§9.prop.13ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ μετὰ τὴν μονάδα πρῶτος ᾖ, ὁ μέγιστος ὑπʼ οὐδενὸς μετρηθήσεται παρὲξ τῶν ὑπαρχόντων ἐν τοῖς ἀνάλογον ἀριθμοῖς.
If any number of numbers starting from a unit be continuously proportional, and the one after the unit be prime, the greatest will not be measured by any number except those which exist among the proportional numbers.
ἔστωσαν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, Δ, ὁ δὲ μετὰ τὴν μονάδα ὁ Α πρῶτος ἔστω· λέγω, ὅτι ὁ μέγιστος αὐτῶν ὁ Δ ὑπʼ οὐδενὸς ἄλλου μετρηθήσεται παρὲξ τῶν Α, Β, Γ.
εἰ γὰρ δυνατόν, μετρείσθω ὑπὸ τοῦ Ε, καὶ ὁ Ε μηδενὶ τῶν Α, Β, Γ ἔστω ὁ αὐτός.
Let there be any number of numbers starting from a unit continuously proportional, A, B, Γ, Δ, and let A, the one after the unit, be prime; I say that the greatest of them, Δ, will not be measured by any other number except A, B, Γ. For, if possible, let it be measured by E, and let E be the same with none of A, B, Γ.
φανερὸν δή, ὅτι ὁ Ε πρῶτος οὔκ ἐστιν.
It is indeed manifest that E is not prime.
εἰ γὰρ ὁ Ε πρῶτός ἐστι καὶ μετρεῖ τὸν Δ, καὶ τὸν Α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός·
For if E is prime and measures Δ, it will also measure A, which is prime, though not being the same with it; which is impossible.
ὅπερ ἐστὶν ἀδύνατον.
Therefore E is not prime.
οὐκ ἄρα ὁ Ε πρῶτός ἐστιν.
Therefore it is composite.
σύνθετος ἄρα. πᾶς δὲ σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται· ὁ Ε ἄρα ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.
And every composite number is measured by some prime number; therefore E is measured by some prime number.
λέγω δή, ὅτι ὑπʼ οὐδενὸς ἄλλου πρώτου μετρηθήσεται πλὴν τοῦ Α. εἰ γὰρ ὑφʼ ἑτέρου μετρεῖται ὁ Ε, ὁ δὲ Ε τὸν Δ μετρεῖ, κἀκεῖνος ἄρα τὸν Δ μετρήσει· ὥστε καὶ τὸν Α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν ἀδύνατον.
I say indeed that it will not be measured by any other prime except A. For if E is measured by another, and E measures Δ, that indeed will also measure Δ; so that it will also measure A, which is prime, though not being the same with it; which is impossible.
ὁ Α ἄρα τὸν Ε μετρεῖ.
Therefore A measures E.
καὶ ἐπεὶ ὁ Ε τὸν Δ μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Ζ. λέγω, ὅτι ὁ Ζ οὐδενὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.
And since E measures Δ, let it measure it according to Z. I say that Z is the same with none of A, B, Γ.
εἰ γὰρ ὁ Ζ ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτὸς καὶ μετρεῖ τὸν Δ κατὰ τὸν Ε, καὶ εἷς ἄρα τῶν Α, Β, Γ τὸν Δ μετρεῖ κατὰ τὸν Ε. ἀλλὰ εἷς τῶν Α, Β, Γ τὸν Δ μετρεῖ κατά τινα τῶν Α, Β, Γ· καὶ ὁ Ε ἄρα ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός· ὅπερ οὐχ ὑπόκειται.
For if Z is the same with one of A, B, Γ and measures Δ according to E, one of A, B, Γ also measures Δ according to E. But one of A, B, Γ measures Δ according to some one of A, B, Γ; therefore E also is the same with one of A, B, Γ; which is not assumed.
οὐκ ἄρα ὁ Ζ ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.
Therefore Z is not the same with one of A, B, Γ.
ὁμοίως δὴ δείξομεν, ὅτι μετρεῖται ὁ Ζ ὑπὸ τοῦ Α, δεικνύντες πάλιν, ὅτι ὁ Ζ οὔκ ἐστι πρῶτος.
Similarly indeed we will show that Z is measured by A, showing again that Z is not prime.
εἰ γάρ, καὶ μετρεῖ τὸν Δ, καὶ τὸν α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν ἀδύνατον· οὐκ ἄρα πρῶτός ἐστιν ὁ Ζ· σύνθετος ἄρα.
For if [it is], and measures Δ, it will also measure A, which is prime, though not being the same with it; which is impossible; therefore Z is not prime; therefore it is composite.
