§9.prop.12ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὑφʼ ὅσων ἂν ὁ ἔσχατος πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ παρὰ τὴν μονάδα μετρηθήσεται.
If any number of numbers starting from a unit be continuously proportional, by as many prime numbers as the last is measured, by the same will the one after the unit also be measured.
ἔστωσαν ἀπὸ μονάδος ὁποσοιδηποτοῦν ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ· λέγω, ὅτι ὑφʼ ὅσων ἂν ὁ Δ πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ Α μετρηθήσεται.
Let there be any number of numbers starting from a unit proportional, A, B, Γ, Δ; I say that, by as many prime numbers as Δ is measured, by the same will A also be measured.
μετρείσθω γὰρ ὁ Δ ὑπό τινος πρώτου ἀριθμοῦ τοῦ Ε· λέγω, ὅτι ὁ ε τὸν Α μετρεῖ.
For let Δ be measured by some prime number E; I say that E measures A.
μὴ γάρ· καί ἐστιν ὁ Ε πρῶτος, ἅπας δὲ πρῶτος ἀριθμὸς πρὸς ἅπαντα, ὃν μὴ μετρεῖ, πρῶτός ἐστιν· οἱ Ε, Α ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.
For let it not; and E is prime, and every prime number is prime to every number which it does not measure; therefore E, A are prime to one another.
καὶ ἐπεὶ ὁ Ε τὸν Δ μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Ζ· ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν.
And since E measures Δ, let it measure it according to Z; therefore E by multiplying Z has made Δ.
πάλιν, ἐπεὶ ὁ Α τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας, ὁ Α ἄρα τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν.
Again, since A measures Δ according to the units in Γ, therefore A by multiplying Γ has made Δ.
ἀλλὰ μὴν καὶ ὁ Ε τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν·
But indeed E also by multiplying Z has made Δ; therefore the product of A, Γ is equal to the product of E, Z.
ὁ ἄρα ἐκ τῶν Α, Γ ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Ζ. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Ε, ὁ Ζ πρὸς τὸν Γ. οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· μετρεῖ ἄρα ὁ Ε τὸν Γ. μετρείτω αὐτὸν κατὰ τὸν Η·
Therefore, as A is to E, so is Z to Γ. But A, E are prime, and those which are prime are also least, and the least measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent; therefore E measures Γ.
ὁ Ε ἄρα τὸν Η πολλαπλασιάσας τὸν Γ πεποίηκεν.
Let it measure it according to H; therefore E by multiplying H has made Γ.
ἀλλὰ μὴν διὰ τὸ πρὸ τούτου καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν.
But indeed, because of the proposition before this, A also by multiplying B has made Γ.
ὁ ἄρα ἐκ τῶν Α, Β ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Η. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Ε, ὁ Η πρὸς τὸν Β. οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· μετρεῖ ἄρα ὁ Ε τὸν Β. μετρείτω αὐτὸν κατὰ τὸν Θ·
Therefore the product of A, B is equal to the product of E, H. Therefore, as A is to E, so is H to B. But A, E are prime, and those which are prime are also least, and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent; therefore E measures B.
ὁ Ε ἄρα τὸν Θ πολλαπλασιάσας τὸν Β πεποίηκεν.
Let it measure it according to Θ; therefore E by multiplying Θ has made B.
ἀλλὰ μὴν καὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν·
But indeed A also by multiplying itself has made B; therefore the product of E, Θ is equal to the square on A.
ὁ ἄρα ἐκ τῶν Ε, Θ ἴσος ἐστὶ τῷ ἀπὸ τοῦ Α. ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Α, ὁ Α πρὸς τὸν Θ. οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· μετρεῖ ἄρα ὁ Ε τὸν Α ὡς ἡγούμενος ἡγούμενον.
Therefore, as E is to A, so is A to Θ. But A, E are prime, and those which are prime are also least, and the least measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent; therefore E measures A as antecedent the antecedent.
ἀλλὰ μὴν καὶ οὐ μετρεῖ· ὅπερ ἀδύνατον.
But indeed it also does not measure it; which is impossible.
οὐκ ἄρα οἱ Ε, Α πρῶτοι πρὸς ἀλλήλους εἰσίν.
Therefore E, A are not prime to one another.
σύνθετοι ἄρα.
Therefore they are composite.
οἱ δὲ σύνθετοι ὑπὸ ἀριθμοῦ τινος μετροῦνται.
And those which are composite are measured by some number.
καὶ ἐπεὶ ὁ Ε πρῶτος ὑπόκειται, ὁ δὲ πρῶτος ὑπὸ ἑτέρου ἀριθμοῦ οὐ μετρεῖται ἢ ὑφʼ ἑαυτοῦ, ὁ Ε ἄρα τοὺς Α, Ε μετρεῖ· ὥστε ὁ Ε τὸν Α μετρεῖ.
And since E is assumed to be prime, and a prime is not measured by any other number than itself, therefore E measures A, E; so that E measures A.
μετρεῖ δὲ καὶ τὸν Δ· ὁ Ε ἄρα τοὺς Α, Δ μετρεῖ.
But it also measures Δ; therefore E measures A, Δ.
ὁμοίως δὴ δείξομεν, ὅτι ὑφʼ ὅσων ἂν ὁ Δ πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ Α μετρηθήσεται· ὅπερ ἔδει δεῖξαι.
Similarly indeed we will show that, by as many prime numbers as Δ is measured, by the same will A also be measured; which it was required to prove.