Euclid · Phaenomena (alternative proof of b recension) §3#2

Setting Times of Equal Arcs in the Semicircle after Capricorn

Passage 5 of 6 · Greek

Summary

It is proved that even in the semicircle after Capricorn, equal arcs do not leave the visible hemisphere in equal times, but the one closer to the summer tropic takes longer, using parallel circles and the addition of common time.

§3#2ὡσαύτως δὲ καὶ ἐπὶ τοῦ μετὰ τὸν Αἰγόκερων ἡμικυκλίου αἱ ἴσαι περιφέρειαι οὐκ ἐν ἴσοις χρόνοις ἐξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον, ἀλλʼ ἐν πλείονι ἡ ἔγγιον τοῦ θερινοῦ τροπικοῦ τῆς ἀπώτερον, ἐν ἴσῳ δὲ αἱ ἴσον ἀπέχουσαι ὁποτεροσοῦν τῶν συναφῶν.

And likewise also in the case of the semicircle after Capricorn, equal arcs do not leave the visible hemisphere in equal times, but the one closer to the summer tropic [leaves it] in a longer [time] than the one further away, and in an equal [time] those equally distant from either of the conjunctions [leave it].

ἔστω ἐν κόσμῳ ὁρίζων ὁ ΑΒΓ∠, καὶ θερινὸς μὲν τροπικὸς ὁ Α∠, ὁ δὲ τῶν ζῳδίων κύκλος θέσιν ἐχέτω τὴν ΒΕΓ, καὶ ἔστω ἡ μὲν ΒΕ περιφέρεια ἐπὶ τοῦ μετὰ τὸν Αἰγόκερων ἡμικυκλίου, ἡ δὲ ΕΓ ἐπὶ τοῦ μετὰ τὸν Καρκίνον, καὶ ἔστωσαν ἀνατολικὰ μὲν τὰ ∠ μέρη, δυτικὰ δὲ τὰ Β, καὶ ἀπειλήφθωσαν ἴσαι περιφέρειαι αἱ ΖΗ, ΗΘ. λέγω, ὅτι ἡ ΖΗ ἐν πλείονι χρόνῳ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ ΗΘ. γεγράφθωσαν παράλληλοι κύκλοι οἱ ΚΛ, ΜΝ, ΞΟ, καθʼ ὧν φέρεται τὰ Ζ, Η, Θ σημεῖα· ἴση ἄρα ἐστὶν ἡ ΖΗ περιφέρεια τῇ ΠΡ περιφερείᾳ, ἡ δὲ ΗΘ τῇ ΡΣ ἀλλ᾿ ἡ ΖΗ τῇ ΗΘ ἔστιν ἴση· καὶ ἡ Π ἄρα τῇ ΡΣ ἐστιν ἴση.

Let the horizon in the cosmos be ΑΒΓ∠, and let the summer tropic be Α∠, and let the zodiacal circle have the position ΒΕΓ, and let the arc ΒΕ be on the semicircle after Capricorn, and [the arc] ΕΓ on the [semicircle] after Cancer, and let the eastern parts be ∠, and the western parts be Β, and let equal arcs ΖΗ, ΗΘ be cut off. I say that ΖΗ leaves the visible hemisphere in a longer time than ΗΘ. Let parallel circles ΚΛ, ΜΝ, ΞΟ be drawn, on which the points Ζ, Η, Θ are carried; therefore, the arc ΖΗ is equal to the arc ΠΡ, and [the arc] ΗΘ [is equal] to [the arc] ΡΣ. But ΖΗ is equal to ΗΘ; therefore, [the arc] Π also is equal to ΡΣ.

καὶ ἐπεὶ ἐν ᾧ χρόνῳ ἡ ΠΡ δύνει, ἡ ΖΗ ἀνατέλλει, κοινὸς προσκείσθω ὁ χρόνος, ἐν ᾧ τὸ Π σημεῖον τὴν ΚΛ περιφέρειαν διαπορεύεται, ἴσος ὢν τῷ χρόνῳ, ἐν ᾧ τὸ Ζ σημεῖον τὴν ΚΛ περιφέρειαν διαπορεύεται.

And since in the time in which ΠΡ sets, ΖΗ rises, let the common time be added in which the point Π traverses the arc ΚΛ, being equal to the time in which the point Ζ traverses the arc ΚΛ.

