Transcription of folios 150v-153r of the Archimedes Palimpsest. (= Archimedes folio Arch58v, 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 5648 Owner of the Archimedes Palimpsest 2008

Licensed for use under Creative Commons Attribution 3.0 Unported, license http://creativecommons.org/licenses/by/3.0/legalcode.

It is requested that copies of any published articles based on the information in this data set be sent to The Curator of Manuscripts, The Walters Art Museum, 600 North Charles Street, Baltimore MD 21201.

 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
εἰ γὰρ δυνατόν, ἔστω· καὶ ὁμοίως εὑρήσθωσαν αἱ ΒΓ εὐθεῖαι ὥστε τὴν Β πρὸς Γ ἐλάσσονα λόγον ἔχειν τοῦ, ὃν ἔχει ὁ Α κύκλος πρὸς τὴν ἐπιφά νειαν τῆς σφαίρας, καὶ τῶν ΒΓ μέ ση ἀνάλογον ἡ Δ, καὶ ἐγγεγράφθω καὶ περιγεγράφθω πάλιν, ποτὲ τὴ τὴν τοῦ περιγεγραμμένου ἐλάσσονα λόγον ἔχειν τοῦ τῆς Β πρὸς δ· καὶ τὰ διπλάσια ἄρα· ἡ ἐπιφάνεια ἄρα τοῦ περιγεγραμμένου πρὸς τὴν ἐπιφάνειαν τοῦ ἐγγεγραμμέ νου ἐλάσσονα λόγον ἔχει ἤ ἤπερ ἡ Β πρὸς Γ. ἡ δὲ Β πρὸς Γ ἐλάσσονα λόγον ἔ χει ἤ ἤπερ ὁ Α κύκλος πρὸς τὴν ἐπι φάνειαν τῆς σφαίρας· ὅ ὅπερ ἄτο πον· ἡ μὲν γὰρ τοῦ περιγεγραμ μένου ἐπιφάνεια μείζων ἐστὶ τοῦ Α κύκλου, ἡ δὲ τοῦ ἐγγεγραμμένου ἐ λάσσων τῆς ἐπιφανείας τῆς σφ σφαί ρας. οὐκ ἄρα οὐδὲ ἐλάσσων ἡ ἐπιφά νεια τῆς σφαίρας τοῦ Α κύκλου. ἐδείχθη δὲ ὅτι οὐδὲ μείζων· ἡ ἄρα ἐπιφά νεια τῆς σφαίρας ἴση ἐστὶ τῶι Α κύ κλωι, τουτ τουτέστι τῶι τετραπλασίωι τ τοῦ μεγίστου κύκλου.
Πᾶσα σφαῖρα τεπλασία τετραπλασία ἐστὶ κών κώνου τοῦ βάσιν μὲν ἔχοντος ἴσην τῶι μεγίστω κύκλωι τῶν ἐν τῆι σφαί ραι, ὕψος δὲ τὴν ἐκ τοῦ κέντρου τ τῆς σφαίρας. ἔστω γὰρ σφαῖρά τις καὶ ἐν αὐτῆι μέγιστος κύκλος ὁ ΑΒ ΓΔ. εἰ οὖν μή ἐστιν ἡ σφαῖρα τετραπλασία τοῦ εἰρημένου κώνου, ἔστω, εἰ δυ νατόν, μείζων ἢ τετραπλασία· ἔστω δὲ ὁ Ξ κῶνος βάσιν μὲν ἔ χων τετραπλασίαν τοῦ ΑΒΓΔ κύκλου, ὕψος δὲ ἴσον τῆι ἐκ τοῦ κέ κέν τρου τῆς σφαίρας· μείζων οὖν ἐστιν ἡ σφαῖρα τοῦ Ξ κώνου. ἔσται δὴ δύο μεγέθη ἄνισα ἥ τε σφαῖρα καὶ ὁ κῶνος· δυνατὸν οὖν δύο εὐθεί εὐθείας λαβεῖν ἀνίσους, ὥστε ἔχειν τὴν μείζονα πρὸς τ τὴν ἐλάσσονα ἐλάσσο να λόγον τοῦ, ὃν ἔχει ἡ σφαῖρα πρὸς τὸν Ξ κῶνον. ἔστωσαν οὖν αἱ ΚΗ, αἱ δὲ ΙΘ εἰλημμέναι, ὥσ τε τῶι ἴσωι ἀλλήλων ὑπερέχειν τὴν Κ τῆς Ι καὶ τὴν Ι τῆς Θ καὶ τὴν Θ τῆς Η, νοείσθω δὲ καὶ εἰς τὸν Α ΒΓΔ κύκλον ἐγγεγραμμένον πο λύγωνον, οὗ τὸ πλῆθος τῶν πλευ ρῶν μετρείσθω ὑπὸ τετράδος, καὶ ἄλλο περιγεγραμμένον ὅμοιον τῶι ἐγγεγραμμένωι, καθάπερ ἐπὶ τῶν πρότερον, ἡ δὲ τοῦ περιγε γραμμένου πολυγώνου πλευρὰ πρὸς τὴν τοῦ ἐγγεγραμμένου ἐλάσ σονα λόγον ἐχέτω τοῦ, ὃν ἔχει ἡ Κ πρὸς Ι, καὶ ἔτωσαν αἱ ΑΒΓΔ διὰ