Measurement of Circle 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 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 Measurement of the Circle 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
καὶ περιγεγράφθω τὸ τετράγωνον, καὶ τετμήσθωσαν αἱ περιφέ ρειαι δίχα, καὶ ἤχθωσαν ἐφα πτόμεναι διὰ τῶν σημείων· ὀρθὴ ἄρα ἡ ὑπὸ ΟΑΡ. ἡ ΟΡ ἄρα τῆς ΜΡ ἐστὶν μεί ζων· ἡ γὰρ ΡΜ τῆι ΡΑ ἴση ἐστί · καὶ τὸ ΡΟΠ τρίγωνον ἄρα τοῦ ΟΖΑΜ σχήμα τος μεῖζόν ἐστιν ἢ τὸ ἥμισυ. λελεί φθωσαν οἱ τῶ ΠΖΑ τομεῖ ὅμοι οι ἐλάσσους τῆς ὑπεροχῆς, ἧ ὑ περέχει τὸ Ε τοῦ ΑΒΓΔ κύκλου πρὸς ἕν, τὸ ΑΓΖ πρὸς τὸ ΑΓΔ ἐστίν , ὡς ΚΒ πρὸς Ζ. ἀλλὰ τὸ ΑΓΔ τετραπλάσιόν ἐστι τὸ ΓΗ τετράγωνον, τὸ δὲ ΑΓΔΖ τρίγωνον τῶι ΑΒ κύκλωι ἴσον ἐστίν ἐπεὶ ἡ μὲν ΑΓ κάθετος ἴση ἐστὶ τῆι ἐκ τοῦ κέντρου, ἡ δὲ βάσις τῆς διαμέτρου τριπλασίω τριπλασίων καὶ τοῦ Ζ ἔγγιστα ὑπερέχουσα δειχθή σεται· ὁ κύκλος οὖν πρὸς τὸ ΓΗ τετράγω νον λόγον ἔχει, ὃν ΙΑ πρὸς ΙΔ .
Figure 1
ἡ ΘΑ ἄρα πρὸς ΘΓ μείζονα λόγον ἔχει ἢ ὃν ΑΡΟΒ Η πρὸς ΡΝΓ. ἔτι δίχα ἡ ὑπὸ ΘΕΓ τῆι ΕΚ· ἡ ΕΓ ἄρα πρὸς ΓΚ μείζονα λόγον ἔχει ἢ ὃν ΒΤΛΔ Δ πρὸς ΡΝΓ· ἡ ΕΚ ἄρα πρὸς ΓΚ μείζων ἢ ὃν ΒΤΛΔ Δ πρὸς ΡΝΓ. ἔτι δίχα ἡ ὑπὸ ΚΕΓ τῆι ΛΕ· ἡ ΕΓ ἄρα πρὸς ΛΓ μείζονα μήκει λόγον ἔχει ἤ ἤπερ ΔΧΟΓ 𐅵 πρὸς ΡΝΓ. ἐπεὶ οὖν ἡ ὑπὸ ΖΕΓ τρί τον οὖσα ὀρθῆς τέτμηται τετράκις δίχα, ἡ ὑπὸ ΛΕΓ ὀρθῆς ἐστι ΜΗ. κείσ θω οὖν αὐτῆ ἴση πρὸς τῶι Ε ἡ ὑπὸ ΓΕ Μ· ἡ ἄρα ὑπὸ ΛΕΜ ὀρθῆς ἐστι ΚΔ. καὶ
Figure 2
