§4## 4.
## 4.
Ad prop. 15.
To prop. 15.
Ἄλλως τὸ αὐτό.
Otherwise the same.
Τοῦ τῶν ζῳδίων κύκλου τῶν ἴσων τε καὶ ἀπεναντίον περιφερειῶν ἐν ᾧ χρόνῳ ἡ μία ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ἑτέρα τὸ ἀφανές, καὶ πάλιν, ἐν ᾧ χρόνῳ ἡ μία τὸ ἀφανές, καὶ ἡ ἑτέρα τὸ φανερόν.
Of the equal and opposite arcs of the zodiacal circle, in whatever time one leaves the visible hemisphere, the other leaves the invisible [hemisphere], and conversely, in whatever time one [leaves] the invisible, the other also [leaves] the visible.
ἔστω ὁρίζων κύκλος ὁ ΑΒΓ∠ καὶ θερινὸς μὲν τροπικὸς ὁ ΒΑ, χειμερινὸς δὲ ὁ Γ∠, ὁ ζῳδιακὸς δὲ κύκλος θέσιν ἐχέτω ὡς τὴν ΑΕΓΖ, καὶ ἀπειλήφθωσαν ἴσαι τε καὶ ἀπεναντίον περιφέρειαι αἱ ΕΗ, ΖΘ· λέγω, ὅτι, ἐν ᾧ χρόνῳ ἡ ΖΘ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἡ ΕΗ τὸ ἀφανές.
Let the horizon circle be ΑΒΓ∠, and the summer tropic ΒΑ, and the winter [tropic] Γ∠, and let the zodiacal circle have the position ΑΕΓΖ, and let equal and opposite arcs ΕΗ, ΖΘ be cut off; I say that, in the time in which ΖΘ leaves the visible hemisphere, ΕΗ [leaves] the invisible [hemisphere].
ἔστωσαν καθʼ ὧν φέρεται τὰ Θ, Ζ, Ε, σημεῖα παρἀλληλοι κύκλοι οἱ ΚΘΛ, Μ Ν Ξ Ζ, Ε Θ Π Ρ, Σ ΗΤ. καὶ μετακεκινήσθω ὁ τῶν ζῳδίων κύκλος καὶ ὅτε μὲν θέσιν ἐχέτω τὴν ΥΛΦ, ὅτε δὲ τὴν ΧΣΨ καὶ ἐπεὶ αἱ ΖΘ, Ε περιφέρειαι ἴσαι τε καὶ ἀπεναντίον εἰσίν, ἴσοι εἰσὶ καὶ οἱ ΜΝΣ, ΟΠΡ κύκλοι, τῶν δὲ ἴσων τε καὶ παραλλήλων κύκλων τὰ ἐναλλὰξ τμήματα ἴσα ἐστὶν ἀλλήλοις· τὸ ἄρα ὑπὲρ γῆν τοῦ ΜΝΞΖ κύκλου τὸ ΜΝΞ ἴσον ἐστὶ τῷ ὑπὸ γῆν τοῦ ΟΕΡΠ κύκλου τῷ ΟΠΡ πάλιν ἐπεὶ αἱ ΖΘ, ΕΗ ἴσαι τε καὶ ἀπεναντίον εἰσίν, ἐν ᾧ χρόνῳ ἡ ΖΘ ἀνατέλλει, ἐν τούτῳ ἡ ΕΗ δύνει.
Let there be parallel circles ΚΘΛ, ΜΝΞΖ, ΕΘΠΡ, ΣΗΤ on which the points Θ, Ζ, E are carried. And let the zodiacal circle be moved, and let it have at one time the position ΥΛΦ, and at another time ΧΣΨ; and since the arcs ΖΘ, ΕΗ are equal and opposite, the circles ΜΝΣ and ΟΠΡ are also equal, and of equal and parallel circles, the alternate segments are equal to one another; therefore, the [segment] ΜΝΞ above the earth of the circle ΜΝΞΖ is equal to the [segment] ΟΠΡ below the earth of the circle ΟΕΡΠ. Again, since the arcs ΖΘ, ΕΗ are equal and opposite, in the time in which ΖΘ rises, ΕΗ sets in this [same time].
