§10.prop1.3δύο μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον αὐτῶν κοινὸν μέτρον εὑρεῖν.
Two commensurable magnitudes being given, to find their greatest common measure.
ἔστω τὰ δοθέντα δύο μεγέθη σύμμετρα τὰ ΑΒ, ΓΔ, ὧν ἔλασσον τὸ ΑΒ· δεῖ δὴ τῶν ΑΒ, ΓΔ τὸ μέγιστον κοινὸν μέτρον εὑρεῖν.
Let the two given commensurable magnitudes be ΑΒ, ΓΔ, of which ΑΒ is the lesser; it is required to find the greatest common measure of ΑΒ, ΓΔ.
τὸ ΑΒ γὰρ μέγεθος ἤτοι μετρεῖ τὸ ΓΔ ἢ οὔ.
For the magnitude ΑΒ either measures ΓΔ or does not.
εἰ μὲν οὖν μετρεῖ, μετρεῖ δὲ καὶ ἑαυτό, τὸ ΑΒ ἄρα τῶν ΑΒ, ΓΔ κοινὸν μέτρον ἐστίν· καὶ φανερόν, ὅτι καὶ μέγιστον.
If then it measures it, and it also measures itself, ΑΒ is a common measure of ΑΒ, ΓΔ; and it is manifest that it is also the greatest.
μεῖζον γὰρ τοῦ ΑΒ μεγέθους τὸ ΑΒ οὐ μετρήσει.
For a magnitude greater than the magnitude ΑΒ will not measure ΑΒ.
μὴ μετρείτω δὴ τὸ ΑΒ τὸ ΓΔ. καὶ ἀνθυφαιρουμένου ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, τὸ περιλειπόμενον μετρήσει ποτὲ τὸ πρὸ ἑαυτοῦ διὰ τὸ μὴ εἶναι ἀσύμμετρα τὰ ΑΒ, ΓΔ·
Let then ΑΒ not measure ΓΔ.
καὶ τὸ μὲν ΑΒ τὸ ΕΔ καταμετροῦν λειπέτω ἑαυτοῦ ἔλασσον τὸ ΕΓ, τὸ δὲ ΕΓ τὸ ΖΒ καταμετροῦν λειπέτω ἑαυτοῦ ἔλασσον τὸ ΑΖ, τὸ δὲ ΑΖ τὸ ΓΕ μετρείτω.
And, the lesser being continually subtracted from the greater, that which is left will sometime measure that which is before it, because ΑΒ, ΓΔ are not incommensurable; and let ΑΒ, measuring ΕΔ, leave ΕΓ less than itself, and let ΕΓ, measuring ΖΒ, leave ΑΖ less than itself, and let ΑΖ measure ΓΕ.
ἐπεὶ οὖν τὸ ΑΖ τὸ ΓΕ μετρεῖ, ἀλλὰ τὸ ΓΕ τὸ ΖΒ μετρεῖ, καὶ τὸ ΑΖ ἄρα τὸ ΖΒ μετρήσει.
Since then ΑΖ measures ΓΕ, but ΓΕ measures ΖΒ, therefore ΑΖ will also measure ΖΒ.
μετρεῖ δὲ καὶ ἑαυτό·
But it also measures itself; therefore ΑΖ will also measure the whole ΑΒ.
καὶ ὅλον ἄρα τὸ ΑΒ μετρήσει τὸ ΑΖ. ἀλλὰ τὸ ΑΒ τὸ ΔΕ μετρεῖ· καὶ τὸ ΑΖ ἄρα τὸ ΕΔ μετρήσει.
But ΑΒ measures ΔΕ; therefore ΑΖ will also measure ΕΔ.
μετρεῖ δὲ καὶ τὸ ΓΕ· καὶ ὅλον ἄρα τὸ ΓΔ μετρεῖ· τὸ ΑΖ ἄρα τῶν ΑΒ, ΓΔ κοινὸν μέτρον ἐστίν.
But it also measures ΓΕ; therefore it also measures the whole ΓΔ; therefore ΑΖ is a common measure of ΑΒ, ΓΔ.
λέγω δή, ὅτι καὶ μέγιστον.
I say then that it is also the greatest.
εἰ γὰρ μή, ἔσται τι μέγεθος μεῖζον τοῦ ΑΖ, ὃ μετρήσει τὰ ΑΒ, ΓΔ. ἔστω τὸ Η. ἐπεὶ οὖν τὸ Η τὸ ΑΒ μετρεῖ, ἀλλὰ τὸ ΑΒ τὸ ΕΔ μετρεῖ, καὶ τὸ Η ἄρα τὸ ΕΔ μετρήσει.
For, if not, there will be some magnitude greater than ΑΖ which will measure ΑΒ, ΓΔ. Let it be Η. Since then Η measures ΑΒ, but ΑΒ measures ΕΔ, therefore Η will also measure ΕΔ.
μετρεῖ δὲ καὶ ὅλον τὸ ΓΔ· καὶ λοιπὸν ἄρα τὸ ΓΕ μετρήσει τὸ Η. ἀλλὰ τὸ ΓΕ τὸ ΖΒ μετρεῖ· καὶ τὸ Η ἄρα τὸ ΖΒ μετρήσει.
But it also measures the whole ΓΔ; therefore Η will also measure the remainder ΓΕ. But ΓΕ measures ΖΒ; therefore Η will also measure ΖΒ.
μετρεῖ δὲ καὶ ὅλον τὸ ΑΒ, καὶ λοιπὸν τὸ ΑΖ μετρήσει, τὸ μεῖζον τὸ ἔλασσον· ὅπερ ἐστὶν ἀδύνατον.
But it also measures the whole ΑΒ; therefore it will also measure the remainder ΑΖ, the greater measuring the less; which is impossible.
οὐκ ἄρα μεῖζόν τι μέγεθος τοῦ ΑΖ τὰ ΑΒ, ΓΔ μετρήσει· τὸ ΑΖ ἄρα τῶν ΑΒ, ΓΔ τὸ μέγιστον κοινὸν μέτρον ἐστίν.
Therefore no magnitude greater than ΑΖ will measure ΑΒ, ΓΔ; therefore ΑΖ is the greatest common measure of ΑΒ, ΓΔ.
δύο ἄρα μεγεθῶν συμμέτρων δοθέντων τῶν ΑΒ, ΓΔ τὸ μέγιστον κοινὸν μέτρον ηὕρηται· ὅπερ ἔδει δεῖξαι.
Therefore, the greatest common measure of two given commensurable magnitudes ΑΒ, ΓΔ has been found; which it was required to prove.
Πόρισμα
ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν μέγεθος δύο μεγέθη μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει.
Porism From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure.