§9.prop.27ἐὰν ἀπὸ περισσοῦ ἀριθμοῦ ἄρτιος ἀφαιρεθῇ, ὁ λοιπὸς περισσὸς ἔσται.
If an even number be subtracted from an odd number, the remainder will be odd.
ἀπὸ γὰρ περισσοῦ τοῦ ΑΒ ἄρτιος ἀφῃρήσθω ὁ ΒΓ· λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ περισσός ἐστιν.
For let the even number BΓ be subtracted from the odd number AB; I say that the remainder ΓA is odd.
ἀφῃρήσθω μονὰς ἡ ΑΔ· ὁ ΔΒ ἄρα ἄρτιός ἐστιν.
Let the unit AΔ be subtracted; therefore ΔB is even.
ἔστι δὲ καὶ ὁ ΒΓ ἄρτιος· καὶ λοιπὸς ἄρα ὁ ΓΔ ἄρτιός ἐστιν.
But BΓ is also even; therefore the remainder ΓΔ is also even.
περισσὸς ἄρα ὁ ΓΑ· ὅπερ ἔδει δεῖξαι.
Therefore ΓA is odd; which it was required to prove.
§9.prop.28ἐὰν περισσὸς ἀριθμὸς ἄρτιον πολλαπλασιάσας ποιῇ τινα, ὁ γενόμενος ἄρτιος ἔσται.
If an odd number by multiplying an even number make some number, the product will be even.
περισσὸς γὰρ ἀριθμὸς ὁ Α ἄρτιον τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω· λέγω, ὅτι ὁ Γ ἄρτιός ἐστιν.
For let the odd number A by multiplying the even number B make Γ; I say that Γ is even.
ʼἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ Γ ἄρα σύγκειται ἐκ τοσούτων ἴσων τῷ Β, ὅσαι εἰσὶν ἐν τῷ Α μονάδες.
For since A by multiplying B has made Γ, therefore Γ is composed of as many numbers equal to B as there are units in A.
καί ἐστιν ὁ Β ἄρτιος· ὁ Γ ἄρα σύγκειται ἐξ ἀρτίων.
And B is even; therefore Γ is composed of evens.
ἐὰν δὲ ἄρτιοι ἀριθμοὶ ὁποσοιοῦν συντεθῶσιν, ὁ ὅλος ἄρτιός ἐστιν.
And if any number of even numbers be added together, the whole is even.
ἄρτιος ἄρα ἐστὶν ὁ Γ· ὅπερ ἔδει δεῖξαι.
Therefore Γ is even; which it was required to prove.
§9.prop.29ἐὰν περισσὸς ἀριθμὸς περισσὸν ἀριθμὸν πολλαπλασιάσας ποιῇ τινα, ὁ γενόμενος περισσὸς ἔσται.
If an odd number by multiplying an odd number make some number, the product will be odd.
περισσὸς γὰρ ἀριθμὸς ὁ Α περισσὸν τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω· λέγω, ὅτι ὁ Γ περισσός ἐστιν.
For let the odd number A by multiplying the odd number B make Γ; I say that Γ is odd.
ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ Γ ἄρα σύγκειται ἐκ τοσούτων ἴσων τῷ Β, ὅσαι εἰσὶν ἐν τῷ Α μονάδες.
For since A by multiplying B has made Γ, therefore Γ is composed of as many numbers equal to B as there are units in A.
καί ἐστιν ἑκάτερος τῶν Α, Β περισσός· ὁ Γ ἄρα σύγκειται ἐκ περισσῶν ἀριθμῶν, ὧν τὸ πλῆθος περισσόν ἐστιν.
And each of A, B is odd; therefore Γ is composed of odd numbers, the multitude of which is odd.
ὥστε ὁ Γ περισσός ἐστιν· ὅπερ ἔδει δεῖξαι.
So that Γ is odd; which it was required to prove.
§9.prop.30ἐὰν περισσὸς ἀριθμὸς ἄρτιον ἀριθμὸν μετρῇ, καὶ τὸν ἥμισυν αὐτοῦ μετρήσει.
If an odd number measure an even number, it will also measure its half.
περισσὸς γὰρ ἀριθμὸς ὁ Α ἄρτιον τὸν Β μετρείτω· λέγω, ὅτι καὶ τὸν ἥμισυν αὐτοῦ μετρήσει.
For let the odd number A measure the even number B; I say that it will also measure its half.
ἐπεὶ γὰρ ὁ Α τὸν Β μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Γ· λέγω, ὅτι ὁ Γ οὐκ ἔστι περισσός.
For since A measures B, let it measure it according to Γ; I say that Γ is not odd.
εἰ γὰρ δυνατόν, ἔστω.
For, if possible, let it be.
καὶ ἐπεὶ ὁ Α τὸν Β μετρεῖ κατὰ τὸν Γ, ὁ Α ἄρα τὸν Γ πολλαπλασιάσας τὸν Β πεποίηκεν.
And since A measures B according to Γ, therefore A by multiplying Γ has made B.
ὁ Β ἄρα σύγκειται ἐκ περισσῶν ἀριθμῶν, ὧν τὸ πλῆθος περισσόν ἐστιν.
Therefore B is composed of odd numbers, the multitude of which is odd.
ὁ Β ἄρα περισσός ἐστιν· ὅπερ ἄτοπον· ὑπόκειται γὰρ ἄρτιος.
Therefore B is odd; which is absurd, for it is assumed to be even.
οὐκ ἄρα ὁ Γ περισσός ἐστιν·
Therefore Γ is not odd; therefore Γ is even.
ἄρτιος ἄρα ἐστὶν ὁ Γ. ὥστε ὁ Α τὸν Β μετρεῖ ἀρτιάκις.
So that A measures B an even number of times.
διὰ δὴ τοῦτο καὶ τὸν ἥμισυν αὐτοῦ μετρήσει· ὅπερ ἔδει δεῖξαι.
For this reason indeed it will also measure its half; which it was required to prove.
§9.prop.31ἐὰν περισσὸς ἀριθμὸς πρός τινα ἀριθμὸν πρῶτος ᾖ, καὶ πρὸς τὸν διπλασίονα αὐτοῦ πρῶτος ἔσται.
If an odd number be prime to some number, it will also be prime to its double.
περισσὸς γὰρ ἀριθμὸς ὁ Α πρός τινα ἀριθμὸν τὸν Β πρῶτος ἔστω, τοῦ δὲ Β διπλασίων ἔστω ὁ Γ· λέγω, ὅτι ὁ Α πρὸς τὸν Γ πρῶτός ἐστιν.
For let the odd number A be prime to some number B, and let Γ be double of B; I say that A is prime to Γ.
εἰ γὰρ μή εἰσιν πρῶτοι, μετρήσει τις αὐτοὺς ἀριθμός.
For if they are not prime, some number will measure them.
μετρείτω, καὶ ἔστω ὁ Δ. καί ἐστιν ὁ Α περισσός·
Let it measure them, and let it be Δ.
περισσὸς ἄρα καὶ ὁ Δ. καὶ ἐπεὶ ὁ Δ περισσὸς ὢν τὸν Γ μετρεῖ, καί ἐστιν ὁ Γ ἄρτιος, καὶ τὸν ἥμισυν ἄρα τοῦ Γ μετρήσει.
And A is odd; therefore Δ is also odd.
τοῦ δὲ Γ ἥμισύ ἐστιν ὁ Β·
And since Δ being odd measures Γ, and Γ is even, therefore it will also measure the half of Γ.
ὁ Δ ἄρα τὸν Β μετρεῖ.
And B is the half of Γ; therefore Δ measures B.
μετρεῖ δὲ καὶ τὸν Α. ὁ Δ ἄρα τοὺς Α, Β μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους·
And it also measures A.
ὅπερ ἐστὶν ἀδύνατον.
Therefore Δ measures A, B which are prime to one another; which is impossible.
οὐκ ἄρα ὁ Α πρὸς τὸν Γ πρῶτος οὔκ ἐστιν.
Therefore it is not the case that A is not prime to Γ.
οἱ Α, Γ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.
Therefore A, Γ are prime to one another; which it was required to prove.
§9.prop.32τῶν ἀπὸ δυάδος διπλασιαζομένων ἀριθμῶν ἕκαστος ἀρτιάκις ἄρτιός ἐστι μόνον.
Each of the numbers doubled from a dyad is even-times even only.
ἀπὸ γὰρ δυάδος τῆς Α δεδιπλασιάσθωσαν ὁσοιδηποτοῦν ἀριθμοὶ οἱ Β, Γ, Δ· λέγω, ὅτι οἱ Β, Γ, Δ ἀρτιάκις ἄρτιοί εἰσι μόνον.
For let any number of numbers, B, Γ, Δ, be doubled from the dyad A; I say that B, Γ, Δ are even-times even only.
ὅτι μὲν οὖν ἕκαστος ἀρτιάκις ἄρτιός ἐστιν, φανερόν· ἀπὸ γὰρ δυάδος ἐστὶ διπλασιασθείς.
Now that each is even-times even is manifest; for it is doubled from a dyad.
λέγω, ὅτι καὶ μόνον.
I say that it is also only so.
ἐκκείσθω γὰρ μονάς.
For let a unit be set out.
ἐπεὶ οὖν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογόν εἰσιν, ὁ δὲ μετὰ τὴν μονάδα ὁ Α πρῶτός ἐστιν, ὁ μέγιστος τῶν Α, Β, Γ, Δ ὁ Δ ὑπʼ οὐδενὸς ἄλλου μετρηθήσεται παρὲξ τῶν Α, Β, Γ. καί ἐστιν ἕκαστος τῶν Α, Β, Γ ἄρτιος·
Since then any number of numbers from a unit are in continued proportion, and A after the unit is prime, therefore the greatest of A, B, Γ, Δ, namely Δ, will be measured by no other except A, B, Γ.
ὁ Δ ἄρα ἀρτιάκις ἄρτιός ἐστι μόνον.
And each of A, B, Γ is even; therefore Δ is even-times even only.
ὁμοίως δὴ δείξομεν, ὅτι ἑκάτερος τῶν Β, Γ ἀρτιάκις ἄρτιός ἐστι μόνον· ὅπερ ἔδει δεῖξαι.
In like manner indeed we shall prove that each of B, Γ is even-times even only; which it was required to prove.