Transcription of folios 031r-032v of the Archimedes Palimpsest. (= Archimedes folio Arch44r, Spiral Lines) 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 5535 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 Spiral Lines 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
εἰς τὸ χωρίον ἐγγράψαι σχᾶμα ἐπίπεδον ἐξ ὁμοίων τομέων συγκείμενον, ὥστε τὸ εἰρημένο εἰρημένον χωρίον μείζονι μὲν τὸ τοῦ ἐγ γραφέντος σχάματος ἐλάσσο νι ἢ ἁλίκω ὑπερέχει τὸ αὐτὸ χω ρίον τοῦ Χ τομέως. ἐγγεγράφθ ἐγγεγράφθω οὖν, καὶ ἔστω τῶν τομέων, ἐξ ὧν σύγκειται τὸ ἐγγεγραμμένον σχᾶ μα, μείζων μὲν ὁ ΘΒΓ, ἐλάσσων δὲ ὁ ΘΕ· δῆλον οὖν, ὅτι τὸ ἐγγεγραμμέ νον σχᾶμα μεῖζόν ἐστι τοῦ Χ τομέ ως. πάλιν οὖν ἐντί τινες γραμ μαὶ τῶι ἴσωι ἀλλαλᾶν ὑπερέ χουσαι ἀπὸ τοῦ Θ ποτὶ τὰν ἕλι κα ποτιπίπτουσαι, ὧν ἐστιν με γίστα μὲν ἁ ΘΑ, ἐλαχίστα δὲ ἁ ΘΕ, ἐντὶ δὲ καὶ ἄλλαι γραμμ γραμμαὶ ἀπὸ τοῦ Θ ποτὶ τὰν τοῦ ΘΑΖ το μέως περιφέρειαν ποτιπί πτουσαι χωρὶς τᾶς ΘΑ τῶι μὲ μὲν πλήθει μιᾶ ἐλάσσονες τᾶν τῶι ἴσωι ἀλλαλᾶν ὑπερεχουσᾶν, τῶι δὲ μεγέθει ἀλλάλαις τε καὶ τᾶ μεγίστα ἴσαι, καὶ ἀναγεγρά φεται ἀπὸ ἑκάστας ὁμοῖοι το μέες, ἀπὸ δὲ τᾶς μεγίστας τᾶν τῶι ἴσωι ἀλλαλᾶν ὑπερεχσ ὑπερεχουσᾶν οὐκ ἀναγέγραπται· οἱ τομέες οὖν οἱ ἀπὸ τᾶν ἰσᾶν ἀλλάλαις τε καὶ τᾶ μεγίστα ποτὶ τοὺς το μέας τοὺς ἀπὸ τᾶν τῶι ἴσωι ἀλλαλᾶν ὑπερεχουσᾶν χωρ χωρὶς τοῦ ἀπὸ τᾶς μεγίστας μείζονα λόγον ἔχοντι ἢ τὸ τετράγωνον τὸ ἀπὸ τᾶς ΘΑ ποτὶ τὸ ὑπὸ τᾶ τᾶν ΘΑ ΘΕ καὶ τὸ τρίτον μέρος τοῦ ἀ πὸ τᾶς ΕΖ· ὥστε καὶ ὁ ΘΑΖ τομεὺς ποτὶ τὸ ἐγγεγραμμένον σχᾶμα μείζονα λόγον ἔχει ἤπερ πρὸς τὸ τὸν Χ τομέα· ὥστε μεῖζον ὁ Χ τομεὺς τοῦ ἐγγεγραμμένου σχάματος. οὐκ ἔστι δέ, ἀλλὰ ἐλάσσων· οὐκ ἄρα ἐστὶν οὐδὲ ἐλάσσων ὁ Χ τομεὺς τοῦ περιεχομένου χωρίου ὑπό τε τᾶς ΑΒ ΓΔ ἕλικος καὶ τᾶν ΑΘ ΘΕ εὐ θειᾶν· ἴσα ἄρα. ΛΑ Τῶν χωρίων τῶν περιεχομένων ὑπό τε τᾶν ἑλίκων καὶ τᾶν εὐ θειᾶν τᾶν ἐν τᾶι περιφορᾶι τὸ μὲν Γ τοῦ Β διπλάσιόν ἐστι, τὸ δὲ Δ τριπλάσιον, τὸ δὲ Ε τετραπλά σιον, καὶ ἀεὶ τὸ ἑπόμενον κατὰ τοὺς ἑξῆς ἀριθμοὺς πολλαπλά σιον τοῦ δευτέρου χωρίου, τὸ δὲ Α χωρίον ἕκτον μέρος ἐστὶν τοῦ δευτέρου. ἔστω ἁ προκειμένα ἕλιξ ἔν τε τᾶι πρώται περιφορᾶι γεγραμ μένα καὶ ἐν τᾶι δευτέρα καὶ ἐν ταῖς ἑπομέναις ὁποσαισοῦν, ἔστω ἀρχὰ μὲν τᾶς ἕλικος τὸ Θ σαμεῖον, ἁ δὲ ΘΕ εὐθεῖα ἀρχὰ τᾶς