§5.prop.19ἐὰν ᾖ ὡς ὅλον πρὸς ὅλον, οὕτως ἀφαιρεθὲν πρὸς ἀφαιρεθέν, καὶ τὸ λοιπὸν πρὸς τὸ λοιπὸν ἔσται ὡς ὅλον πρὸς ὅλον.
If as a whole is to a whole, so is a magnitude subtracted to a magnitude subtracted, then the remainder to the remainder will also be as the whole to the whole.
ἔστω γὰρ ὡς ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ, οὕτως ἀφαιρεθὲν τὸ ΑΕ πρὸς ἀφαιρεθὲν τὸ ΓΖ· λέγω, ὅτι καὶ λοιπὸν τὸ ΕΒ πρὸς λοιπὸν τὸ ΖΔ ἔσται ὡς ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ.
ἐπεὶ γάρ ἐστιν ὡς τὸ ΑΒ πρὸς τὸ ΓΔ, οὕτως τὸ ΑΕ πρὸς τὸ ΓΖ, καὶ ἐναλλὰξ ὡς τὸ ΒΑ πρὸς τὸ ΑΕ, οὕτως τὸ ΔΓ πρὸς τὸ ΓΖ. καὶ ἐπεὶ συγκείμενα μεγέθη ἀνάλογόν ἐστιν, καὶ διαιρεθέντα ἀνάλογον ἔσται, ὡς τὸ ΒΕ πρὸς τὸ ΕΑ, οὕτως τὸ ΔΖ πρὸς τὸ ΓΖ· καὶ ἐναλλάξ, ὡς τὸ ΒΕ πρὸς τὸ ΔΖ, οὕτως τὸ ΕΑ πρὸς τὸ ΖΓ. ὡς δὲ τὸ ΑΕ πρὸς τὸ ΓΖ, οὕτως ὑπόκειται ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ. καὶ λοιπὸν ἄρα τὸ ΕΒ πρὸς λοιπὸν τὸ ΖΔ ἔσται ὡς ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ.
ἐὰν ἄρα ᾖ ὡς ὅλον πρὸς ὅλον, οὕτως ἀφαιρεθὲν πρὸς ἀφαιρεθέν, καὶ τὸ λοιπὸν πρὸς τὸ λοιπὸν ἔσται ὡς ὅλον πρὸς ὅλον. .
For let as whole AB is to whole ΓΔ, so is subtracted AE to subtracted ΓΖ; I say that remainder EB to remainder ΖΔ will also be as whole AB is to whole ΓΔ. For since as AB is to ΓΔ, so is AE to ΓΖ, also alternando, as BA is to AE, so is ΔΓ to ΓΖ. And since compounded magnitudes are proportional, they will also be proportional separated, so that as BE is to EA, so is ΔΖ to ΓΖ; and alternando, as BE is to ΔΖ, so is EA to ΖΓ. But as AE is to ΓΖ, so whole AB is assumed to be to whole ΓΔ; therefore remainder EB to remainder ΖΔ will also be as whole AB is to whole ΓΔ. If therefore as a whole is to a whole, so is a magnitude subtracted to a magnitude subtracted, then the remainder to the remainder will also be as the whole to the whole..
Πόρισμα
ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν συγκείμενα μεγέθη ἀνάλογον ᾖ, καὶ ἀναστρέψαντι ἀνάλογον ἔσται· ὅπερ ἔδει δεῖξαι.
Corollary From this it is manifest that if compounded magnitudes are proportional, they will also be proportional by conversion; which was to be proved.
§5.prop.20ἐὰν ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος, σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, διʼ ἴσου δὲ τὸ πρῶτον τοῦ τρίτου μεῖζον ᾖ, καὶ τὸ τέταρτον τοῦ ἕκτου μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον.
If there are three magnitudes, and others equal to them in multitude, taken two by two and in the same ratio, and ex aequali the first is greater than the third, then the fourth will also be greater than the sixth, and if equal, equal, and if less, less.
ἔστω τρία μεγέθη τὰ Α, Β, Γ, καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος τὰ Δ, Ε, Ζ, σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, ὡς μὲν τὸ Α πρὸς τὸ Β, οὕτως τὸ Δ πρὸς τὸ Ε, ὡς δὲ τὸ Β πρὸς τὸ Γ, οὕτως τὸ Ε πρὸς τὸ Ζ, διʼ ἴσου δὲ μεῖζον ἔστω τὸ Α τοῦ Γ·
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, so that as A is to B, so is Δ to E, and as B is to Γ, so is E to Z, and ex aequali let A be greater than Γ; I say that Δ will also be greater than Z, and if equal, equal, and if less, less.
λέγω, ὅτι καὶ τὸ Δ τοῦ Ζ μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον. ἐπεὶ γὰρ μεῖζόν ἐστι τὸ Α τοῦ Γ, ἄλλο δέ τι τὸ Β, τὸ δὲ μεῖζον πρὸς τὸ αὐτὸ μείζονα λόγον ἔχει ἤπερ τὸ ἔλαττον, τὸ Α ἄρα πρὸς τὸ Β μείζονα λόγον ἔχει ἤπερ τὸ Γ πρὸς τὸ Β. ἀλλʼ ὡς μὲν τὸ Α πρὸς τὸ Β, τὸ Δ πρὸς τὸ Ε, ὡς δὲ τὸ Γ πρὸς τὸ Β, ἀνάπαλιν οὕτως τὸ Ζ πρὸς τὸ Ε· καὶ τὸ Δ ἄρα πρὸς τὸ Ε μείζονα λόγον ἔχει ἤπερ τὸ Ζ πρὸς τὸ Ε. τῶν δὲ πρὸς τὸ αὐτὸ λόγον ἐχόντων τὸ μείζονα λόγον ἔχον μεῖζόν ἐστιν.
For since A is greater than Γ, and B is some other magnitude, and the greater has to the same a greater ratio than the less, therefore A has to B a greater ratio than Γ has to B. But as A is to B, so is Δ to E, and as Γ is to B, so, inversely, is Z to E; therefore Δ has to E a greater ratio than Z has to E. And of those having a ratio to the same, that having the greater ratio is greater; therefore Δ is greater than Z.
μεῖζον ἄρα τὸ Δ τοῦ Ζ. ὁμοίως δὴ δείξομεν, ὅτι κἂν ἴσον ᾖ τὸ Α τῷ Γ, ἴσον ἔσται καὶ τὸ Δ τῷ Ζ, κἂν ἔλαττον, ἔλαττον.
In like manner we shall also prove that, even if A is equal to Γ, Δ will also be equal to Z, and if less, less.
ἐὰν ἄρα ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος, σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, διʼ ἴσου δὲ τὸ πρῶτον τοῦ τρίτου μεῖζον ᾖ, καὶ τὸ τέταρτον τοῦ ἕκτου μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον· ὅπερ ἔδει δεῖξαι.
If therefore there are three magnitudes, and others equal to them in multitude, taken two by two and in the same ratio, and ex aequali the first is greater than the third, the fourth will also be greater than the sixth, and if equal, equal, and if less, less; which was to be proved.