§7.prop.32ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται. ἔστω ἀριθμὸς ὁ Α· λέγω, ὅτι ὁ Α ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.
Any number is either prime or is measured by some prime number. Let A be a number; I say that A is either prime or is measured by some prime number. Let A be a number; I say that A is either prime or is measured by some prime number.
εἰ μὲν οὖν πρῶτός ἐστιν ὁ Α, γεγονὸς ἂν εἴη τὸ ἐπιταχθέν. εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν πρῶτος ἀριθμός. ἅπας ἄρα ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται· ὅπερ ἔδει δεῖξαι.
If then A be prime, what was enjoined would be done. But if composite, some prime number will measure it. If then A be prime, what was enjoined would be done. But if composite, some prime number will measure it. Therefore any number is either prime or is measured by some prime number; which was to be proved. Therefore any number is either prime or is measured by some prime number; which was to be proved.
§7.prop.33ἀριθμῶν δοθέντων ὁποσωνοῦν εὑρεῖν τοὺς ἐλαχίστους τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. ἔστωσαν οἱ δοθέντες ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ· δεῖ δὴ εὑρεῖν τοὺς ἐλαχίστους τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ.
οἱ Α, Β, Γ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ. εἰ μὲν οὖν οἱ Α, Β, Γ πρῶτοι πρὸς ἀλλήλους εἰσίν, ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.
Given as many numbers as we please, to find the least of those which have the same ratio with them. Given as many numbers as we please, to find the least of those which have the same ratio with them. Let the given numbers, as many as we please, be A, B, Γ; it is then required to find the least of those which have the same ratio with A, B, Γ. For A, B, Γ are either prime to one another or not. Let the given numbers, as many as we please, be A, B, Γ; it is then required to find the least of those which have the same ratio with A, B, Γ. For A, B, Γ are either prime to one another or not. If then A, B, Γ be prime to one another, they are the least of those which have the same ratio with them. For A, B, Γ are either prime to one another or not. If then A, B, Γ be prime to one another, they are the least of those which have the same ratio with them.
εἰ δὲ οὔ, εἰλήφθω τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον ὁ Δ, καὶ ὁσάκις ὁ Δ ἕκαστον τῶν Α, Β, Γ μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν ἑκάστῳ τῶν Ε, Ζ, Η. καὶ ἕκαστος ἄρα τῶν Ε, Ζ, Η ἕκαστον τῶν Α, Β, Γ μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας.
But if not, let the greatest common measure of A, B, Γ be taken, and let it be Δ, and as many times as Δ measures each of A, B, Γ, so many units let there be in each of E, Z, H. Therefore also each of E, Z, H measures each of A, B, Γ according to the units in Δ.
οἱ Ε, Ζ, Η ἄρα τοὺς Α, Β, Γ ἰσάκις μετροῦσιν· οἱ Ε, Ζ, Η ἄρα τοῖς Α, Β, Γ ἐν τῷ αὐτῷ λόγῳ εἰσίν.
Therefore E, Z, H measure A, B, Γ the same number of times; therefore E, Z, H are in the same ratio with A, B, Γ.
λέγω δή, ὅτι καὶ ἐλάχιστοι.
I say then that they are also the least.
εἰ γὰρ μή εἰσιν οἱ Ε, Ζ, Η ἐλάχιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ, ἔσονται τῶν Ε, Ζ, Η ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β, Γ. ἔστωσαν οἱ Θ, Κ, Λ·
For if E, Z, H be not the least of those which have the same ratio with A, B, Γ, there will be numbers less than E, Z, H which are in the same ratio with A, B, Γ.
ἰσάκις ἄρα ὁ Θ τὸν Α μετρεῖ καὶ ἑκάτερος τῶν Κ, Λ ἑκάτερον τῶν Β, Γ. ὁσάκις δὲ ὁ Θ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Μ·
Let them be Θ, K, Λ; therefore Θ measures A the same number of times as each of K, Λ measures each of B, Γ.
καὶ ἑκάτερος ἄρα τῶν Κ, Λ ἑκάτερον τῶν Β, Γ μετρεῖ κατὰ τὰς ἐν τῷ Μ μονάδας.
And as many times as Θ measures A, so many units let there be in M; therefore also each of K, Λ measures each of B, Γ according to the units in M.
καὶ ἐπεὶ ὁ Θ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Μ μονάδας, καὶ ὁ Μ ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Θ μονάδας.
And since Θ measures A according to the units in M, therefore M also measures A according to the units in Θ.
διὰ τὰ αὐτὰ δὴ ὁ Μ καὶ ἑκάτερον τῶν Β, Γ μετρεῖ κατὰ τὰς ἐν ἑκατέρῳ τῶν Κ, Λ μονάδας· ὁ Μ ἄρα τοὺς Α, Β, Γ μετρεῖ.
For the same reason indeed, M also measures each of B, Γ according to the units in each of K, Λ; therefore M measures A, B, Γ.
καὶ ἐπεὶ ὁ Θ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Μ μονάδας, ὁ Θ ἄρα τὸν Μ πολλαπλασιάσας τὸν Α πεποίηκεν.
And since Θ measures A according to the units in M, therefore Θ by multiplying M has made A.
διὰ τὰ αὐτὰ δὴ καὶ ὁ Ε τὸν Δ πολλαπλασιάσας τὸν Α πεποίηκεν.
For the same reason indeed, E also by multiplying Δ has made A.
ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Ε, Δ τῷ ἐκ τῶν Θ, Μ. ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Μ πρὸς τὸν Δ. μείζων δὲ ὁ Ε τοῦ Θ·
Therefore the product of E, Δ is equal to the product of Θ, M. Therefore, as E is to Θ, so is M to Δ.
μείζων ἄρα καὶ ὁ Μ τοῦ Δ. καὶ μετρεῖ τοὺς Α, Β, Γ·
But E is greater than Θ; therefore M is also greater than Δ.
ὅπερ ἐστὶν ἀδύνατον· ὑπόκειται γὰρ ὁ Δ τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον.
And it measures A, B, Γ; which is impossible, for Δ is assumed to be the greatest common measure of A, B, Γ.
οὐκ ἄρα ἔσονταί τινες τῶν Ε, Ζ, Η ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β, Γ. οἱ Ε, Ζ, Η ἄρα ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ· ὅπερ ἔδει δεῖξαι.
Therefore there will not be any numbers less than E, Z, H which are in the same ratio with A, B, Γ. Therefore E, Z, H are the least of those which have the same ratio with A, B, Γ; which was to be proved.