Page 284 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

τρείτω τὸ πρὸ ἑαυτοῦ· λέγω, ὅτι ἀσύμμετρά ἐστι τὰ ΑΒ, the (magnitude) before it, (when) the lesser (magnitude
ΓΔ μεγέθη. is) continually subtracted in turn from the greater. I say
that the magnitudes AB and CD are incommensurable.
Α Η Β A G B

Ε E

Γ Ζ ∆ C F D
Εἰ γάρ ἐστι σύμμετρα, μετρήσει τι αὐτὰ μέγεθος. με- For if they are commensurable then some magnitude
τρείτω, εἰ δυνατόν, καὶ ἔστω τὸ Ε· καὶ τὸ μὲν ΑΒ τὸ ΖΔ will measure them (both). If possible, let it (so) measure
καταμετροῦν λειπέτω ἑαυτοῦ ἔλασσον τὸ ΓΖ, τὸ δὲ ΓΖ τὸ (them), and let it be E. And let AB leave CF less than
ΒΗ καταμετροῦν λειπέτω ἑαυτοῦ ἔλασσον τὸ ΑΗ, καὶ τοῦτο itself (in) measuring FD, and let CF leave AG less than
ἀεὶ γινέσθω, ἕως οὗ λειφθῇ τι μέγεθος, ὅ ἐστιν ἔλασσον τοῦ itself (in) measuring BG, and let this happen continually,
Ε. γεγονέτω, καὶ λελείφθω τὸ ΑΗ ἔλασσον τοῦ Ε. ἐπεὶ οὖν until some magnitude which is less than E is left. Let
†
τὸ Ε τὸ ΑΒ μετρεῖ, ἀλλὰ τὸ ΑΒ τὸ ΔΖ μετρεῖ, καὶ τὸ Ε ἄρα (this) have occurred, and let AG, (which is) less than
τὸ ΖΔ μετρήσει. μετρεῖ δὲ καὶ ὅλον τὸ ΓΔ· καὶ λοιπὸν ἄρα E, have been left. Therefore, since E measures AB, but
τὸ ΓΖ μετρήσει. ἀλλὰ τὸ ΓΖ τὸ ΒΗ μετρεῖ· καὶ τὸ Ε ἄρα AB measures DF, E will thus also measure FD. And it
τὸ ΒΗ μετρεῖ. μετρεῖ δὲ καὶ ὅλον τὸ ΑΒ· καὶ λοιπὸν ἄρα τὸ also measures the whole (of) CD. Thus, it will also mea-
ΑΗ μετρήσει, τὸ μεῖζον τὸ ἔλασσον· ὅπερ ἐστὶν ἀδύνατον. sure the remainder CF. But, CF measures BG. Thus, E
οὐκ ἄρα τὰ ΑΒ, ΓΔ μεγέθη μετρήσει τι μέγεθος· ἀσύμμετρα also measures BG. And it also measures the whole (of)
gþ magnitudes AB and CD. Thus, the magnitudes AB and
ἄρα ἐστὶ τὰ ΑΒ, ΓΔ μεγέθη. AB. Thus, it will also measure the remainder AG, the
᾿Εὰν ἄρα δύο μεγεθῶν ἀνίσων, καὶ τὰ ἑξῆς. greater (measuring) the lesser. The very thing is impos-
sible. Thus, some magnitude cannot measure (both) the

CD are incommensurable [Def. 10.1].
Thus, if . . . of two unequal magnitudes, and so on . . . .

† The fact that this will eventually occur is guaranteed by Prop. 10.1.

.
Proposition 3
Δύο μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον αὐτῶν To find the greatest common measure of two given
κοινὸν μέτρον εὑρεῖν. commensurable magnitudes.
Α Ζ Β A F B

Γ Ε ∆ C E D

Η G
῎Εστω τὰ δοθέντα δύο μεγέθη σύμμετρα τὰ ΑΒ, ΓΔ, Let AB and CD be the two given magnitudes, of
ὧν ἔλασσον τὸ ΑΒ· δεῖ δὴ τῶν ΑΒ, ΓΔ τὸ μέγιστον κοινὸν which (let) AB (be) the lesser. So, it is required to find
μέτρον εὑρεῖν. the greatest common measure of AB and CD.
Τὸ ΑΒ γὰρ μέγεθος ἤτοι μετρεῖ τὸ ΓΔ ἢ οὔ. εἰ μὲν For the magnitude AB either measures, or (does) not
οὖν μετρεῖ, μετρεῖ δὲ καὶ ἑαυτό, τὸ ΑΒ ἄρα τῶν ΑΒ, ΓΔ (measure), CD. Therefore, if it measures (CD), and
κοινὸν μέτρον ἐστίν· καὶ φανερόν, ὅτι καὶ μέγιστον. μεῖζον (since) it also measures itself, AB is thus a common mea-
γὰρ τοῦ ΑΒ μεγέθους τὸ ΑΒ οὐ μετρήσει. sure of AB and CD. And (it is) clear that (it is) also (the)
Μὴ μετρείτω δὴ τὸ ΑΒ τὸ ΓΔ. καὶ ἀνθυφαιρουμένου greatest. For a (magnitude) greater than magnitude AB
ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, τὸ περιλειπόμενον cannot measure AB.
μετρήσει ποτὲ τὸ πρὸ ἑαυτοῦ διὰ τὸ μὴ εἶναι ἀσύμμετρα τὰ So let AB not measure CD. And continually subtract-
ΑΒ, ΓΔ· καὶ τὸ μὲν ΑΒ τὸ ΕΔ καταμετροῦν λειπέτω ἑαυτοῦ ing in turn the lesser (magnitude) from the greater, the

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