματος ἀγομένηι ἐπὶ τὴν περιφέρειαν τ τοῦ κύκλου, ὅς ἐστι βάσις τοῦ τμάματος·