Ἔστω κύκλος ΗΑΓ, καὶ ἡ ὑπὸ ΗΓΒ τῆι ὑπὸ ΗΑΓ ἐστὶν ἴση. καὶ κοινὴ ἡ ὑπὸ ΑΗΓ ὀρθή· καὶ τρίτη ἄρα ἡ ὑπὸ ΗΖΓ τρίτη τῆι ὑπὸ ΑΓΗ. ἰσογώνιον ἄρα τὸ ΑΗΓ τῶι ΓΗΖ τριγώνωι· ἔστιν ἄρα, ὡς ἡ ΑΗ πρὸς ΗΓ, ἡ ΓΗ πρὸς ΗΖ καὶ ἡ ΑΓ πρὸς ΓΖ. ἀλλ’ ὡς ἡ ΑΓ πρὸς ΓΖ, καὶ συναμ φότερος ἡ ΓΑΒ πρὸς ΒΓ· καὶ ὡς συ ναμφότερος ἄρα ἡ ΒΑΓ πρὸς ΒΓ, ἡ ΑΗ πρὸς ΗΓ. διὰ τοῦτο οὖν ἡ ΑΗ πρὸς τὴν ΗΓ ἐλάσσονα λόγον ἔχει ἤ ἤπερ ΒϡΙΑ πρὸς ΨΠ, ἡ δὲ ΑΓ πρὸς τὴν ΓΗ ἐλάσσονα ἢ ὃν ΓΙΓ Γ Δ πρὸς ΨΠ. δίχα ἡ ὑπὸ Γ ΑΗ τῆι ΑΘ· ἡ ΑΘ ἄρα διὰ τὰ αὐτὰ πρὸς τὴν ΘΓ ἐλάσσονο λόγον ἔχει ἢ ὃν ΕΤΚΔ Ε Δ πρὸς ΨΠ ἢ ὃν ΑΩΚΓ πρὸς ΣΜ· ἑκατέρα γὰρ ἑκατέρας ΔΙ ΓΑ· ὥστε ἡ ΑΓ πρὸς τὴν ΓΘ ἢ ὃν ΑΩ ΛΑ Θ πρὸς ΣΜ. ἔτι δίχα ἡ ὑπὸ ΘΑΓ τῆι ΚΑ· καὶ ἡ ΑΚ πρὸς τὴν ΚΓ ἐλάσσονα ἄρα λόγον ἔχει ἢ ὃν ΑΖ πρὸς ΞϚ· ἑκα τέρα γὰρ ἑκάτερα οι Μ ἄρα πρ πρὸς τὴν λόγον ΑΟΣ πρὸς ΞϚ. ἔτι δίχα ἡ ὑπὸ Κ ΑΓ τῆι ΛΑ· ἡ ΑΛ ἄρα πρὸς τὴν ΛΓ ἐλάσ σονα λόγον ἔχει ἢ ὃν τὰ ΒΙϚ Ϛ πρὸς ΞϚ, ἡ δὲ ΑΓ πρὸς ΓΛ ἐλάσσονα ἢ τὰ ΒΙΖ Δ πρὸς ΞϚ. ἀνάπαλιν ἄρα ἡ περίμε τρος τ τοῦ πολυγώνου πρὸς τὴν δμετρ διάμετρον μείζονα λόγον ἔχει ἤ ἤπερ ϚΤΑ Ϛ πρὸς ΖΙΖ Δ, ἅ ἅπερ τῶν ΒΙΖ Δ μείζον μείζονά ἐστιν ἢ τριπλασίονα καὶ δέκα ΟΑ· καὶ ἡ περίμετρος ἄρα τοῦ ϘϚγώ νου τοῦ ἐν τῶι κύκλωι τῆς δμέ διαμέ τρου τριπλασίων ἐστὶ καὶ μείζω μείζων ἢ ὃν ΟΙΑ· ὥστε καὶ ὁ κύκλος ἔτι μᾶλλον τριπλασίων ἐστὶ καὶ μείζω μείζων ἢ ΘΙ Α. ἡ ἄρα τοῦ κύκλου περίμετρος τῆς διαμέτρου τριπλασίων ἐστὶ καὶ ἐλάσ σων μὲν ἢ ἑβδόμωι μέρει, μείζων δὲ ἢ
Figure 3