# Book 7 Proposition 36

Τριῶν ἀριθμῶν δοθέντων εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν ἀριθμόν. Ἔστωσαν οἱ δοθέντες τρεῖς ἀριθμοὶ οἱ Α, Β, Γ: δεῖ δὴ εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν ἀριθμόν. Εἰλήφθω γὰρ ὑπὸ δύο τῶν Α, Β ἐλάχιστος μετρούμενος ὁ Δ. ὁ δὴ Γ τὸν Δ ἤτοι μετρεῖ ἢ οὐ μετρεῖ. μετρείτω πρότερον. μετροῦσι δὲ καὶ οἱ Α, Β τὸν Δ: οἱ Α, Β, Γ ἄρα τὸν Δ μετροῦσιν. λέγω δή, ὅτι καὶ ἐλάχιστον. εἰ γὰρ μή, μετρήσουσιν [τινα] ἀριθμὸν οἱ Α, Β, Γ ἐλάσσονα ὄντα τοῦ Δ. μετρείτωσαν τὸν Ε. ἐπεὶ οἱ Α, Β, Γ τὸν Ε μετροῦσιν, καὶ οἱ Α, Β ἄρα τὸν Ε μετροῦσιν. καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Α, Β μετρούμενος [τὸν Ε] μετρήσει. ἐλάχιστος δὲ ὑπὸ τῶν Α, Β μετρούμενός ἐστιν ὁ Δ: ὁ Δ ἄρα τὸν Ε μετρήσει ὁ μείζων τὸν ἐλάσσονα: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα οἱ Α, Β, Γ μετρήσουσί τινα ἀριθμὸν ἐλάσσονα ὄντα τοῦ Δ: οἱ Α, Β, Γ ἄρα ἐλάχιστον τὸν Δ μετροῦσιν. Μὴ μετρείτω δὴ πάλιν ὁ Γ τὸν Δ, καὶ εἰλήφθω ὑπὸ τῶν Γ, Δ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Ε. ἐπεὶ οἱ Α, Β τὸν Δ μετροῦσιν, ὁ δὲ Δ τὸν Ε μετρεῖ, καὶ οἱ Α, Β ἄρα τὸν Ε μετροῦσιν. μετρεῖ δὲ καὶ ὁ Γ [τὸν Ε: καὶ] οἱ Α, Β, Γ ἄρα τὸν Ε μετροῦσιν. λέγω δή, ὅτι καὶ ἐλάχιστον. εἰ γὰρ μή, μετρήσουσί τινα οἱ Α, Β, Γ ἐλάσσονα ὄντα τοῦ Ε. μετρείτωσαν τὸν Ζ. ἐπεὶ οἱ Α, Β, Γ τὸν Ζ μετροῦσιν, καὶ οἱ Α, Β ἄρα τὸν Ζ μετροῦσιν: καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Α, Β μετρούμενος τὸν Ζ μετρήσει. ἐλάχιστος δὲ ὑπὸ τῶν Α, Β μετρούμενός ἐστιν ὁ Δ: ὁ Δ ἄρα τὸν Ζ μετρεῖ. μετρεῖ δὲ καὶ ὁ Γ τὸν Ζ: οἱ Δ, Γ ἄρα τὸν Ζ μετροῦσιν: ὥστε καὶ ὁ ἐλάχιστος ὑπὸ τῶν Δ, Γ μετρούμενος τὸν Ζ μετρήσει. ὁ δὲ ἐλάχιστος ὑπὸ τῶν Γ, Δ μετρούμενός ἐστιν ὁ Ε: ὁ Ε ἄρα τὸν Ζ μετρεῖ ὁ μείζων τὸν ἐλάσσονα: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα οἱ Α, Β, Γ μετρήσουσί τινα ἀριθμὸν ἐλάσσονα ὄντα τοῦ Ε. ὁ Ε ἄρα ἐλάχιστος ὢν ὑπὸ τῶν Α, Β, Γ μετρεῖται: ὅπερ ἔδει δεῖξαι.

Given three numbers, to find the least number which they measure. Let A, B, C be the three given numbers; thus it is required to find the least number which they measure. Let D, the least number measured by the two numbers A, B, be taken. [VII. 34] Then C either measures, or does not measure, D. First, let it measure it. But A, B also measure D; therefore A, B, C measure D. I say next that it is also the least that they measure. For, if not, A, B, C will measure some number which is less than D. Let them measure E. Since A, B, C measure E, therefore also A, B measure E. Therefore the least number measured by A, B will also measure E. [VII. 35] But D is the least number measured by A, B; therefore D will measure E, the greater the less: which is impossible. Therefore A, B, C will not measure any number which is less than D; therefore D is the least that A, B, C measure. Again, let C not measure D, and let E, the least number measured by C, D, be taken. [VII. 34] Since A, B measure D, and D measures E, therefore also A, B measure E. But C also measures E; therefore also A, B, C measure E. I say next that it is also the least that they measure. For, if not, A, B, C will measure some number which is less than E. Let them measure F. Since A, B, C measure F, therefore also A, B measure F; therefore the least number measured by A, B will also measure F. [VII. 35] But D is the least number measured by A, B; therefore D measures F. But C also measures F; therefore D, C measure F, so that the least number measured by D, C will also measure F. But E is the least number measured by C, D; therefore E measures F, the greater the less: which is impossible. Therefore A, B, C will not measure any number which is less than E.