§7.prop.29ἅπας πρῶτος ἀριθμὸς πρὸς ἅπαντα ἀριθμόν, ὃν μὴ μετρεῖ, πρῶτός ἐστιν.
Any prime number is prime to any number which it does not measure.
ἔστω πρῶτος ἀριθμὸς ὁ Α καὶ τὸν Β μὴ μετρείτω· λέγω, ὅτι οἱ Β, Α πρῶτοι πρὸς ἀλλήλους εἰσίν.
Let A be a prime number, and let it not measure B; I say that B, A are prime to one another.
εἰ γὰρ μή εἰσιν οἱ Β, Α πρῶτοι πρὸς ἀλλήλους, μετρήσει τις αὐτοὺς ἀριθμός.
For if B, A be not prime to one another, some number will measure them.
μετρείτω ὁ Γ. ἐπεὶ ὁ Γ τὸν Β μετρεῖ, ὁ δὲ Α τὸν Β οὐ μετρεῖ, ὁ Γ ἄρα τῷ Α οὔκ ἐστιν ὁ αὐτός.
Let Γ measure them. Since Γ measures B, and A does not measure B, therefore Γ is not the same with A. Let Γ measure them. Since Γ measures B, and A does not measure B, therefore Γ is not the same with A. Let Γ measure them. Since Γ measures B, and A does not measure B, therefore Γ is not the same with A.
καὶ ἐπεὶ ὁ Γ τοὺς Β, Α μετρεῖ, καὶ τὸν Α ἄρα μετρεῖ πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα τοὺς Β, Α μετρήσει τις ἀριθμός. οἱ Α, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.
And since Γ measures B, A, therefore it also measures A, which is prime, though it is not the same with it; which is impossible. Therefore no number will measure B, A. Therefore A, B are prime to one another; which was to be proved. Therefore no number will measure B, A. Therefore A, B are prime to one another; which was to be proved.
§7.prop.30ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει.
If two numbers by multiplying one another make some number, and any prime number measure the product from them, it will also measure one of the original numbers.
δύο γὰρ ἀριθμοὶ οἱ Α, Β πολλαπλασιάσαντες ἀλλήλους τὸν Γ ποιείτωσαν, τὸν δὲ Γ μετρείτω τις πρῶτος ἀριθμὸς ὁ Δ· λέγω, ὅτι ὁ Δ ἕνα τῶν Α, Β μετρεῖ. τὸν γὰρ Α μὴ μετρείτω· καί ἐστι πρῶτος ὁ Δ· οἱ Α, Δ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.
For let two numbers A, B by multiplying one another make Γ, and let any prime number Δ measure Γ; I say that Δ measures one of A, B. For let it not measure A; and Δ is prime; therefore A, Δ are prime to one another.
καὶ ὁσάκις ὁ Δ τὸν Γ μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ε. ἐπεὶ οὖν ὁ Δ τὸν Γ μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, ὁ Δ ἄρα τὸν Ε πολλαπλασιάσας τὸν Γ πεποίηκεν.
And as many times as Δ measures Γ, so many units let there be in E. Since then Δ measures Γ according to the units in E, therefore Δ by multiplying E has made Γ.
ἀλλὰ μὴν καὶ ὁ Α τὸν β πολλαπλασιάσας τὸν Γ πεποίηκεν· ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Δ, Ε τῷ ἐκ τῶν Α, Β. ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Α, οὕτως ὁ Β πρὸς τὸν Ε. οἱ δὲ Δ, Α πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· ὁ Δ ἄρα τὸν Β μετρεῖ.
But indeed A also by multiplying B has made Γ; therefore the product of Δ, E is equal to the product of A, B. Therefore, as Δ is to A, so is B to E. But Δ, A are prime, and those which are prime are also least, and the least measure those which have the same ratio the same number of times, both the greater the greater and the less the less, that is, both the antecedent the antecedent and the consequent the consequent; Therefore, as Δ is to A, so is B to E. Therefore the product of Δ, E is equal to the product of A, B. Therefore, as Δ is to A, so is B to E. But Δ, A are prime, and those which are prime are also least, and the least measure those which have the same ratio the same number of times, both the greater the greater and the less the less, that is, both the antecedent the antecedent and the consequent the consequent; therefore Δ measures B.
ὁμοίως δὴ δείξομεν, ὅτι καὶ ἐὰν τὸν Β μὴ μετρῇ, τὸν Α μετρήσει.
Similarly indeed we shall show that, even if it does not measure B, it will measure A.
ὁ Δ ἄρα ἕνα τῶν Α, Β μετρεῖ· ὅπερ ἔδει δεῖξαι.
Therefore Δ measures one of A, B; which was to be proved.
§7.prop.31ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.
Any composite number is measured by some prime number.
ἔστω σύνθετος ἀριθμὸς ὁ Α· λέγω, ὅτι ὁ Α ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.
Let A be a composite number; I say that A is measured by some prime number. Let A be a composite number; I say that A is measured by some prime number.
ʼἐπεὶ γὰρ σύνθετός ἐστιν ὁ Α, μετρήσει τις αὐτὸν ἀριθμός.
\nFor since A is composite, some number will measure it.
μετρείτω, καὶ ἔστω ὁ Β. καὶ εἰ μὲν πρῶτός ἐστιν ὁ Β, γεγονὸς ἂν εἴη τὸ ἐπιταχθέν.
Let it measure it, and let it be B. And if B be prime, what was enjoined would be done.
εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν ἀριθμός.
But if composite, some number will measure it.
μετρείτω, καὶ ἔστω ὁ Γ. καὶ ἐπεὶ ὁ Γ τὸν Β μετρεῖ, ὁ δὲ Β τὸν Α μετρεῖ, καὶ ὁ Γ ἄρα τὸν Α μετρεῖ.
Let it measure it, and let it be Γ. And since \nΓ measures B, and B measures A, therefore Γ also measures A.
καὶ εἰ μὲν πρῶτός ἐστιν ὁ Γ, γεγονὸς ἂν εἴη τὸ ἐπιταχθέν.
And if Γ be prime, what was enjoined would be done.
εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν ἀριθμός.
But if composite, some number will measure it.
τοιαύτης δὴ γινομένης ἐπισκέψεως ληφθήσεταί τις πρῶτος ἀριθμός, ὃς μετρήσει.
If then such an investigation be made, some prime number will be reached which will measure.
εἰ γὰρ οὐ ληφθήσεται, μετρήσουσι τὸν Α ἀριθμὸν ἄπειροι ἀριθμοί, ὧν ἕτερος ἑτέρου ἐλάσσων ἐστίν· ὅπερ ἐστὶν ἀδύνατον ἐν ἀριθμοῖς.
For if it be not reached, infinite numbers will measure the number A, each of which is less than the other; which is impossible in numbers.
ληφθήσεταί τις ἄρα πρῶτος ἀριθμός, ὃς μετρήσει τὸν πρὸ ἑαυτοῦ, ὃς καὶ τὸν Α μετρήσει. ἅπας ἄρα σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται· ὅπερ ἔδει δεῖξαι.
Therefore some prime number will be reached which will measure the one before itself, which will also measure A. Therefore any composite number is measured by some prime number. Therefore any composite number is measured by some prime number; which was to be proved.