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Euclid · Elements §8.prop.15-8.prop.17

Divisibility of Cubes and Squares and Their Sides

Passage 139 of 316 · Greek

Summary

This section proves that a cube number measures another if and only if its side measures the other's side (Proposition 15), and similarly shows through proof by contradiction that one number does not measure another if and only if their sides do not measure each other for both squares (Proposition 16) and cubes (Proposition 17).

§8.prop.15ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μετρῇ, καὶ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ ὁ κύβος τὸν κύβον μετρήσει.
If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.
κύβος γὰρ ἀριθμὸς ὁ Α κύβον τὸν Β μετρείτω, καὶ τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι ὁ Γ τὸν Δ μετρεῖ.
For let the cube number Α measure the cube Β, and let the side of Α be Γ, and that of Β be Δ; I say that Γ also measures Δ.
ὁ Γ γὰρ ἑαυτὸν πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Δ ἑαυτὸν πολλαπλασιάσας τὸν Η ποιείτω, καὶ ἔτι ὁ Γ τὸν Δ πολλαπλασιάσας τὸν Ζ, ἑκάτερος δὲ τῶν Γ, Δ τὸν Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ ποιείτω.
For let Γ by multiplying itself make Ε, and let Δ by multiplying itself make Η, and further let Γ by multiplying Δ make Ζ, and let each of Γ, Δ by multiplying Ζ make each of Θ, Κ.
φανερὸν δή, ὅτι οἱ Ε, Ζ, Η καὶ οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ.
It is then manifest that Ε, Ζ, Η and Α, Θ, Κ, Β are in continued proportion in the ratio of Γ to Δ.
καὶ ἐπεὶ οἱ Α, Θ, κ, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ ὁ Α τὸν Β, μετρεῖ ἄρα καὶ τὸν Θ. καί ἐστιν ὡς ὁ Α πρὸς τὸν Θ, οὕτως ὁ Γ πρὸς τὸν Δ· μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ. ἀλλὰ δὴ μετρείτω ὁ Γ τὸν Δ· λέγω, ὅτι καὶ ὁ Α τὸν Β μετρήσει.
And since Α, Θ, Κ, Β are in continued proportion, and Α measures Β, therefore it also measures Θ. And as Α is to Θ, so is Γ to Δ; therefore Γ also measures Δ. Next, let Γ measure Δ; I say that Α will also measure Β.
τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δὴ δείξομεν, ὅτι οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ.
For with the same construction, we shall similarly prove that Α, Θ, Κ, Β are in continued proportion in the ratio of Γ to Δ.
καὶ ἐπεὶ ὁ Γ τὸν Δ μετρεῖ, καί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Θ, καὶ ὁ Α ἄρα τὸν Θ μετρεῖ· ὥστε καὶ τὸν Β μετρεῖ ὁ Α· ὅπερ ἔδει δεῖξαι.
And since Γ measures Δ, and as Γ is to Δ, so is Α to Θ, therefore Α also measures Θ; so that Α also measures Β; which was to be proved.
§8.prop.16ἐὰν τετράγωνος ἀριθμὸς τετράγωνον ἀριθμὸν μὴ μετρῇ, οὐδὲ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· κἂν ἡ πλευρὰ τὴν πλευρὰν μὴ μετρῇ, οὐδὲ ὁ τετράγωνος τὸν τετράγωνον μετρήσει.
If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.
῎ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, πλευραὶ δὲ αὐτῶν ἔστωσαν οἱ Γ, Δ, καὶ μὴ μετρείτω ὁ Α τὸν Β· λέγω, ὅτι οὐδὲ ὁ Γ τὸν Δ μετρεῖ.
Let Α, Β be square numbers, and let their sides be Γ, Δ, and let Α not measure Β; I say that Γ does not measure Δ either.
εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, μετρήσει καὶ ὁ Α τὸν Β. οὐ μετρεῖ δὲ ὁ Α τὸν Β· οὐδὲ ἄρα ὁ Γ τὸν Δ μετρήσει.
For if Γ measures Δ, Α will also measure Β. But Α does not measure Β; therefore Γ will not measure Δ either.
μὴ μετρείτω πάλιν ὁ Γ τὸν Δ· λέγω, ὅτι οὐδὲ ὁ Α τὸν Β μετρήσει.
Next, let Γ not measure Δ; I say that Α will not measure Β either.
εἰ γὰρ μετρεῖ ὁ Α τὸν Β, μετρήσει καὶ ὁ Γ τὸν Δ. οὐ μετρεῖ δὲ ὁ Γ τὸν Δ· οὐδʼ ἄρα ὁ Α τὸν Β μετρήσει· ὅπερ ἔδει δεῖξαι.
For if Α measures Β, Γ will also measure Δ. But Γ does not measure Δ; therefore Α will not measure Β either; which was to be proved.
§8.prop.17ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μὴ μετρῇ, οὐδὲ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· κἂν ἡ πλευρὰ τὴν πλευρὰν μὴ μετρῇ, οὐδὲ ὁ κύβος τὸν κύβον μετρήσει.
If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.
κύβος γὰρ ἀριθμὸς ὁ Α κύβον ἀριθμὸν τὸν Β μὴ μετρείτω, καὶ τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι ὁ Γ τὸν Δ οὐ μετρήσει.
For let the cube number Α not measure the cube number Β, and let the side of Α be Γ, and that of Β be Δ; I say that Γ will not measure Δ.
εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, καὶ ὁ Α τὸν Β μετρήσει.
For if Γ measures Δ, Α will also measure Β.
οὐ μετρεῖ δὲ ὁ Α τὸν Β· οὐδʼ ἄρα ὁ Γ τὸν Δ μετρεῖ.
But Α does not measure Β; therefore Γ does not measure Δ either.
ἀλλὰ δὴ μὴ μετρείτω ὁ Γ τὸν Δ· λέγω, ὅτι οὐδὲ ὁ Α τὸν Β μετρήσει.
Next, let Γ not measure Δ; I say that Α will not measure Β either.
εἰ γὰρ ὁ Α τὸν Β μετρεῖ, καὶ ὁ Γ τὸν Δ μετρήσει.
For if Α measures Β, Γ will also measure Δ.
οὐ μετρεῖ δὲ ὁ Γ τὸν Δ· οὐδʼ ἄρα ὁ Α τὸν Β μετρήσει· ὅπερ ἔδει δεῖξαι.
But Γ does not measure Δ; therefore Α will not measure Β either; which was to be proved.

Notes

  1. §8.prop.15ἑκάτερος δὲ τῶν Γ, Δ τὸν Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ ποιείτω — The double use of "each" (nominative ἑκάτερος and accusative ἑκάτερον) expresses symmetrical multiplication. It means that Γ by multiplying Ζ makes Θ (Γ × Ζ = Θ), and Δ by multiplying Ζ makes Κ (Δ × Ζ = Κ).
  2. §8.prop.15μετρεῖ ἄρα καὶ τὸν Θ — The subject of the verb μετρεῖ (measures) is omitted but is understood from the context to be 'Α (ὁ Α)' from the preceding clause, meaning 'therefore Α also measures Θ.'
  3. §8.prop.16κἂν ἡ πλευρὰ τὴν πλευρὰν μὴ μετρῇ — The word κἂν is a crasis of καὶ ἐάν (and if), introducing a conditional clause.

Cite this passage

Euclid, Elements §8.prop.15-8.prop.17. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:8.prop.15-8.prop.17

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