Book X, Proposition 7

Incommensurable magnitudes have not to one another the ratio which a number has to a number.

Τὰ ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν. Ἔστω ἀσύμμετρα μεγέθη τὰ Α, Β: λέγω, ὅτι τὸ Α πρὸς τὸ Β λόγον οὐκ ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν. εἰ γὰρ ἔχει τὸ Α πρὸς τὸ Β λόγον, ὃν ἀριθμὸς πρὸς ἀριθμόν, σύμμετρον ἔσται τὸ Α τῷ Β. οὐκ ἔστι δέ: οὐκ ἄρα τὸ Α πρὸς τὸ Β λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν. Τὰ ἄρα ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, καὶ τὰ ἑξῆς.

Incommensurable magnitudes have not to one another the ratio which a number has to a number. Let A, B be incommensurable magnitudes; I say that A has not to B the ratio which a number has to a number. For, if A has to B the ratio which a number has to a number, A will be commensurable with B. [X. 6] But it is not; therefore A has not to B the ratio which a number has to a number.

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