Page 199 - Euclid's Elements of Geometry

ST EW zþ.

ELEMENTS BOOK 7

Οἱ Α, ΒΓ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ. For A and BC are either prime to one another, or not.
ἔστωσαν πρότερον οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους. διαι- Let A and BC, first of all, be prime to one another. So
ρεθέντος δὴ τοῦ ΒΓ εἰς τὰς ἐν αὐτῷ μονάδας ἔσται ἑκάστη separating BC into its constituent units, each of the units
μονὰς τῶν ἐν τῷ ΒΓ μέρος τι τοῦ Α· ὥστε μέρη ἐστὶν ὁ ΒΓ in BC will be some part of A. Hence, BC is parts of A.
τοῦ Α.

Β B

Ε E

Ζ F

Γ C
Α ∆ A D
Μὴ ἔστωσαν δὴ οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους· ὁ δὴ So let A and BC be not prime to one another. So BC
ΒΓ τὸν Α ἤτοι μετρεῖ ἢ οὐ μετρεῖ. εἰ μὲν οὖν ὁ ΒΓ τὸν either measures, or does not measure, A. Therefore, if
Α μετρεῖ, μέρος ἐστὶν ὁ ΒΓ τοῦ Α. εἰ δὲ οὔ, εἰλήφθω τῶν BC measures A then BC is part of A. And if not, let the
Α, ΒΓ μέγιστον κοινὸν μέτρον ὁ Δ, καὶ διῃρήσθω ὁ ΒΓ εἰς greatest common measure, D, of A and BC have been
eþ
τοὺς τῷ Δ ἴσους τοὺς ΒΕ, ΕΖ, ΖΓ. καὶ ἐπεὶ ὁ Δ τὸν Α taken [Prop. 7.2], and let BC have been divided into BE,
μετρεῖ, μέρος ἐστὶν ὁ Δ τοῦ Α· ἴσος δὲ ὁ Δ ἑκάστῳ τῶν EF, and FC, equal to D. And since D measures A, D is
ΒΕ, ΕΖ, ΖΓ· καὶ ἕκαστος ἄρα τῶν ΒΕ, ΕΖ, ΖΓ τοῦ Α μέρος a part of A. And D is equal to each of BE, EF, and FC.
ἐστίν· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α. Thus, BE, EF, and FC are also each part of A. Hence,
῞Απας ἄρα ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ BC is parts of A.
Thus, any number is either part or parts of any (other)
μείζονος ἤτοι μέρος ἐστὶν ἢ μέρη· ὅπερ ἔδει δεῖξαι. number, the lesser of the greater. (Which is) the very
thing it was required to show.

.
†
Proposition 5
᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ If a number is part of a number, and another (num-
αὐτὸ μέρος ᾖ, καὶ συναμφότερος συναμφοτέρου τὸ αὐτὸ ber) is the same part of another, then the sum (of the
μέρος ἔσται, ὅπερ ὁ εἷς τοῦ ἑνός. leading numbers) will also be the same part of the sum
(of the following numbers) that one (number) is of an-
other.
Β B

Ε E

Η G
Θ H

Γ Ζ C F
Α ∆ A D
᾿Αριθμὸς γὰρ ὁ Α [ἀριθμοῦ] τοῦ ΒΓ μέρος ἔστω, καὶ For let a number A be part of a [number] BC, and

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