Transcription of folios 003r-006v of the Archimedes Palimpsest. (= Archimedes folio Arch46r, 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 5521 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
ἐπὶ τὰ αὐτὰ ἔσονται ἢ οὐδὲν ἔχουσιν ἐπὶ τὰ ἕτερα ἐπὶ τὰ αὐτὰ δὴ κοίλας καλῶ τὰς τοιαύτας ἐπιφα νείας, ἐν αἷς ἂν δύο σημείων λαμ βανομένων αἱ μεταξὺ τῶν σημεί ων εὐθεῖαι ἤτοι πᾶσαι ἐπὶ τὰ αὐ τὰ πίπτουσιν τῆς ἐπιφανείας, ἢ τινὲς μὲν ἐπὶ τὰ αὐτὰ, τινὲς δὲ κατ’ αὐτῆς, ἐπὶ τὰ ἕτερα δὲ μέρη μη δεμία. τομέα δὲ στερεὸν καλῶ, ἐπειδὰν σφαῖραν κῶνος τέμηι κορυφὴν ἔχων πρὸς τῶι κέντρωι τῆς σφαίρας, τὸ ἐμπεριεχόμε νον σχῆμα ὑπό τε τῆς ἐπιφα νείας τοῦ κώνου καὶ τῆς ἐπιφα νείας τῆς σφαίρας ἐντὸς τοῦ κώνου. ῥόμβον δὲ καλῶ στερε όν, ἐπειδὰν δύο κῶνοι τὴν αὐτὴν βάσιν ἔχοντες τὰς κορυφὰς ἔχωσιν ἐφ’ ἑκάτερα τοῦ ἐπιπέ δου τῆς βάσεως, ὅπως οἱ ἄξο νες αὐτῶν ἐπ’ εὐθείας ὦσι κείμε νοι, τὸ ἐξ ἀμφοῖν τοῖν κώνοιν συγκείμενον τὸ στερεὸν σχῆμα. λαμβάνω δὲ ταῦτα· τῶν τὰ αὐτὰ πέρατα ἐχουσῶν γραμμῶν ἐ λαχίστην εἶναι τὴν εὐθεῖαν. τῶν δὲ ἄλλων γραμμῶν, εἶναι τὰς τοιαύ τας, ἐπειδ’ ἂν ὦσιν ἀμφότεραι ἐπὶ τὰ αὐτὰ κοῖλαι, καὶ ἤτοι ὅλη περιλαμβάνηται ἡ ἑτέρα αὐτ αὐτῶν ὑπὸ τῆς ἑτέρας ἐπιφανείας καὶ τῆς εὐθείας τῆς τὰ αὐτὰ πέρα τα ἐχούσης αὐτῆ, ἢ τινὰ μὲν περι λαμβάνηται, τινὰ δὲ κοινὰ ἔχη, καὶ ἐλάσσονα εἶναι τὴν περιλαμ βανομένην. ὁμοίως δὲ καὶ τῶν ἐπιφα νειῶν τῶν τὰ αὐτὰ πέρατα ἐχου σῶν, ἐὰν ἐν ἐπιπέδωι τὰ πέρα τα ἔχωσιν, ἐλάσσονα εἶναι τὴν ἐπίπεδον. τῶν δὲ ἄλλων ἐπιφα νειῶν καὶ τὰ αὐτὰ πέρατα ἐχουσῶν, ἐὰν ἐν ἐπιπέδωι τὰ πέρατα ἦ, ἀ νίσους εἶναι τὰς τοιαύτας, ἐπειδὰν ὦσιν ἀμφότεραι ἐπὶ τὰ αὐτὰ κοῖ λαι, καὶ ἤτοι ὅλη περιλαμβάνηται ὑπὸ τῆς ἑτέρας ἐπιφανείας καὶ τῆς ἐπιπέδου τῆς τὰ αὐτὰ πέρα τα ἐχούσης αὐτῆ, ἢ τινὰ μὲν πε ριλαμβάνηται, τινὰ δὲ κοινὰ ἔ χη, καὶ ἐλάσσονα εἶναι τὴν περι λαμβανομένην. ἔτι δὲ τῶν ἀνίσων γραμμῶν καὶ τῶν ἀνίσων ἐπιφα νειῶν καὶ τῶν ἀνίσων στερεῶν τὸ μεῖζον τοῦ ἐλάσσονος ὑπερέχει ὑπερέχειν τοιούτω, ὃ συντιθέμενον ἑαυτὸ ἑ αυτῶι δυνατόν ἐστιν ὑπερέχειν παν τὸς τοῦ προτεθέντος τῶν πρὸς ἄλλη λα λεγομένων. τούτων δὲ ὑποκει μένων, ἐὰν εἰς κύκλον πολύγωνον ἐγγραφῆ, φανερὸν ὅτι ἡ περίμετρος τοῦ ἐγγραφέντος πολυγώνου ἐλάσ σων ἐστὶν τῆς τοῦ κύκλου περιφερεί περιφερείας · ἑκάστη γὰρ τῶν τοῦ πολυγώνου πλευ ρῶν ἐλάσσων ἐστὶ τῆς τοῦ κύκλου περι φερείας τῆς ὑπὸ τῆς αὐτῆς ἀπο τεμνομένης. Α ἐάνπερ κύκλ κύκλον