previous next

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

ἐὰν μονὰς ἀριθμόν τινα μετρῇ, ἰσάκις δὲ ἕτερος ἀριθμὸς ἄλλον τινὰ ἀριθμὸν μετρῇ, καὶ ἐναλλὰξ ἰσάκις μονὰς τὸν τρίτον ἀριθμὸν μετρήσει καὶ δεύτερος τὸν τέταρτον.

μονὰς γὰρ Α ἀριθμόν τινα τὸν ΒΓ μετρείτω, ἰσάκις δὲ ἕτερος ἀριθμὸς Δ ἄλλον τινὰ ἀριθμὸν τὸν ΕΖ μετρείτω: λέγω, ὅτι καὶ ἐναλλὰξ ἰσάκις Α μονὰς τὸν Δ ἀριθμὸν μετρεῖ καὶ ΒΓ τὸν ΕΖ.

ἐπεὶ γὰρ ἰσάκις Α μονὰς τὸν ΒΓ ἀριθμὸν μετρεῖ
10καὶ Δ τὸν ΕΖ, ὅσαι ἄρα εἰσὶν ἐν τῷ ΒΓ μονάδες, τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἀριθμοὶ ἴσοι τῷ Δ. διῃρήσθω μὲν ΒΓ εἰς τὰς ἐν ἑαυτῷ μονάδας τὰς ΒΗ, ΗΘ, ΘΓ, δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους τοὺς ΕΚ, ΚΛ, ΛΖ. ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΘ, ΘΓ τῷ πλήθει τῶν
15ΕΚ, ΚΛ, ΛΖ. καὶ ἐπεὶ ἴσαι εἰσὶν αἱ ΒΗ, ΗΘ, ΘΓ μονάδες ἀλλήλαις, εἰσὶ δὲ καὶ οἱ ΕΚ, ΚΛ, ΛΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΘ, ΘΓ μονάδων τῷ πλήθει τῶν ΕΚ, ΚΛ, ΛΖ ἀριθμῶν, ἔσται ἄρα ὡς ΒΗ μονὰς πρὸς τὸν ΕΚ ἀριθμόν, οὕτως
20 ΗΘ μονὰς πρὸς τὸν ΚΛ ἀριθμὸν καὶ ΘΓ μονὰς πρὸς τὸν ΛΖ ἀριθμόν. ἔσται ἄρα καὶ ὡς εἷς τῶν ἡγουμένων πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες οἱ ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους: ἔστιν ἄρα ὡς ΒΗ μονὰς πρὸς τὸν ΕΚ ἀριθμόν, οὕτως ΒΓ πρὸς τὸν ΕΖ. ἴση δὲ
25 ΒΗ μονὰς τῇ Α μονάδι, δὲ ΕΚ ἀριθμὸς τῷ Δ ἀριθμῷ. ἔστιν ἄρα ὡς Α μονὰς πρὸς τὸν Δ ἀριθμόν, οὕτως ΒΓ πρὸς τὸν ΕΖ. ἰσάκις ἄρα Α μονὰς τὸν Δ ἀριθμὸν μετρεῖ καὶ ΒΓ τὸν ΕΖ: 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: