# De utilitate mathematicae 19

Theon Smyrnaeus, De utilitate mathematicae, Paragraph 19. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/theon-smyrnaeus/de-utilitate-mathematicae/19

ἀρτιοπέριττοι δέ εἰσιν οἱ ὑπὸ δυάδος καὶ περιττοῦ οὑτινοσοῦν μετρούμενοι, οἵτινες ἐκ παντὸς περιττὰ μέρη ἔχουσι τὰ ἡμίσεα κατὰ τὴν εἰς ἴσα διαίρεσιν· ὡς τὰ δὶς ζ΄ ιδ΄. ἀρτιάκις μὲν γὰρ οὗτοι καλοῦνται περιττοί, ἐπεὶ ὑπὸ τῆς δυάδος ἀρτίας οὔσης μετροῦνται καὶ περισσοῦ τινος, ὁ μὲν δύο τοῦ ἑνός, ὁ δὲ ϛʹ τοῦ γʹ, ὁ δὲ ιʹ τοῦ ε΄, ὁ δὲ ιδʹ τοῦ ζ΄. διαιροῦνται δὲ οὗτοι τὴν πρώτην διαίρεσιν εἰς περιττόν, μετὰ δὲ τὴν πρώτην εἰς ἴσα διαίρεσιν οὐκ ἔτι διαιροῦνται. τῶν γὰρ ϛ΄ τὰ μὲν γ΄ ἥμισυ, τὰ δὲ γ΄ οὐκ ἔτι εἰς ἴσα διαιρεῖται· μονὰς γὰρ ἀδιαίρετος.

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Theon Smyrnaeus, De utilitate mathematicae, ed. Eduardus Hiller. Text: The Greek Library, from the Perseus Digital Library and other open sources, CC BY-SA 4.0. Changes: apostrophes normalised and the text divided into reading sections by Interlinea.
License: CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Source: https://thegreeklibrary.pages.lisn.upsaclay.fr/works/tlg1724.tlg001.local001/.