§5.prop.23ἐὰν ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, ᾖ δὲ τεταραγμένη αὐτῶν ἡ ἀναλογία, καὶ διʼ ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται.
If there are three magnitudes, and others equal to them in multitude, taken two by two in the same ratio, and their proportion is perturbed, they will also be ex aequali in the same ratio.
ἔστω τρία μεγέθη τὰ Α, Β, Γ καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ τὰ Δ, Ε, Ζ, ἔστω δὲ τεταραγμένη αὐτῶν ἡ ἀναλογία, ὡς μὲν τὸ Α πρὸς τὸ Β, οὕτως τὸ Ε πρὸς τὸ Ζ, ὡς δὲ τὸ Β πρὸς τὸ Γ, οὕτως τὸ Δ πρὸς τὸ Ε· λέγω, ὅτι ἐστὶν ὡς τὸ Α πρὸς τὸ Γ, οὕτως τὸ Δ πρὸς τὸ Ζ.
εἰλήφθω τῶν μὲν Α, Β, Δ ἰσάκις πολλαπλάσια τὰ Η, Θ, Κ, τῶν δὲ Γ, Ε, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια τὰ Λ, Μ, Ν.
καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσια τὰ Η, Θ τῶν Α, Β, τὰ δὲ μέρη τοῖς ὡσαύτως πολλαπλασίοις τὸν αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Η πρὸς τὸ Θ. διὰ τὰ αὐτὰ δὴ καὶ ὡς τὸ Ε πρὸς τὸ Ζ, οὕτως τὸ Μ πρὸς τὸ Ν· καί ἐστιν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Ε πρὸς τὸ Ζ·
Let there be three magnitudes A, B, Γ, and others equal to them in multitude, Δ, E, Z, taken two by two in the same ratio, and let their proportion be perturbed, so that as A is to B, so is E to Z, and as B is to Γ, so is Δ to E; I say that as A is to Γ, so is Δ to Z. For let there be taken of A, B, Δ equimultiples H, Θ, K, and of Γ, E, Z other, chance, equimultiples Λ, M, N. And since H, Θ are equimultiples of A, B, and parts have the same ratio as their equimultiples, therefore as A is to B, so is H to Θ. For the same reasons indeed, as E is to Z, so is M to N; and as A is to B, so is E to Z; therefore also, as H is to Θ, so is M to N.
καὶ ὡς ἄρα τὸ Η πρὸς τὸ Θ, οὕτως τὸ Μ πρὸς τὸ Ν. καὶ ἐπεί ἐστιν ὡς τὸ Β πρὸς τὸ Γ, οὕτως τὸ δ πρὸς τὸ Ε, καὶ ἐναλλὰξ ὡς τὸ Β πρὸς τὸ Δ, οὕτως τὸ Γ πρὸς τὸ Ε. καὶ ἐπεὶ τὰ Θ, Κ τῶν Β, Δ ἰσάκις ἐστὶ πολλαπλάσια, τὰ δὲ μέρη τοῖς ἰσάκις πολλαπλασίοις τὸν αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ Β πρὸς τὸ Δ, οὕτως τὸ Θ πρὸς τὸ Κ. ἀλλʼ ὡς τὸ Β πρὸς τὸ Δ, οὕτως τὸ Γ πρὸς τὸ Ε· καὶ ὡς ἄρα τὸ Θ πρὸς τὸ Κ, οὕτως τὸ Γ πρὸς τὸ Ε. πάλιν, ἐπεὶ τὰ Λ, Μ τῶν Γ, Ε ἰσάκις ἐστι πολλαπλάσια, ἔστιν ἄρα ὡς τὸ Γ πρὸς τὸ Ε, οὕτως τὸ Λ πρὸς τὸ Μ. ἀλλʼ ὡς τὸ Γ πρὸς τὸ Ε, οὕτως τὸ Θ πρὸς τὸ Κ· καὶ ὡς ἄρα τὸ Θ πρὸς τὸ Κ, οὕτως τὸ Λ πρὸς τὸ Μ, καὶ ἐναλλὰξ ὡς τὸ Θ πρὸς τὸ Λ, τὸ Κ πρὸς τὸ Μ. ἐδείχθη δὲ καὶ ὡς τὸ Η πρὸς τὸ Θ, οὕτως τὸ Μ πρὸς τὸ Ν. ἐπεὶ οὖν τρία μεγέθη ἐστὶ τὰ Η, Θ, Λ, καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος τὰ Κ, Μ, Ν σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, καί ἐστιν αὐτῶν τεταραγμένη ἡ ἀναλογία, διʼ ἴσου ἄρα, εἰ ὑπερέχει τὸ Η τοῦ Λ, ὑπερέχει καὶ τὸ Κ τοῦ Ν, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.
And since as B is to Γ, so is Δ to E, therefore, alternately, as B is to Δ, so is Γ to E. And since Θ, K are equimultiples of B, Δ, and parts have the same ratio as their equimultiples, therefore as B is to Δ, so is Θ to K. But as B is to Δ, so is Γ to E; therefore also, as Θ is to K, so is Γ to E. Again, since Λ, M are equimultiples of Γ, E, therefore as Γ is to E, so is Λ to M. But as Γ is to E, so is Θ to K; therefore also, as Θ is to K, so is Λ to M, and alternately, as Θ is to Λ, so is K to M. But it was also proved that as H is to Θ, so is M to N. Since then there are three magnitudes H, Θ, Λ, and others equal to them in multitude, K, M, N, taken two by two in the same ratio, and their proportion is perturbed, therefore ex aequali, if H exceeds Λ, K also exceeds N, and if equal, equal, and if less, less.
καί ἐστι τὰ μὲν Η, Κ τῶν Α, Δ ἰσάκις πολλαπλάσια, τὰ δὲ Λ, Ν τῶν Γ, Ζ. ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως τὸ Δ πρὸς τὸ Ζ.
ἐὰν ἄρα ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, ᾖ δὲ τεταραγμένη αὐτῶν ἡ ἀναλογία, καὶ διʼ ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται· ὅπερ ἔδει δεῖξαι.
And H, K are equimultiples of A, Δ, and Λ, N of Γ, Z. Therefore, as A is to Γ, so is Δ to Z. If therefore there are three magnitudes, and others equal to them in multitude, taken two by two in the same ratio, and their proportion is perturbed, they will also be ex aequali in the same ratio; which was to be proved.