§3#1## 3.
## 3.
Ad prop. XIV Ἄλλως τὸ ιδ΄.
Another proof for Proposition XIV.
Ἔστι δὲ καὶ αὕτη ἔκθεσις σαφεστέρα τῆς προτέρας.
And this exposition is clearer than the former.
ἔστω ἐν κόσμῳ ὁρίζων ὁ ΑΒΓ, καὶ μέγιστος μὲν τῶν ἀεὶ φανερῶν ὁ Α∠Ε, μέγιστος δὲ τῶν ἀεὶ ἀφανῶν ὁ ΗΘ, καὶ θερινὸς μὲν τροπικὸς ἔστω ὁ ΚΛ, χειμερινὸς δὲ τροπικὸς ὁ ΒΓ, καὶ ἔστωο ὁ τοῦ ΑΒΓ κύκλου πόλος μεταξὺ τῶν Α∠Ε, ΚΛ κύκλων, καὶ ἔστω ἀνατολικὰ μὲν τὰ Λ, Γ μέρη, δυτικὰ δὲ τὰ Κ, Β, ζῳδιακοῦ ὁδὲ θέσεις ἔστωσαν τοῦ μετὰ τὸν Καρκίνον ἡμικυκλίου αἰ ΝΞ, ΟΠ. καὶ ἀπειλήφθω ἡ Ο Π περιφέρεια μὴ μείζων ἡμικυκλίου οὖσα, καὶ γεγράφθω διὰ τοῦ Π μέγιστος κύκλος ἐφαπτόμενος τοῦ Α∠Ε· ἐφάψεται ἄρα καὶ τοῦ ΖΗΘ. ἤτοι δὴ διὰ τοῦ Ο σημείου ἥξει ἢ ὑπερπεσεῖται τὸ Ο σημεῖον.
Let the horizon in the cosmos be ΑΒΓ, and let the greatest of the always visible [circles] be Α∠Ε, and the greatest of the always invisible [circles] be ΗΘ, and let the summer tropic be ΚΛ, and the winter tropic be ΒΓ, and let the pole of the circle ΑΒΓ be between the circles Α∠Ε, ΚΛ, and let the eastern parts be Λ, Γ, and the western parts be Κ, Β, and let the positions of the semicircle of the zodiac after Cancer be ΝΞ, ΟΠ. And let the arc ΟΠ be cut off, being not greater than a semicircle, and let a great circle be drawn through Π tangent to Α∠Ε; therefore it will also be tangent to ΖΗΘ. It will either pass through the point Ο or go beyond the point Ο.
γεγράφθω καὶ ἔστω ὁ ΕΘΠ, ὥστε ἀσύμπτωτον εἶναι τὸ ἀπὸ τοῦ Ε ἡμικύκλιον ὡς ἐπὶ τὰ Ε, Ξ, Π μέρη τῷ ἀπὸ τοῦ Α ἡμικυκλίῳ ὡς ἐπὶ τὰ Α, Κ, Ρ μέρη, καὶ προςαναπεπληρώσθωσαν οἱ ΞΝ Ϛ, ΠΟΡ κύκλοι.
Let it be drawn and let it be ΕΘΠ, so that the semicircle from Ε towards the parts Ε, Ξ, Π is non-secant to the semicircle from Α towards the parts Α, Κ, Ρ, and let the circles ΞΝϚ, ΠΟΡ be further completed.
ἐπεὶ ἐν σφαίρᾳ μέγιστος κύκλος ἐστὶν ὁ ΑΒΓ καὶ τέμνουσι δύο μέγιστοι κύκλοι ἀλλήλους οἱ Ϛ ΝΣ, ΡΟ ΤΠ καί ἐστιν ὁ τοῦ ΑΒΓ κύκλου πόλος μεταξὺ τῶν Α∠Ε, ΚΛ περιφερειῶν, μείζων ἄρα ἐστὶν ἡ ΣΝΥ περιφέρεια τῆς ΥΤ περιφερείας· ἡ ΤΥ ἄρα περιφέρεια τῆς ΥΝΣ ἐλάσσων ἐστίν.
Since in a sphere the great circle is ΑΒΓ and two great circles ϚΝΣ, ΡΟΤΠ intersect each other, and the pole of the circle ΑΒΓ is between the arcs Α∠Ε, ΚΛ, therefore the arc ΣΝΥ is greater than the arc ΥΤ; therefore the arc ΤΥ is smaller than ΥΝΣ.
καὶ ἐπεὶ ἐν σφαίρᾳ δύο μέγιστοι κύκλοι οἱ ΑΒΓ, ΕΘΠ τοῦ αὐτοῦ κύκλου ἐφάπτονται τοῦ Α∠Ε καὶ τῷ Α∠Ε παράλληλον ὄντα τὸν ΚΛ τέμνουσι καί ἔστιν ὁ τοῦ ΑΒΓ πόλος μεταξὺ τῶν Α∠Ε, ΚΛ κύκλων, καὶ ὁ τοῦ ΕΘΠ ἄρα πόλος μεταξύ ἐστι τῶν Α∠Ε, ΚΛ κύκλων.
