Page 296 - Euclid's Elements of Geometry

ST EW iþ.

Α Β Γ A ELEMENTS BOOK 10
C
B
∆ D
᾿Επεὶ γὰρ σύμμετρά ἐστι τὰ ΑΒ, ΒΓ, μετρήσει τι αὐτὰ For since AB and BC are commensurable, some mag-
μέγεθος. μετρείτω, καὶ ἔστω τὸ Δ. ἐπεὶ οὖν τὸ Δ τὰ ΑΒ, nitude will measure them. Let it (so) measure (them),
ΒΓ μετρεῖ, καὶ ὅλον τὸ ΑΓ μετρήσει. μετρεῖ δὲ καὶ τὰ ΑΒ, and let it be D. Therefore, since D measures (both) AB
ΒΓ. τὸ Δ ἄρα τὰ ΑΒ, ΒΓ, ΑΓ μετρεῖ· σύμμετρον ἄρα ἐστὶ and BC, it will also measure the whole AC. And it also
τὸ ΑΓ ἑκατέρῳ τῶν ΑΒ, ΒΓ. measures AB and BC. Thus, D measures AB, BC, and
᾿Αλλὰ δὴ τὸ ΑΓ ἔστω σύμμετρον τῷ ΑΒ· λέγω δή, ὅτι AC. Thus, AC is commensurable with each of AB and
καὶ τὰ ΑΒ, ΒΓ σύμμετρά ἐστιν. BC [Def. 10.1].
᾿Επεὶ γὰρ σύμμετρά ἐστι τὰ ΑΓ, ΑΒ, μετρήσει τι αὐτὰ And so let AC be commensurable with AB. I say that
μέγεθος. μετρείτω, καὶ ἔστω τὸ Δ. ἐπεὶ οὖν τὸ Δ τὰ ΓΑ, AB and BC are also commensurable.
iþ And it also measures AB. Thus, D will measure (both)
ΑΒ μετρεῖ, καὶ λοιπὸν ἄρα τὸ ΒΓ μετρήσει. μετρεῖ δὲ καὶ For since AC and AB are commensurable, some mag-
τὸ ΑΒ· τὸ Δ ἄρα τὰ ΑΒ, ΒΓ μετρήσει· σύμμετρα ἄρα ἐστὶ nitude will measure them. Let it (so) measure (them),
τὰ ΑΒ, ΒΓ. and let it be D. Therefore, since D measures (both) CA
᾿Εὰν ἄρα δύο μεγέθη, καὶ τὰ ἑξῆς. and AB, it will thus also measure the remainder BC.

AB and BC. Thus, AB and BC are commensurable
[Def. 10.1].
Thus, if two magnitudes, and so on . . . .

.
Proposition 16
᾿Εὰν δύο μεγέθη ἀσύμμετρα συντεθῇ, καὶ τὸ ὅλον If two incommensurable magnitudes are added to-
ἑκατέρῳ αὐτῶν ἀσύμμετρον ἔσται· κἂν τὸ ὅλον ἑνὶ αὐτῶν gether then the whole will also be incommensurable with
ἀσύμμετρον ᾖ, καὶ τὰ ἐξ ἀρχῆς μεγέθη ἀσύμμετρα ἔσται. each of them. And if the whole is incommensurable with
one of them then the original magnitudes will also be in-
commensurable (with one another).
Α Β Γ A B C

∆ D
Συγκείσθω γὰρ δύο μεγέθη ἀσύμμετρα τὰ ΑΒ, ΒΓ· For let the two incommensurable magnitudes AB and
λέγω, ὅτι καὶ ὅλον τὸ ΑΓ ἑκατέρῳ τῶν ΑΒ, ΒΓ ἀσύμμετρόν BC be laid down together. I say that that the whole AC
ἐστιν. is also incommensurable with each of AB and BC.
Εἰ γὰρ μή ἐστιν ἀσύμμετρα τὰ ΓΑ, ΑΒ, μετρήσει τι For if CA and AB are not incommensurable then
[αὐτὰ] μέγεθος. μετρείτω, εἰ δυνατόν, καὶ ἔστω τὸ Δ. ἐπεὶ some magnitude will measure [them]. If possible, let it
οὖν τὸ Δ τὰ ΓΑ, ΑΒ μετρεῖ, καὶ λοιπὸν ἄρα τὸ ΒΓ μετρήσει. (so) measure (them), and let it be D. Therefore, since
μετρεῖ δὲ καὶ τὸ ΑΒ· τὸ Δ ἄρα τὰ ΑΒ, ΒΓ μετρεῖ. σύμμετρα D measures (both) CA and AB, it will thus also mea-
ἄρα ἐστὶ τὰ ΑΒ, ΒΓ· ὑπέκειντο δὲ καὶ ἀσύμμετρα· ὅπερ sure the remainder BC. And it also measures AB. Thus,
ἐστὶν ἀδύνατον. οὐκ ἄρα τὰ ΓΑ, ΑΒ μετρήσει τι μέγεθος· D measures (both) AB and BC. Thus, AB and BC are
ἀσύμμετρα ἄρα ἐστὶ τὰ ΓΑ, ΑΒ. ὁμοίως δὴ δείξομεν, ὅτι καὶ commensurable [Def. 10.1]. But they were also assumed
τὰ ΑΓ, ΓΒ ἀσύμμετρά ἐστιν. τὸ ΑΓ ἄρα ἑκατέρῳ τῶν ΑΒ, (to be) incommensurable. The very thing is impossible.
ΒΓ ἀσύμμετρόν ἐστιν. Thus, some magnitude cannot measure (both) CA and
᾿Αλλὰ δὴ τὸ ΑΓ ἑνὶ τῶν ΑΒ, ΒΓ ἀσύμμετρον ἔστω. AB. Thus, CA and AB are incommensurable [Def. 10.1].
ἔστω δὴ πρότερον τῷ ΑΒ· λέγω, ὅτι καὶ τὰ ΑΒ, ΒΓ So, similarly, we can show that AC and CB are also
ἀσύμμετρά ἐστιν. εἰ γὰρ ἔσται σύμμετρα, μετρήσει τι αὐτὰ incommensurable. Thus, AC is incommensurable with
μέγεθος. μετρείτω, καὶ ἔστω τὸ Δ. ἐπεὶ οὖν τὸ Δ τὰ ΑΒ, each of AB and BC.
ΒΓ μετρεῖ, καὶ ὅλον ἄρα τὸ ΑΓ μετρήσει. μετρεῖ δὲ καὶ τὸ And so let AC be incommensurable with one of AB
ΑΒ· τὸ Δ ἄρα τὰ ΓΑ, ΑΒ μετρεῖ. σύμμετρα ἄρα ἐστὶ τὰ and BC. So let it, first of all, be incommensurable with

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