Transcription of folios 014v-019r of the Archimedes Palimpsest. (= Archimedes folio Arch02v, 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 5526 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
πρὸς ΔΗ δυνάμει, οὕτως ἡ ΖΒ πρὸς ΒΗ μήκει, τουτέστιν ἡ ΜΝ πρὸς ΝΘ. ὡς δὲ ἡ ΜΝ πρὸς ΝΘ μήκει, οὕτως ἡ ΜΝ πρὸς ΝΞ δυνάμει· καὶ ὡς ἄρα ἡ ΑΖ πρὸς ΔΗ δυνάμει, οὕτως ἡ ΜΝ πρὸς ΝΞ δυνάμει· ὥστε καὶ μήκει ἐν τῶι αὐτῶ λόγωι. καὶ ὡς ἄρα ὁ ἀπὸ ΑΖ κύβος πρὸς τὸν ἀπὸ ΔΗ κύβον, οὕ τως ἀπὸ ΜΝ κύβος πρὸς τὸν ἀ πὸ ΝΞ κύβον. ἀλλ’ ὡς μὲν ὁ ἀπὸ ΝΞ κύβος πρὸς τὸν ἀπὸ ΔΗ, οὕτως ΑΒΓ τμᾶμα πρὸς τὸ ΔΒΕ τμᾶμα, ὡς δὲ ἀπὸ ΜΝ κύβος πρὸς τὸν ἀπὸ ΝΞ κύβον. ἀλλ’ ὡς μὲν ὁ ἀπὸ ΑΖ κύβος πρὸς τὸν ἀπὸ ΔΗ κύβον, οὕ τως ΑΒ ΓΑ τμᾶμα πρὸς τὸ ΔΒΕ τμᾶμα, ὡς δὲ ὁ ἀπὸ ΜΝ κύβος πρὸς τὸν ἀπὸ ΝΗΞ κύβον, οὕτως ἡ ΜΝ πρὸς ΝΤ· ὥστε καὶ διελόν τι ἐστὶν ὡς ὁ ΑΔ ΕΓ τόμος πρὸς τὸ ΔΒ τμᾶμα, οὕτως ἡ ΜΤ πρὸς ΝΤ, τουτέστι τὰ ΓΕ τῆς ΗΖ πρὸς ΤΝ. καὶ ἐπὶ τὸ στερεὸν τὸ βάσιν μὲν ἔχον τὸ ἀπὸ ΑΖ τετράγωνον, ὕψος δὲ τὴν συγκειμένην ἔκ τε τῆς Β τ τᾶς ΔΗ καὶ τῆς ΑΖ, πρὸς τὸν ἀπὸ ΑΖ κύβον λόγον ἔχει, ὃν ἡ Β τῆς ΔΗ μετὰ τᾶς ΑΖ πρὸς ΖΑ, ὥστε καὶ ἡ Σ τῆς ΝΞ μετὰ τῆς ΝΜ πρὸς ΝΜ, ἔστι δὲ καὶ ὡς ὁ ἀπὸ ΑΖ κύβος πρὸς τὸν ἀπὸ ΔΗ κύβον, οὕτως ἡ ΜΝ πρὸς ΝΤ, ὥστε ἀπὸ ΔΗ κύβος πρὸς τὸ στερεὸν τὸ βάσιν μὲν ἔ χον τὸ ἀπὸ ΔΗ τετράγωνον, ὕψος δὲ τὴν συγκειμένην ἔκ τε τῆς Β τῆς ΑΖ μετὰ τῆς ΔΗ, οὕ τως ἡ ΔΗ πρὸς τὴν συγκειμένη συγκειμένην ἔκ τε τῆς Β τῆς ΑΖ καὶ τῆς ΔΗ, ὥστε καὶ ἡ ΤΝ πρὸς τὴν συγκειμένη συγκειμένην ἔκ τε τῆς Β τῆς ΟΝ καὶ τῆς ΤΝ, γέγονεν οὖν τάσσαρα μεγέθεα, τὸ στερεὸν τὸ βάσιν μὲν ἔχον τὸ ἀπὸ ΑΖ τετράγωνον, ὕψος δὲ τὴν συγκειμένην ἔκ τε τῆς Β τῆς ΔΗ καὶ τῆς ΑΖ, κύβος καὶ ὁ ἀπὸ ΔΗ κύβος καὶ τὸ στερεὸν τὸ βά σιν μὲν ἔχον τὸ ἀπὸ ΔΗ τετράγω νον, ὕψος δὲ τὴν συγκειμένην ἔκ τε τῆς Β τῆς ΑΖ καὶ τῆς ΔΗ, τέταρσι μεγέθεσιν ἀνάλογον σύν δυο λαμβανομένοις, τῆι τε συγ κειμένηι ἔκ τε τῆς Β τῆς ΝΞ καὶ τῆς ΝΜ καὶ ἑτέρου μεγέθεος τ τῆς ΜΝ καὶ ἄλλο ἑξῆς ἡ ΝΓ καὶ τελευ ταῖον ἡ συγκειμένη ἔκ τε τῆς ΒΗ τῆς ΝΟ καὶ τῆς ΝΤ· δι’ ἴσου ἄρα γενήσεται ὡς τὸ στερεὸν τὸ βάσιν μὲν ἔχον τὸ ἀπὸ ΑΖ τετράγωνος, ὕψος δὲ τὴν συγκειμένη συγκειμένην ἔκ τε τῆς Β τῆς ΔΗ καὶ τῆς ΑΖ, πρὸς τὸ στε ρεὸν τὸ βάσιν μὲν ἔχον τὸ ἀπὸ ΔΗ τετράγωνον, ὕψος δὲ τὴν συγκειμέ νην ἔκ τε τῆς Β τῆς ΑΖ καὶ τῆς ΔΗ, οὕτως ἡ συγκειμένη ἔκ τε τῆς Β τῆς ΝΞ καὶ τῆς ΜΝ πρὸς τὴν συγκειμένην ἔκ τε τῆς Β τῆς ΝΟ καὶ τῆς ΝΤ. ἀλλ’ ὡς τὸ εἰρημένον στερεὸν πρὸς τὸ εἰρημένον στερε όν, οὕτως ἡ ΘΙ πρὸς ΙΚ· οὕτως ἡ συγκειμένη πρὸς τὴν συγκειμέν συγκειμένην · ὥστε καὶ συνθέντι καὶ τῶν ἡγου μένων τὰ Ε· ἔστιν ἄρα ὡς ἡ ΖΗ πρὸς ΙΚ, οὕτως ἡ Ε συναμφοτέ ρου τῆς ΜΝΤ καὶ Ι συναμφοτέρ συναμφοτέρου τῆς ΝΞ ΝΟ πρὸς τὴν Β τῆς ΟΝ καὶ ΝΤ. καὶ ὡς ἡ ΖΗ πρὸς ΖΚ οὖσα οὖσαν