Transcription of folios 037v-036r of the Archimedes Palimpsest. (= Archimedes folio Arch54v, 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 5540 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
μιᾶι ὁποιαοῦν τῶν ὑπὸ δύο πλευ ρὰς τοῦ πολυγώνου ὑποτεινου σῶν αἱ ἐπιζευγνύουσαι πᾶσαι πρὸς τ τὴν τ τοῦ κύκλου διάμετρον τοῦτον ἔ χουσι τὸν λόγον ὃν ἔχει ἡ ὑποτεί νουσα τὰς μιᾶ ἐλάσσονα τῶν ἡμίσεων πρὸς τὴν πλευρὰν τοῦ πο λυγώνου. ἔστω κύκλος ὁ ΑΒ ΓΔ καὶ ἐν αὐτῶι πολύγωνον ἐγγεγρά φθω τὸ ΑΕΖ ΒΗΘ ΓΜ ΝΔ ΛΚΑ καὶ ἐπεζεύχθωσαν αἱ ΕΚ ΖΛ ΒΔ ΗΝ ΘΜ· δῆλον δὴ ὅτι παράλληλ παράλληλοί εἰσιν τῆι ὑπὸ δύο πλευρὰς τοῦ πολυ γώνου ὑποτεινούσηι· λέγω οὖν ὅτι αἱ εἰρημέν εἰρημέναι πᾶσ πᾶσαι πρὸς τ τὴν τ τοῦ κύκλου διά μετρον τὴν ΑΓ τὸν αὐτὸν λόγον ἔχουσι τῶι τῆς ΓΕ πρὸς ΕΑ. ἐπεζεύ χθωσαν γὰρ αἱ ΖΚ ΑΒ ΗΔ ΘΝ· παράλληλος ἄρα ἡ μὲν ΖΚ τῆι ΕΑ ἡ δὲ ΒΛ τῆι ΖΚ, καὶ ἔτι ἡ μὲν ΔΗ τῆι ΒΛ ἡ δὲ ΘΝ τῆι ΔΗ καὶ ἡ ΓΜ τῆι ΘΝ· καὶ ἐπεὶ δύο παράλληλοί εἰσιν αἱ ΕΑ ΚΖ καὶ δύο διηγμέναι εἰσὶν αἱ ΕΚ ΑΟ· ἔστιν ἄρα ὡς ἡ ΕΞ πρὸς ΞΑ ἡ ΚΞ πρὸς ΞΟ. ὡς δὲ ἡ ΚΞ πρὸς ΞΟ ἡ ΖΠ πρὸς ΠΟ ὡς δὲ ἡ ΖΠ πρὸς ΠΟ ἡ ΛΠ πρὸς ΠΡ ὡς δὲ ἡ ΛΠ πρὸς ΠΡ οὕτως ἡ ΒΣ πρὸς ΡΣ, καὶ ἔτι ὡς ἡ μὲν ΒΣ πρὸς ΣΡ ἡ ΔΣ πρὸς ΣΤ ὡς δὲ ἡ ΔΣ πρὸς ΣΤ ἡ ΗΥ πρὸς ΥΤ, καὶ ἔτι ὡς ἡ μὲν ΗΥ πρὸς ΥΤ ἡ ΝΥ πρὸς ΥΦ ὡς δὲ ἡ ΝΥ πρὸς ΥΦ ἡ ΘΧ πρὸς ΧΦ, καὶ ἔτι ὡς μὲν ἡ ΘΧ πρὸς ΧΦ ἡ ΜΧ πρὸς ΗΓ, καὶ πάν τα ἄρα πρὸς πάντα ἐστὶν ὡς εἷς τῶν λόγων πρὸς ἕνα· ὡς ἄρα ἡ ΕΞ πρὸς ΞΑ οὕτως αἱ ΕΚ ΖΛ ΒΔ ΗΝ ΘΜ πρὸς τὴν ΑΓ διάμετρον. ὡς δὲ ἡ ΕΞ πρὸς ΞΑ οὕτως ἡ ΓΕ πρὸς ΕΑ· ἔστ ἔσται ἄρα καὶ ὡ ὡς ἡ ΓΕ πρὸς ΕΑ οὕτως πᾶσαι αἱ ΕΚ ΖΛ ΒΔ ΗΝ ΘΜ πρὸς τὴν ΑΓ διάμετρον. ἑξ ἑξῆς Η ΚΑΤΑΓΡΑΦΗ. Ἐὰν εἰς τμῆμα κύκλου πολύγωνον ἐγ γραφῆι τὰς πλευρὰς ἔχον χωρὶς τῆς βάσεως ἴσας καὶ ἀρτίους, ἀ χθῶσι δὲ εὐθεῖαι π παρὰ τὴν βάσιν τ τοῦ τμήματος αἱ τὰς πλευρὰς ἐ πιζευγνύουσαι τοῦ πολυγώνου αἱ ἀχθεῖσαι πᾶσαι καὶ ἡμίσεια τῆς βάσεως πρὸς τὸ ὕψος τοῦ τμή ματος τὸν αὐτὸν λόγον ἔχουσιν ὃν ἡ ἀπὸ τῆς διαμέτρου τοῦ κύκλ κύκλου ἐπὶ τὴν πλευρὰν τοῦ πολυγώνου ἐπιζευγνυμένη πρὸς τὴν τοῦ πολυγώ νου πλευράν. εἰς γὰρ κύκλον τὸν ΑΒΓΔ καὶ ἐπὶ τῆς ΑΓ πολύγων πολύγωνον ἐγγεγράφθω εἰς τὸ ΑΒΓ τμῆμα ἀρ τιόπλευρόν τε καὶ ἴσας ἔχον τὰς πλευρὰς χωρὶς τῆς βάσεως τῆς ΑΓ καὶ ἐπεζεύχθωσαν αἱ ΖΗ ΕΘ αἵ εἰσιν παράλληλοι τῆι βάσει τοῦ τμή ματος· λέγω ὅτι ἐστὶν ὡς ἡ ΖΗ ΕΘ ΑΞ