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Euclid · Phaenomena (alternative proof of b recension) §1

Proof on the Rising and Setting of Opposite Points

Passage 1 of 6 · Greek

Summary

Using geometric methods and proof by contradiction, it is proved that for two diametrically opposite points on the circle of the zodiac, when one point rises, the other sets.

§1## 1.
## 1.
Ad prop.
Ad prop.
VI. Ἁλίως τὸ αὐτό.
VI. Ἁλίως τὸ αὐτό.
Ἔστω ὁ ὁρίζων κύκλος ὁ ΑΒΓ∠ καὶ θερινὸς μὲν τροπικὸς ἔστω ὁ Α∠, χειμερινὸς δὲ ὁ ΒΓ, ὁ δὲ τῶν ζῳδίων κύκλος θέσιν ἐχέτω ὡς τὴν ∠ΗΒΖ, καὶ ἔστω ἐπὶ τοῦ ∠ΗΒΖ κατὰ διάμετρον τὰ Ζ, Η σημεῖα·
Let the circle of the horizon be ΑΒΓ∠, and let the summer tropic be Α∠, and the winter tropic be ΒΓ, and let the circle of the zodiac have the position ∠ΗΒΖ, and let the points Ζ and Η be diametrically opposite on ∠ΗΒΖ.
λέγω, ὅτι τοῦ Ζ ἀνατέλλοντος τὸ Η δύνει.
I say that when Ζ rises, Η sets.
εἰ γὰρ δυνατόν, μὴ δυνέτω, ἀλλʼ ἔστω τὸ Θ δῦνον, καὶ διὰ τῶν Ζ, Θ παράλληλοι κύκλοι γεγράφθωσαν οἱ ΝΘ, ΖΚ. ὥστε τοῦ Ζ ἀνατέλλοντος κατὰ τὸ Κ σημεῖον τὸ Θ δύσεται κατὰ τὸ Ν καὶ ὁ τῶν ζῳδίων κύκλος θέσιν ἔξει τὴν ΜΝΛΚ. καὶ ἐπεὶ ἑκάτερος τῶν ΑΒΓ∠, ΜΝΛΚ. κύκλων μέγιστός ἐστιν, κατὰ διάμετρόν ἐστι τὸ Κ τῷ Ν ἀλλὰ τὸ μὲν Κ τῷ Ζ ἔστι τὸ αὐτό, τὸ δὲ Ν τῷ Θ· καὶ τὸ Ζ ἄρα τῷ Θ ἔστι κατὰ διάμετρον·
For, if possible, let it not set, but let Θ be the setting point, and let parallel circles ΝΘ and ΖΚ be drawn through Ζ and Θ. Thus, when Ζ rises at point Κ, Θ will set at Ν, and the circle of the zodiac will have the position ΜΝΛΚ. And since each of the circles ΑΒΓ∠ and ΜΝΛΚ is a great circle, Κ is diametrically opposite to Ν. But Κ is the same as Ζ, and Ν is the same as Θ; therefore Ζ also is diametrically opposite to Θ.
ἀλλὰ καὶ τῷ ὅπερ ἀδύνατον.
But it is also [diametrically opposite] to [Η], which is impossible.
οὐκ ἄρα τοῦ Ζ ἀνατέλλοντος τὸ Η οὐ δύνει· δύνει ἄρα.
Therefore, it is not the case that when Ζ rises, Η does not set; therefore it sets.

Notes

  1. §1τοῦ Ζ ἀνατέλλοντος — A genitive absolute construction ("while Ζ is rising"), expressing a temporal condition. The later occurrence "τοῦ Ζ ἀνατέλλοντος τὸ Η οὐ δύνει" shares the same structure.
  2. §1εἰ γὰρ δυνατόν, μὴ δυνέτω — A formulaic expression introducing a proof by contradiction (reductio ad absurdum). The negative particle "μή" is used because it governs the imperative "δυνέτω" ("let it not set").
  3. §1ἀλλὰ καὶ τῷ ὅπερ ἀδύνατον — The dative article "τῷ" is likely missing its head noun, presumably the point "Η" (or an infinitive phrase like "κατὰ διάμετρον εἶναι"), either due to ellipsis or a scribal omission. Understood as "to [Η]" (i.e., [Ζ is also diametrically opposite] to [Η]), which is required by the geometrical logic.

Cite this passage

Euclid, Phaenomena (alternative proof of b recension) §1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg014.humanitext-grc1:1

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