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.