§10.prop1.4τριῶν μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον αὐτῶν κοινὸν μέτρον εὑρεῖν.
Three commensurable magnitudes being given, to find their greatest common measure.
ἔστω τὰ δοθέντα τρία μεγέθη σύμμετρα τὰ Α, Β, Γ· δεῖ δὴ τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον εὑρεῖν.
Let the three given commensurable magnitudes be Α, Β, Γ; it is required to find the greatest common measure of Α, Β, Γ.
εἰλήφθω γὰρ δύο τῶν Α, Β τὸ μέγιστον κοινὸν μέτρον, καὶ ἔστω τὸ Δ· τὸ δὴ Δ τὸ Γ ἤτοι μετρεῖ ἢ οὔ.
For let the greatest common measure of two of them, Α, Β, be taken, and let it be Δ; then Δ either measures Γ or does not.
μετρείτω πρότερον.
Let it measure it first.
ἐπεὶ οὖν τὸ Δ τὸ Γ μετρεῖ, μετρεῖ δὲ καὶ τὰ Α, Β, τὸ Δ ἄρα τὰ Α, Β, Γ μετρεῖ· τὸ Δ ἄρα τῶν Α, Β, Γ κοινὸν μέτρον ἐστίν.
Since then Δ measures Γ, and it also measures Α, Β, therefore Δ measures Α, Β, Γ; therefore Δ is a common measure of Α, Β, Γ.
καὶ φανερόν, ὅτι καὶ μέγιστον· μεῖζον γὰρ τοῦ Δ μεγέθους τὰ Α, Β οὐ μετρεῖ.
And it is manifest that it is also the greatest; for a magnitude greater than the magnitude Δ will not measure Α, Β.
μὴ μετρείτω δὴ τὸ Δ τὸ Γ. λέγω πρῶτον, ὅτι σύμμετρά ἐστι τὰ Γ, Δ. ἐπεὶ γὰρ σύμμετρά ἐστι τὰ Α, Β, Γ, μετρήσει τι αὐτὰ μέγεθος, ὃ δηλαδὴ καὶ τὰ Α, Β μετρήσει· ὥστε καὶ τὸ τῶν Α, Β μέγιστον κοινὸν μέτρον τὸ Δ μετρήσει.
Let then Δ not measure Γ. I say first that Γ, Δ are commensurable. For, since Α, Β, Γ are commensurable, some magnitude will measure them, which will clearly also measure Α, Β; so that it will also measure Δ, the greatest common measure of Α, Β.
μετρεῖ δὲ καὶ τὸ Γ· ὥστε τὸ εἰρημένον μέγεθος μετρήσει τὰ Γ, Δ· σύμμετρα ἄρα ἐστὶ τὰ Γ, Δ. εἰλήφθω οὖν αὐτῶν τὸ μέγιστον κοινὸν μέτρον, καὶ ἔστω τὸ Ε. ἐπεὶ οὖν τὸ Ε τὸ Δ μετρεῖ, ἀλλὰ τὸ Δ τὰ Α, Β μετρεῖ, καὶ τὸ Ε ἄρα τὰ Α, Β μετρήσει.
But it also measures Γ; so that the said magnitude will measure Γ, Δ; therefore Γ, Δ are commensurable. Let then their greatest common measure be taken, and let it be Ε. Since then Ε measures Δ, but Δ measures Α, Β, therefore Ε will also measure Α, Β.
μετρεῖ δὲ καὶ τὸ Γ. τὸ Ε ἄρα τὰ Α, Β, Γ μετρεῖ· τὸ Ε ἄρα τῶν Α, Β, Γ κοινόν ἐστι μέτρον.
But it also measures Γ. Therefore Ε measures Α, Β, Γ; therefore Ε is a common measure of Α, Β, Γ.
λέγω δή, ὅτι καὶ μέγιστον.
I say then that it is also the greatest.
εἰ γὰρ δυνατόν, ἔστω τι τοῦ Ε μεῖζον μέγεθος τὸ Ζ, καὶ μετρείτω τὰ Α, Β, Γ. καὶ ἐπεὶ τὸ Ζ τὰ Α, Β, Γ μετρεῖ, καὶ τὰ Α, Β ἄρα μετρήσει καὶ τὸ τῶν Α, Β μέγιστον κοινὸν μέτρον μετρήσει.
For, if possible, let there be some magnitude Ζ greater than Ε, and let it measure Α, Β, Γ. And, since Ζ measures Α, Β, Γ, therefore it will also measure Α, Β, and it will measure the greatest common measure of Α, Β.
τὸ δὲ τῶν Α, Β μέγιστον κοινὸν μέτρον ἐστὶ τὸ Δ· τὸ Ζ ἄρα τὸ Δ μετρεῖ.
But the greatest common measure of Α, Β is Δ; therefore Ζ measures Δ.
μετρεῖ δὲ καὶ τὸ Γ· τὸ Ζ ἄρα τὰ Γ, Δ μετρεῖ· καὶ τὸ τῶν Γ, Δ ἄρα μέγιστον κοινὸν μέτρον μετρήσει τὸ Ζ. ἔστι δὲ τὸ ε· τὸ Ζ ἄρα τὸ Ε μετρήσει, τὸ μεῖζον τὸ ἔλασσον· ὅπερ ἐστὶν ἀδύνατον.
But it also measures Γ; therefore Ζ measures Γ, Δ; therefore Ζ will also measure the greatest common measure of Γ, Δ. But it is Ε; therefore Ζ will measure Ε, the greater the less; which is impossible.
οὐκ ἄρα μεῖζόν τι τοῦ Ε μεγέθους τὰ Α, Β, Γ μετρεῖ· τὸ Ε ἄρα τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον ἐστίν, ἐὰν μὴ μετρῇ τὸ Δ τὸ Γ, ἐὰν δὲ μετρῇ, αὐτὸ τὸ Δ.
τριῶν ἄρα μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον κοινὸν μέτρον ηὕρηται.
Therefore no magnitude greater than the magnitude Ε measures Α, Β, Γ; therefore Ε is the greatest common measure of Α, Β, Γ, if Δ do not measure Γ, and, if it measure it, Δ itself. Therefore, the greatest common measure of three given commensurable magnitudes has been found.
Πόρισμα
ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν μέγεθος τρία μεγέθη μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει.
Porism From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure.
ὁμοίως δὴ καὶ ἐπὶ πλειόνων τὸ μέγιστον κοινὸν μέτρον ληφθήσεται, καὶ τὸ πόρισμα προχωρήσει.
And similarly also in the case of more magnitudes the greatest common measure will be taken, and the porism will hold.
ὅπερ ἔδει δεῖξαι.
Which it was required to prove.