Page 285 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ἔλασσον τὸ ΕΓ, τὸ δὲ ΕΓ τὸ ΖΒ καταμετροῦν λειπέτω remaining (magnitude) will (at) some time measure the
ἑαυτοῦ ἔλασσον τὸ ΑΖ, τὸ δὲ ΑΖ τὸ ΓΕ μετρείτω. (magnitude) before it, on account of AB and CD not be-
᾿Επεὶ οὖν τὸ ΑΖ τὸ ΓΕ μετρεῖ, ἀλλὰ τὸ ΓΕ τὸ ΖΒ μετρεῖ, ing incommensurable [Prop. 10.2]. And let AB leave EC
καὶ τὸ ΑΖ ἄρα τὸ ΖΒ μετρήσει. μετρεῖ δὲ καὶ ἑαυτό· καὶ less than itself (in) measuring ED, and let EC leave AF
ὅλον ἄρα τὸ ΑΒ μετρήσει τὸ ΑΖ. ἀλλὰ τὸ ΑΒ τὸ ΔΕ μετρεῖ· less than itself (in) measuring FB, and let AF measure
καὶ τὸ ΑΖ ἄρα τὸ ΕΔ μετρήσει. μετρεῖ δὲ καὶ τὸ ΓΕ· καί CE.
ὅλον ἄρα τὸ ΓΔ μετρεῖ· τὸ ΑΖ ἄρα τῶν ΑΒ, ΓΔ κοινὸν Therefore, since AF measures CE, but CE measures
μέτρον ἐστίν. λέγω δή, ὅτι καὶ μέγιστον. εἰ γὰρ μή, ἔσται FB, AF will thus also measure FB. And it also mea-
τι μέγεθος μεῖζον τοῦ ΑΖ, ὃ μετρήσει τὰ ΑΒ, ΓΔ. ἔστω τὸ sures itself. Thus, AF will also measure the whole (of)
Η. ἐπεὶ οὖν τὸ Η τὸ ΑΒ μετρεῖ, ἀλλὰ τὸ ΑΒ τὸ ΕΔ μετρεῖ, AB. But, AB measures DE. Thus, AF will also mea-
καὶ τὸ Η ἄρα τὸ ΕΔ μετρήσει. μετρεῖ δὲ καὶ ὅλον τὸ ΓΔ· sure ED. And it also measures CE. Thus, it also mea-
καὶ λοιπὸν ἄρα τὸ ΓΕ μετρήσει τὸ Η. ἀλλὰ τὸ ΓΕ τὸ ΖΒ sures the whole of CD. Thus, AF is a common measure
μετρεῖ· καὶ τὸ Η ἄρα τὸ ΖΒ μετρήσει. μετρεῖ δὲ καὶ ὅλον of AB and CD. So I say that (it is) also (the) greatest
τὸ ΑΒ, καὶ λοιπὸν τὸ ΑΖ μετρήσει, τὸ μεῖζον τὸ ἔλασσον· (common measure). For, if not, there will be some mag-
ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα μεῖζόν τι μέγεθος τοῦ ΑΖ nitude, greater than AF, which will measure (both) AB
τὰ ΑΒ, ΓΔ μετρήσει· τὸ ΑΖ ἄρα τῶν ΑΒ, ΓΔ τὸ μέγιστον and CD. Let it be G. Therefore, since G measures AB,
κοινὸν μέτρον ἐστίν. but AB measures ED, G will thus also measure ED. And
Δύο ἄρα μεγεθῶν συμμέτρων δοθέντων τῶν ΑΒ, ΓΔ it also measures the whole of CD. Thus, G will also mea-
τὸ μέγιστον κοινὸν μέτρον ηὕρηται· ὅπερ ἔδει δεῖξαι. sure the remainder CE. But CE measures FB. Thus, G
will also measure FB. And it also measures the whole
ìri sma possible. Thus, some magnitude greater than AF cannot
(of) AB. And (so) it will measure the remainder AF,
the greater (measuring) the lesser. The very thing is im-

measure (both) AB and CD. Thus, AF is the greatest
common measure of AB and CD.
Thus, the greatest common measure of two given
dþ
commensurable magnitudes, AB and CD, has been
found. (Which is) the very thing it was required to show.

.
Corollary
᾿Εκ δὴ τούτου φανερόν, ὅτι, ἐὰν μέγεθος δύο μεγέθη So (it is) clear, from this, that if a magnitude measures
μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει. two magnitudes then it will also measure their greatest
common measure.

Proposition 4
.
Τριῶν μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον To find the greatest common measure of three given
αὐτῶν κοινὸν μέτρον εὑρεῖν. commensurable magnitudes.
Α A

Β B
Γ C

∆ Ε Ζ D E F
῎Εστω τὰ δοθέντα τρία μεγέθη σύμμετρα τὰ Α, Β, Γ· Let A, B, C be the three given commensurable mag-
δεῖ δὴ τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον εὑρεῖν. nitudes. So it is required to find the greatest common
Εἰλήφθω γὰρ δύο τῶν Α, Β τὸ μέγιστον κοινὸν μέτρον, measure of A, B, C.
καὶ ἔστω τὸ Δ· τὸ δὴ Δ τὸ Γ ἤτοι μετρεῖ ἢ οὔ [μετρεῖ]. For let the greatest common measure of the two (mag-
μετρείτω πρότερον. ἐπεὶ οὖν τὸ Δ τὸ Γ μετρεῖ, μετρεῖ δὲ nitudes) A and B have been taken [Prop. 10.3], and let it

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