Transcription of folios 014r-019v of the Archimedes Palimpsest. (= Archimedes folio Arch02r, 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 5525 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
φοτέρου τᾶς ΑΒΔ καὶ Δ τᾶς ΓΒ λόγον ἔχει, ὃν τρία πρὸς δύο, πρ πρὸς δὲ τὰ τρία πέμπτα τᾶς αὐτᾶς λόγον ἔχει, ὃν πέντε πρὸς δύο· ἐ δείχθη δὲ καὶ ὁ Α πρὸς ΗΘ λόγον ἔχοντα, ὃν πέντε πρὸς δύο· καὶ ὅλα ἄρα ἁ ΒΘ πρὸς ὅλαν τὰν ΖΘ λόγον ἔχει, ὃν πέντε πρὸς τὰ δύο. εἰ δὲ τοῦτο, πεμπταμόρια ἐ ναντία ΖΘ τᾶς ΔΒ· ΟΙ.
ΙΑ Παντὸς τόμου ἀπὸ ὀρθογωνί ὀρθογωνίου κώνου τομᾶς ἀφαιρουμέν ἀφαιρουμένου τὸ κέντρον τοῦ βάρεος ἐπὶ τᾶς εὐθείας ἐστίν, διάμετρός ἐστιν τοῦ τόμου, τὸν δὲ τρόπον κείμεν κείμενον · διαιρεθείσας τᾶς εὐθείας εἰς ἴσα πέντε ἐπὶ μέσου πεμπτα μορίου, ὥστε τὸ τμᾶμα αὐτοῦ τὸ ἐγγύτερον τᾶς ἐλάσσονος βάσι βάσιος τοῦ τόμου ποτὶ τὸ λοιπὸν τμᾶ μα τὸν αὐτὸν ἔχει λόγον, ὃν ἔ χει τὸ στερεὸν τὸ βάσιν μὲν ἔχ ἔχον τὸ τετράγωνον τὸ ἀπὸ τᾶς μεί ζονος τᾶν βάσεων τοῦ τόμ τόμου , ὕψος δὲ τὰν ἴσαν συναμφοτέρα τᾶ τε διπλασία τᾶς ἐλάσσον ἐλάσσονος τῶν βασίων καὶ τᾶ μείζονι, πο τὶ τὸ στερεὸν τὸ βάσιν μὲν ἔχο ἔχον τὸ τετράγωνον τὸ ἀπὸ τᾶς ἐλάσ σονος τᾶν βάσεων τοῦ τόμ τόμου , ὕψος δὲ τὰν ἴσαν ἀμφοτέρας τᾶ τε διπλασία τᾶς μείζονος καὶ τᾶ ἐλάσσονι αὐτᾶν. ἔστωσα ἔστωσαν ὀρθογωνίου τομᾶ δύο εὐθεῖαι αἱ ΑΓ ΔΕ διάμετρος δὲ ἔστω τοῦ Α ΒΓ τμάματος ἁ ΒΖ· φανερὸν δὲ ὅτι καὶ τοῦ ΑΓΔΕ παράλληλοί ἐντι τᾶ κατὰ τὸ Β ἐφαπτομένα τᾶς τομᾶς· καὶ τᾶς ΗΖ εὐθείας διαιρεθείσας εἰς πέντε ἴσα μέσο μέσον ἔστω πεμπταμόριον ἁ ΘΚ, ἁ δὲ ΘΙ πρὸς τὰν ΙΚ τὸν αὐτὸν ἐχέτω λόγον, ὃν ἔχει τὸ στερεὸν τὸ βάσι βάσιν μὲν ἔχον τὸ ἀπὸ τῆς ΛΖ τετρά γωνον, ὕψος δὲ τὰν ἴσαν ἀμφο τέραις τᾶ τε Β τᾶς ΖΗ καὶ τᾶ ΑΖ, ποτὶ τὸ στερεὸν τὸ βάσιν ἔχο ἔχον τὸ ἀπὸ τᾶς ΔΗ τετράγωνον, ὕ ψος δὲ τὰ ἴσαν ἀμφοτέραις τε Β τᾶς ΑΖ καὶ τᾶ ΔΗ. δεικτέον ὅτι τοῦ ΑΔΕΓ τόμου κέντρον ἐστὶ τοῦ βάρεος τὸ Ι σαμεῖον.   ἔστω δὴ τᾶ μὲν ΖΒ ἴσα ἁ ΜΝ, τᾶ δὲ ΗΒ ἴσα ἁ ΝΘ, καὶ εἰλήφθω τᾶν μὲν ΜΝΘ μέσα ἀνάλογον ἁ ΝΞ, τετάρτη δὲ ἀνάλογον ἁ ΤΝ, καὶ ὡς ἡ ΤΜ πρὸς ΤΝ, οὕτως ἁ ΖΘ πρός τινα ἀπὸ τοῦ Ι, ὅπου ἐὰν ἔρχηται τὸ ἕτερον σαμεῖον· οὐ δὲν γὰρ φέρει διαφέρει εἴτε καὶ μεταξὺ τῶν ΗΒ τὴν ΙΡ. καὶ ἐπεὶ ἐν ὀρθογωνί ου κωνίου τομᾶ διάμετρός ἐστι τοῦ τμάματος ΑΖ ΒΗ ΒΖ ἤτοι ἀρχική ἐστιν τῆς τομῆς ἢ παρὰ τὴν διάμετρον ἦκται, αἱ δὲ ΑΖ ΔΗ εἰς αὐτὴν τεταγμένως εἰσὶν κα τηγμένη, ἐπειδὴ παράλληλοί εἰ σι τῆι ἀπὸ τοῦ Β τῆς τομῆς ἐφα πτομένη. εἰ δὲ τοῦτο, ἔστιν ὡς ἡ ΑΖ