ὁ ἄρα χρόνος, ἐν ᾧ τὸ Π σημεῖον τὴν ΚΛ περιφέρειαν διαπορεύεται καὶ ἡ ΠΡ δύνει, ἴσος ἐστὶ τῷ χρόνῳ, ἐν ᾧ ἡ ΖΗ περιφέρεια ἀνατέλλει καὶ τὸ Ζ σημεῖον τὴν ΚΛ περιφέρειαν διαπορεύεται.

Therefore, the time in which the point Π traverses the arc ΚΛ and ΠΡ sets is equal to the time in which the arc ΖΗ rises and the point Ζ traverses the arc ΚΛ.

ἀλλʼ ὁ μὲν χρόνος, ἐν ᾧ τὸ Π σημεῖον τὴν ΚΛ περιφέρειαν διαπορεύεται καὶ ἡ Π Ρ δύνει, ὁ χρόνος ἐστίν, ἐν ᾧ ἡ ΠΡ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον· ὁ δὲ χρόνος, ἐν ᾧ ἡ ΖΗ ἀνατέλλει καὶ τὸ Ζ σημεῖον τὴν ΚΛ περιφέρειαν διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ ἡ ΖΗ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον· αἱ ΖΗ, ΠΡ ἄρα ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον.

But the time in which the point Π traverses the arc ΚΛ and ΠΡ sets is the time in which ΠΡ leaves the visible hemisphere; and the time in which ΖΗ rises and the point Ζ traverses the arc ΚΛ is the time in which ΖΗ leaves the visible hemisphere; therefore, ΖΗ, ΠΡ leave the visible hemisphere in equal time.

ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ΗΘ, ΡΣ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον.

Indeed, we shall show similarly that ΗΘ, ΡΣ also leave the visible hemisphere in equal time.

ἡ δὲ ΠP τῆς ΡΣ ἐν πλείονι χρόνῳ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον, καὶ ἀπεδείχθησαν αἱ ΖΗ, Π Ρ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσαι τὸ φανερὸν ἡμισφαίριον·

But ΠΡ leaves the visible hemisphere in a longer time than ΡΣ, and ΖΗ, ΠΡ were proved to leave the visible hemisphere in equal time; therefore ΖΗ also leaves the visible hemisphere in a longer time than ΗΘ.

καὶ ἡ ΖΗ ἄρα ἐν πλείονι χρόνῳ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ ΗΘ. τοῦ ἄρα τῶν ζῳδίων κύκλου αἱ ἴσαι περιφέρειαι οὐκ ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσι τὸ φανερὸν ἡμισφαίριον, ἀλλ᾿ ἐν πλείονι ἡ ἔγγιον τοῦ θερινοῦ τροπικοῦ τῆς ἀπώτερον.

Therefore, equal arcs of the zodiacal circle do not leave the visible hemisphere in equal time, but the one closer to the summer tropic [leaves it] in a longer [time] than the one further away.

καὶ συναποδέδεικται, ὅτι αἰ ἴσον ἀπέχουσαι ἐν ἴσῳ χρόνῳ ἐξαλλάσσουσιν.

And it has been proved along with this, that those equally distant leave [it] in equal time.

Notes

15τῶν συναφῶν — "of the conjunctions" or "of the intersections". It refers to the points where the zodiac intersects or touches the tropics or the horizon.
p.128ἡ Π — A scribal error or an abbreviation for "the arc ΠΡ". Based on the context and geometric logic, it clearly refers to the arc ΠΡ rather than just the point Π.
5ἴσος ὢν τῷ χρόνῳ — An agreeing participle (nominative, masculine singular) modifying the subject "the time" (ὁ χρόνος) in the main clause. It indicates a logical operation based on Euclidean common notions, where a common time is introduced and added to both sides.
25συναποδέδεικται — A compound verb with double prefixes meaning "has been proved together [with this]", used in an impersonal perfect passive construction. It indicates that through the proof of the main proposition, the secondary claim (that equally distant arcs leave in equal time) is also simultaneously established due to symmetry.

Cite this passage

Euclid, Phaenomena (alternative proof of b recension) §3#2. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg014.humanitext-grc1:3%232

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