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Euclid · Phaenomena (b recension) §1

Times of Zodiac Arcs Traversing the Invisible Hemisphere

Passage 1 of 1 · Greek

Summary

The text geometrically proves the relationship between the lengths of time that equal arcs of the zodiac take to pass through the invisible hemisphere, depending on their distance from the solstices and the equator.

§1## Recensio b. ιϚ ΄.
Recension b 16.
Τοῦ τῶν ζῳδίων κύκλου αἰ ἴσαι περιφέρειαι οὐκ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ ἀφανὲς ἡμισφαίριον, ἀλλʼ ἐν πλείονι χρόνῳ ἀεὶ ἡ ἔγγιον τοῦ χειμερινοῦ τροπικοῦ τῆς ἀπώτερον, ἐν ἴσῳ δὲ αἰ ἴσον ἀπέχουσαι ὁποτεροσοῦν τῶν συναφῶν.
Equal arcs of the zodiacal circle do not pass through the invisible hemisphere in equal time, but that which is closer to the winter solstice always passes through in a longer time than that which is further, and equal arcs equally distant from either of the points of connection pass through in equal time.
ἔστω ἐν κόσμῳ ὁρίζων ὁ ΑΒΓ, καὶ θερινὸς μὲν τροπικὸς ἔστω ὁ ΑΒ, χειμερινὸς δὲ τροπικὸς ἔστω ὁ ΓΤ, ὁ δὲ τῶν ζῳδίων κύκλος θέσιν ἐχέτω τὴν ΑΓΕ, καὶ ἀπειλήφθωσαν ἴσαι περιφέρειαι αἰ ∠Ε, ΕΖ λέγω, ὅτι αἱ ∠Ε, ΕΖ οὐκ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ ἀφανὲς ἡμισφαίριον, ἀλλʼ ἐν πλείονι ἡ ∠Ε περιφέρεια τῆς ΕΖ περιφερείας.
Let the horizon in the cosmos be ΑΒΓ, and let the summer solstice be ΑΒ, and the winter solstice be ΓΤ, and let the zodiacal circle have the position ΑΓΕ, and let equal arcs ΔΕ, ΕΖ be cut off. I say that ΔΕ, ΕΖ do not pass through the invisible hemisphere in equal time, but the arc ΔΕ in a longer time than the arc ΕΖ.
ἀπειλήφθωσαν γὰρ ταῖς ∠Ε, ΕΖ περιφερείαις ἴσαι τε καὶ ἀπεναντίον περιφέρειαι αἱ ΗΘ, ΘΚ·
For let there be cut off arcs ΗΘ, ΘΚ equal and opposite to the arcs ΔΕ, ΕΖ.
αἱ ΗΘ, ΘΚ ἄρα περιφέρειαι οὐκ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον, ἀλλʼ ἐν πλείονι ἡ ΗΘ τῆς ΚΘ ἀλλʼ ἐν ᾧ μὲν χρόνῳ ἡ ΗΘ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ∠Ε τὸ ἀφανές, ἐν ᾧ δὲ ἡ ΘΚ τὸ φανερὸν ἡμισφαίριον ἐξαλλάσσει, ἡ ΕΖ τὸ ἀφανές·
Therefore, the arcs ΗΘ, ΘΚ do not pass through the visible hemisphere in equal time, but ΗΘ in a longer time than ΚΘ. But in whatever time ΗΘ passes through the visible hemisphere, ΔΕ passes through the invisible, and in whatever time ΘΚ passes through the visible hemisphere, ΕΖ passes through the invisible.
αἱ ∠Ε, ΕΖ ἄρα περιφέρειαι οὐκ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ ἀφανὲς ἡμισφαίριον, ἀλλʼ ἐν πλείονι ἡ ∠Ε τῆς ΕΖ. λέγω, ὅτι καὶ ἐν ἴσῳ χρόνῳ αἱ ἴσον ἀπέχουσαι ὁποτεροσοῦν τῶν τροπικῶν συναφῶν ἐξαλλάσσουσι τὸ ἀφανὲς ἡμισφαίριον.
Therefore, the arcs ΔΕ, ΕΖ do not pass through the invisible hemisphere in equal time, but ΔΕ in a longer time than ΕΖ. I say that also equal arcs equally distant from either of the solstitial points of connection pass through the invisible hemisphere in equal time.
ἔστωσαν γὰρ καθʼ ὧν φέρεται τὰ Δ, Ε, Ζ, Κ, Θ, Η σημεῖα παράλληλοι κύκλοι οἱ ∠Ο, ΚΞ, ΖΡ, ΚΝ, ΘΜ, ΗΛ. αἱ ΗΘ, ΛΜ ἄρα περιφέρειαι ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον·
For let the parallel circles through which the points Δ, Ε, Ζ, Κ, Θ, Η are carried be ΔΟ, ΚΞ, ΖΡ, ΚΝ, ΘΜ, ΗΛ. Therefore, the arcs ΗΘ, ΛΜ pass through the visible hemisphere in equal time.
ἀλλʼ ἐν μὲν χρόνῳ ἡ ΗΘ τὸ φανερὸν ἡμισφαίριον ἐξαλλάσσει, ἡ ∠Ε τὸ ἀφανὲς ἐξαλλάσσει·
But in whatever time ΗΘ passes through the visible hemisphere, ΔΕ passes through the invisible hemisphere.
ἔν ᾧ δὲ ἡ ΛΜ τὸ φανερὸν ἡμισφαίριον ἐξαλλάσσει, ἡ ΞΟ τὸ ἀφανὲς ἐξαλλάσσει·
And in whatever time ΛΜ passes through the visible hemisphere, ΞΟ passes through the invisible hemisphere.
αἱ ∠Ε ΟΞ ἄρα περιφέρειαι ἐν ἴσῳ χρόνῳ ἔξαλλάσσουσι τὸ ἀφανὲς ἡμισφαίριον. ιζ΄.
Therefore, the arcs ΔΕ, ΟΞ pass through the invisible hemisphere in equal time. 17.
Τῶν ἐφʼ ἑκάτερα τοῦ ἰσημερινοῦ περιφερειῶν ἴσων τε καὶ ἴσον ἀπεχουσῶν τοῦ ἰσημερινοῦ ἔν ᾦ χρόνῳ ἡ ἑτέρα ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ἑτέρα τὸ ἀφανές, καὶ ἔν ᾧ χρόνῳ ἡ ἑτέρα ἐξαλλάσσει τὸ ἀφανὲς ἡμισφαίριον, ἡ ἑτέρα τὸ φανερόν.
Of arcs on either side of the equator, equal and equally distant from the equator, in whatever time one passes through the visible hemisphere, the other passes through the invisible, and in whatever time one passes through the invisible hemisphere, the other passes through the visible.
