§7.prop.19ἐὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, ὁ ἐκ πρώτου καὶ τετάρτου γενόμενος ἀριθμὸς ἴσος ἔσται τῷ ἐκ δευτέρου καὶ τρίτου γενομένῳ ἀριθμῷ· καὶ ἐὰν ὁ ἐκ πρώτου καὶ τετάρτου γενόμενος ἀριθμὸς ἴσος ᾖ τῷ ἐκ δευτέρου καὶ τρίτου, οἱ τέσσαρες ἀριθμοὶ ἀνάλογον ἔσονται.
If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to the second and third, the four numbers will be proportional.
ἔστωσαν τέσσαρες ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ, καὶ ὁ μὲν Α τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Β τὸν Γ πολλαπλασιάσας τὸν Ζ ποιείτω·
Let there be four proportional numbers A, B, Γ, Δ, so that, as A is to B, so is Γ to Δ, and let A by multiplying Δ make E, and B by multiplying Γ make Z; I say that E is equal to Z.
λέγω, ὅτι ἴσος ἐστὶν ὁ Ε τῷ Ζ.
ὁ γὰρ Α τὸν Γ πολλαπλασιάσας τὸν Η ποιείτω.
For let A by multiplying Γ make H.
ἐπεὶ οὖν ὁ Α τὸν Γ πολλαπλασιάσας τὸν Η πεποίηκεν, τὸν δὲ Δ πολλαπλασιάσας τὸν Ε πεποίηκεν, ἀριθμὸς δὴ ὁ Α δύο ἀριθμοὺς τοὺς Γ, Δ πολλαπλασιάσας τοὺς Η, Ε πεποίηκεν.
Since therefore A by multiplying Γ has made H, and by multiplying Δ has made E, therefore the number A by multiplying two numbers Γ, Δ has made H, E.
ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Η πρὸς τὸν Ε. ἀλλʼ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Ε. πάλιν, ἐπεὶ ὁ Α τὸν Γ πολλαπλασιάσας τὸν Η πεποίηκεν, ἀλλὰ μὴν καὶ ὁ Β τὸν Γ πολλαπλασιάσας τὸν Ζ πεποίηκεν, δύο δὴ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν Γ πολλαπλασιάσαντες τοὺς Η, Ζ πεποιήκασιν.
Therefore, as Γ is to Δ, so is H to E. But, as Γ is to Δ, so is A to B; therefore also, as A is to B, so is H to E. Again, since A by multiplying Γ has made H, but indeed B also by multiplying Γ has made Z, therefore two numbers A, B by multiplying any number Γ have made H, Z.
ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Ζ. ἀλλὰ μὴν καὶ ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Ε· καὶ ὡς ἄρα ὁ Η πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Ζ. ὁ Η ἄρα πρὸς ἑκάτερον τῶν Ε, Ζ τὸν αὐτὸν ἔχει λόγον· ἴσος ἄρα ἐστὶν ὁ Ε τῷ Ζ.
ἔστω δὴ πάλιν ἴσος ὁ Ε τῷ Ζ· λέγω, ὅτι ἐστὶν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ.
τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ ἴσος ἐστὶν ὁ Ε τῷ Ζ, ἔστιν ἄρα ὡς ὁ Η πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Ζ. ἀλλʼ ὡς μὲν ὁ Η πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν δ, ὡς δὲ ὁ Η πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Β. καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· ὅπερ ἔδει δεῖξαι.
Therefore, as A is to B, so is H to Z. But indeed, as A is to B, so is H to E; therefore also, as H is to E, so is H to Z. Therefore H has the same ratio to each of E, Z; therefore E is equal to Z. Now let E again be equal to Z; I say that, as A is to B, so is Γ to Δ. For, with the same construction, since E is equal to Z, therefore, as H is to E, so is H to Z. But, as H is to E, so is Γ to Δ, and, as H is to Z, so is A to B. Therefore also, as A is to B, so is Γ to Δ; which was to be proved.
§7.prop.20οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα.
The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater, and the less the less.
ἔστωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β οἱ ΓΔ, ΕΖ· λέγω, ὅτι ἰσάκις ὁ ΓΔ τὸν Α μετρεῖ καὶ ὁ ΕΖ τὸν Β.
ὁ ΓΔ γὰρ τοῦ Α οὔκ ἐστι μέρη.
For let ΓΔ, EZ be the least numbers of those which have the same ratio with A, B; I say that ΓΔ measures A the same number of times that EZ measures B. For ΓΔ is not parts of A.
εἰ γὰρ δυνατόν, ἔστω· καὶ ὁ ΕΖ ἄρα τοῦ Β τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὁ ΓΔ τοῦ Α. ὅσα ἄρα ἐστὶν ἐν τῷ ΓΔ μέρη τοῦ Α, τοσαῦτά ἐστι καὶ ἐν τῷ ΕΖ μέρη τοῦ Β. διῃρήσθω ὁ μὲν ΓΔ εἰς τὰ τοῦ Α μέρη τὰ ΓΗ, ΗΔ, ὁ δὲ ΕΖ εἰς τὰ τοῦ Β μέρη τὰ ΕΘ, ΘΖ·
For, if possible, let it be; therefore EZ also is the same parts of B that ΓΔ is of A. Therefore, as many parts of A as there are in ΓΔ, so many parts of B also are there in EZ. Let ΓΔ be divided into the parts of A, namely ΓH, HΔ, and EZ into the parts of B, namely EΘ, ΘZ; therefore the multitude of ΓH, HΔ will be equal to the multitude of EΘ, ΘZ.
ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΓΗ, ΗΔ τῷ πλήθει τῶν ΕΘ, ΘΖ. καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΓΗ, ΗΔ ἀριθμοὶ ἀλλήλοις, εἰσὶ δὲ καὶ οἱ ΕΘ, ΘΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ πλῆθος τῶν ΓΗ, ΗΔ τῷ πλήθει τῶν ΕΘ, ΘΖ, ἔστιν ἄρα ὡς ὁ ΓΗ πρὸς τὸν ΕΘ, οὕτως ὁ ΗΔ πρὸς τὸν ΘΖ. ἔσται ἄρα καὶ ὡς εἷς τῶν ἡγουμένων πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες οἱ ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους.
And since the numbers ΓH, HΔ are equal to one another, and the numbers EΘ, ΘZ are also equal to one another, and the multitude of ΓH, HΔ is equal to the multitude of EΘ, ΘZ, therefore, as ΓH is to EΘ, so is HΔ to ΘZ. Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
ἔστιν ἄρα ὡς ὁ ΓΗ πρὸς τὸν ΕΘ, οὕτως ὁ ΓΔ πρὸς τὸν ΕΖ· οἱ ΓΗ, ΕΘ ἄρα τοῖς ΓΔ, ΕΖ ἐν τῷ αὐτῷ λόγῳ εἰσὶν ἐλάσσονες ὄντες αὐτῶν· ὅπερ ἐστὶν ἀδύνατον· ὑπόκεινται γὰρ οἱ ΓΔ, ΕΖ ἐλάχιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.
Therefore, as ΓH is to EΘ, so is ΓΔ to EZ; therefore ΓH, EΘ are in the same ratio with ΓΔ, EZ, being less than they; which is impossible, for ΓΔ, EZ were assumed to be the least of those which have the same ratio with them.
οὐκ ἄρα μέρη ἐστὶν ὁ ΓΔ τοῦ Α· μέρος ἄρα.
Therefore ΓΔ is not parts of A; therefore it is a part.
καὶ ὁ ΕΖ τοῦ Β τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὁ ΓΔ τοῦ Α· ἰσάκις ἄρα ὁ ΓΔ τὸν Α μετρεῖ καὶ ὁ ΕΖ τὸν Β· ὅπερ ἔδει δεῖξαι.
And EZ is the same part of B that ΓΔ is of A; therefore ΓΔ measures A the same number of times that EZ measures B; which was to be proved.