§8.2#1β΄.
β΄.
Ἐὰν ᾖ θέσει εὐθεῖα ἡ ΑΒ καὶ δοθὲν τὸ Γ ἐν τῷ αὐτῷ ἐπιπέδῳ, καὶ διαχθῇ ἡ ∠Γ, καὶ παρὰ θέσει ἀχθῇ ἡ ∠Ε, λόγος δὲ ᾖ τῆς Γ∠ πρὸς ∠Ε, τὸ ∠ ἅπτεται θέσει κωνικῆς τομῆς· δείκνυται δέ, ὅτι γραμμῆς· δειχθήσεται δὲ οὕτως προγραφέντος τόπου τοῦδε· Δύο δοθέντων τῶν Α, Β καὶ ὀρθῆς τῆς Γ∠ λόγος ἔστω τοῦ ἀπὸ Α∠ πρὸς τὰ ἀπὸ Γ∠, ∠Β. λέγω, ὅτι τὸ Γ ἄπτεται κώνου τομῆς, ἐάν τε ᾖ λόγος ἴσος πρὸς ἴσον ἢ μείζων πρὸς ἐλάσσονα ἢ ἐλάσσων πρὸς μείζονα.
If AB is a straight line given in position, and Γ is a given point in the same plane, and ∠Γ is drawn through, and ∠Ε is drawn parallel [to a line] given in position, and there is a ratio of Γ∠ to ∠Ε, [then] ∠ lies on a conic section given in position; and it is shown that [it is] a curve; and it will be shown thus, after this locus has been written beforehand: Two [points] A, B being given, and Γ∠ being perpendicular, let there be a ratio of the square on Α∠ to the [sum of the] squares on Γ∠, ∠Β. I say that Γ lies on a conic section, whether the ratio is of equal to equal, or of greater to lesser, or of lesser to greater.
ἔστω γὰρ πρότερον ὁ λόγος ἴσος πρὸς ἴσον.
For let the ratio first be of equal to equal.
καὶ ἐπεὶ ἴσον ἐστὶν τὸ ἀπὸ Α∠ τοῖς ἀπὸ Γ∠, ∠Β, κείσθω τῇ Β∠ ἴση ἡ ∠Ε ἴσον ἄρα ἐστὶ τὸ ὑπὸ ΒΑΕ τῷ ἀπὸ ∠Γ. τετμήσθω δίχα ἡ ΑΒ τῷ Ζ· δοθὲν ἄρα τὸ Ζ. καὶ ἔσται διπλῆ ἡ ΑΕ τῆς ΖΔ· ὥστε τὸ ὑπὸ ΒΑΕ τὸ δίς ἐστιν ὑπὸ τῶν ΑΒ, Ζ∠.
And since the square on Α∠ is equal to the [sum of the] squares on Γ∠, ∠Β, let ∠Ε be laid down equal to Β∠; therefore, the rectangle under Β, Α, Ε is equal to the square on ∠Γ. Let ΑΒ be bisected at Ζ; therefore, Ζ is given. And ΑΕ will be double of Ζ∠; so that the rectangle under Β, Α, Ε is twice the rectangle contained by ΑΒ, Ζ∠.
καί ἐστιν ἡ διπλῆ τῆς ΑΒ δοθεῖσα· τὸ ἄρα ὑπὸ δοθείσης καὶ τῆς Ζ∠ ἴσον ἐστὶν τῷ ἀπὸ τῆς ∠Γ. τὸ Γ ἄρα ἅπτεται θέσει παραβολῆς ἐρχομένης διὰ τοῦ Ζ.
συντεθήσεται δὴ ὁ τόπος οὕτως·
And the double of ΑΒ is given; therefore, the rectangle contained by the given [line] and Ζ∠ is equal to the square on ∠Γ. Therefore, Γ lies on a parabola given in position passing through Ζ.
ἔστω τὰ δοθέντα Α, Β, ὁ δὲ λόγος ἔστω ἴσος πρὸς ἴσον, καὶ τετμήσθω ἡ ΑΒ δίχα τῷ Ζ, τῆς δὲ ΑΒ διπλῆ ἔστω ἡ Ρ, καὶ θέσει οὔσης εὐθείας τῆς ΖΒ πεπερασμένης κατὰ τὸ Ζ, τῆς δὲ P δεδομένης τῷ μεγέθει, γεγράφθω περὶ ἄξονα τὸν ΖΒ παραβολὴ ἡ ΗΖ, ὥστε, οἷον ἐὰν ἐπ᾿ αὐτῆς σημεῖον ληφθῇ ὡς τὸ Γ, κάθετος δὲ ἀχθῇ ἡ ΓΔ, ἴσον εἶναι τὸ ὑπὸ Ρ Ζ∠ τῷ ἀπὸ ∠.
The locus will indeed be synthesized thus: Let A, B be the given [points], and let the ratio be of equal to equal, and let AB be bisected at Ζ, and let P be double of AB, and ΖB being a straight line given in position, bounded at Ζ, and P being given in magnitude, let there be described about the axis ΖB a parabola HΖ, so that, if any point is taken on it, such as Γ, and a perpendicular ΓΔ is drawn, the rectangle contained by P and Ζ∠ is equal to the square on ∠ [Γ].
καὶ ἤχθω ὀρθὴ ΒΗ. λέγω, ὅτι τὸ ΓΗ. μέρος τῆς παραβολῆς ἐστιν.
And let the perpendicular BH be drawn. I say that ΓH is a part of the parabola.
ἤχθω γὰρ κάθετος ἡ Γ∠, καὶ τῇ Β∠ ἴση κείσθω ἡ ∠Ε. ἐπεὶ οὖν διπλῆ ἐστιν ἡ μὲν ΑΒ τῆς ΒΖ, ἡ δὲ ΕΒ τῆς Β∠, διπλῆ ἄρα καὶ ἡ ΑΕ τῆς Ζ∠· τὸ ἄρα ὑπὸ ΒΑΕ ἴσον ἐστὶν τῷ δὶς ὑπὸ τῶν ΑΒ, Ζ∠, τουτέστιν τῷ ἀπὸ ∠Γ. κοινὸν προσκείσθω τὸ ἀπὸ Ε∠ ἴσον ὄν τῷ ἀπὸ ∠Β·
For let the perpendicular Γ∠ be drawn, and let ∠Ε be laid down equal to Β∠. Since, then, AB is double of BΖ, and EB is double of B∠, therefore AE is also double of Ζ∠; therefore the rectangle under Β, Α, Ε is equal to twice the rectangle contained by ΑΒ, Ζ∠, that is, to the square on ∠Γ. Let there be added in common the square on Ε∠, which is equal to the square on ∠Β; therefore, the whole square on Α∠ is equal to the [sum of the] squares on Γ∠, ∠Β.
ὅλον ἄρα τὸ ἀπὸ Α∠ ἴσον ἐστὶν τοῖς ἀπὸ τῶν Γ∠, ∠Β. ἡ ΖΓΗ ἄρα γραμμὴ ποιεῖ τὸν τόπον.
Therefore, the line ΖΓH forms the locus.
Ἔστω δὴ πάλιν τὰ δύο δοθέντα σημεῖα τὰ Α, Β καὶ εὐθεῖά τε ἡ ∠Γ καὶ ὀρθή, λόγος δὲ ἔστω τοῦ ἀπὸ Α∠ πρὸς τὰ ἀπὸ B∠, ∠Γ ἐπὶ μὲν τῆς πρώτης πτώσεως ἐλάσσων πρὸς μείζονα, ἐπὶ δὲ τῆς δευτέρας μείζων πρὸς ἐλάσσονα.
Let indeed again the two given points be A, B, and the straight line ∠Γ be perpendicular, and let there be a ratio of the square on Α∠ to the [sum of the] squares on B∠, ∠Γ, in the first case being of lesser to greater, and in the second case of greater to lesser.
λέγω, ὅτι τὸ ἅπτεται κώνου τομῆς, ἐπὶ μὲν τῆς πρώτης πτώσεως ἐλλείψεως, ἐπὶ δὲ τῆς δευτέρας ὑπερβολῆς.
I say that Γ lies on a conic section, in the first case an ellipse, and in the second case a hyperbola.
ἐπεὶ γὰρ λόγος ἐστὶν τοῦ ἀπὸ Α∠ πρὸς τὰ ἀπὸ Β∠, ∠Γ, ὁ αὐτὸς αὐτῷ γεγονέτω ὁ τοῦ ἀπὸ Β∠ πρὸς τὸ ἀπὸ ΔΕ·
For since there is a ratio of the square on Α∠ to the [sum of the] squares on Β∠, ∠Γ, let the ratio of the square on Β∠ to the square on ΔΕ be made the same as it.
ἐπὶ μὲν οὖν τῆς πρώτης πτώσεως ἐλάσσων ἐστὶν ἡ Β∠ τῆς ∠Ε, ἐπὶ δὲ τῆς δευτέρας μείζων ἐστὶν ἡ Β τῆς ∠Ε. κείσθω οὖν τῇ Ε∠ ἴση ἡ ∠Ζ. ἐπεῖ λόγος ἐστὶν τοῦ ἀπὸ Α∠ πρὸς τὰ ἀπὸ Γ∠, ∠Β, καί ἐστιν αὐτῷ ὁ αὐτὸς ὁ τοῦ ἀπὸ Ε∠ πρὸς τὸ ἀπὸ ∠Β, καὶ λοιπὸς ἄρα τοῦ ὑπὸ ΖΑΕ πρὸς τὸ ἀπὸ ∠Γ λόγος ἐστὶν δοθείς.