ἀλλʼ ὁ μὲν χρόνος, ἐν ᾧ ἡ ΖΘ ἀνατέλλει, τουτέστιν ἡ ΥΛ, ὁ χρόνος ἐστίν, ἐν ᾧ τὸ Υ σημεῖον ἀρξάμενον ἀπὸ τοῦ τὴν ΥΞ περιφέρειαν διελθὸν ἐπὶ τὸ Ξ παραγίγνται, ὁ δὲ χρόνος, ἐν ᾧ ἡ ΕΗ δύνει, τουτέστιν ἡ ΧΣ, ὁ χρόνος ἐστίν, ἐν ᾧ τὸ Χ ἀρξάμενον ἀπὸ τοῦ Χ τὴν ΧΟ περιφέρειαν διελθὸν ἐπὶ τὸ Ο παραγίγνεται· ὁ ἄρα χρόνος, ἐν ᾧ τὸ ἀρξάμενον ἀπὸ τοῦ Υ τὴν ΥΞ περιφέρειαν διελθὸν ἐπὶ τὸ παραγίγνεται, ἴσος ἐστὶ τῷ χρόνῳ, ἐν ᾧ τὸ Χ ἀρξάμενον ἀπὸ τοῦ Χ τὴν ΧΟ περιφέρειαν δεελθὸν ἐπὶ τὸ Ο παραγίγνεται.
But the time in which ΖΘ rises, that is [the arc] ΥΛ, is the time in which the point Υ, starting from Υ, having traversed the arc ΥΞ, arrives at Ξ. And the time in which ΕΗ sets, that is [the arc] ΧΣ, is the time in which the point Χ, starting from Χ, having traversed the arc ΧΟ, arrives at Ο; therefore, the time in which the point Υ, starting from Υ, having traversed the arc ΥΞ, arrives at Ξ, is equal to the time in which the point Χ, starting from Χ, having traversed the arc ΧΟ, arrives at Ο.
κοινὸς προσκείσθω ὁ χρόνος, ἐν ᾧ τὸ Υ ἀρξάμενον ἀπὸ τοῦ Ξ τὴν ΞΝΜ περιφέρειαν διελθὸν ἐπὶ τὸ Μ παραγίγνεται, ἴσος ὢν τῷ χρόνῳ, ἐν τὸ Χ ἀρξάμενον ἀπὸ τοῦ Ο τὴν ΟΠΡ περιφέρειαν διελθὸν ἐπὶ τὸ Ρπαραγίγνεται· ὁ ἄρα χρόνος, ἐν ᾧ τὸ Υ ἀρξάμενον ἀπὸ τοῦ Υ τὴν Υ Ξ Ν Μ περιφέρειαν διελθὸν ἐπὶ τὸ Μ παραγίγνεται, ἴσος ἐστὶ τῷ χρόνῳ, ἐν ᾧ τὸ ἀρξάμενον ἀπὸ τοῦ Χ τὴν ΧΟΠΡ περιφέρειαν διελθὸν ἐπὶ τὸ Ρ παραγίγνεται.
Let the common time be added, in which the point Υ, starting from Ξ, having traversed the arc ΞΝΜ, arrives at Μ, being equal to the time in which the point Χ, starting from Ο, having traversed the arc ΟΠΡ, arrives at Ρ; therefore, the time in which the point Υ, starting from Υ, having traversed the arc ΥΞΝΜ, arrives at Μ, is equal to the time in which the point Χ, starting from Χ, having traversed the arc ΧΟΠΡ, arrives at Ρ.
ἀλλʼ ὁ μὲν χρόνος, ἐν ᾧ τὸ Υ ἀρξάμενον ἀπὸ τοῦ Υ τὴν Υ Ξ Ν Μ περιφέρειαν διελθὸν ἐπὶ τὸ Μ παραγίγνεται, ὁ χρόνος ἐστίν, ἐν ᾧ ἡ ΥΛ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, τουτέστιν ἡ Θ Ζ ὁ δὲ χρόνος, ἐν ᾧ τὸ Χ ἀρξάμενον ἀπὸ τοῦ Χ τὴν ΧΟΠΡ περιφέρειαν διελθὸν ἐπὶ τὸ Ρ παραγίγνεται, ται, ὁ χρόνος ἐστίν, ἐνᾧ ἡ ΧΣ ἐξαλλάσσει τὸ ἀφανὲς ἡμισφαίριον, τουτέστιν ἡ ΕΗ ἐν ᾧ ἄρα χρόνῳ ἡ ΘΖ ἐξαλλάσσει τὸ φανερὸν ἡμισφαίριον, ἐν τούτῳ ἡ ΕΗ τὸ ἀφανές.
But the time in which the point Υ, starting from Υ, having traversed the arc ΥΞΝΜ arrives at Μ, is the time in which ΥΛ, that is ΘΖ, leaves the visible hemisphere; and the time in which the point Χ, starting from Χ, having traversed the arc ΧΟΠΡ, arrives at Ρ, is the time in which ΧΣ, that is ΕΗ, leaves the invisible hemisphere; therefore, in the time in which ΘΖ leaves the visible hemisphere, ΕΗ [leaves] the invisible [hemisphere] in this [same time].