Transcription of folios 030v-033r of the Archimedes Palimpsest. (= Archimedes folio Arch37v, 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 5534 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
ἐπ’ αὐτὰν ἀγμέναν· δυνατὸν οὖν ἐστιν ἀπὸ τοῦ Α ποτιβαλεῖν τὰν ΑΣ ποτὶ τὰν ΤΜΝ ἐκβεβλημέναν, ὥσ τε τὰν μεταξὺ τᾶς περιφερείας καὶ τᾶς ἐκβεβλημένας τὰν ΡΣ ποτὶ τὰν ΤΡ τὸν αὐτὸν ἔχειν λόγο λόγον , ὃν ἁ ΤΑ ποτὶ τὰν ΑΛ· τεμεῖ δὴ ἁ ΑΣ τὸν μὲ μὲν κύκλον κατὰ τὸ Ρ, τὰν δὲ ἕ λικα κατὰ τὸ Χ· καὶ ἐναλλὰξ τὸ τὸν αὐτὸν ἕξει λόγον ἁ ΡΣ ποτὶ τὰν ΤΑ, ὃν ἁ ΤΡ ποτὶ τὰν ΑΛ. ἐλάσσο να λόγον ἔχει ἁ ΤΡ περιφέρεια πο τὶ τὰν διπλασίαν τοῦ ΜΝ κύκλ κύκλου περιφέρειαν· ἔστι γὰρ ἁ μὲν ΤΡ εὐθεῖα ἐλάσσων τᾶς ΤΡ περιφε ρείας, ἁ δὲ ΑΛ εὐθεῖα μείζων ἢ διπλασία τᾶς τοῦ ΤΜΝ περι φερείας· ἐλάσσονα ἄρα λόγον ἔχει ἁ ΡΣ ποτὶ τὰν ΑΡ ἢ ἁ ΤΡ πε ριφέρεια ποτὶ τὰν διπλασίαν τᾶς τοῦ ΤΜΝ κύκλου περιφερεί περιφερείας · ὅλα οὖν ἁ ΣΑ ποτὶ τὰν ΑΡ ἐλάσσ ονα λόγον ἔχει ἢ ἁ ΤΡ περιφέ ρεια μετὰ τᾶς τοῦ ΤΜΝ κύκλου περιφερείας δὶς εἰρημένας ποτὶ τὰν τοῦ ΤΜΝ κύκλου πε ριφέρειαν δὶς εἰρημέναν. ὃν δὲ λόγον ἔχουσιν αἱ εἰρημέναι περι φέρειαι, τοῦτον ἔχει τὸν λόγον ἁ ΧΑ ποτὶ τὰν ΑΤ· δέδεικται γὰρ τοῦτο· ἐλάσσονα ἄρα λόγον ἔχει ἁ ΑΣ ποτὶ τὰν ΑΡ ἢ ἁ ΧΑ ποτὶ τὰν ΤΑ· ὅ ὅπερ ἀδύνατον. οὐκ ἄρα μείζων ἐστὶν ἢ διπλασία ἁ ΖΑ εὐθεῖα τᾶς τοῦ ΤΜΝ κύκλου περι φερείας. ὁμοίως δὲ δειχθήσεται, οὐδὲ ἐλάσσων ἢ διπλασία. δῆ λον οὖν, ὅτι διπλασία ἐστίν. διὰ δὲ τοῦ αὐτοῦ τρόπου δεικτέον, καὶ εἴ κα τᾶς ἐν ὁποιαοῦν περιφορᾶι γεγραμμένας ἕλικος ἐπιψαύηι τις εὐθεῖα κατὰ τὸ πέρας τᾶς ἕλικος, καὶ ἀπὸ τᾶς ἀρχᾶς τᾶς ἕλικος ποτ’ ὀρθὰς ἀχθεῖσα τᾶι ἀρχᾶι τᾶς περιφορᾶς συμπίπτει ποτὶ τὰν ἐπιψαύουσαν, πολλα πλασίαν ἐστὶν τᾶς τοῦ κύκλου πε ριφερείας τοῦ κατὰ τὸν ἀριθμ ἀριθμὸν τᾶς περιφορᾶς λεγομένου τῶι αὐτῶι ἀριθμῶι.
ΕΙ κα τᾶς ἕλικος τᾶς ἐν τᾶι πρώ ται περιφορᾶι γεγραμμένας εὐθεῖα γραμμὰ ἐπιψαύη μὴ κατὰ τὸ πέρας τᾶς ἕλικος, ἀπὸ δὲ τᾶς ἁφᾶς ἐπὶ τὰν ἀρχὰν τᾶς ἕλικος εὐθεῖα ἐπιζευχθῆι, καὶ κέ κέν τρωι μὲν τᾶι ἀρχᾶι τᾶς ἕλικος, διαστάματι δὲ τᾶι ἐπιζευχθείσαι κύκλος γραφῆι, ἀπὸ δὲ τᾶς ἀρχᾶς τᾶς ἕλικος ἀχθῆ τις ποτ’ ὀρθὰς τᾶ ἀπὸ τᾶς ἁφᾶς ἐπὶ τὰν ἀρχ ἀρχὰν