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Euclid: Elementa

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ς῾
PROPOSITION 6.

Ἐὰν ἀριθμὸς ἑαυτὸν πολλαπλασιάσας κύβον ποιῇ, καὶ αὐτὸς κύβος ἔσται.
If a number by multiplying itself make a cube number, it will itself also be cube.
Si numerus se ipsum multiplicans cubum facit, et ipse cubus erit.

Ἀριθμὸς γὰρ ὁ Α ἑαυτὸν πολλαπλασιάσας κύβον τὸν Β ποιείτω:  λέγω, ὅτι καὶ ὁ Α κύβος ἐστίν.
For let the number A by multiplying itself make the cube number B;  I say that A is also cube.
Numerus enim A se ipsum multiplicans cubum B faciat.  Dico quoniam et A cubus est.

῾Ο γὰρ Α τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω.  ἐπεὶ οὖν ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Β πεποίηκεν, τὸν δὲ Β πολλαπλασιάσας τὸν Γ πεποίηκεν,  ὁ Γ ἄρα κύβος ἐστίν.  καὶ ἐπεὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν,  ὁ Α ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν αὑτῷ μονάδας.  μετρεῖ δὲ καὶ ἡ μονὰς τὸν Α κατὰ τὰς ἐν αὐτῷ μονάδας.  ἔστιν ἄρα ὡς ἡ μονὰς πρὸς τὸν Α, οὕτως ὁ Α πρὸς τὸν Β.  καὶ ἐπεὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν,  ὁ Β ἄρα τὸν Γ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας.  μετρεῖ δὲ καὶ ἡ μονὰς τὸν Α κατὰ τὰς ἐν αὐτῷ μονάδας.  ἔστιν ἄρα ὡς ἡ μονὰς πρὸς τὸν Α, οὕτως ὁ Β πρὸς τὸν Γ.  ἀλλ᾽ ὡς ἡ μονὰς πρὸς τὸν Α, οὕτως ὁ Α πρὸς τὸν Β:  καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, ὁ Β πρὸς τὸν Γ.  καὶ ἐπεὶ οἱ Β, Γ κύβοι εἰσίν, ὅμοιοι στερεοί εἰσιν.  τῶν Β, Γ ἄρα δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοί.  καί ἐστιν ὡς ὁ Β πρὸς τὸν Γ, ὁ Α πρὸς τὸν Β.  καὶ τῶν Α, Β ἄρα δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοί.  καί ἐστι κύβος ὁ Β:  κύβος ἄρα ἐστὶ καὶ ὁ Α:  ὅπερ ἔδει δεῖξαι.
For let A by multiplying B make C.  Since, then, A by multiplying itself has made B, and by multiplying B has made C,  therefore C is cube.  And, since A by multiplying itself has made B,  therefore A measures B according to the units in itself.  But the unit also measures A according to the units in it.  Therefore, as the unit is to A, so is A to B. [VII. Def. 20]  And, since A by multiplying B has made C,  therefore B measures C according to the units in A.  But the unit also measures A according to the units in it.  Therefore, as the unit is to A, so is B to C. [VII. Def. 20]  But, as the unit is to A, so is A to B;  therefore also, as A is to B, so is B to C.  And, since B, C are cube, they are similar solid numbers.  Therefore there are two mean proportional numbers between B, C. [VIII. 19]  And, as B is to C, so is A to B.  Therefore there are two mean proportional numbers between A, B also. [VIII. 8]  And B is cube;  therefore A is also cube. [cf. VIII. 23]  Q. E. D.
Numerus enim A numerum B multiplicans numerum G faciat.  Quoniam ergo numerus A se ipsum quidem multiplicans numerum B fecit, numerum vero B multiplicans numerum G fecit,  numerus ergo G cubus est.  Et quoniam numerus A se ipsum quidem multiplicans numerum B fecit,        numerum vero B multiplicans numerum G fecit,          sicut ergo A ad B ita B ad G.  Et quoniam numeri B, G cubi sunt, similes solidi sunt.  Numerorum ergo B, G duo medii proportionaliter sunt numeri  et est sicut B ad G ita A ad B.  Et numerorum ergo A, B duo medii proportionaliter sunt numeri  et est cubus B,  cubus ergo est et A.  Quod oportet ostendere.

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