Transcription of folios 028v-021r of the Archimedes Palimpsest. (= Archimedes folio Arch08v, On Floating Bodies) 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 5532 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 Floating Bodies 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
αὐτοῦ ὅλην εἶναι ἐν τῶι ὑγρῶι, τε θὲν κεκλιμένον οὐ μενεῖ κεκλι μένον, ἀλλ’ ἀποκαταστησεῖται οὕτως, ὥστε τὸν ἄξονα αὐτοῦ κα τὰ κάθετον εἶναι. ἀφείσθω γάρ τι τμᾶμα εἰς τὸ ὑγρόν, οἷον εἴρητ εἴρηται , καὶ ἔσται αὐτοῦ ἡ βάσει ἐν τῶι ὑ γρῶι, τμηθέντος δὲ αὐτοῦ ἐπιπέ δωι διὰ τοῦ ἄξονος ὀρθῶι πρὸς τὴν ἐπιφάνειαν τοῦ ὑγροῦ το μὴ ἔστω ἡ ΑΠΟΛ ὀρθογωνίου κώνου τομή, ἄξων δὲ τοῦ τμά ματος καὶ διὰ τῆς τομῆς ἡ ΠΦ, τῆς δ’ ἐπιφανείας τοῦ ὑγροῦ το μὴ ἡ ΙΣ. ἔπειθ’ οὖν κεκλιμένον κεῖται τὸ τμᾶμα, οὐκ ἔσται κα τὰ κάθετον ὁ ἄξων· οὐκ ἄρα ποιήσει ἡ ΠΦ ἴσας γωνίας πρὸς τῆι ΙΣ η η ΧΘ ω δή τις ἡ ΚΩ παρὰ τὴν ΙΣ ἐφαπτομένα κατὰ τὸ Ο τᾶς ΑΠΟΛ τομῆς, καὶ τοῦ μ μὲν ΑΠΟΛ στερεοῦ ἔστω τοῦ βάρους τὸ Ρ, τοῦ δὲ ΙΠΟΣ στερεοῦ τὸ Β, καὶ ἐπιζευχθεῖσα δὴ ΒΡ ἐκβεβλήσ θω, καὶ ἔστω κέντρον τοῦ βάρους τὸ Γ τοῦ ΙΣ ΛΑ. ὁμοίως δὲ δειχθήσεται ἡ μὲν ὑπὸ τᾶν ΡΟ ΟΚ γωνίαν ὀξεῖ α, ἡ δὲ ἀπὸ τοῦ Ρ κάθετος ἐπὶ τὴν ΚΩ ἀγομένα μεταξὺ πίπτουσα τῶν ΚΩ· ἔστω ἡ ΡΘ. ἐὰν δὴ ἀπὸ τῶν ΓΒ ἀχθὴν ἔσται παρὰ τὴν ΡΘ, τὸ μὲν ἐν τῶι ὑγρῶι ἀποληφθὲ ἀποληφθὲν ἐνεχθήσεται ἄνω κατὰ τὴν δι ὰ τοῦ Γ ἀγομέναν, τὸ δ’ ἐκτὸς τοῦ ὑγροῦ κατὰ τὴν διὰ τοῦ Β εἰς τὸ κά τω, καὶ οὐ μενεῖ τὸ ΑΠΟΛ στερεὸν οὕτως ἔχον ἐν τῶι ὑγρῶι, ἀλλὰ τὸ μὲν κατὰ τὸ Α ἄνω τὴν φορὰν ἕξει, τὸ δὲ κατὰ τὸ Λ κάτω, ἕως ἂν γένηται ἡ ΠΦ κατὰ κά θετον.
ΤΟ ὀρθὸν τμᾶμα τοῦ ὀρθογωνί ὀρθογωνίου κωνοειδοῦς, ὁπόταν κουφότε ρον ἦ τοῦ ὑγροῦ καὶ τὸν ἄξονα ἔχη μεῖζον ἡμιόλιον τῆς μέ χρι τοῦ ἄξονος, ἐὰν τῶι βάρει πρὸς τὸ ἴσογκον ὑγρὸν μὴ ἐλάσ σονα λόγον ἔχη τοῦ ὃν ἔχει τὸ τετράγωνον τὸ ἀπὸ τῆς ὑπε ροχῆς, ἧ μεῖζόν ἐστιν ὁ ἄξων ἢ ἡμιόλιος τῆς μέχρι τοῦ ἄξον ἄξονος , πρὸς τὸ τετράγωνον τὸ ἀπὸ τοῦ ἄξονος, ἀφεθὲν εἰς τὸ ὑγρὸν οὕτως, ὥστε τὴν βάσιν αὐτοῦ μὴ ἅπτεσθαι τοῦ ὑγροῦ, τεθὲν κεκλιμένον οὐ μενεῖ κεκλιμέ νον, ἀλλὰ ἀποκαταστήσεται εἰς ὀρθόν. ἔστω τμᾶμα ὀρθο γωνίου κωνοειδοῦς, οἷον εἴρη ται, καὶ ἀφεθὲν εἰς τὸ ὑγρόν, εἰ δυ νατόν, ἔστω μὴ ὀρθόν, ἀλλὰ ἐκκλιθέν, τμηθέντος δὲ αὐτοῦ ἐπιπέδωι διὰ τοῦ ἄξονος ὀρ θῶ πρὸς τὴν ἐπιφάνειαν τοῦ ὑγροῦ τοῦ μὲν τμάματος τομὴ.