In the first case, then, Β∠ is lesser than ∠Ε, and in the second case, Β is greater than ∠Ε. Let, then, ∠Ζ be laid down equal to Ε∠. Since there is a ratio of the square on Α∠ to the [sum of the] squares on Γ∠, ∠Β, and the same is the ratio of the square on Ε∠ to the square on ∠Β, therefore also the remaining ratio of the rectangle under Ζ, Α, Ε to the square on ∠Γ is given.
ἐπεὶ δὲ λόγος ἐστὶν τῆς Ε∠ πρὸς ∠Β καὶ τῆς Ζ∠ πρὸς ∠Β καὶ τῆς ΖΒ πρὸς Β∠, ὁ αὐτὸς αὐτῷ γεγονέτω ὁ τῆς ΑΒ πρὸς ΒΗ· καὶ ὅλης ἄρα τῆς ΑΖ πρὸς ∠Η λόγος ἐστὶν δοθείς.
And since there is a ratio of Ε∠ to ∠Β, and of Ζ∠ to ∠Β, and of ΖΒ to Β∠, let the ratio of ΑΒ to ΒΗ be made the same as it; therefore, also the ratio of the whole ΑΖ to ∠Η is given.
πάλιν, ἐπεὶ λόγος ἐστὶν τῆς Ε∠ πρὸς ∠Β δοθείς, καὶ τῆς ΕΒ ἄρα πρὸς Β∠ λόγος ἐστὶν δοθείς. ὁ αὐτὸς αὐτῷ γεγονέτω ὁ τῆς ΑΒ πρὸς ΒΘ·
Again, since the ratio of Ε∠ to ∠Β is given, therefore also the ratio of ΕΒ to Β∠ is given.
λόγος ἄρα καὶ τῆς ΑΒ πρὸς ΒΘ ἐστιν δοθείς· δοθὲν ἄρα τὸ Θ. καὶ λοιπὸς τῆς ΑΕ πρὸς Θ∠ λόγος ἐστὶν δοθείς· καὶ τοῦ ὑπὸ ΖΑΕ ἄρα πρὸς τὸ ὑπὸ Θ∠Η λόγος ἐστὶ δοθείς. τοῦ δὲ ὑπὸ ΖΑΕ πρὸς τὸ ἀπὸ Γ∠ λόγος ἐστὶν δοθείς· καὶ τοῦ ὑπὸ Η∠Θ ἄρα πρὸς τὸ ἀπὸ ∠Γ λόγος ἐστὶν δοθείς.
Let the ratio of ΑΒ to ΒΘ be made the same as it; therefore, the ratio of ΑΒ to ΒΘ is also given; therefore, Θ is given. And the remaining ratio of ΑΕ to Θ∠ is given; therefore, also the ratio of the rectangle under Ζ, Α, Ε to the rectangle under Θ, ∠, Η is given. And the ratio of the rectangle under Ζ, Α, Ε to the square on Γ∠ is given; therefore, also the ratio of the rectangle under Η, ∠, Θ to the square on ∠Γ is given.
καί ἐστιν δύο δοθέντα τὰ Θ, Η· ἐπὶ μὲν ἄρα τῆς πρώτης πτώσεως τὸ Γ ἅπτεται ἐλλείψψεως, ἐπὶ δὲ τῆς δευτέρας ὑπερβολῆς.
And Θ, Η are two given [points]; therefore, in the first case, Γ lies on an ellipse, and in the second case, on a hyperbola.
συντεθήσεται δὲ ὁ τόπος οὕτως· ἔστω τὰ μὲν δύο δοθέντα σημεῖα τὰ Α, Β, ὁ δὲ δοθεὶς λόγος ὁ τῆς ΡΤ πρὸς ΤΣ, ἐπὶ μὲν τῆς πρώτης πτώσεως ἐλάσσων πρὸς μείζονα, ἐπὶ δὲ τῆς δευτέρας μείζων πρὸς ἐλάσσονα, καὶ τῇ ΡΤ ἴση κείσθω ἡ ΤΥ.
And the locus will be synthesized thus: Let the two given points be A, B, and the given ratio be that of ΡΤ to ΤΣ, in the first case being of lesser to greater, and in the second case of greater to lesser, and let ΤΥ be laid down equal to ΡΤ.