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Euclid: Elementa

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ις῾ 
PROPOSITION 16. 
16 
 
 
 
Ἐὰν 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. 
Si tetragonus numerus tetragonum numerum non metitur, neque latus latus metietur. Et si latus latus non metitur, neque tetragonus tetragonum metietur. 
 
 
 
῎εστωσαν 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 A, B be square numbers, and let C, D be their sides; and let A not measure B;  I say that neither does C measure D. 
Sint tetragoni numeri A, B, latera vero ipsorum sint D, G. Et non metiatur numerus A numerum B.  Dico quoniam neque numerus G numerum D metitur. 
   
   
   
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 C measures D, A will also measure B. [VIII. 14]  But A does not measure B; therefore neither will C measure D. 
Si enim metitur numerus G numerum D, metietur et numerus A numerum B.  Non metitur autem numerus A numerum B, non ergo numerus G numerum D metietur. 
   
   
   
Μὴ μετρείτω [δὴ] πάλιν ὁ Γ τὸν Δ:  λέγω, ὅτι οὐδὲ ὁ Α τὸν Β μετρήσει. 
Again, let C not measure D;  I say that neither will A measure B. 
Non metiatur ergo rursum numerus G numerum D.  Dico quoniam neque numerus A numerum B metietur. 
   
   
   
Εἰ γὰρ μετρεῖ ὁ Α τὸν Β, μετρήσει καὶ ὁ Γ τὸν Δ.  οὐ μετρεῖ δὲ ὁ Γ τὸν Δ:  οὐδ᾽ ἄρα ὁ Α τὸν Β μετρήσει:  ὅπερ ἔδει δεῖξαι. 
For, if A measures B, C will also measure D. [VIII. 14]  But C does not measure D;  therefore neither will A measure B.  Q. E. D. 
Si enim metitur numerus A numerum B, metietur et numerus G numerum D.  Non meiitur autem numerus G numerum D,  neque ergo numerus A numerum B metietur.  Quod oportet ostendere. 
       
       
       
 
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