Transcription of folios 177r-172v of the Archimedes Palimpsest. (= Archimedes folio Arch69r, The Measurement of the Circle, Stomachion) 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 5669 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 Measurement of the Circle Content: Stomachion 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
ΗΑΓ, καὶ ἡ ὑπὸ ΗΓΒ τῆι ὑπὸ ΗΑΓ ἐστὶν ἴση. καὶ κοινὴ ἡ ὑπὸ ΑΗΓ ὀρθή· καὶ τρίτη ἄρα ἡ ὑπὸ ΗΖΓ τρίτη τῆι ὑπὸ ΑΓΗ. ἰσογώνιον ἄρα τὸ ΑΗΓ τῶι ΓΗΖ τριγώνωι· ἔστιν ἄρα, ὡς ἡ ΑΗ πρὸς ΗΓ, ἡ ΓΗ πρὸς ΗΖ καὶ ἡ ΑΓ πρὸς ΓΖ. ἀλλ’ ὡς ἡ ΑΓ πρὸς ΓΖ, καὶ συναμ φότερος ἡ ΓΑΒ πρὸς ΒΓ· καὶ ὡς συ ναμφότερος ἄρα ἡ ΒΑΓ πρὸς ΒΓ, ἡ ΑΗ πρὸς ΗΓ. διὰ τοῦτο οὖν ἡ ΑΗ πρὸς τὴν ΗΓ ἐλάσσονα λόγον ἔχει ἤ ἤπερ ΒϡΙΑ πρὸς ΨΠ, ἡ δὲ ΑΓ πρὸς τὴν ΓΗ ἐλάσσονα ἢ ὃν ΓΙΓ Γ Δ πρὸς ΨΠ. δίχα ἡ ὑπὸ Γ ΑΗ τῆι ΑΘ· ἡ ΑΘ ἄρα διὰ τὰ αὐτὰ πρὸς τὴν ΘΓ ἐλάσσονο λόγον ἔχει ἢ ὃν ΕΤΚΔ Ε Δ πρὸς ΨΠ ἢ ὃν ΑΩΚΓ πρὸς ΣΜ· ἑκατέρα γὰρ ἑκατέρας ΔΙ ΓΑ· ὥστε ἡ ΑΓ πρὸς τὴν ΓΘ ἢ ὃν ΑΩ ΛΑ Θ πρὸς ΣΜ. ἔτι δίχα ἡ ὑπὸ ΘΑΓ τῆι ΚΑ· καὶ ἡ ΑΚ πρὸς τὴν ΚΓ ἐλάσσονα ἄρα λόγον ἔχει ἢ ὃν ΑΖ πρὸς ΞϚ· ἑκα τέρα γὰρ ἑκάτερα οι Μ ἄρα πρ πρὸς τὴν λόγον ΑΟΣ πρὸς ΞϚ. ἔτι δίχα ἡ ὑπὸ Κ ΑΓ τῆι ΛΑ· ἡ ΑΛ ἄρα πρὸς τὴν ΛΓ ἐλάσ σονα λόγον ἔχει ἢ ὃν τὰ ΒΙϚ Ϛ πρὸς ΞϚ, ἡ δὲ ΑΓ πρὸς ΓΛ ἐλάσσονα ἢ τὰ ΒΙΖ Δ πρὸς ΞϚ. ἀνάπαλιν ἄρα ἡ περίμε τρος τ τοῦ πολυγώνου πρὸς τὴν δμετρ διάμετρον μείζονα λόγον ἔχει ἤ ἤπερ ϚΤΑ Ϛ πρὸς ΖΙΖ Δ, ἅ ἅπερ τῶν ΒΙΖ Δ μείζον μείζονά ἐστιν ἢ τριπλασίονα καὶ δέκα ΟΑ· καὶ ἡ περίμετρος ἄρα τοῦ ϘϚγώ νου τοῦ ἐν τῶι κύκλωι τῆς δμέ διαμέ τρου τριπλασίων ἐστὶ καὶ μείζω μείζων ἢ ὃν ΟΙΑ· ὥστε καὶ ὁ κύκλος ἔτι μᾶλλον τριπλασίων ἐστὶ καὶ μείζω μείζων ἢ ΘΙ Α. ἡ ἄρα τοῦ κύκλου περίμετρος τῆς διαμέτρου τριπλασίων ἐστὶ καὶ ἐλάσ σων μὲν ἢ ἑβδόμωι μέρει, μείζων δὲ ἢ
ΑΡΧΙΜΗΔΟΥΣ ΣΤΟΜΑ ΧIΟΝ Τοῦ λεγομένου στομαχίου ποικί λαν ἔχοντος τα ἐξ ὧν συνέστακε σχημάτων μεταθέσεως θεωρία θεωρίαν ἀναγκαῖον ἡγησάμην πράον πράττον του ὅλου σχάματος μέγεθος θεω ρῶν ἐκθέσθαι, εἴς τε ἃ διαιρεῖται, ἕκαστόν τε αὐτῶν τίνι ἐστὶ ἴσον καὶ ὅ μοιον, ἔτι δὲ καὶ ποῖαι γωνίαι συ νοδοιο λαμβανόμεναι καὶ κα θ’ ἃς εἴρηται πρὸς τὸ τὰς ἐναρμόσεις τῶν ἐξ αὐτῶν γεννωμένων σχα μάτων γιγνώσκεσθαι, εἴτε ἐπ’ εὐ θείας εἰσὶν αἱ γεννώμεναι ἐν τοῖς σχάμασι πλευραί, εἴτε καὶ μικρῶς λιποῦσαι τᾶι θεωρίαι λανθά νουσιν· τὰ γὰρ τοιαῦτα φιλότεχνα· καὶ ἐὰν ἐλάχιστον μὲν λίπηται, τᾶ δὲ θεωρίαι λανθάνηι, οὐ παρὰ τοῦ τ’ ἔστιν ἔκβλητα ἃ συνίσταται. ἔστι μὲν οὖν ἐξ αὐτῶν οὐκ ὀλίγων σχαμάτ σχαμάτων