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ἐὰν ἀριθμὸς μήτε τῶν ἀπὸ δυάδος διπλασιαζομένων μήτε τὸν ἥμισυν ἔχῃ περισσόν, ἀρτιάκις τε ἄρτιός ἐστι καὶ ἀρτιάκις περισσός.

ἀριθμὸς γὰρ Α μήτε τῶν ἀπὸ δυάδος διπλασιαζομένων
5ἔστω μήτε τὸν ἥμισυν ἐχέτω περισσόν: λέγω, ὅτι Α ἀρτιάκις τέ ἐστιν ἄρτιος καὶ ἀρτιάκις περισσός.

ὅτι μὲν οὖν Α ἀρτιάκις ἐστὶν ἄρτιος, φανερόν: τὸν γὰρ ἥμισυν οὐκ ἔχει περισσόν. λέγω δή, ὅτι καὶ ἀρτιάκις περισσός ἐστιν. ἐὰν γὰρ τὸν Α τέμνωμεν δίχα καὶ τὸν
10ἥμισυν αὐτοῦ δίχα καὶ τοῦτο ἀεὶ ποιῶμεν, καταντήσομεν εἴς τινα ἀριθμὸν περισσόν, ὃς μετρήσει τὸν Α κατὰ ἄρτιον ἀριθμόν. εἰ γὰρ οὔ, καταντήσομεν εἰς δυάδα, καὶ ἔσται Α τῶν ἀπὸ δυάδος διπλασιαζομένων: ὅπερ οὐχ ὑπόκειται. ὥστε Α ἀρτιάκις περισσός ἐστιν. ἐδείχθη δὲ
15καὶ ἀρτιάκις ἄρτιος. Α ἄρα ἀρτιάκις τε ἄρτιός ἐστι καὶ ἀρτιάκις περισσός: ὅπερ ἔδει δεῖξαι.

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load focus English (Thomas L. Heath, Sir Thomas Little Heath, 1956)
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