§2#1## 2.
## 2.
Ad prop.
Ad prop.
XII. Ἄλλως τὸ ιβ΄.
XII. Ἄλλως τὸ ιβ΄.
Αὕτη δέ ἐστιν ἡ σαφεστέρα ἔκθεσις.
And this is the clearer exposition.
ἔστω ἐν κόσμῳ ὁρίζων ὁ ΑΒΓ∠, καὶ θερινὸς μὲν τροπικὸς ἔστω ὁ Α∠, χειμερινὸς δὲ τροπικὸς ὁ ΒΓ, ζῳδιακοῦ δὲ ἡμικύκλιον τὸ μετὰ τὸν Καρκίνον ἔστω ὑπὲρ γῆν τὸ ΑΓ καὶ ἔστω ἀνατολικὰ μὲν μέρη τὰ ∠, Γ, δυτικὰ δὲ τὰ Α, Β, ἰσημερινὸς δὲ κύκλος ἔστω ὁ ΕΖ, καὶ διῃρήσθω τὸ ΑΓ ἡμικύκλιον εἰς τὰ ἐν αὐτῷ ζῴδια κατὰ τὰ Η, Θ, Λ, Μ σημεῖα, καὶ γεγράφθωσαν παράλληλοι κύκλοι οἱ ΝΗΞ, ΟΘΠ, ΡΛΣ, ΤΜΥ, καθʼ ὧν φέρεται τὰ Η, Θ, Λ, Μ σημεῖα·
Let the horizon in the universe be ΑΒΓ∠, and let the summer tropic be Α∠, and the winter tropic be ΒΓ, and let the semicircle of the zodiac after Cancer which is above the earth be ΑΓ, and let the eastern parts be ∠, Γ, and the western parts Α, Β, and let the equator circle be ΕΖ, and let the semicircle ΑΓ be divided at points Η, Θ, Λ, Μ according to the zodiac signs on it, and let the parallel circles ΝΗΞ, ΟΘΠ, ΡΛΣ, ΤＭΥ, along which the points Η, Θ, Λ, Μ are carried, be drawn.
λέγω, ὅτι ἐν πλείστῳ μὲν χρόνῳ δύνουσιν αἰ ΑΗ, ΜΓ περιφέρειαι, ἐν ἐλάσσονι δὲ αἰ ΗΘ, ΛΜ, ἐν ἐλαχίστῳ δὲ αἰ ΘΚ, ΛΚ, ἐν ἴσῳ δὲ χρόνῳ αἱ ἴσον ἀπέχουσαι τοῦ ἰσημερινοῦ.
I say that the arcs ΑΗ, ΜΓ set in the longest time, and [the arcs] ΗΘ, ΛＭ in a shorter [time], and [the arcs] ΘΚ, ΛΚ in the shortest [time], and those equidistant from the equator in an equal time.
ἔστω μέγιστος τῶν ἀεὶ φανερῶν ὁ τυ Φ, καὶ γεγράφθωσαν διὰ τῶν Η, Θ μέγιστοι κύκλοι οἱ υ ΗΩΧ, ΦΘΨ ἐφαπτόμενοι τοῦ τυ Φ κύκλου, ὥστε ἀσύμπτωτα εἶναι τὰ ἀπὸ τῶν υ, Φ ἡμικύκλια ὡς ἐπὶ τὰ ΗΧ, ΘΨ μέρη τῷ ἀπὸ τοῦ τ ἡμικυκλίῳ ὡς ἐπὶ τὰ τ, Α μέρη·
Let the circle of the always-visible [stars] (the greatest of those always visible) be τυ Φ, and let great circles υ ΗΩΧ, ΦΘΨ be drawn through Η, Θ, tangent to the circle τυ Φ, so that the semicircles from υ, Φ towards the parts ΗΧ, ΘΨ are non-intersecting with the semicircle from τ towards the parts τ, Α.
ὁμοία ἄρα ἐστὶν ἡ μὲν ΗΝ περιφέρεια ἑκατέρᾳ τῶν ΩΟ, ΧΕ, ἡ δὲ ΘΩ τῇ ΨΧ·
Therefore, the arc ΗΝ is similar to each of ΩΟ, ΧΕ, and ΘΩ to ΨΧ.
ἐν ἴσῳ ἄρα χρόνῳ τὸ Η τὴν ΗΝ περιφέρειαν διαπορεύεται καὶ τὸ Ω τὴν ΩΟ. ἀλλʼ ὁ χρόνος, ἐν ᾧ τὸ τὴν ΗΝ περιφέρειαν διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ ΗΑ περιφέρεια·
Therefore, in an equal time, [the point] Η traverses the arc ΗΝ, and [the point] Ω [traverses] ΩΟ.
καὶ ὁ χρόνος ἄρα, ἐν ᾧ τὸ Ω τὴν ΩΟ διαπορεύεται, ὁ αὐτός ἐστι τῷ χρόνῳ, ἐν ᾧ δύνει ἡ ΗΑ περιφέρεια.
But the time in which [the point] traverses the arc ΗΝ is the time in which the arc ΗΑ sets; therefore, the time in which Ω traverses ΩΟ is the same as the time in which the arc ΗΑ sets.
πάλιν ἐπεὶ ὁ χρόνος, ἐν ᾧ τὸ Θ τὴν ΘΟ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ ΘΑ περιφέρεια, ὧν ἀφαιρεῖται ὁ χρόνος, ἐν ᾧ τὸ Ω τὴν ΩΟ διαπορεύεται, ὁ αὐτὸς ὢν τῷ χρόνῳ, ἐν ᾧ δύνει ἡ ΗΑ περιφέρεια, λοιπὸς ἄρα ὁ χρόνος, ἐν τὸ Θ τὴν ΘΩ διαπορεύεται, ὁ αὐτός ἐστι τῷ χρόνῳ, ἐν ᾧ δύνει ἡ ΗΘ περιφέρεια·
Again, since the time in which Θ traverses ΘΟ is the same as the time in which the arc ΘΑ sets, from which [times] there is subtracted the time in which Ω traverses ΩΟ, which is the same as the time in which the arc ΗΑ sets, therefore the remaining time, in which Θ traverses ΘΩ, is the same as the time in which the arc ΗΘ sets.
ὁμοία δέ ἐστιν ἡ μὲν ΟΩ τῇ ΕΧ, ἡ δὲ ΩΘ τῇ ΧΨ· καὶ ὁ χρόνος ἄρα, ἐν ᾧ τὸ Χ τὴν ΧΕ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ ΗΑ περιφέρεια· ὁ δὲ χρόνος, ἐν ᾧ τὸ Ψ τὴν ΨΧ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ ΘΗ περιφέρεια.
And the arc ΟΩ is similar to ΕΧ, and ΩΘ to ΧΨ; therefore also the time in which Χ traverses ΧΕ is the time in which the arc ΗΑ sets; and the time in which Ψ traverses ΨΧ is the time in which the arc ΘΗ sets.
διὰ τὰ αὐτὰ δὴ καὶ ὁ χρόνος, ἐν ᾧ τὸ Κ τὴν ΚΨ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ ΚΘ περιφέρεια.
Therefore, for the same reasons, the time in which Κ traverses ΚΨ is also the time in which the arc ΚΘ sets.