And since in a sphere two great circles ΑΒΓ, ΕΘΠ are tangent to the same circle Α∠Ε and intersect ΚΛ, which is parallel to Α∠Ε, and the pole of ΑΒΓ is between the circles Α∠Ε, ΚΛ, therefore the pole of ΕΘΠ also is between the circles Α∠Ε, ΚΛ.
ὁ ἄρα ἕτερος αὐτοῦ πόλος ἐστὶ μεταξὑ τῶν ΗΖΘ, ΒΓ κύκλων.
Therefore its other pole is between the circles ΗΖΘ, ΒΓ.
ἐπεὶ οὖν ἐν σφαίρᾳ μέγιστος κύκλος ἐστὶν ὁ ΕΘ Π καὶ τὸν ΕΘΠ τέμνουσι δύο μέγιστοι κύκλοι οἱ ΡΟ Π, Ϛ ΝΞ καί ἐστιν ὁ τοῦ ΕΘ Π πόλος μεταξὺ τῶν ΒΓ, ΗΘΖ, μείζων ἐστὶν ἡ ΠΥ περιφέρεια τῆς ΥΝΞ περιφερείας, ὧν ἡ ΥΤ τῆς ΥΝΣ ἐλάσσσων ἐστίν· λοιπὴ ἄρα ἡ ΤΠ λοιπῆς τῆς ΣΞ μείζων ἐστίν.
Since, therefore, in a sphere the great circle is ΕΘΠ and two great circles ΡΟΠ, ϚΝΞ intersect ΕΘΠ, and the pole of ΕΘΠ is between [the circles] ΒΓ, ΗΘΖ, the arc ΠΥ is greater than the arc ΥΝΞ, of which ΥΤ is smaller than ΥΝΣ; therefore the remaining ΤΠ is greater than the remaining ΣΞ.
κείσθω τῇ ΣΞ περιφερείᾳ ἴση περιφέρεια ἡ ΤΧ καὶ γεγράφθωσαν παράλληλοι κύκλοι, καθʼ ὧν φέρεται τὰ Ξ σημεῖα, οἱ ΞΨ, ΩϚ· ὁμοία ἄρα ἐστὶν ἡ Ξ Ψ περιφέρεια τῇ Ω Ϛ περιφερείᾳ ἡ Ξ Ψ ἄρα τῆς Χ Ϛ μείζων ἐστὶν ἢ ὁμοία·
Let the arc ΤΧ be placed equal to the arc ΣΞ, and let parallel circles, on which the points Ξ are carried, be drawn, namely ΞΨ, ΩϚ; therefore the arc ΞΨ is similar to the arc ΩϚ; therefore the arc ΞΨ is greater than or similar to ΧϚ; therefore in a longer time the point [Ξ] traverses the arc ΞΨ than [the point] Χ [traverses] ΧϚ.
ἐν πλείονι ἄρα χρόνῳ τὸ σημεῖον τὴν Ξ Ψ περιφέρειαν διαπορεύεται ἤπερ τὸ Χ τὴν Χ Ϛ. ἀλλʼ ὁ μὲν χρόνος, ἐν ᾧ τὸ Ξ σημεῖον τὴν ΞΨ περιφέρειαν διαπορεύεται, ὁ χρόνος ἐστίν, ἔν ᾧ ἡ ΣΞ περιφέρεια ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον· ὁ δὲ χρόνος, ἔν ᾧ τὸ Χ σημεῖον τὴν ΧϚ περιφέρειαν διαπορεύεται, ὁ χρόνος ἐστίν, ἐν ᾧ ἡ ΤΧ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον·
But the time in which the point Ξ traverses the arc ΞΨ is the time in which the arc ΣΞ leaves the visible hemisphere; and the time in which the point Χ traverses the arc ΧϚ is the time in which ΤΧ leaves the visible hemisphere; therefore in a longer time ΣΞ leaves the visible hemisphere than ΤΧ.
ἔν πλείονι ἄρα χρόνῳ ἡ ΣΞ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον ἤπερ ἡ ΤΧ. καί ἐστιν ἡ ΣΞ ἔγγιον τοῦ θερινοῦ τροπικοῦ ἤπερ ἡ ΤΧ. ἐν πλείονι ἄρα χρόνῳ ἐξαλλάττει τὸ φανερὸν ἡμισφαίριον ἡ ἔγγιον τοῦ θερινοῦ τροπικοῦ τῆς ἀπώτερον.
And ΣΞ is closer to the summer tropic than ΤΧ. Therefore, in a longer time the [arc] closer to the summer tropic leaves the visible hemisphere than the one further away.