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ἐὰν ἀριθμὸς τὸν ἥμισυν ἔχῃ περισσόν, ἀρτιάκις περισσός ἐστι μόνον. ἀριθμὸς γὰρ ὁ Α τὸν ἥμισυν ἐχέτω περισσόν: λέγω, ὅτι ὁ Α ἀρτιάκις περισσός ἐστι μόνον.
5ὅτι μὲν οὖν ἀρτιάκις περισσός ἐστιν, φανερόν: ὁ γὰρ ἥμισυς αὐτοῦ περισσὸς ὢν μετρεῖ αὐτὸν ἀρτιάκις. λέγω δή, ὅτι καὶ μόνον. εἰ γὰρ ἔσται ὁ Α καὶ ἀρτιάκις ἄρτιος, μετρηθήσεται ὑπὸ ἀρτίου κατὰ ἄρτιον ἀριθμόν: ὥστε καὶ ὁ ἥμισυς αὐτοῦ μετρηθήσεται ὑπὸ ἀρτίου ἀριθμοῦ
10περισσὸς ὤν: ὅπερ ἐστὶν ἄτοπον. ὁ Α ἄρα ἀρτιάκις περισσός ἐστι μόνον: ὅπερ ἔδει δεῖξαι.
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