Transcription of folios 027r-022v of the Archimedes Palimpsest. (= Archimedes folio Arch01r, Planes In Equilibrium) 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 5529 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 Equilibrium of Planes 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
τᾶς μεγίστας. ἔστωσαν τέσσαρες γραμμαὶ ἀνάλογον αἱ ΑΒ ΒΓ ΔΒ καὶ ὃν μὲν ἔχει λόγον ἁ ΒΕ ποτὶ ΕΑ, τοῦτον ἐχέτω ἁ ΖΑ ποτὶ τὰ τρί α Ε τᾶς ΑΔ, ὃν δὲ λόγον ἔχει ἁ ἴ σα τᾶ Β τᾶς ΑΒ καὶ Δ τᾶς ΒΓ καὶ Ϛ τᾶς ΒΔ κ καὶ Γ τᾶς ΒΕ ποτὶ τὰν ἴσαν τᾶ Ε τᾶς ΑΒ καὶ Ι τᾶς ΓΒ καὶ Ι τᾶς ΒΔ καὶ Ε τᾶς ΒΕ, τοῦτον ἐχέτω τὸ τὸν λόγον ἁ ΗΕ ποτὶ τὰν ΑΔ. δεικτ δεικτέον , ὅτι τὰ ΑΖΕ δύο πεμπταμόριά ἐν τι τᾶς ΑΒ. ἐπεὶ γὰρ ἀνάλογόν ἐντι αἱ ΑΒ ΒΓ ΒΔ ΔΕ καὶ αἱ ΑΓ ΓΔ ΔΕ ἐν τῶι αὐτῶι λόγωι ἐντί, καὶ συναμ φότερος ἁ ΑΒ ΒΓ πρὸς τὰν Β τ τᾶς ΒΔ, ἔχει τὸν αὐτὸν λόγον, ὃν ἁ ΑΔ ποτὶ τὰν ΔΕ, καὶ συναμφότερος ἁ ΔΒ ΒΓ ποτὶ τὰν Ε, καὶ πάντα πρὸς πάντα· τὸν αὐτὸν ἄρα λόγο λόγον ἔχει ἁ ΑΔ ποτὶ τὰν ΔΕ, ὃν ἁ ἴσα τᾶ τε Β τᾶς ΑΒ καὶ τᾶ Γ τᾶς ΓΒ καὶ ἁ ΔΒ καὶ ἁ ΔΒ ποτὶ τὰν ἴσαν τᾶ τε Β τᾶς ΒΔ καὶ τὰν ΒΕ, ὃν δὲ λόγον ἔχει ἁ ἴσα τᾶ τε Β τᾶς Β καὶ τᾶ Δ καὶ τᾶ Δ τᾶς ΒΓ καὶ τᾶ Δ τᾶς ΒΔ καὶ τᾶ Β τᾶς ΒΕ πο τὶ τὰν ἴσαν τᾶ τε Β τᾶς ΔΒ καὶ τὰν ΕΒ, τοῦτον ἕξει ἁ ΔΑ πρὸς ἐλάσσονα τᾶς ΔΕ. ἐχέτω οὖν πρὸς ΔΘ. καὶ ἀμφότεραι ποτὶ τὰς πρώ τας τὸν αὐτὸν ἑξοῦντι λόγον· ἕξει οὖν ἁ ΕΑ ποτὶ ΑΔ τὸν αὐτὸν λόγον, ὃν ἁ ἴσα τᾶ τε Β τᾶς ΑΒ καὶ Ϛ τᾶς ΒΔ καὶ Γ τᾶς ΒΕ πρὸς τὰν συγκειμέναν ἔκ τε τᾶς Β συναμ φοτέρας τᾶς ΑΒ ΕΒ καὶ Δ συναμ φοτέρου τᾶς ΓΒ ΒΔ. ἔχει δὲ καὶ ἁ ΑΔ πρ πρὸς ΗΘ τὸν αὐτὸν λόγον, ὃν Α Ε συναμφοτέρου τᾶς ΑΒ ΒΕ μετὰ τ τᾶς Ι συναμφοτέρου τᾶς ΓΒ ΒΔ πρὸς τὰν συγκειμέναν ἔκ τε τᾶς Β τὰς ΑΒ καὶ τᾶς Δ τᾶς ΓΒ καὶ τᾶς Γ τᾶς ΕΒ καὶ Ϛ τᾶς ΒΔ· ἀνομοίως δὲ τῶν λόγ λόγων τεταγμένων, τουτέστιν ἐν τεταραγμέ να ἀναλογία, δι’ ἴσου τὸν αὐτὸν ἔ χει λόγον ὁ Α πρὸς ΗΘ, ὃν ΑΕ σιναμ φοτέρου τᾶς ΑΒ ΒΕ μετὰ τᾶς Ι τᾶς ΓΒ ΒΔ πρὸς τὰν κειμέναν ἔκ τε τᾶς Β συναμφοτέρου τᾶς ΑΒ ΒΕ καὶ τᾶς Δ συναμφοτέρου τᾶς ΓΒ ΒΔ. ἀλλὰ ἁ συγκειμένα ἔκ τε τᾶς Ε συναμφο τέρου τᾶς ΑΒ ΒΕ μετὰ τᾶς Ι συναμφο τέρου τᾶς ΓΒ ΒΔ πρ πρὸς τὰν συγκειμένα συγκειμέναν ἔκ τε τᾶς Β συναμφοτέρου τᾶς ΑΒ ΒΕ καὶ Δ συναμφοτέρου τᾶς ΓΒ ΒΔ λόγον ἔχει, ὃν πέντε πρὸς δύο· καὶ ἁ ΑΟ ἄρα πρὸς ΗΘ λόγον ἔχει, ὃν πέν τε πρὸς δύο. πάλιν, ἐπεὶ ἁ ΟΔ πρὸς ΔΑ τὸν αὐτὸν ἔχει λόγον, ὃν ἁ ΕΒ με τὰ τᾶς Β τᾶς ΒΔ πρὸς τὰν ἴσαν τὰ τὰν συγκειμέναν ἔκ τε τᾶς Β συναμφο τέρου τᾶς ΑΒ ΒΕ μετὰ τᾶς Δ συναμ φοτέρου τᾶς ΓΒ ΒΔ, ἔστιν δὲ καὶ ὡς ἁ ΔΒ πρὸς ΔΕ, οὕτως ἁ συγκειμένα ἔκ τε τᾶς Β τᾶς ΑΒ καὶ Γ τᾶς ΓΒ καὶ τᾶς ΒΔ ποτὶ τὰν ἴσαν τᾶ τε ΕΒ καὶ τᾶ Β τᾶς ΗΒΔ, ἀνομοίως οὖν τῶν λόγων τετμημένων, τουτέστιν τεταραγμένας οὔσας τᾶς ἀνα λογίας, δι’ ἴσου ὡς ἁ ΟΔ πρὸς ΔΕ, οὕτως ἁ ΑΒ τᾶς ΑΒ μετὰ τᾶς Γ τ τᾶς ΒΓ καὶ ἁ ΒΔ πρὸς τὰν συγκειμέναν ἐκ τᾶς Β συναμφοτέρου τᾶς ΑΒ ΒΕ καὶ τᾶς Δ τᾶς ΓΒ ΒΔ· ὥστε καὶ ὡς ἁ ΟΕ πρὸς ΕΔ ἐστίν, ὡς ἁ ΓΒ με τὰ τᾶς Γ τᾶς ΒΔ καὶ Β τᾶς ΕΒ πρὸς