§2#2καὶ ἐπεὶ ἐν σφαίρᾳ μέγιστος ὁ ΑΒΓ ἐφάπτεταί τινος κύκλου τῶν παραλλήλων τοῦ τυ Φ καὶ τὸν ΑΒΓ τέμνουσιν οἱ μέγιστοι κύκλοι οἱ ΕΖ, ΑΓ, ὧν ὁ μὲν ΕΖ μέγιστος τῶν παραλλήλων, ὁ δὲ ΑΓ λοξὸς πρὸς τοὺς παραλλήλους, καὶ ἀπειλημμέναι εἴσὶ περιφέρειαι αἱ Α Η, Η Θ, ΘΚ ἐπὶ τῆς τοῦ λοξοῦ κύκλου περιφερείας ἴσαι ἐξῆς ἐπὶ τὰ αὐτὰ μέρη τοῦ μεγίστου τῶν παραλλήλων, καὶ διὰ τῶν Η, Θ σημείων γεγραμμένοι εἰσὶ μέγιστοι κύκλοι οἱ υ ΗΧ, ΦΘΨ ἐφαπτόμενοι τοῦ τυΦ κύκλου, μείζων ἐστὶν ἡ μὲν ΕΧ περιφέρεια τῆς Χ Ψ περιφερείας, ἡ δὲ Χ Ψ τῆς Ψ Κ ἐν πλείονι ἄρα χρόνῳ τὸ Χ τὴν ΧΕ διαπορεύεται ἤπερ τὸ Ψ τήν Φ Σʼ καὶ τὸ Ψ τήν Ψ Χ ἐν πλείονι χρόνῳ διαπορεύεται ἤπερ τὸ Κ τὴν Κ Ψ ἀλλʼ ὁ μὲν χρόνος, ἐν ᾧ τὸ τὴν ΧΕ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ τὸ τὴν ΗΝ περιφέρειαν διαπορεύεται, τουτέστιν ἐν ᾧ δύνει ἡ Α περιφέρεια· ὁ δὲ χρόνος, ἐν ᾧ τὸ Ψ τὴν ΨΧ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ τὸ Θ τὴν ΘΩ διαπορεύεται, τουτέστιν ὁ χρόνος, ἐν ᾧ δύνει ἡ ΘΗ περιφέρεια· ὁ δὲ χρόνος, ἐν ᾧ τὸ Κ τὴν Κ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ ΚΘ περιφέρεια·
And since in a sphere the great circle ΑΒΓ is tangent to one of the parallel circles, namely τυ Φ, and the great circles ΕΖ, ΑΓ intersect ΑΒΓ, of which ΕΖ is the greatest of the parallel circles, and ΑΓ is inclined to the parallel circles, and there are cut off successive equal arcs ΑΗ, ΗΘ, ΘΚ on the circumference of the inclined circle towards the same parts of the greatest of the parallel circles, and through the points Η, Θ the great circles υ ΗΧ, ΦΘΨ have been drawn tangent to the circle τυ Φ, the arc ΕΧ is greater than the arc ΧΨ, and ΧΨ than ΨΚ. Therefore, in a longer time [the point] Χ traverses ΧΕ than [the point] Ψ [traverses] ΦΣ; and [the point] Ψ traverses ΨΧ in a longer time than [the point] Κ [traverses] ΚΨ. But the time in which [the point] traverses ΧΕ is the time in which [the point] traverses the arc ΗΝ, that is, in which the arc Α[Η] sets; and the time in which Ψ traverses ΨΧ is the time in which Θ traverses ΘΩ, that is, the time in which the arc ΘΗ sets; and the time in which Κ traverses Κ[Ψ] is the time in which the arc ΚΘ sets.
ἐν πλείονι ἄρα χρόνῳ δύνει ἡ μὲν ΑΗ περιφέρεια τῆς ΗΘ περιφερείας. ἡ δὲ ΗΘ τῆς ΘΚ.
λέγω δή, ὅτι ἐν ἴσοις χρόνοις αἰ ἴσον ἀπέχουσαι τοῦ ἰσημερινοῦ δύνουσιν.
Therefore, in a longer time the arc ΑΗ sets than the arc ΗΘ, and ΗΘ than ΘΚ. I say indeed that the [arcs] equidistant from the equator set in equal times.
γενομένου γὰρ δὴ τοῦ Κ σημείου ἐπὶ τὸ Ε σημεῖον, ὁ τῶν ζῳδίων κύκλος θέσιν ἐχέτω τὴν Ϛ Ελ.
For indeed, when the point Κ has come to the point Ε, let the zodiac circle have the position ϚΕλ.
καὶ ἐπεὶ ἴση ἐστὶν ἡ ϛ Ε περιφέρεια τῇ Ε λ περιφερείᾳ καί ἐστι μέγιστος τῶν παραλλήλων ὁ ΕΖ, ἴσος ἄρα ἐστὶν ὁ Ϛ ΘΠ κύκλος τῷ ΡΛΣ κύκλῳ· ἴση ἄρα ἐστὶ καὶ ἡ ΟΕ περνφέρεια τῇ ΕΡ περιφερείᾳ.
