Euclid · Elements §10.prop1.7-10.prop1.8

Incommensurability and Magnitudes Lacking a Ratio of Numbers

Passage 165 of 316 · Greek

Summary

This section proves, by contradiction, that incommensurable magnitudes do not have a ratio of a number to a number (Proposition 7), and that magnitudes not having a ratio of a number to a number are incommensurable (Proposition 8).

§10.prop1.7τὰ ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.

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

ἔστω ἀσύμμετρα μεγέθη τὰ Α, Β· λέγω, ὅτι τὸ Α πρὸς τὸ Β λόγον οὐκ ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.

Let Α, Β be incommensurable magnitudes; I say that Α does not have to Β the ratio which a number has to a number.

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.

For, if Α has to Β the ratio which a number has to a number, Α will be commensurable with Β. But it is not; therefore Α does not have to Β the ratio which a number has to a number.

τὰ ἄρα ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, καὶ τὰ ἑξῆς.

Therefore incommensurable magnitudes do not have to one another a ratio, and so on.

§10.prop1.8ἐὰν δύο μεγέθη πρὸς ἄλληλα λόγον μὴ ἔχῃ, ὃν ἀριθμὸς πρὸς ἀριθμόν, ἀσύμμετρα ἔσται τὰ μεγέθη.

If two magnitudes do not have to one another the ratio which a number has to a number, the magnitudes will be incommensurable.

δύο γὰρ μεγέθη τὰ Α, Β πρὸς ἄλληλα λόγον μὴ ἐχέτω, ὃν ἀριθμὸς πρὸς ἀριθμόν· λέγω, ὅτι ἀσύμμετρά ἐστι τὰ Α, Β μεγέθη.

For let two magnitudes Α, Β not have to one another the ratio which a number has to a number; I say that the magnitudes Α, Β are incommensurable.

εἰ γὰρ ἔσται σύμμετρα, τὸ Α πρὸς τὸ Β λόγον ἕξει, ὃν ἀριθμὸς πρὸς ἀριθμόν.

For, if they be commensurable, Α will have to Β the ratio which a number has to a number.

οὐκ ἔχει δέ. ἀσύμμετρα ἄρα ἐστὶ τὰ Α, Β μεγέθη.

But it does not; therefore the magnitudes Α, Β are incommensurable.

ἐὰν ἄρα δύο μεγέθη πρὸς ἄλληλα, καὶ τὰ ἑξῆς.

If then two magnitudes to one another, and so on.

Notes

§10.prop1.7οὐκ ἔστι δέ — Meaning "But it is not." This is a denial of the consequence of the preceding conditional clause, "Α will be commensurable with Β" (σύμμετρον ἔσται). The third-person singular present `ἔστι` is used here without an explicit subject to directly negate the assumed situation.
§10.prop1.8εἰ γὰρ ἔσται σύμμετρα — The conditional particle `εἰ` followed by the future indicative `ἔσται` (third-person singular, agreeing with the neuter plural subject `τὰ Α, Β μεγέθη`). In Greek mathematical texts, this future indicative construction is regularly used in conditional clauses to introduce an assumption for a proof by contradiction (reductio ad absurdum).

Cite this passage

Euclid, Elements §10.prop1.7-10.prop1.8. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:10.prop1.7-10.prop1.8

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