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Euclid · Elements §7.prop.4-7.prop.5

Numbers as Parts and the Addition of Equal Parts

Passage 113 of 316 · Greek

Summary

Proposition 4 shows that any smaller number is either a part or parts of a larger number, and Proposition 5 proves that if two numbers are the same parts of two other numbers respectively, their sum will also be the same part of the sum of the others.

§7.prop.4ἅπας ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος ἤτοι μέρος ἐστὶν ἢ μέρη.
Any number is either a part or parts of any number, the less of the greater.
ἔστωσαν δύο ἀριθμοὶ οἱ Α, ΒΓ, καὶ ἔστω ἐλάσσων ὁ ΒΓ· λέγω, ὅτι ὁ ΒΓ τοῦ Α ἤτοι μέρος ἐστὶν ἢ μέρη.
Let there be two numbers A, BΓ, and let BΓ be the less; I say that BΓ is either a part or parts of A.
οἱ Α, ΒΓ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ.
For A, BΓ are either prime to one another or not.
ἔστωσαν πρότερον οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους.
First, let A, BΓ be prime to one another.
διαιρεθέντος δὴ τοῦ ΒΓ εἰς τὰς ἐν αὐτῷ μονάδας ἔσται ἑκάστη μονὰς τῶν ἐν τῷ ΒΓ μέρος τι τοῦ Α· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α. μὴ ἔστωσαν δὴ οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους· ὁ δὴ ΒΓ τὸν Α ἤτοι μετρεῖ ἢ οὐ μετρεῖ.
Since BΓ is divided into the units in it, each unit of those in BΓ will be some part of A; so that BΓ is parts of A. Next, let A, BΓ not be prime to one another; then BΓ either measures A or does not measure (it).
εἰ μὲν οὖν ὁ ΒΓ τὸν Α μετρεῖ, μέρος ἐστὶν ὁ ΒΓ τοῦ Α. εἰ δὲ οὔ, εἰλήφθω τῶν Α, ΒΓ μέγιστον κοινὸν μέτρον ὁ Δ, καὶ διῃρήσθω ὁ ΒΓ εἰς τοὺς τῷ Δ ἴσους τοὺς ΒΕ, ΕΖ, ΖΓ. καὶ ἐπεὶ ὁ Δ τὸν Α μετρεῖ, μέρος ἐστὶν ὁ Δ τοῦ Α· ἴσος δὲ ὁ Δ ἑκάστῳ τῶν ΒΕ, ΕΖ, ΖΓ· καὶ ἕκαστος ἄρα τῶν ΒΕ, ΕΖ, ΖΓ τοῦ Α μέρος ἐστίν· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α. ἅπας ἄρα ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος ἤτοι μέρος ἐστὶν ἢ μέρη· ὅπερ ἔδει δεῖξαι.
If then BΓ measures A, BΓ is a part of A. But if not, let the greatest common measure of A, BΓ be taken, (namely) Δ, and let BΓ be divided into BE, EZ, ZΓ, which are equal to Δ. And since Δ measures A, Δ is a part of A; and Δ is equal to each of BE, EZ, ZΓ; therefore each of BE, EZ, ZΓ is also a part of A; so that BΓ is parts of A. Therefore any number is either a part or parts of any number, the less of the greater; which was to be proved.
§7.prop.5ἐὰν ἀριθμός ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ αὐτὸ μέρος ᾖ, καὶ συναμφότερος συναμφοτέρου τὸ αὐτὸ μέρος ἔσται, ὅπερ ὁ εἷς τοῦ ἑνός.
If a number be a part of a number, and another be the same part of another, then both together will also be the same part of both together that the one is of the one.
ἀριθμὸς γὰρ ὁ Α τοῦ ΒΓ μέρος ἔστω, καὶ ἕτερος ὁ Δ ἑτέρου τοῦ ΕΖ τὸ αὐτὸ μέρος, ὅπερ ὁ Α τοῦ ΒΓ· λέγω, ὅτι καὶ συναμφότερος ὁ Α, Δ συναμφοτέρου τοῦ ΒΓ, ΕΖ τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὁ Α τοῦ ΒΓ. ἐπεὶ γάρ, ὃ μέρος ἐστὶν ὁ Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Δ τοῦ ΕΖ, ὅσοι ἄρα εἰσὶν ἐν τῷ ΒΓ ἀριθμοὶ ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἀριθμοὶ ἴσοι τῷ Δ. διῃρήσθω ὁ μὲν ΒΓ εἰς τοὺς τῷ Α ἴσους τοὺς ΒΗ, ΗΓ, ὁ δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους τοὺς ΕΘ, ΘΖ·
For let the number A be a part of BΓ, and another Δ the same part of another EZ that A is of BΓ; I say that both together A, Δ are also the same part of both together BΓ, EZ that A is of BΓ. For since, what part A is of BΓ, the same part Δ is also of EZ, as many numbers as there are in BΓ equal to A, so many are there also in EZ equal to Δ. Let BΓ be divided into BH, HΓ which are equal to A, and EZ into EΘ, ΘZ which are equal to Δ; then the multitude of BH, HΓ will be equal to the multitude of EΘ, ΘZ.
ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ. καὶ ἐπεὶ ἴσος ἐστὶν ὁ μὲν ΒΗ τῷ Α, ὁ δὲ ΕΘ τῷ Δ, καὶ οἱ ΒΗ, ΕΘ ἄρα τοῖς Α, Δ ἴσοι.
And since BH is equal to A, and EΘ is equal to Δ, both BH, EΘ are also equal to A, Δ.
διὰ τὰ αὐτὰ δὴ καὶ οἱ ΗΓ, ΘΖ τοῖς Α, Δ. ὅσοι ἄρα ἐν τῷ ΒΓ ἀριθμοὶ ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τοῖς ΒΓ, ΕΖ ἴσοι τοῖς α, Δ. ὁσαπλασίων ἄρα ἐστὶν ὁ ΒΓ τοῦ Α, τοσαυταπλασίων ἐστὶ καὶ συναμφότερος ὁ ΒΓ, ΕΖ συναμφοτέρου τοῦ Α, Δ. ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ Α, Δ συναμφοτέρου τοῦ ΒΓ, ΕΖ· ὅπερ ἔδει δεῖξαι.
For the same reasons also HΓ, ΘZ are equal to A, Δ. Therefore, as many numbers as there are in BΓ equal to A, so many are there also in BΓ, EZ equal to α, Δ. Therefore, of whatever multiple BΓ is of A, of the same multiple both together BΓ, EZ are also of both together A, Δ. Therefore, what part A is of BΓ, the same part both together A, Δ are also of both together BΓ, EZ; which was to be proved.