ἔστω ἐν κόσμῳ ὁρίζων ὁ ΑΒΓ, ἰσημερινὸς δὲ κύκλος ἔστω ὁ Β∠Γ, ὁ δὲ τῶν ζῳδίων κύκλος θέσιν ἐχέτω ὡς τὴν ΑΕΘ, καὶ τοῦ Β∠Γ ἰσημερινοῦ ἔφʼ ἑκάτερα ἴσαι τε καὶ ἴσον ἀπέχουσαι περιφέρειαι ἔστωσαν αἱ ΘΚ, ΖΗ. λέγω, ὅτι, ἐν ᾧ χρόνῳ ἡ ΘΚ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ΖΗ τὸ ἀφανές.
Let the horizon in the cosmos be ΑΒΓ, and let the equinoctial circle be ΒΔΓ, and let the zodiacal circle have the position ΑΕΘ, and let ΘΚ, ΖΗ be arcs on either side of the equator ΒΔΓ, equal and equally distant from it. I say that, in whatever time ΘΚ passes through the visible hemisphere, ΖΗ passes through the invisible.
κείσθω γὰρ τῇ ΖΗ ἴση τε καὶ ἀπεναντίον περιφέρεια ἡ ΛΜ. αἱ ΜΛ, ΘΚ ἄρα περιφέρειαι ἐν ἴσῳ χρόνῳ ἔξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον·
For let there be placed an arc ΛΜ equal and opposite to ΖΗ. Therefore, the arcs ΜΛ, ΘΚ pass through the visible hemisphere in equal time.
ἀλλʼ ἐν ᾦ χρόνῳ ἡ ΜΛ τὸ φανερὸν ἡμισφαίριον ἐξαλλάσσει, ἡ ΖΗ τὸ ἀφανὲς ἐξαλλάσσει·
But in whatever time ΜΛ passes through the visible hemisphere, ΖΗ passes through the invisible.
καὶ ἐν ᾦ ἄρα χρόνῳ ἡ ΘΚ περιφέρεια ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ΗΖ περιφέρεια τὸ ἀφανὲς ἐξαλλάσσει.
Therefore, in whatever time the arc ΘΚ passes through the visible hemisphere, the arc ΗΖ passes through the invisible.
διὰ τὰ αὐτὰ δὴ καὶ ἐν ᾦ χρόνῳ ἡ ΘΚ περιφέρεια ἐξαλλάσσει τὸ ἀφανὲς ἡμισφαίριον, ἡ ΗΖ τὸ φανερὸν ἐξαλλάσσει.
For the same reasons, indeed, in whatever time the arc ΘΚ passes through the invisible hemisphere, ΗΖ passes through the visible.
ιη΄.
18.
Τῶν δὲ ἐν τῷ ἡμικυκλίῳ τῷ ἀπολαμβανομένῳ ὑπὸ τοῦ ἰσημερινοῦ πρὸς τῷ θερινῷ τροπικῷ ἴσων περιφερειῶν ἐν πλείονι χρόνῳ ἡ ἑτέρα αὐτῶν ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ λοιπὴ τὸ ἀφανὲς καὶ ἡ τυχοῦσα τῆς τυχούσης.
And of equal arcs in the semicircle cut off by the equator toward the summer solstice, one of them passes through the visible hemisphere in a longer time than the remaining one passes through the invisible, and any one than any other.
ἔστω ἐν κόσμῳ ὁρίζων ὁ ΑΒΓ καὶ θερινὸς μὲν τροπικὸς ἔστω ὁ ΑΞ, χειμερινὸς δὲ ὁ #8736; Ο, ἰσημερινὸς δὲ κύκλος ἔστω ὁ ΒΕΓ, ὁ δὲ τῶν ζῳδίων κύκλος θέσιν ἐχέτω τὴν ΑΕΟ, καὶ ἐν τῷ ΤΑΕ ἡμικυκλίῳ ἴσαι περιφέρειαι ἔστωσαν αἰ ΖΗ, ΘΚ, ἔγγιον δὲ ἔστω τοῦ θερινοῦ τροπικοῦ ἡ ΖΗ λέγω, ὅτι ἔν πλείονι χρόνῳ ἡ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ ΘΚ τὸ ἀφανὲς καὶ ἡ τυχοῦσα τῆς τυχούσης.
Let the horizon in the cosmos be ΑΒΓ, and let the summer solstice be ΑΞ, and the winter solstice be ΔΟ, and let the equinoctial circle be ΒΕΓ, and let the zodiacal circle have the position ΑΕΟ, and let there be equal arcs ΖΗ, ΘΚ in the semicircle [Τ]ΑΕ, and let ΖΗ be closer to the summer solstice. I say that [ΖΗ] passes through the visible hemisphere in a longer time than ΘΚ passes through the invisible, and any one than any other.
κείσθω γὰρ τῇ ΘΚ περιφερείᾳ ἴση τε καὶ ἀπεναντίον περιφέρεια ἡ ΜΝ. ἔγγιον ἄρα ἡ Ζ τοῦ θερινοῦ τροπικοῦ ἤπερ ἡ ΜΝ. ἐν πλείονι ἄρα χρόνῳ ἡ ΖΗ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ ΜΝ. τὸ φανερόν.
For let there be placed an arc ΜΝ equal and opposite to the arc ΘΚ. Therefore, Ζ[Η] is closer to the summer solstice than ΜΝ is. Therefore, in a longer time ΖΗ passes through the visible hemisphere than ΜΝ passes through the visible.
ἀλλʼ ἐν ᾧ χρόνῳ ἡ ΜΝ περιφέρεια ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ΘΚ τὸ ἀφανές·
But in whatever time the arc ΜΝ passes through the visible hemisphere, ΘΚ passes through the invisible.
ἐν πλείονι ἄρα χρόνῳ ἡ περιφέρεια ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ ΘΚ τὸ ἀφανές.
Therefore, in a longer time the arc [ΖΗ] passes through the visible hemisphere than ΘΚ passes through the invisible.
ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ τυχοῦσα τῆς τυχούσης ἐν πλείονι χρόνῳ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ λοιπὴ τὸ ἀφανές.
Similarly, indeed, we shall show that any one also passes through the visible hemisphere in a longer time than the remaining one passes through the invisible.
ὡσαύτως δὲ καὶ τῶν ἐν τῷ ἑτέρῳ ἡμικυκλίῳ τῷ ἀπολαμβανομένῳ ὑπὸ τοῦ ἰσημερινοῦ πρὸς τῷ χειμερινῷ τροπικῷ ἴσων περιφερειῶν ἐν πλείονι χρόνῳ ἡ ἑτέρα αὐτῶν ἐξαλλάσσει τὸ ἀφανὲς ἡμισφαίριον ἤπερ ἡ λοιπὴ τὸ φανερὸν καὶ ἡ τυχοῦσα τῆς τυχούσης.
And likewise also of equal arcs in the other semicircle cut off by the equator toward the winter solstice, one of them passes through the invisible hemisphere in a longer time than the remaining one passes through the visible, and any one than any other.

