previous next

Click on a word to bring up parses, dictionary entries, and frequency statistics

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.

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 United States License.

An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.

load focus English (Thomas L. Heath, Sir Thomas Little Heath, 1956)
load Vocabulary Tool
hide Display Preferences
Greek Display:
Arabic Display:
View by Default:
Browse Bar: