Transcription of folios 037r-036v of the Archimedes Palimpsest. (= Archimedes folio Arch54r, Sphere and Cylinder) 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 5539 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 Sphere and Cylinder 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
κορυφῆς τοῦ ἑτέρου κώνου ἐπὶ τὴν πλευρὰν τοῦ ἑτέρου κώνου κα θέτωι ἠγμένηι. ἔστω ῥόμβος ἐξ ἰσο σκελῶν κώνων συγκείμενος ὁ ΑΒΓΔ, καὶ τμηθήτω ὁ ἕτερος κῶνος ἐπιπέδωι παραλλήλωι τῆι βάσει, καὶ ποιείτω τομὴν τὴ τὴν ΕΖ, ἀπὸ δὲ τοῦ περὶ διάμετρον τὴ τὴν ΕΖ κύκλον κῶνος ἀναγεγράφθω τὴν κορυφὴν ἔχων τὸ Δ σημεῖον· ἔσται δὴ γεγονὼς ῥόμβος ὁ ΕΒΔΖ. καὶ νοείσθω ἀφηρημένος ἀπὸ τοῦ ὅλου ῥόμβου, ἐκκείσθω δέ τις κῶνος ὁ ΘΚΛ τὴν μὲν βάσιν ἴ σην ἔχων τῆι ἐπιφανείαι τῆι με ταξὺ τῶν ΑΓ ΕΖ, τὸ δὲ ὕψος ἴσον τῆι ἀπὸ τοῦ Δ σημείου καθέτω ἀγομένη ἐπὶ τὴν ΒΔ ἢ τὴν ἐπ’ εὐ θείας αὐτῆι· λέγω ὅτι ὁ ΘΚΛ κῶν κῶνος ἴσος ἐστὶ τῶι εἰρημένῶι περιλείμ ματι. ἐκκείσθωσαν γὰρ δύο κῶ νοι οἱ ΜΝΞ ΟΠΡ, καὶ ἡ μὲν βάσις τοῦ ΜΝΞ κώνου ἴση ἔστω τῆι ἐπι φανείαι τοῦ ΑΒΓ, τὸ δὲ ὕψος ἴσ ἴσον τῆι ΔΗ διὰ δὴ τὰ προδειχθέντα ἴ σος ἐστὶν ὁ ΜΝΞ κῶνος τῶι ΑΒΓΔ ῥόμβωι, τοῦ δὲ ΟΠΡ κώνου ἡ μ μὲν βάσις ἴση ἔστω τῆι ἐπιφανείαι τοῦ ΕΒΖ κώνου, τὸ δὲ ὕψος ἴσον τῆι ΔΗ ὁμοίως δὴ ἴσος ἐστὶν ὁ ΟΠΡ κῶνος τῶ ΕΒΔΖ ῥόμβῶι. ἐπεὶ δὲ ὁμοίως ἡ ἐπιφάνεια τοῦ ΑΒΓ κώνου σύγκειται ἔκ τε τῆς τοῦ ΕΒΖ καὶ τῆς μεταξὺ τῶν ΕΖ ΑΓ, ἀλλὰ ἡ μὲν τοῦ ΑΒΓ κώνου ἐπιφάνεια ἴση ἐστὶ τῆι βάσει τοῦ ΜΝΞ, ἡ δὲ τοῦ ΕΒΖ κώνου ἐπιφάνεια ἴση ἐστὶ τῆι βά σει τοῦ ΟΠΡ κώνου, ἡ δὲ μεταξὺ τῶν ΕΖ ΑΓ ἴση ἐστὶ τῆι βάσει τ τοῦ ΘΚΛ, ἡ ἄρα βάσις τοῦ ΜΝΞ ἴση ἐστὶ τς ταῖς βάσεσι τῶν ΟΠΡ ΘΚΛ. καί εἰσιν οἱ κῶνοι ὑ πὸ τὸ αὐτὸ ὕψος· καὶ ὁ ΜΝΞ ἄρα κῶ νος ἴσος ἐστὶ τοῖς ΘΚΛ ΟΠΡ κώνοις. ἀλλ’ ὁ μὲν ΜΝΞ κῶνος ἴσος ἐστὶ τῶι Α ΒΓΔ ῥόμβωι, ὁ δὲ ΟΠΡ κῶνος τῶ Ε ΒΔΖ ῥόμβωι· λοιπὸς ἄρα ὁ κῶνος ὁ ΘΚΛ ἴσος ἐστὶ τῶι περιλείμματι τῶι λοιπῶι. ΚΒ Ἐὰν εἰς κύκλον πολύγωνον ἐγγραφῆι ἀρτιόπλευρόν τε καὶ ἰσόπλευρο ἰσόπλευρον καὶ διαχθῶσιν εὐθεῖαι ἐπιζευγ ἐπιζευγνύ ουσαι τὰς πλευρὰς τοῦ κώνου ὥστε αὐτὰς παραλλήλους εἶναι