Transcription of folios 003v-006r of the Archimedes Palimpsest. (= Archimedes folio Arch46v, 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 5522 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
πολύγωνον περιγραφῆι, ἡ τοῦ περιγραφέντος πολυγώνου περί μετρος μείζων ἐστὶν τῆς περιμέ τρου τοῦ κύκλου. περὶ γὰρ κύκλο κύκλον πολύγωνον περιγεγράφθω τὸ ὑ ποκείμενον. λέγω ὅτι ἡ περίμε τρος τοῦ πολυγώνου μείζων ἐστὶν τῆς περιμέτρου τοῦ κύκλου. ἐπεὶ γὰρ συναμφότερος ἡ ΒΑΛ μείζων ἐστὶ τῆς ΒΛ περιφερείας διὰ τὸ τὰ αὐτὰ πέρατα ἔχουσαν πε ριλαμβάνειν τὴν περιφέρεια περιφέρειαν , ὁμοίως καὶ συναμφότερος μὲν ἡ ΔΓ ΓΒ τῆς ΔΒ, συναμφότερος δὲ ἡ ΛΚ ΚΘ τῆς ΛΘ, συναμφό τερος δὲ ἡ ΖΗΘ τῆς ΘΖ, ἔτι δὲ συναμ φότερος ἡ ΔΕ ΕΖ τῆς ΔΖ, ὅλη ἄρα ἡ περίμετρος τοῦ πολυγώνου μεί ζων ἐστὶν τῆς περιφερείας τ τοῦ κύκλ κύκλου . Β Δύο μεγεθῶν ἀνίσων δοθέντων δυνατόν ἐστιν εὑρεῖν δύο εὐθείας ἀνίσους, ὥστε τὴν μείζονα εὐθεῖα εὐθεῖαν πρὸς τὴν ἐλάσσονα λόγον ἔχειν ἐλάσ σονα ἤτοι μείζων μέγεθος πρὸς τὸ ἔλασσον. ἔστω δύο μεγέθη ἄνισα τὰ ΑΒΔ, καὶ ἔστω μείζων τὸ ΑΒ. λέγω ὅτι δυνατόν ἐστι δύο εὐθείας ἀνί σους εὑρεῖν τὸ εἰρημένον ἐπίταγ μα ποιούσας. κείσθω διὰ τοῦ Β τοῦ Α Εὐκλείδου τῶι Δ ἴσον τὸ ΒΓ, καὶ κείσθω τις εὐθεῖα γραμμὴ ἡ ΖΗ· τὸ δὴ ΓΑ ἑαυτῶι ἐπισυντιθέμεν ἐπισυντιθέμενον ὑπερέξει τοῦ Δ. πεπολλαπλασι άσθω οὖν, καὶ ἔστω τὸ ΑΘ, καὶ ὁσαπλά σιόν ἐστι τὸ ΑΘ τοῦ ΑΓ, τοσαυταπλά σιος ἔστω ἡ ΖΗ τῆς ΖΕ· ἔστιν ἄρα, ὡς τὸ ΘΑ πρὸς ΑΓ, οὕτως τὸ ΖΗ πρὸς ΗΕ· καὶ ἀνά παλίν ἐστιν, ὡς ἡ ΗΕ πρὸς ΗΖ, οὕτως τὸ ΑΓ πρὸς ΑΘ. καὶ ἐπεὶ μεῖζόν ἐστιν τὸ ΑΘ τοῦ Δ, τουτέστι τοῦ ΓΒ, τὸ ἄρα ΓΑ πρὸς τὸ ΑΘ λόγον ἐλάσσονα ἔχει ἤ ἤπερ τὸ ΓΑ πρὸς ΓΒ. ἀλλ’ ὡς τὸ ΓΑ πρὸς ΑΘ, οὕτως ἡ ΕΗ πρὸς ΗΖ· ἡ ΕΗ ἄρα πρὸς ΗΖ ἐλάσσονα λόγον ἔχει ἤπερ τὸ ΓΑ πρὸς τὸ ΓΒ· καὶ συνθέντι ἡ ΕΖ ἄρα πρὸς ΖΗ ἐλάσσονα λόγον ἔχει ἤ ἤπερ τὸ ΑΒ πρὸς ΒΓ διάλλη μα. ἴσον δὲ τὸ ΒΓ τῶι Δ· ἡ ΕΖ ἄρα πρὸς ΖΗ ἐλάσσονα λόγον ἔχει ἤπερ τὸ ΑΒ πρὸς τὸ Δ. εὑρημέναι εἰσὶν ἄρα δύο εὐ θεῖαι ἄνισοι ποιοῦσαι τὸ εἰρημέ νον ἐπίταγμα τουτέστιν τὴν μείζο να πρὸς τὴν ἐλάσσονα λόγον ἔχoν ἐλάσσονα ἢ τὸ μεῖζον μέγεθος πρὸς τὸ ἔλασσον.