Page 286 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

καὶ τὰ Α, Β, τὸ Δ ἄρα τὰ Α, Β, Γ μετρεῖ· τὸ Δ ἄρα τῶν Α, be D. So D either measures, or [does] not [measure], C.
Β, Γ κοινὸν μέτρον ἐστίν. καὶ φανερόν, ὅτι καὶ μέγιστον· Let it, first of all, measure (C). Therefore, since D mea-
μεῖζον γὰρ τοῦ Δ μεγέθους τὰ Α, Β οὐ μετρεῖ. sures C, and it also measures A and B, D thus measures
Μὴ μετρείτω δὴ τὸ Δ τὸ Γ. λέγω πρῶτον, ὅτι σύμμετρά A, B, C. Thus, D is a common measure of A, B, C. And
ἐστι τὰ Γ, Δ. ἐπεὶ γὰρ σύμμετρά ἐστι τὰ Α, Β, Γ, μετρήσει (it is) clear that (it is) also (the) greatest (common mea-
τι αὐτὰ μέγεθος, ὃ δηλαδὴ καὶ τὰ Α, Β μετρήσει· ὥστε sure). For no magnitude larger than D measures (both)
καὶ τὸ τῶν Α, Β μέγιστον κοινὸν μέτρον τὸ Δ μετρήσει. A and B.
μετρεῖ δὲ καὶ τὸ Γ· ὥστε τὸ εἰρημένον μέγεθος μετρήσει τὰ So let D not measure C. I say, first, that C and D are
Γ, Δ· σύμμετρα ἄρα ἐστὶ τὰ Γ, Δ. εἰλήφθω οὖν αὐτῶν τὸ commensurable. For if A, B, C are commensurable then
μέγιστον κοινὸν μέτρον, καὶ ἔστω τὸ Ε. ἐπεὶ οὖν τὸ Ε τὸ some magnitude will measure them which will clearly
Δ μετρεῖ, ἀλλὰ τὸ Δ τὰ Α, Β μετρεῖ, καὶ τὸ Ε ἄρα τὰ Α, Β also measure A and B. Hence, it will also measure D, the
μετρήσει. μετρεῖ δὲ καὶ τὸ Γ. τὸ Ε ἄρα τὰ Α, Β, Γ μετρεῖ· greatest common measure of A and B [Prop. 10.3 corr.].
τὸ Ε ἄρα τῶν Α, Β, Γ κοινόν ἐστι μέτρον. λέγω δή, ὅτι καὶ And it also measures C. Hence, the aforementioned mag-
μέγιστον. εἰ γὰρ δυνατόν, ἔστω τι τοῦ Ε μεῖζον μέγεθος nitude will measure (both) C and D. Thus, C and D are
τὸ Ζ, καὶ μετρείτω τὰ Α, Β, Γ. καὶ ἐπεὶ τὸ Ζ τὰ Α, Β, Γ commensurable [Def. 10.1]. Therefore, let their greatest
μετρεῖ, καὶ τὰ Α, Β ἄρα μετρήσει καὶ τὸ τῶν Α, Β μέγιστον common measure have been taken [Prop. 10.3], and let
κοινὸν μέτρον μετρήσει. τὸ δὲ τῶν Α, Β μέγιστον κοινὸν it be E. Therefore, since E measures D, but D measures
μέτρον ἐστὶ τὸ Δ· τὸ Ζ ἄρα τὸ Δ μετρεῖ. μετρεῖ δὲ καὶ τὸ (both) A and B, E will thus also measure A and B. And
Γ· τὸ Ζ ἄρα τὰ Γ, Δ μετρεῖ· καὶ τὸ τῶν Γ, Δ ἄρα μέγιστον it also measures C. Thus, E measures A, B, C. Thus, E
κοινὸν μέτρον μετρήσει τὸ Ζ. ἔστι δὲ τὸ Ε· τὸ Ζ ἄρα τὸ Ε is a common measure of A, B, C. So I say that (it is) also
μετρήσει, τὸ μεῖζον τὸ ἔλασσον· ὅπερ ἐστὶν ἀδύνατον. οὐκ (the) greatest (common measure). For, if possible, let F
ἄρα μεῖζόν τι τοῦ Ε μεγέθους [μέγεθος] τὰ Α, Β, Γ μετρεῖ· be some magnitude greater than E, and let it measure A,
τὸ Ε ἄρα τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον ἐστίν, ἐὰν B, C. And since F measures A, B, C, it will thus also
μὴ μετρῇ τὸ Δ τὸ Γ, ἐὰν δὲ μετρῇ, αὐτὸ τὸ Δ. measure A and B, and will (thus) measure the greatest
Τριῶν ἄρα μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον common measure of A and B [Prop. 10.3 corr.]. And D
κοινὸν μέτρον ηὕρηται [ὅπερ ἔδει δεῖξαι]. is the greatest common measure of A and B. Thus, F
measures D. And it also measures C. Thus, F measures
(both) C and D. Thus, F will also measure the greatest
common measure of C and D [Prop. 10.3 corr.]. And it is
E. Thus, F will measure E, the greater (measuring) the
ìri sma B, C. Thus, if D does not measure C then E is the great-
lesser. The very thing is impossible. Thus, some [magni-
tude] greater than the magnitude E cannot measure A,

est common measure of A, B, C. And if it does measure
(C) then D itself (is the greatest common measure).
Thus, the greatest common measure of three given
commensurable magnitudes has been found. [(Which is)
the very thing it was required to show.]

Corollary
.
᾿Εκ δὴ τούτου φανερόν, ὅτι, ἐὰν μέγεθος τρία μεγέθη So (it is) clear, from this, that if a magnitude measures
μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει. three magnitudes then it will also measure their greatest
῾Ομοίως δὴ καὶ ἐπὶ πλειόνων τὸ μέγιστον κοινὸν μέτρον common measure.
ληφθήσεται, καὶ τὸ πόρισμα προχωρήσει. ὅπερ ἔδει δεῖξαι. So, similarly, the greatest common measure of more
(magnitudes) can also be taken, and the (above) corol-
lary will go forward. (Which is) the very thing it was
required to show.

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