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Euclid: Elementa

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ιδ῾ 
PROPOSITION 14. 
14 
 
 
 
Ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ καὶ ἄλλοι αὐτοῖς ἴσοι τὸ πλῆθος σύνδυο λαμβανόμενοι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ δι᾽ ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσονται. 
If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali. 
Si fuerint quotlibet numeri et alii ipsis equales multitudine bini sumpti et in eadem proportione, et per equale in eadem proportione erunt. 
 
 
 
Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ καὶ ἄλλοι αὐτοῖς ἴσοι τὸ πλῆθος σύνδυο λαμβανόμενοι ἐν τῷ αὐτῷ λόγῳ οἱ Δ, Ε, Ζ, ὡς μὲν ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε, ὡς δὲ ὁ Β πρὸς τὸν Γ, οὕτως ὁ Ε πρὸς τὸν Ζ:  λέγω, ὅτι καὶ δι᾽ ἴσου ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ζ. 
Let there be as many numbers as we please A, B, C, and others equal to them in multitude D, E, F, which taken two and two are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F;  I say that, ex aequali, as A is to C, so also is D to F. 
Sint quotlibet numeri A, B, G, et alii ipsis equales multitudine bini sumpti in eadem proportione D, E, Z, sicut quidem est numerus A ad numerum B ita numerus D ad numerum E, sicut autem numerus B ad numerum G ita numerus E ad numerum Z.  Dico quoniam et per equale est sicut numerus A ad numerum G ita numerus D ad numerum Z. 
   
   
   
Ἐπεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε,  ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, οὕτως ὁ Β πρὸς τὸν Ε.  πάλιν, ἐπεί ἐστιν ὡς ὁ Β πρὸς τὸν Γ, οὕτως ὁ Ε πρὸς τὸν Ζ,  ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Β πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν Ζ.  ὡς δὲ ὁ Β πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Δ:  καὶ ὡς ἄρα ὁ Α πρὸς τὸν Δ, οὕτως ὁ Γ πρὸς τὸν Ζ:  ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ζ:  ὅπερ ἔδει δεῖξαι. 
For, since, as A is to B, so is D to E,  therefore, alternately, as A is to D, so is B to E. [VII. 13]  Again, since, as B is to C, so is E to F,  therefore, alternately, as B is to E, so is C to F. [VII. 13]  But, as B is to E, so is A to D;  therefore also, as A is to D, so is C to F.  Therefore, alternately, as A is to C, so is D to F. [id.]   
Quoniam enim est sicut numerus A ad numerum B ita numerus D ad numerum E,  permutatim ergo est sicut numerus A ad numerum D ita numerus B ad numerum E.  Rursum quoniam est sicut numerus B ad numerum G ita numerus E ad numerum Z.  Permutatim ergo est sicut numerus B ad numerum E ita numerus G ad numerum Z.  Sicut autem numerus B ad numerum E ita numerus A ad numerum D,  et sicut numerus A ad numerum D ita numerus G ad numerum Z.  Permutatim ergo est sicut numerus A ad numerum G ita numerus D ad numerum Z.  Quod oportebat ostendere. 
               
               
               
 
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