Transcription of folios 031v-032r of the Archimedes Palimpsest. (= Archimedes folio Arch44v, 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 5536 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
περιφορᾶς, τῶν δὲ χωρίων ἔστω τὸ μὲν Κ τὸ Α, τὸ δὲ Λ τὸ Β, τὸ δὲ Μ τὸ Γ, τὸ δὲ Ν τὸ Δ, τὸ δὲ Ξ τὸ Ε. δει κτέον ὅτι τὸ μὲν Κ χωρίον ἕκτον μέρος ἐστὶ τοῦ ἑπομένου, τὸ δὲ Μ δι πλάσιον τοῦ Γ, τὸ δὲ Ν Γ Γπλασιον τοῦ Λ, κατὰ τοὺς ἑξῆς ἀριθμούς. ὅτι μὲν οὖ οὖν τὸ Κ Ϛ μέρος ἐστὶ τοῦ Λ, ὧδε δείκνυται. ἐπεὶ τὸ ΚΛ χωρίον ποτὶ τὸν Β κύκλ κύκλον δέδεικται τοῦτον ἔχειν τὸν λόγον, ὃν ἔχει τὰ Ζ ποτὶ τὰ ΙΒ, ὁ δὲ Β κύ κλος ποτὶ τὸν Α πρῶτον κύκλον ὡς ΙΒ πο τὶ τὰ Γ· δῆλον γάρ ἐστιν · ὁ δὲ Α κύκλος ποτὶ τὸ Κ χωρίον ἔχει ὡς Γ πρὸς Α, Ϛ ἄρα ἐστὶν τὸ Κ χωρίον τοῦ Λ. πά λιν δὲ καὶ τὸ ΚΛΜ χωρίον ποτὶ τὸν Γ κύκλον δέδεικται ὅτι τοῦτον ἔ χει τὸν λόγον, ὃν ἔχει συναμφότε ρον τό τε ὑπὸ ΓΘΒ καὶ τὸ τρίτον μέ ρος τοῦ ἀπὸ ΓΒ τετραγώνου ποτὶ τὸ ἀπὸ ΓΘ τετράγωνον. ὁ δὲ Γ κύ κλος ἔχει ποτὶ τὸν Β κύκλον ὃν τὸ ἀπὸ τῆς ΓΘ τετράγωνον ποτὶ τὸ ἀπὸ τᾶς ΘΒ, ὁ δὲ Β κύκλος ἔχει πο τὶ τὸ ΚΛ χωρίον ὃν τὸ ἀπὸ ΒΘ τετράγωνον ποτὶ τὰ συναμφότε ρα τό τε ὑπὸ τᾶν ΒΘ ΘΑ καὶ τὸ τρί τον μέρος τοῦ ἀπὸ τᾶς ΑΒ τετρα γώνου. ταῦτα δὲ ἔχει πρὸς ἄλληλα λόγον, ὃν ΙΘ ποτὶ τὰ Ζ· ὥστε καὶ τὸ ΚΛΜ χωρίον ποτὶ τὸ ΛΚ χωρίο χωρίον τοῦτον ἔχει τὸν λόγον, ὃν ΙΘ ποτὶ τὰ Ζ· αὐτὸ οὖν τὸ Μ ποτὶ τὸ ΚΛ λό γον ἕξει, ὃν τὰ ΙΒ ποτὶ τὰ Ζ. τὸ δὲ ΚΛ ποτὶ τὸ Λ λόγον ἔχει, ὃν τὰ ΙΒ ποτὶ τὰ Ζ . τὸ δὲ ΚΛ ποτὶ τὸ Λ λό γον ἔχει, ὃν τὰ Ζ ποτὶ τὰ Ϛ · δῆλον οὖν ὅτι διπλάσιόν ἐστι τὸ Μ τοῦ Λ. ὅτι δὲ τὰ ἑπόμενα τὸν τῶν ἑξῆς ἀριθμὸν λόγον ἔχει, δειχθήσεται. τὸ γὰρ ΗΚΜ ΜΝΞ ποτὶ τὸν κύκλο κύκλον , οὗ ἐστιν ἐκ τοῦ κέντρου ἁ ΘΕ, τοῦ τον ἔχει τὸν λόγον, ὃν ἔχει συναμ φότερον τό τε ὑπὸ τᾶν ΕΘ ΔΘ πε ριεχόμενον καὶ τὸ τρίτον μέρος τοῦ ὑπὸ τᾶς ΔΘ τετράγωνον πο τὶ τὸ ἀπὸ τᾶς ΘΕ τετράγωνον. ὁ δὲ κύκλος, οὗ ἐστιν ἐκ τοῦ κέντρου ἁ ΘΕ, ποτὶ τὸν κύκλον, οὗ ἐστιν ἐκ τοῦ κέντρου ἁ ΘΔ, τοῦτον ἔχει τὸν λό γον, ὃν τὸ ἀπὸ τᾶς ΘΕ τετράγων τετράγωνον ποτὶ τὸ ἀπὸ τᾶς ΘΔ τετράγωνο τετράγωνον , ὁ δὲ κύκλος, οὗ ἐστιν ἀπὸ τοῦ κέν τρου ΔΘ, ποτὶ τὸ ΚΛ ΜΝ χωρίον τοῦτον ἔχει τὸν λόγον, ὃν τὸ ἀπὸ τ τᾶς ΘΔ τετράγωνον ποτὶ τὰ συναμ φότερά τε τὸ ὑπὸ τᾶν ΘΔ ΘΓ καὶ τὸ τρίτον μέρος τοῦ ἀπὸ τᾶς ΔΕ ποτὶ τὸ ὑπὸ τῶν ΑΘ ΘΓ καὶ τὸ τρί τον μέρος τοῦ ἀπὸ τᾶς ΛΓ· διελόν τι καὶ τὸ Ξ χωρίον ποτὶ τὸ ΚΛ ΜΝ λόγον ἔχει, ὃν ἁ ὑπεροχὰ τοῦ τε ὑ πὸ ΕΘ ΘΔ μετὰ τοῦ τρίτου μέρους τοῦ ἀπὸ τᾶς ΕΔ πρός τε τὸ ὑπὸ τᾶ τᾶν ΔΘ ΘΓ καὶ τὸ τρίτον μέρος τοῦ ἀ πὸ τᾶς ΔΓ· ὑπερέχει δὲ τὰ συναμ φότερα τῶν συναμφοτέρων, ὧι καὶ τὸ ὑπὸ τῶν ΕΘΔ τοῦ ὑπὸ τῶν ΔΘΓ, ὑπερέχει δὲ τῶι ὑπὸ τῶν ΔΘΓ ὑπερέχει δὲ τῶι ὑπὸ τῶ τῶν