§5.prop.9τὰ πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχοντα λόγον ἴσα ἀλλήλοις ἐστίν· καὶ πρὸς ἃ τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον, ἐκεῖνα ἴσα ἐστίν.
Magnitudes which have the same ratio to the same are equal to one another; and those to which the same has the same ratio are equal.
ἐχέτω γὰρ ἑκάτερον τῶν Α, Β πρὸς τὸ Γ τὸν αὐτὸν λόγον·
For let each of A, B have the same ratio to Γ.
λέγω, ὅτι ἴσον ἐστὶ τὸ Α τῷ Β.
εἰ γὰρ μή, οὐκ ἂν ἑκάτερον τῶν Α, Β πρὸς τὸ Γ τὸν αὐτὸν εἶχε λόγον·
I say that A is equal to B.
ἔχει δέ·
For if not, each of A, B would not have had the same ratio to Γ; but it has.
ἴσον ἄρα ἐστὶ τὸ Α τῷ Β.
ἐχέτω δὴ πάλιν τὸ Γ πρὸς ἑκάτερον τῶν Α, Β τὸν αὐτὸν λόγον·
Therefore A is equal to B.
λέγω, ὅτι ἴσον ἐστὶ τὸ Α τῷ Β.
εἰ γὰρ μή, οὐκ ἂν τὸ Γ πρὸς ἑκάτερον τῶν Α, Β τὸν αὐτὸν εἶχε λόγον·
Let again Γ have the same ratio to each of A, B. I say that A is equal to B.
ἔχει δέ·
For if not, Γ would not have had the same ratio to each of A, B; but it has.
ἴσον ἄρα ἐστὶ τὸ Α τῷ Β.
τὰ ἄρα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχοντα λόγον ἴσα ἀλλήλοις ἐστίν·
Therefore A is equal to B.
καὶ πρὸς ἃ τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον, ἐκεῖνα ἴσα ἐστίν· ὅπερ ἔδει δεῖξαι.
Therefore magnitudes which have the same ratio to the same are equal to one another; and those to which the same has the same ratio are equal; which was to be proved.
§5.prop.10τῶν πρὸς τὸ αὐτὸ λόγον ἐχόντων τὸ μείζονα λόγον ἔχον ἐκεῖνο μεῖζόν ἐστιν· πρὸς ὃ δὲ τὸ αὐτὸ μείζονα λόγον ἔχει, ἐκεῖνο ἔλαττόν ἐστιν.
Of magnitudes which have a ratio to the same, that which has a greater ratio is itself greater; and that to which the same has a greater ratio is less.
ἐχέτω γὰρ τὸ Α πρὸς τὸ Γ μείζονα λόγον ἤπερ τὸ Β πρὸς τὸ Γ·
For let A have a greater ratio to Γ than B has to Γ.
λέγω, ὅτι μεῖζόν ἐστι τὸ Α τοῦ Β.
εἰ γὰρ μή, ἤτοι ἴσον ἐστὶ τὸ Α τῷ Β ἢ ἔλασσον.
I say that A is greater than B. For if not, A is either equal to B or less.
ἴσον μὲν οὖν οὔκ ἐστι τὸ Α τῷ Β· ἑκάτερον γὰρ ἂν τῶν Α, Β πρὸς τὸ Γ τὸν αὐτὸν εἶχε λόγον.
Now indeed, A is not equal to B; for each of A, B would have had the same ratio to Γ.
οὐκ ἔχει δέ· οὐκ ἄρα ἴσον ἐστὶ τὸ Α τῷ Β. οὐδὲ μὴν ἔλασσόν ἐστι τὸ Α τοῦ Β·
But it has not; therefore A is not equal to B.
τὸ Α γὰρ ἂν πρὸς τὸ Γ ἐλάσσονα λόγον εἶχεν ἤπερ τὸ Β πρὸς τὸ Γ. οὐκ ἔχει δέ· οὐκ ἄρα ἔλασσόν ἐστι τὸ Α τοῦ Β. ἐδείχθη δὲ οὐδὲ ἴσον· μεῖζον ἄρα ἐστὶ τὸ Α τοῦ Β.
ἐχέτω δὴ πάλιν τὸ Γ πρὸς τὸ Β μείζονα λόγον ἤπερ τὸ Γ πρὸς τὸ Α·
Nor indeed is A less than B; for A would have had a less ratio to Γ than B has to Γ. But it has not; therefore A is not less than B. And it was shown to be not equal; therefore A is greater than B. Let again Γ have a greater ratio to B than Γ has to A.
λέγω, ὅτι ἔλασσόν ἐστι τὸ Β τοῦ Α.
εἰ γὰρ μή, ἤτοι ἴσον ἐστὶν ἢ μεῖζον.
I say that B is less than A. For if not, it is either equal or greater.
ἴσον μὲν οὖν οὔκ ἐστι τὸ Β τῷ Α· τὸ Γ γὰρ ἂν πρὸς ἑκάτερον τῶν Α, Β τὸν αὐτὸν εἶχε λόγον.
Now indeed, B is not equal to A; for Γ would have had the same ratio to each of A, B.
οὐκ ἔχει δέ· οὐκ ἄρα ἴσον ἐστὶ τὸ Α τῷ Β. οὐδὲ μὴν μεῖζόν ἐστι τὸ Β τοῦ Α·
But it has not; therefore A is not equal to B.
τὸ Γ γὰρ ἂν πρὸς τὸ Β ἐλάσσονα λόγον εἶχεν ἤπερ πρὸς τὸ Α. οὐκ ἔχει δέ· οὐκ ἄρα μεῖζόν ἐστι τὸ Β τοῦ Α. ἐδείχθη δέ, ὅτι οὐδὲ ἴσον· ἔλαττον ἄρα ἐστὶ τὸ Β τοῦ Α.
τῶν ἄρα πρὸς τὸ αὐτὸ λόγον ἐχόντων τὸ μείζονα λόγον ἔχον μεῖζόν ἐστιν· καὶ πρὸς ὃ τὸ αὐτὸ μείζονα λόγον ἔχει, ἐκεῖνο ἔλαττόν ἐστιν· ὅπερ ἔδει δεῖξαι.
Nor indeed is B greater than A; for Γ would have had a less ratio to B than to A. But it has not; therefore B is not greater than A. And it was shown that it is not equal; therefore B is less than A. Therefore, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less; which was to be proved.