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Euclid: Elementa

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κβ῾ 
PROPOSITION 22. 
22 
 
 
 
Ἐὰν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ πρῶτος τετράγωνος ᾖ, καὶ ὁ τρίτος τετράγωνος ἔσται. 
If three numbers be in continued proportion, and the first be square, the third will also be square. 
Si tres numeri deinceps proportionales fuerint, primus autem tetragonus erit, et tertius tetragonus erit. 
 
 
 
Ἔστωσαν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, ὁ δὲ πρῶτος ὁ Α τετράγωνος ἔστω:  λέγω, ὅτι καὶ ὁ τρίτος ὁ Γ τετράγωνός ἐστιν. 
Let A, B, C be three numbers in continued proportion, and let A the first be square;  I say that C the third is also square. 
Sint tres numeri deinceps proportionales A, B, G. Primus autem A tetragonus esto.  Dico quoniam et tertius G tetragonus est. 
   
   
   
Ἐπεὶ γὰρ τῶν Α, Γ εἷς μέσος ἀνάλογόν ἐστιν ἀριθμὸς ὁ Β,  οἱ Α, Γ ἄρα ὅμοιοι ἐπίπεδοί εἰσιν.  τετράγωνος δὲ ὁ Α:  τετράγωνος ἄρα καὶ ὁ Γ:  ὅπερ ἔδει δεῖξαι. 
For, since between A, C there is one mean proportional number, B,  therefore A, C are similar plane numbers. [VIII. 20]  But A is square;  therefore C is also square.  Q. E. D. 
Quoniam enim numerorum A, G unus medius proportionalis est numerus B,  numeri ergo A, G similes epipedi sunt.  Est autem tetragonus A,  tetragonus ergo et G.  Quod oportet ostendere. 
         
         
         
 
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