Page 289 - Euclid's Elements of Geometry

ST EW iþ.

Α A ELEMENTS BOOK 10
hþ
Β B
Εἰ γὰρ ἔχει τὸ Α πρὸς τὸ Β λόγον, ὃν ἀριθμὸς πρὸς For if A has to B the ratio which (some) number (has)
ἀριθμόν, σύμμετρον ἔσται τὸ Α τῷ Β. οὐκ ἔστι δέ· οὐκ ἄρα to (some) number then A will be commensurable with B
τὸ Α πρὸς τὸ Β λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν. [Prop. 10.6]. But it is not. Thus, A does not have to B
Τὰ ἄρα ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, the ratio which (some) number (has) to (some) number.
Thus, incommensurable numbers do not have to one
καὶ τὰ ἑξῆς. another, and so on . . . .

.
Proposition 8
᾿Εὰν δύο μεγέθη πρὸς ἄλληλα λόγον μὴ ἔχῃ, ὃν ἀριθμὸς If two magnitudes do not have to one another the ra-
πρὸς ἀριθμόν, ἀσύμμετρα ἔσται τὰ μεγέθη. tio which (some) number (has) to (some) number then
the magnitudes will be incommensurable.
Α A
Β B
Δύο γὰρ μεγέθη τὰ Α, Β πρὸς ἄλληλα λόγον μὴ ἐχέτω, For let the two magnitudes A and B not have to one
ὃν ἀριθμὸς πρὸς ἀριθμόν· λέγω, ὅτι ἀσύμμετρά ἐστι τὰ Α, another the ratio which (some) number (has) to (some)
jþ the magnitudes A and B are incommensurable.
Β μεγέθη. number. I say that the magnitudes A and B are incom-
Εἰ γὰρ ἔσται σύμμετρα, τὸ Α πρὸς τὸ Β λόγον ἕξει, ὃν mensurable.
ἀριθμὸς πρὸς ἀριθμόν. οὐκ ἔχει δέ. ἀσύμμετρα ἄρα ἐστὶ τὰ For if they are commensurable, A will have to B
Α, Β μεγέθη. the ratio which (some) number (has) to (some) number
᾿Εὰν ἄρα δύο μεγέθη πρὸς ἄλληλα, καὶ τὰ ἑξῆς. [Prop. 10.5]. But it does not have (such a ratio). Thus,
Thus, if two magnitudes . . . to one another, and so on
. . . .

Proposition 9
.
Τὰ ἀπὸ τῶν μήκει συμμέτρων εὐθειῶν τετράγωνα Squares on straight-lines (which are) commensurable
πρὸς ἄλληλα λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς in length have to one another the ratio which (some)
τετράγωνον ἀριθμόν· καὶ τὰ τετράγωνα τὰ πρὸς ἄλληλα square number (has) to (some) square number. And
λόγον ἔχοντα, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον squares having to one another the ratio which (some)
ἀριθμόν, καὶ τὰς πλευρὰς ἕξει μήκει συμμέτρους. τὰ square number (has) to (some) square number will also
δὲ ἀπὸ τῶν μήκει ἀσυμμέτρων εὐθειῶν τετράγωνα πρὸς have sides (which are) commensurable in length. But
ἄλληλα λόγον οὐκ ἔχει, ὅνπερ τετράγωνος ἀριθμὸς πρὸς squares on straight-lines (which are) incommensurable
τετράγωνον ἀριθμόν· καὶ τὰ τετράγωνα τὰ πρὸς ἄλληλα in length do not have to one another the ratio which
λόγον μὴ ἔχοντα, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον (some) square number (has) to (some) square number.
ἀριθμόν, οὐδὲ τὰς πλευρὰς ἕξει μήκει συμμέτρους. And squares not having to one another the ratio which
(some) square number (has) to (some) square number
will not have sides (which are) commensurable in length
either.
Α Β A B

Γ ∆ C D
῎Εστωσαν γὰρ αἱ Α, Β μήκει σύμμετροι· λέγω, ὅτι τὸ For let A and B be (straight-lines which are) commen-
ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β τετράγωνον surable in length. I say that the square on A has to the
λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον square on B the ratio which (some) square number (has)
ἀριθμόν. to (some) square number.

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