Transcription of folios 017v-016r of the Archimedes Palimpsest. (= Archimedes folio Arch07v, 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 5528 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
ὅτι τὸ ΦΑ μέγεθος τῶι βάρει πρὸς τὸ ὑγρὸν τὸ ἰσόογκον τοῦτον ἔχει τὸν λόγον, ὃν τὸ Α πρὸς τὸ ΦΑ. εἰλήφθω γάρ τι τοῦ ὑγροῦ μέγεθος ἰσόογκ ἰσόογκον τῶι ΦΑ, τὸ ΝΙ καὶ τῶι μὲν Φ ἴσον ἔ στω τὸ Ν, τῶι δὲ Α τὸ Ι, καὶ ἔτι τὸ μὲν τοῦ ΦΑ μεγέθους βάρος ἔστω τὸ Β, τοῦ δὲ ΝΙ τὸ ΡΟ, τοῦ δὲ Ι τὸ Ρ· τὸ ΦΑ ἄρα πρὸς τὸ ΝΙ τοῦτον ἔχει τὸν λό γον, ὃν τὸ Β πρὸς τὸ ΡΟ. ἀλλ’ ἐπὶ τὸ ΦΑ μέγεθος ἐς τὸ ὑγρὸν ἀφίηται κου φότερον ὑπάρχον τοῦ ὑγροῦ, δῆ λον, ὡς ὁ τοῦ δεδυκότος μεγέ θους ὄγκος ἴσον βάρος ἔχει τῶι ΦΑ μεγέθει· δέδεικται γὰρ τοῦτο· ἴ σον ἄρα τὸ Β βάρος τῶι Ρ, ἐπει ἐπειδὴ τὸ μὲν Β βάρος τοῦ ὅλου τοῦ ΦΑ μεγέθους, τὸ δὲ Ρ τοῦ Ι ὑγροῦ οὗ περ ἐγίγνετο ἴσον τὸ ἴσον ὄγκο ὄγκον ἔχοντι τῶι δεδυκότι μεγέθει τῶι Α· ἔχει ἄρα τὸ ΦΑ μέγεθος τῶι βάρει πρὸς τὸ ΝΙ, ὃν τὸ Ρ πρὸς τὸν ΡΟ. ὃν δὲ λόγον ἔχει τὸ Ρ πρὸς τὸν ΡΟ, τοῦτον ἔχει τὸν λόγον τὸ Ι πρὸς τὸ ΙΝ καὶ τὸ Α πρὸς τὸ ΦΑ· δέδεικτ δέδεικται τὸ ὀρθόν.
Β Τὸ ὀρθὸν τμᾶμα τοῦ ὀρθογωνίου κωνοειδοῦς, ὅταν τὸν ἄξονα σχῆι μὴ μείζονα ἢ ἡμιόλιον τῆς μέ χρι τοῦ ἄξονος, πάντα λόγον ἔχο ἔχον πρὸς τὸ ὑγρὸν τῶι βάρει, ἀφεθὲν εἰς τὸ ὑγρὸν οὕτως, ὥστε τὴν βάσιν αὐτοῦ μὴ ἅπτεσθαι τοῦ ὑγροῦ, τεθὲ τεθὲν κεκλιμένον οὐ μενεῖ κεκλιμέ νον, ἀλλὰ ἀποκαταστήσεται ὀρθ ὀρθόν . ὀρθὸν δὲ λέγω καθεστηκέναι τὸ τοιοῦτο τμᾶμα, ὁπόταν τὸ ἀπο τετμηκὸς αὐτὸ ἐπίπεδον ἦι π παρὰ τὴν ἐπιφάνειαν ἦι τοῦ ὑγροῦ. ἔστω τμᾶμα ὀρθογωνίου κωνοει δοῦς, οἷον εἴρηται, καὶ κείσθω κεκλιμένον δεικτέον, ὅτι οὐ με νεῖ, ἀλλ’ ἀποκαταστήσεται ὀρθό ὀρθόν . τμηθέντος δὴ αὐτοῦ ἐπιπέδωι διὰ τοῦ ἄξονος ὀρθῶι πρὸς τὸ ἐπίπεδον τὸ ἐν τῆι ἐπιφανείαι τοῦ ὑγροῦ τμάματος ἔστω τὸ μὴ ΑΠ ΟΛ ὀρθογωνίου κών κώνου τομή, ἄξων δὲ τοῦ τμάματος καὶ διάμετρος τῆς τομῆς ἡ ΝΟ, τῆς δὲ τοῦ ὑγροῦ ἐπιφανεί ἐπιφανείας τομὴ ἡ ΙΣ. ἐπεὶ οὖν τὸ τμᾶμα οὐ κ ἔστιν ὀρθόν, οὐκ ἂν εἴη παράλ ληλος ἡ ΩΛ τῆς ΙΣ· ὥστε οὐ ποι ήσει ὀρθὴν γωνίαν ἡ ΝΘ πρ πρὸς τ τὴν ΙΣ. ἤχθω οὖν παράλληλος ἡ ἐ φαπτομένη ΙΣ ΚΩ τῆι τῆς τοῦ κώνου τομῆς κατὰ τὸ Π, καὶ ἀπὸ τοῦ Π παρὰ τὸ ΝΟ ἤχθω· τέ μνει δὲ ἡ ΠΦ δίχα τὴν ΙΣ· δέδει κται γὰρ ἐν τοῖς κωνικοῖς. τετμήσ θω ἡ ΠΦ, ὥστε εἶναι διπλῆ τὴν ΠΒ τῆς ΒΦ, καὶ ἡ ΝΟ κατὰ τὸ