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Euclid · Elements §1.comm_not.1-1.comm_not.9

Common Notions on Quantities and Figures

Passage 3 of 316 · Greek

Summary

Presents five common notions (axioms) of Euclidean geometry (notions 4-6 are blank). They establish rules for the equality of quantities under addition and subtraction, the coincidence of figures, the relation between whole and part, and a property of straight lines.

§1.comm_not.1τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα.
Things which are equal to the same thing are also equal to one another.
§1.comm_not.2καὶ ἐὰν ἴσοις ἴσα προστεθῇ, τὰ ὅλα ἐστὶν ἴσα.
And if equals be added to equals, the wholes are equal.
§1.comm_not.3καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ, τὰ καταλειπόμενά ἐστιν ἴσα.
And if equals be subtracted from equals, the remainders are equal.
§1.comm_not.7καὶ τὰ ἐφαρμόζοντα ἐπʼ ἄλληλα ἴσα ἀλλήλοις ἐστίν.
And things which coincide with one another are equal to one another.
§1.comm_not.8καὶ τὸ ὅλον τοῦ μέρους μεῖζον.
And the whole is greater than the part.
§1.comm_not.9καὶ δύο εὐθεῖαι χωρίον οὐ περιέχουσιν.
And two straight lines do not enclose a space.

Notes

  1. §1.comm_not.2προστεθῇ — The verb προστεθῇ (3rd person singular aorist passive subjunctive of προστίθημι) agrees with the neuter plural subject ἴσα. This is a standard grammatical rule in Classical Greek (known as schema Atticum), where a neuter plural subject takes a singular verb.
  2. §1.comm_not.8τοῦ μέρους — This is a genitive of comparison dependent on the comparative adjective μεῖζον (neuter singular comparative of μέγας), meaning 'than the part'. The copula verb ἐστίν is omitted.

Cite this passage

Euclid, Elements §1.comm_not.1-1.comm_not.9. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:1.comm_not.1-1.comm_not.9

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