Humanitext Reader

Euclid · Fragments §8.1

Extension from a Conic Section to a Surface Locus

Passage 22 of 29 · Greek

Summary

This section discusses the first proposition of Euclid's lost work *Surface Loci*, showing the process by which a planar geometric relationship between points and lines is extended to a locus on a surface in three-dimensional space.

§8.1## Εἰς τοὺς πρὸς ἐπιφανείᾳ. α΄.
## On the [Loci] on a Surface. α΄.
Ἐὰν ᾖ εὐθεῖα ἡ ΑΒ καὶ παρὰ θέσει ἡ Γ∠, καὶ ᾖ λόγος τοῦ ὑπὸ Α∠Β πρὸς τὸ ἀπὸ ∠Γ, τὸ Γ ἅπτεται κωνικῆς γραμμῆς.
If AB is a straight line and Γ∠ is given in position, and there is a ratio of the rectangle contained by A∠, ∠B to the square on ∠Γ, Γ lies on a conic section.
ἐὰν οὖν ἡ μὲν ΑΒ στερηθῇ τῆς θέσεως, καὶ τὰ Α, Β στερηθῇ τοῦ δοθέντα εἶναι, γένηται δὲ πρὸς θέσει εὐθείαις ταῖς ΑΕ, ΕΒ, τὸ Γ μετεωρισθὲν γίνεται πρὸς θέσει ἐπιφανείᾳ.
Therefore, if AB is deprived of its position, and A, B are deprived of being given, but come to be on the straight lines AE, EB which are given in position, Γ, being raised in space, comes to be on a surface given in position.
τοῦτο δὲ ἐδείχθη.
And this has been demonstrated.

Notes

  1. §8.1Εἰς τοὺς πρὸς ἐπιφανείᾳ. — The masculine plural accusative article τούς implies the omitted noun 'loci' (τόπους). This refers to Euclid's lost work, *Surface Loci* (Πρὸς ἐπιφανείᾳ τόποι).
  2. 8.1τοῦ ὑπὸ Α∠Β πρὸς τὸ ἀπὸ ∠Γ — In Greek mathematical terminology, 'τοῦ ὑπὸ Α∠Β' refers to 'the rectangle contained by A∠ and ∠B', and 'τὸ ἀπὸ ∠Γ' refers to 'the square on ∠Γ'. The symbol '∠' in the manuscript is a variant or corruption of 'Δ' (Delta) representing a point.
  3. 8.1τοῦ δοθέντα εἶναι — A construction featuring an articular infinitive in the genitive (τοῦ εἶναι) accompanied by an accusative neuter plural complement (δοθέντα). Since the subject of the clause, 'τὰ Α, Β' (neuter plural), serves as the logical subject of the infinitive, the complement agrees with it in gender, number, and case (accusative by attraction or as the subject/complement of an infinitive).

Cite this passage

Euclid, Fragments §8.1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg016.humanitext-grc1:8.1

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