ἅπας δὲ σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται· ὁ Ζ ἄρα ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.
And every composite number is measured by some prime number; therefore Z is measured by some prime number.
λέγω δή, ὅτι ὑφʼ ἑτέρου πρώτου οὐ μετρηθήσεται πλὴν τοῦ α.
I say indeed that it will not be measured by another prime except A.
εἰ γὰρ ἕτερός τις πρῶτος τὸν Ζ μετρεῖ, ὁ δὲ Ζ τὸν Δ μετρεῖ, κἀκεῖνος ἄρα τὸν Δ μετρήσει· ὥστε καὶ τὸν Α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν ἀδύνατον.
For if some other prime measures Z, and Z measures Δ, that indeed will also measure Δ; so that it will also measure A, which is prime, though not being the same with it; which is impossible.
ὁ Α ἄρα τὸν Ζ μετρεῖ.
Therefore A measures Z.
καὶ ἐπεὶ ὁ Ε τὸν Δ μετρεῖ κατὰ τὸν Ζ, ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν.
And since E measures Δ according to Z, therefore E by multiplying Z has made Δ.
ἀλλὰ μὴν καὶ ὁ Α τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν· ὁ ἄρα ἐκ τῶν Α, Γ ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Ζ. ἀνάλογον ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Ε, οὕτως ὁ Ζ πρὸς τὸν Γ. ὁ δὲ Α τὸν Ε μετρεῖ· καὶ ὁ Ζ ἄρα τὸν Γ μετρεῖ.
But indeed A also by multiplying Γ has made Δ; therefore the product of A, Γ is equal to the product of E, Z. Therefore, proportionally, as A is to E, so is Z to Γ. But A measures E; therefore Z also measures Γ.
μετρείτω αὐτὸν κατὰ τὸν Η. ὁμοίως δὴ δείξομεν, ὅτι ὁ Η οὐδενὶ τῶν Α, Β ἐστιν ὁ αὐτός, καὶ ὅτι μετρεῖται ὑπὸ τοῦ Α. καὶ ἐπεὶ ὁ Ζ τὸν Γ μετρεῖ κατὰ τὸν Η, ὁ Ζ ἄρα τὸν Η πολλαπλασιάσας τὸν Γ πεποίηκεν.
Let it measure it according to H. Similarly indeed we will show that H is the same with none of A, B, and that it is measured by A. And since Z measures Γ according to H, therefore Z by multiplying H has made Γ.
ἀλλὰ μὴν καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν· ὁ ἄρα ἐκ τῶν Α, Β ἴσος ἐστὶ τῷ ἐκ τῶν Ζ, Η. ἀνάλογον ἄρα ὡς ὁ Α πρὸς τὸν Ζ, ὁ Η πρὸς τὸν Β. μετρεῖ δὲ ὁ Α τὸν Ζ· μετρεῖ ἄρα καὶ ὁ Η τὸν Β. μετρείτω αὐτὸν κατὰ τὸν Θ. ὁμοίως δὴ δείξομεν, ὅτι ὁ Θ τῷ Α οὐκ ἔστιν ὁ αὐτός.
But indeed A also by multiplying B has made Γ; therefore the product of A, B is equal to the product of Z, H. Therefore, proportionally, as A is to Z, so is H to B. But A measures Z; therefore H also measures B. Let it measure it according to Θ. Similarly indeed we will show that Θ is not the same with A.
καὶ ἐπεὶ ὁ Η τὸν Β μετρεῖ κατὰ τὸν Θ, ὁ Η ἄρα τὸν Θ πολλαπλασιάσας τὸν Β πεποίηκεν.
And since H measures B according to Θ, therefore H by multiplying Θ has made B.
ἀλλὰ μὴν καὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν· ὁ ἄρα ὑπὸ Θ, Η ἴσος ἐστὶ τῷ ἀπὸ τοῦ Α τετραγώνῳ.
But indeed A also by multiplying itself has made B; therefore the product of Θ, H is equal to the square on A.
ἔστιν ἄρα ὡς ὁ Θ πρὸς τὸν Α, ὁ Α πρὸς τὸν Η. μετρεῖ δὲ ὁ Α τὸν Η· μετρεῖ ἄρα καὶ ὁ Θ τὸν Α πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἄτοπον.
Therefore, as Θ is to A, so is A to H. But A measures H; therefore Θ also measures A, which is prime, though not being the same with it; which is absurd.
οὐκ ἄρα ὁ μέγιστος ὁ Δ ὑπὸ ἑτέρου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν Α, Β, Γ· ὅπερ ἔδει δεῖξαι.
Therefore the greatest, Δ, will not be measured by another number except A, B, Γ; which it was required to prove.