§9.prop.36#1ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἐκτεθῶσιν ἐν τῇ διπλασίονι ἀναλογίᾳ, ἕως οὗ ὁ σύμπας συντεθεὶς πρῶτος γένηται, καὶ ὁ σύμπας ἐπὶ τὸν ἔσχατον πολλαπλασιασθεὶς ποιῇ τινα, ὁ γενόμενος τέλειος ἔσται.
If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all of them when added together becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.
ἀπὸ γὰρ μονάδος ἐκκείσθωσαν ὁσοιδηποτοῦν ἀριθμοὶ ἐν τῇ διπλασίονι ἀναλογίᾳ, ἕως οὗ ὁ σύμπας συντεθεὶς πρῶτος γένηται, οἱ Α, Β, Γ, Δ, καὶ τῷ σύμπαντι ἴσος ἔστω ὁ Ε, καὶ ὁ Ε τὸν Δ πολλαπλασιάσας τὸν ΖΗ ποιείτω.
For let as many numbers as we please, A, B, Γ, Δ, beginning from a unit, be set out in double proportion, until the sum of all of them when added together becomes prime, and let E be equal to the sum, and let E by multiplying Δ make ZH.
λέγω, ὅτι ὁ ΖΗ τέλειός ἐστιν.
I say that ZH is perfect.
ὅσοι γάρ εἰσιν οἱ Α, Β, Γ, Δ τῷ πλήθει, τοσοῦτοι ἀπὸ τοῦ Ε εἰλήφθωσαν ἐν τῇ διπλασίονι ἀναλογίᾳ οἱ Ε, ΘΚ, Λ, Μ· διʼ ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς τὸν Μ. ὁ ἄρα ἐκ τῶν Ε, Δ ἴσος ἐστὶ τῷ ἐκ τῶν Α, Μ. καί ἐστιν ὁ ἐκ τῶν Ε, Δ ὁ ΖΗ· καὶ ὁ ἐκ τῶν Α, Μ ἄρα ἐστὶν ὁ ΖΗ. ὁ Α ἄρα τὸν Μ πολλαπλασιάσας τὸν ΖΗ πεποίηκεν· ὁ Μ ἄρα τὸν ΖΗ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας.
For as many as A, B, Γ, Δ are in multitude, let so many, E, ΘK, Λ, M, be taken beginning from E in double proportion; therefore, ex aequali, as A is to Δ, so is E to M. Therefore the product of E, Δ is equal to the product of A, M. And the product of E, Δ is ZH; therefore the product of A, M is also ZH. Therefore A by multiplying M has made ZH; therefore M measures ZH according to the units in A.
καί ἐστι δυὰς ὁ Α· διπλάσιος ἄρα ἐστὶν ὁ ΖΗ τοῦ Μ. εἰσὶ δὲ καὶ οἱ Μ, Λ, ΘΚ, Ε ἑξῆς διπλάσιοι ἀλλήλων· οἱ Ε, ΘΚ, Λ, Μ, ΖΗ ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τῇ διπλασίονι ἀναλογίᾳ.
And A is a dyad; therefore ZH is double of M. But M, Λ, ΘK, E are also continuously double of one another; therefore E, ΘK, Λ, M, ZH are in continued proportion in double proportion.
ἀφῃρήσθω δὴ ἀπὸ τοῦ δευτέρου τοῦ ΘΚ καὶ τοῦ ἐσχάτου τοῦ ΖΗ τῷ πρώτῳ τῷ Ε ἴσος ἑκάτερος τῶν ΘΝ, ΖΞ· ἔστιν ἄρα ὡς ἡ τοῦ δευτέρου ἀριθμοῦ ὑπεροχὴ πρὸς τὸν πρῶτον, οὕτως ἡ τοῦ ἐσχάτου ὑπεροχὴ πρὸς τοὺς πρὸ ἑαυτοῦ πάντας.
Let there be subtracted from the second ΘK and the last ZH each of ΘN, ZΞ equal to the first E; therefore, as the excess of the second number is to the first, so is the excess of the last to all those before itself.
ἔστιν ἄρα ὡς ὁ ΝΚ πρὸς τὸν Ε, οὕτως ὁ ΞΗ πρὸς τοὺς Μ, Λ, ΚΘ, Ε. καί ἐστιν ὁ ΝΚ ἴσος τῷ Ε· καὶ ὁ ΞΗ ἄρα ἴσος ἐστὶ τοῖς Μ, Λ, ΘΚ, Ε. ἔστι δὲ καὶ ὁ ΖΞ τῷ Ε ἴσος, ὁ δὲ Ε τοῖς Α, Β, Γ, Δ καὶ τῇ μονάδι.
Therefore, as NK is to E, so is ΞH to M, Λ, KΘ, E. And NK is equal to E; therefore ΞH is also equal to M, Λ, ΘK, E. And ZΞ is also equal to E, while E is equal to A, B, Γ, Δ and the unit.
ὅλος ἄρα ὁ ΖΗ ἴσος ἐστὶ τοῖς τε Ε, ΘΚ, Λ, Μ καὶ τοῖς Α, Β, Γ, Δ καὶ τῇ μονάδι· καὶ μετρεῖται ὑπʼ αὐτῶν.
Therefore the whole ZH is equal to E, ΘK, Λ, M and A, B, Γ, Δ and the unit; and it is measured by them.
λέγω, ὅτι καὶ ὁ ΖΗ ὑπʼ οὐδενὸς ἄλλου μετρηθήσεται παρὲξ τῶν Α, Β, Γ, Δ, Ε, ΘΚ, Λ, Μ καὶ τῆς μονάδος.
I say that ZH will also not be measured by any other number except A, B, Γ, Δ, E, ΘK, Λ, M and the unit.
εἰ γὰρ δυνατόν, μετρείτω τις τὸν ΖΗ ὁ Ο, καὶ ὁ Ο μηδενὶ τῶν Α, Β, Γ, Δ, Ε, ΘΚ, Λ, Μ ἔστω ὁ αὐτός.
For, if possible, let some number O measure ZH, and let O not be the same with any of A, B, Γ, Δ, E, ΘK, Λ, M.
καὶ ὁσάκις ὁ Ο τὸν ΖΗ μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Π· ὁ Π ἄρα τὸν Ο πολλαπλασιάσας τὸν ΖΗ πεποίηκεν.
And as many times as O measures ZH, let so many units be in Π; therefore Π by multiplying O has made ZH.