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ἐκ δύο ὀνομάτων κατὰ ἓν μόνον σημεῖον διαιρεῖται εἰς τὰ ὀνόματα.

ἔστω ἐκ δύο ὀνομάτων ΑΒ διῃρημένη εἰς τὰ ὀνόματα κατὰ τὸ Γ: αἱ ΑΓ, ΓΒ ἄρα ῥηταί εἰσι δυνάμει μόνον
5σύμμετροι. λέγω, ὅτι ΑΒ κατ᾽ ἄλλο σημεῖον οὐ διαιρεῖται εἰς δύο ῥητὰς δυνάμει μόνον συμμέτρους.

εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε καὶ τὰς ΑΔ, ΔΒ ῥητὰς εἶναι δυνάμει μόνον συμμέτρους. φανερὸν δή, ὅτι ΑΓ τῇ ΔΒ
10οὐκ ἔστιν αὐτή. εἰ γὰρ δυνατόν, ἔστω. ἔσται δὴ καὶ ΑΔ τῇ ΓΒ αὐτή: καὶ ἔσται ὡς ΑΓ πρὸς τὴν ΓΒ, οὕτως ΒΔ πρὸς τὴν ΔΑ, καὶ ἔσται ΑΒ κατὰ τὸ αὐτὸ τῇ κατὰ τὸ Γ διαιρέσει διαιρεθεῖσα καὶ κατὰ τὸ Δ: ὅπερ οὐχ
15ὑπόκειται. οὐκ ἄρα ΑΓ τῇ ΔΒ ἐστιν αὐτή.
15διὰ δὴ τοῦτο καὶ τὰ Γ, Δ σημεῖα οὐκ ἴσον ἀπέχουσι τῆς διχοτομίας. ἄρα διαφέρει τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ, τούτῳ διαφέρει καὶ τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ διὰ τὸ καὶ τὰ ἀπὸ
20τῶν ΑΓ, ΓΒ μετὰ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ καὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ μετὰ τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ ἴσα εἶναι τῷ ἀπὸ τῆς ΑΒ. ἀλλὰ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ διαφέρει ῥητῷ: ῥητὰ γὰρ ἀμφότερα: καὶ τὸ δὶς ἄρα ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ,
25ΓΒ διαφέρει ῥητῷ μέσα ὄντα: ὅπερ ἄτοπον: μέσον γὰρ μέσου οὐχ ὑπερέχει ῥητῷ.

οὐκ ἄρα ἐκ δύο ὀνομάτων κατ᾽ ἄλλο καὶ ἄλλο σημεῖον διαιρεῖται: καθ᾽ ἓν ἄρα μόνον: ὅπερ ἔδει δεῖξαι.

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