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Euclid · Elements §1.post.1-1.post.5

The Five Postulates of Plane Geometry

Passage 2 of 316 · Greek

Summary

Presents the five postulates of Euclidean geometry, defining the constructs of straight lines and circles, the equality of all right angles, and the parallel postulate.

§1.post.1Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν.
Let it be postulated to draw a straight line from any point to any point.
§1.post.2καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπʼ εὐθείας ἐκβαλεῖν.
And to produce a finite straight line continuously in a straight line.
§1.post.3καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι.
And to describe a circle with any centre and distance.
§1.post.4καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι.
And that all right angles are equal to one another.
§1.post.5καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς δύο εὐθείας ἐπʼ ἄπειρον συμπίπτειν, ἐφʼ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες.
And that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Notes

  1. 1.post.1Ἠιτήσθω — The third-person singular perfect passive imperative of the verb αἰτέω, meaning "let it be postulated." All subsequent infinitives in the postulates (ἀγαγεῖν, ἐκβαλεῖν, γράφεσθαι, εἶναι, συμπίπτειν) function as the grammatical subjects of this verb.
  2. 1.post.3παντὶ κέντρῳ καὶ διαστήματι — Dative of means or instrument. The word διάστημα ("interval" or "distance") here refers to the radius of the circle.
  3. 1.post.5ἐφʼ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες — This clause exhibits antecedent attraction, where the nominal antecedent μέρη is drawn into the relative clause (standing for ἐπὶ τὰ μέρη, ἐφʼ οἷς...). In the subject phrase αἱ ... ἐλάσσονες, the feminine plural noun γωνίαι ("angles") is omitted.

Cite this passage

Euclid, Elements §1.post.1-1.post.5. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:1.post.1-1.post.5

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