§10.prop1.9#2εἰ γὰρ ἔχει τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β λόγον, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, σύμμετρος ἔσται ἡ Α τῇ Β. οὐκ ἔστι δέ·
For if the square on A has to the square on B the ratio which a square number has to a square number, A will be commensurable with B.
οὐκ ἄρα τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
But it is not; therefore the square on A does not have to the square on B the ratio which a square number has to a square number.
πάλιν δὴ τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β λόγον μὴ ἐχέτω, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· λέγω, ὅτι ἀσύμμετρός ἐστιν ἡ Α τῇ Β μήκει.
Next, let the square on A not have to the square on B the ratio which a square number has to a square number; I say that A is incommensurable in length with B.
εἰ γάρ ἐστι σύμμετρος ἡ Α τῇ Β, ἕξει τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς Β λόγον, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
For if A is commensurable with B, the square on A will have to the square on B the ratio which a square number has to a square number.
οὐκ ἔχει δέ· οὐκ ἄρα σύμμετρός ἐστιν ἡ Α τῇ Β μήκει.
But it does not have it; therefore A is not commensurable in length with B.
τὰ ἄρα ἀπὸ τῶν μήκει συμμέτρων, καὶ τὰ ἑξῆς.
Therefore, the [squares] on [straight lines] commensurable in length, and the rest.
Πόρισμα
καὶ φανερὸν ἐκ τῶν δεδειγμένων ἔσται, ὅτι αἱ μήκει σύμμετροι πάντως καὶ δυνάμει, αἱ δὲ δυνάμει οὐ πάντως καὶ μήκει [εἴπερ τὰ ἀπὸ τῶν μήκει συμμέτρων εὐθειῶν τετράγωνα λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, τὰ δὲ λόγον ἔχοντα, ὃν ἀριθμὸς πρὸς ἀριθμόν, σύμμετρά ἐστιν.
Porism And it will be manifest from what has been proved that straight lines commensurable in length are always commensurable in square also, but those commensurable in square are not always commensurable in length [if indeed the squares on straight lines commensurable in length have the ratio which a square number has to a square number, and those which have a ratio which a number has to a number are commensurable.
ὥστε αἱ μήκει σύμμετροι εὐθεῖαι οὐ μόνον μήκει σύμμετροι, ἀλλὰ καὶ δυνάμει.
So that straight lines commensurable in length are not only commensurable in length, but also in square.
πάλιν ἐπεί, ὅσα τετράγωνα πρὸς ἄλληλα λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, μήκει ἐδείχθη σύμμετρα καὶ δυνάμει ὄντα σύμμετρα τῷ τὰ τετράγωνα λόγον ἔχειν, ὃν ἀριθμὸς πρὸς ἀριθμόν, ὅσα ἄρα τετράγωνα λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ἀλλὰ ἁπλῶς, ὃν ἀριθμὸς πρὸς ἀριθμόν, σύμμετρα μὲν ἔσται αὐτὰ τὰ τετράγωνα δυνάμει, οὐκέτι δὲ καὶ μήκει· ὥστε τὰ μὲν μήκει σύμμετρα πάντως καὶ δυνάμει, τὰ δὲ δυνάμει οὐ πάντως καὶ μήκει, εἰ μὴ καὶ λόγον ἔχοιεν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
Again, since, as many squares as have to one another the ratio which a square number has to a square number, have been proved commensurable in length, and since those commensurable in square are so by the squares having the ratio which a number has to a number, as many squares therefore as do not have the ratio which a square number has to a square number, but simply that which a number has to a number, the squares themselves will indeed be commensurable in square, but no longer also in length; so that those commensurable in length are always and in square, but those in square not always also in length, unless they should also have the ratio which a square number has to a square number.
λέγω δή, ὅτι αἱ μήκει ἀσύμμετροι οὐ πάντως καὶ δυνάμει, ἐπειδήπερ αἱ δυνάμει σύμμετροι δύνανται λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, καὶ διὰ τοῦτο δυνάμει οὖσαι σύμμετροι μήκει εἰσὶν ἀσύμμετροι.
And I say that those incommensurable in length are not always and in square, since those commensurable in square can not have the ratio which a square number has to a square number, and for this reason, being commensurable in square, they are incommensurable in length.
ὥστε οὐχ αἱ τῷ μήκει ἀσύμμετροι πάντως καὶ δυνάμει, ἀλλὰ δύνανται μήκει οὖσαι ἀσύμμετροι δυνάμει εἶναι καὶ ἀσύμμετροι καὶ σύμμετροι.
So that those incommensurable in length are not always in square also, but those being incommensurable in length can be in square both incommensurable and commensurable.
αἱ δὲ δυνάμει ἀσύμμετροι πάντως καὶ μήκει ἀσύμμετροι· εἰ γὰρ μήκει σύμμετροι, ἔσονται καὶ δυνάμει σύμμετροι.
But those incommensurable in square are always incommensurable in length also; for if they are commensurable in length, they will also be commensurable in square.
ὑπόκεινται δὲ καὶ ἀσύμμετροι· ὅπερ ἄτοπον.
But they are also assumed to be incommensurable; which is absurd.
αἱ ἄρα δυνάμει ἀσύμμετροι πάντως καὶ μήκει].
Therefore those incommensurable in square are always incommensurable in length also].
λῆμμα
δέδεικται ἐν τοῖς ἀριθμητικοῖς, ὅτι οἱ ὅμοιοι ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, καὶ ὅτι, ἐὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὅμοιοί εἰσιν ἐπίπεδοι.
Lemma It has been proved in the arithmetical books that similar plane numbers have to one another the ratio which a square number has to a square number, and that, if two numbers have to one another the ratio which a square number has to a square number, they are similar plane numbers.
καὶ δῆλον ἐκ τούτων, ὅτι οἱ μὴ ὅμοιοι ἐπίπεδοι ἀριθμοί, τουτέστιν οἱ μὴ ἀνάλογον ἔχοντες τὰς πλευράς, πρὸς ἀλλήλους λόγον οὐκ ἔχουσιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
And it is clear from these things that plane numbers which are not similar, that is, those which do not have their sides proportional, do not have to one another the ratio which a square number has to a square number.
εἰ γὰρ ἕξουσιν, ὅμοιοι ἐπίπεδοι ἔσονται· ὅπερ οὐχ ὑπόκειται.
For if they have, they will be similar plane numbers; which is not assumed.
οἱ ἄρα μὴ ὅμοιοι ἐπίπεδοι πρὸς ἀλλήλους λόγον οὐκ ἔχουσιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
Therefore plane numbers which are not similar do not have to one another the ratio which a square number has to a square number.