Transcription of folios 027v-022r of the Archimedes Palimpsest. (= Archimedes folio Arch01v, 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 5530 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
τὰν Β συναμφοτέρου τᾶς ΑΒ ΒΕ καὶ Δ συναμφοτέρου τᾶς ΓΒ ΒΔ. ἔστι δὲ καὶ ὡς ἁ ΔΕ πρὸς ΕΒ, οὕτως ἅ τε ΑΓ πρὸς ΓΒ, ἐπεὶ καὶ κατὰ σύν θεσιν, καὶ ἁ Γ τᾶς ΓΔ πρὸς τὰν Γ τᾶς ΔΒ καὶ ἁ Β τᾶς ΔΕ πρὸς τὰν Β τᾶς ΕΒ· ὥστε καὶ ἁ συγκειμένα ἔκ τε τᾶς ΑΓ καὶ Γ τᾶς ΓΔ καὶ Β τᾶς ΔΕ πρὸς τὰν συγκειμέναν εἴς τε τᾶς ΓΒ καὶ Γ τᾶς ΒΔ καὶ Β τᾶς ΕΒ. ἀ νομοίως οὖν πάλιν τῶν λόγων τε ταγμένων, τουτέστιν ἐν τεταραγ μέναι ἀναλογίαι, δι’ ἴσου τὸν αὐτὸν ἕξει λόγον ἁ ΕΟ πρ πρὸς ΕΒ, ὃν ἁ ΑΓ μετὰ τᾶς Γ τᾶς ΓΔ καὶ Β τᾶς δὲ πρὸς διπλασίαν συναμφοτέρου τᾶς ΑΒ ΒΕ μετὰ τᾶς Δ συναμφο τέρου τᾶς ΓΒ ΒΔ· ὅλα οὖν ἁ ΟΒ πρὸς ΒΕ τὸν αὐτὸν ἔχει λόγον, ὃν ἁ ἴσα τᾶ τε Γ τᾶς ΑΒ μετὰ τᾶς Ϛ τᾶς ΓΒ καὶ τᾶ Γ τᾶς ΒΔ πρὸς τὰν Β συναμφοτέρου τᾶς ΑΒ ΒΕ μετὰ τᾶς Δ συναμφοτέρου τᾶς ΓΒ ΒΔ. καὶ ἐπεὶ αἵ τε ΕΔ ΔΓ ΓΑ ἐν τῶι αὐτῶι λόγωι ἐντὶ καὶ συναμφό τερος ἑκάστα τῶν ΕΒ ΒΔ ΔΒ ΒΓ ΓΒ ΒΑ ἔστ ἔσται καὶ ὡς ἁ ΕΔ πρὸς ΔΑ, οὕ τως συναμφότερος ἁ ΕΒ ΒΔ πρ πρὸς συναμφότερον τὰν ΔΒ ΒΓ με τὰ τᾶς συναμφοτέρου τᾶς ΓΒ ΒΑ. καὶ συνθέντι ἄρα ἐστὶν ὡς ἁ Ε πρὸς ΑΔ, οὕτως συναμφότε ρος ἁ ΕΒ ΒΔ μετὰ συναμφοτέ ρου τᾶς ΔΒ ΒΓ καὶ συναμφοτέρου τᾶς ΓΒΔ, ὅ ἐστι συναμφότερος ἁ ΕΒΑ μετὰ τᾶς Β συναμφοτέ ρου τᾶς ΑΒΓ πρὸς συναμφότερο συναμφότερον τὰν ΒΔ ΒΑ μετὰ τᾶς Β τᾶς ΒΓ· ὥστε καὶ ἁ Β πρὸς τὰν Β τὸν αὐ τὸν ἕξει λόγον, τουτέστιν ὡς ἁ ΕΑ πρὸς ΑΔ, οὕτως ἁ Β συναμφοτέ ρου τᾶς ΕΒΑ μετὰ τᾶς Δ συναμ φοτέρου τᾶς ΓΒΔ πρὸς τὰν Β συναμφοτέρου τῆς ΑΒΔ μετὰ τᾶς Δ τᾶς ΓΒ· ὥστε καὶ ὡς ἁ ΕΑ πρὸς τὰ τρία πέμπτα τᾶς ΑΔ, οὕτως ἁ συγκειμένα ἔκ τε τᾶς Β συναμφοτέρου τᾶς ΑΒΕ καὶ Δ συναμφοτέρου τᾶς ΓΒΔ πρὸς τὰ τρία πέμπτα τᾶς συγ κειμένας ἔκ τε τᾶς Β συναμφο τέρου τᾶς ΑΒΔ καὶ Δ τᾶς ΓΒ. ἀλλ’ ὡς ἁ ΕΑ πρὸς τὰ τρία πέμ πτα τᾶς ΑΔ, οὕτως ἐστὶν ἁ ΕΒ πρὸς ΖΗ· καὶ ὡς ἄρα ἁ ΕΒ πρὸς ΖΗ, καὶ ὡς ἄρα ἁ ΕΒ συναμ φοτέρου τᾶς ΔΒΓ πρὸς τὰ τρί α πέμπτα τᾶς συγκειμένας ἔκ τε τᾶς Β συναμφοτέρου τᾶς ΑΒΔ μετὰ τᾶς Δ τᾶς ΓΒ. ἐδείχθη δὲ καὶ ὡς ὁ ΑΒ πρὸς ΕΒ, οὕτως ἁ Γ συναμφοτέρου τᾶς ΑΒΔ με τὰ τᾶς Ϛ τᾶς ΓΒ πρὸς τὰν Β συναμφοτέρου τῆς ΑΒΕ καὶ Δ συναμφοτέρου τῆς ΓΒΔ. καὶ δι’ ἴσου ἄρα ἐστὶν ὡς ἡ ΕΣ πρὸς ΖΗ, οὕτως ἁ συγκειμένα ἔκ τε τᾶς Γ συναμφοτέρου τᾶς Α ΒΔ καὶ Ϛ τᾶς ΓΒ πρὸς τὰ Γ πέμ πτα τᾶς συγκειμένας ἔκ τε τᾶς Β συναμφοτέρου τᾶς ΑΒΔ καὶ Δ τᾶς ΓΒ. ἀλλ’ ἁ συγκειμένα ἔκ τε τᾶς Γ συναμφοτέρου τᾶς ΑΒΔ καὶ Ϛ τᾶς ΓΒ πρὸς μὲν τὰν συγκειμέναν ἔκ τε τᾶς Β συναμ