And since the arc ϚΕ is equal to the arc Ελ, and ΕΖ is the greatest of the parallel circles, therefore the circle ϚΘΠ is equal to the circle ΡΛΣ; therefore also the arc ΟΕ is equal to the arc ΕΡ.
ἔστι δὲ καὶ ἡ ϛ Ε τῇ Ε λ ἴση ἄρα καὶ ἡ ἀπὸ τοῦ ϛ ἐπὶ τὸ Ο τῇ ἀπὸ τοῦ Ρ ἐπὶ τὸ λ.
And [the arc] ϚΕ is [equal] to Ελ; therefore also the [arc] from Ϛ to Ο is equal to that from Ρ to λ.
καί εἴσιν ἴσοι κύκλοι οἱ Ϛ ΘΠ, ΡΛΣ ὁμοία ἄρα ἐστὶν ἡ Ϛ Ο περιφέρεια τῇ Ρ λ περιφερείᾳ·
And the circles ϚΘΠ, ΡΛΣ are equal; therefore the arc ϚΟ is similar to the arc Ρλ.
ἐν ἴσῳ ἄρα χρόνῳ τὸ λ σημεῖον τὴν λ Ρ διαπορεύεται καὶ τὸ Ο τὴν Ο Ϛ. ἀλλʼ ὁ μὲν χρόνος, ἐν ᾧ τὸ λ τὴν λ Ρ διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ δύνει ἡ λ Ε περιφέρεια, ὁ δὲ χρόνος, ἐν ᾧ τὸ Ο τὴν Ο Ϛ διαπορεύεται, ἴσος ἐστὶ τῷ χρόνῳ, ἐν δύνει ἡ Ε Ϛ περιφέρεια· ἐν ἴσῳ ἄρα χρόνῳ δύνουσιν αἰ λ Ε, Ε Ϛ περιφέρειαι.
Therefore, in an equal time the point λ traverses λΡ, and Ο [traverses] ΟϚ. But the time in which λ traverses λΡ is the time in which the arc λΕ sets, and the time in which Ο traverses ΟϚ is equal to the time in which the arc ΕϚ sets; therefore, in an equal time the arcs λΕ, ΕϚ set.
ἴση δὲ ἡ μὲν λ Ε τῇ Λ Κ, ἡ δὲ Ε Ϛ τῇ ΚΘ αἰ ΛΚ, ΚΘ ἄρα ἐν ἴσῳ χρόνῳ δύνουσιν.
And [the arc] λΕ is equal to ΛΚ, and ΕϚ to ΚΘ; therefore ΛΚ, ΚΘ set in an equal time.
ὁμοίως δὲ δείξομεν, ὅτι καὶ αἰ ΜΚ, ΚΗ ἐν ἴσῳ χρόνῳ δύνουσιν, ὧν αἰ ΛΚ, ΚΘ ἐν ἴσῳ χρόνῳ δύνουσιν· λοιπαὶ ἄρα αἱ ΜΛ, ΘΗ ἐν ἴσῳ χρόνω δύνουσιν.
And we shall show similarly that ΜΚ, ΚΗ also set in an equal time, of which ΛΚ, ΚΘ set in an equal time; therefore the remaining [arcs] ΜΛ, ΘΗ set in an equal time.
ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ΜΓ, ΑΗ περιφέρειαι ἐν ἴσῳ χρόνῳ δύνουσιν.
Similarly indeed we shall show that the arcs ΜΓ, ΑΗ also set in an equal time.
καὶ ἐπεὶ ἐν πλείονι χρόνῳ δύνει ἡ ΑΗ περιφέρεια ἤπερ ἡ ΗΘ καὶ ἡ ΗΘ ἤπερ ἡ ΘΚ, ἐν πλείονι ἄρα χρόνῳ δύνει ἡ Γ Μ περιφέρεια ἤπερ ἡ ΜΛ, καὶ ἡ ΜΛ ἤπερ ἡ ΛΚ. ἐν πλείστῳ ἄρα χρόνῳ δύνουσιν αἱ ΑΗ, ΜΓ περιφέρειαι, ἐν ἐλάσσονι δὲ αἱ ΗΘ, ΛΜ ἐν ἐλαχίστῳ δὲ αἱ Θ Κ, ΛΚ, ἐν ἴσῳ δὲ αἰ ἴσον ἀπέχουσαι τοῦ ἰσημερινοῦ καὶ δύνουσι καὶ ἀνατέλλουσιν.
And since in a longer time the arc ΑΗ sets than ΗΘ, and ΗΘ than ΘΚ, therefore in a longer time the arc ΓΜ sets than ΜΛ, and ΜΛ than ΛΚ. Therefore, the arcs ΑΗ, ΜΓ set in the longest time, and ΗΘ, ΛΜ in a shorter [time], and ΘΚ, ΛΚ in the shortest [time], and those equidistant from the equator both set and rise in an equal [time].