Notes

  1. p.106∠Ε — The symbol `∠` in the Greek text (and `#8736;` on p. 112, etc.) is a manuscript variant or typographical corruption of the Greek letter Delta (Δ). Thus, `∠Ε` and `#8736;Ο` are to be read as geometric designations "ΔΕ" and "ΔΟ".
  2. p.106τῶν συναφῶν — Meaning 'points of connection' or 'junctions', in this astronomical context (referring to the contact points between the zodiac and the solstitial or equinoctial circles), it designates the solstitial points (or equinoctial points) where the circles touch.
  3. p.110ἡ ἐξαλλάσσει — A subject (most likely `ΖΗ` or `ΖΗ περιφέρεια`) is missing after the article `ἡ`. Based on the context and the preceding argument, it is the arc 'ΖΗ' that passes through the visible hemisphere, which has been supplied in the translation. A similar omission occurs on p. 112, l. 3 at `ἡ περιφέρεια` where `ΖΗ` is omitted.
  4. p.110ἡ τυχοῦσα τῆς τυχούσης — Literally 'the chance one than the chance one', this is a formulaic Greek mathematical expression meaning 'any arbitrary one than any other'. It indicates that the proposition holds generally for any pair of arcs satisfying the conditions, not just a specific pair.

Cite this passage

Euclid, Phaenomena (b recension) §1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg013.humanitext-grc1:1

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