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Euclid: Elementa

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ιγ῾ 
PROPOSITION 13. 
13 
 
 
 
Ἐὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, καὶ ἐναλλὰξ ἀνάλογον ἔσονται. 
If four numbers be proportional, they will also be proportional alternately. 
Si quattuor numeri proportionales fuerint, et permutatim proportionales erunt. 
 
 
 
Ἔστωσαν τέσσαρες ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ:  λέγω, ὅτι καὶ ἐναλλὰξ ἀνάλογον ἔσονται, ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Β πρὸς τὸν Δ. 
Let the four numbers A, B, C, D be proportional, so that, as A is to B, so is C to D;  I say that they will also be proportional alternately, so that, as A is to C, so will B be to D. 
Sint quattuor numeri proportionales A, B, G, D; sicut A ad B ita G ad D.  Dico quoniam et permutatim proportionales erunt, sicut A ad G ita B ad D. 
   
   
   
Ἐπεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ,  ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ Β ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Δ ἢ τὰ αὐτὰ μέρη.  ἐναλλὰξ ἄρα, ὃ μέρος ἐστὶν ὁ Α τοῦ Γ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Β τοῦ Δ ἢ τὰ αὐτὰ μέρη.  ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Β πρὸς τὸν Δ:  ὅπερ ἔδει δεῖξαι. 
For since, as A is to B, so is C to D,  therefore, whatever part or parts A is of B, the same part or the same parts is C of D also. [VII. Def. 20]  Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. [VII. 10]  Therefore, as A is to C, so is B to D. [VII. Def. 20]  Q. E. D. 
Quoniam enim est sicut A ad B ita G ad D,  que ergo pars est numerus A numeri B vel partes, eadem pars est et numerus G numeri D vel eedem partes.  Permutatim ergo que pars est numerus A numeri G vel partes, eadem pars et numerus B numeri D vel eedem partes.  Est ergo sicut A ad G ita B ad D.  Quod oportet ostendere. 
         
         
         
 
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