Notes

  1. 7.prop.4μέρος ἐστὶν ἢ μέρη — In Greek mathematics, "μέρος" (part, singular) refers to a submultiple or unit fraction (1/n) where the smaller number measures the larger, whereas "μέρη" (parts, plural) refers to a proper fraction (m/n with m > 1) where it does not. This proposition declares that any smaller number must stand in one of these two relations to a larger number.
  2. 7.prop.5συναμφότερος συναμφοτέρου — The adjective "συναμφότερος" (both together) is used substantively to refer to the sum of the two pairs of numbers previously introduced (A + Δ and BΓ + EZ respectively). This is a formulaic expression in Greek geometry and arithmetic to denote a sum.
  3. 7.prop.5ὁσαπλασίων ἄρα ἐστὶν ὁ ΒΓ τοῦ Α, τοσαυταπλασίων ἐστὶ καὶ συναμφότερος — A correlative construction utilizing "ὁσαπλασίων ... τοσαυταπλασίων ..." (as many times multiple ... so many times multiple ...). It structurally matches the proportion, showing that if BΓ is an n-fold multiple of A, then the sum BΓ + EZ is likewise an n-fold multiple of the sum A + Δ.
  4. 7.prop.5τοῖς α, Δ — The lowercase "α" in the text is a transcription variant representing the number A. The plural dative article "τοῖς" syntactically agrees with the implicit plural elements or sets of numbers equal to A and Δ.

Cite this passage

Euclid, Elements §7.prop.4-7.prop.5. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:7.prop.4-7.prop.5

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