Euclid · Elements §3.prop.15

Diameter as the Greatest Chord and Chords by Distance

Passage 44 of 316 · Greek

Summary

This proposition proves that in a circle, the diameter is the longest straight line, and of the other straight lines, those closer to the center are longer than those further away.

§3.prop.15ἐν κύκλῳ μεγίστη μὲν ἡ διάμετρος τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν.

In a circle the greatest is the diameter, and of the others, that which is nearer to the center is always greater than that which is further.

ἔστω κύκλος ὁ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἔστω ἡ ΑΔ, κέντρον δὲ τὸ Ε, καὶ ἔγγιον μὲν τῆς ΑΔ διαμέτρου ἔστω ἡ ΒΓ, ἀπώτερον δὲ ἡ ΖΗ· λέγω, ὅτι μεγίστη μέν ἐστιν ἡ ΑΔ, μείζων δὲ ἡ ΒΓ τῆς ΖΗ. ἤχθωσαν γὰρ ἀπὸ τοῦ Ε κέντρου ἐπὶ τὰς ΒΓ, ΖΗ κάθετοι αἱ ΕΘ, ΕΚ. καὶ ἐπεὶ ἔγγιον μὲν τοῦ κέντρου ἐστὶν ἡ ΒΓ, ἀπώτερον δὲ ἡ ΖΗ, μείζων ἄρα ἡ ΕΚ τῆς ΕΘ. κείσθω τῇ ΕΘ ἴση ἡ ΕΛ, καὶ διὰ τοῦ Λ τῇ ΕΚ πρὸς ὀρθὰς ἀχθεῖσα ἡ ΛΜ διήχθω ἐπὶ τὸ Ν, καὶ ἐπεζεύχθωσαν αἱ ΜΕ, ΕΝ, ΖΕ, ΕΗ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΕΘ τῇ ΕΛ, ἴση ἐστὶ καὶ ἡ ΒΓ τῇ ΜΝ. πάλιν, ἐπεὶ ἴση ἐστὶν ἡ μὲν ΑΕ τῇ ΕΜ, ἡ δὲ ΕΔ τῇ ΕΝ, ἡ ἄρα ΑΔ ταῖς ΜΕ, ΕΝ ἴση ἐστίν.

Let ΑΒΓΔ be a circle, and let its diameter be ΑΔ, and the center Ε, and let ΒΓ be nearer to the diameter ΑΔ, and ΖΗ further; I say that ΑΔ is the greatest, and ΒΓ is greater than ΖΗ. For let perpendiculars ΕΘ, ΕΚ be drawn from the center Ε to ΒΓ, ΖΗ. And since ΒΓ is nearer to the center, and ΖΗ further, ΕΚ is therefore greater than ΕΘ. Let ΕΛ be made equal to ΕΘ, and through Λ let ΛΜ be drawn at right angles to ΕΚ, and let it be produced to Ν, and let ΜΕ, ΕΝ, ΖΕ, ΕΗ be joined. And since ΕΘ is equal to ΕΛ, ΒΓ is also equal to ΜΝ. Again, since ΑΕ is equal to ΕΜ, and ΕΔ to ΕΝ, ΑΔ is therefore equal to ΜΕ, ΕΝ.

ἀλλʼ αἱ μὲν ΜΕ, ΕΝ τῆς ΜΝ μείζονές εἰσιν [καὶ ἡ ΑΔ τῆς ΜΝ μείζων ἐστίν, ἴση δὲ ἡ ΜΝ τῇ ΒΓ· ἡ ΑΔ ἄρα τῆς ΒΓ μείζων ἐστίν.

But ΜΕ, ΕΝ are greater than ΜΝ [and ΑΔ is greater than ΜΝ, and ΜΝ is equal to ΒΓ; therefore ΑΔ is greater than ΒΓ.

καὶ ἐπεὶ δύο αἱ ΜΕ, ΕΝ δύο ταῖς ΖΕ, ΕΗ ἴσαι εἰσίν, καὶ γωνία ἡ ὑπὸ ΜΕΝ γωνίας τῆς ὑπὸ ΖΕΗ μείζων, βάσις ἄρα ἡ ΜΝ βάσεως τῆς ΖΗ μείζων ἐστίν.

And since the two ΜΕ, ΕΝ are equal to the two ΖΕ, ΕΗ, and the angle ΜΕΝ is greater than the angle ΖΕΗ, the base ΜΝ is therefore greater than the base ΖΗ.

ἀλλὰ ἡ ΜΝ τῇ ΒΓ ἐδείχθη ἴση. μεγίστη μὲν ἄρα ἡ ΑΔ διάμετρος, μείζων δὲ ἡ ΒΓ τῆς ΖΗ. ἐν κύκλῳ ἄρα μεγίστη μέν ἐστιν ἡ διάμετρος, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν· ὅπερ ἔδει δεῖξαι.

But ΜΝ was proved equal to ΒΓ; therefore the diameter ΑΔ is the greatest, and ΒΓ is greater than ΖΗ]. Therefore in a circle the greatest is the diameter, and of the others, that which is nearer to the center is always greater than that which is further; which was to be proved.

Notes

3.prop.15τῆς ἀπώτερον — A genitive of comparison governed by the comparative `μείζων` (greater), meaning "than the one further away." The article `τῆς` substantivizes the adverb `ἀπώτερον`, agreeing with the feminine noun `εὐθεῖα` (straight line).
3.prop.15τῆς ΑΔ διαμέτρου — A genitive of relative position governed by the comparative adverb `ἔγγιον` (nearer), meaning "nearer to the diameter ΑΔ."
3.prop.15ταῖς ΜΕ, ΕΝ — A dative plural dependent on the adjective `ἴση` (equal). Although ΑΔ is a single line (the sum of ΑΕ and ΕΔ), its equality to the combined sum of the two lines ΜΕ and ΕΝ is expressed by the plural dative `ταῖς ΜΕ, ΕΝ`.

Cite this passage

Euclid, Elements §3.prop.15. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:3.prop.15

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