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Euclid · Fragments §11.1

Geometric Propositions on the Parabola

Passage 27 of 29 · Greek

Summary

This chunk quotes the first three propositions of Archimedes' "Quadrature of the Parabola," presenting the geometric properties and proportional relations of diameters, tangents, and ordinates in a parabola (section of a right-angled cone).

§11.111.
11.
Archimedes Quadrat. parabol. prop. 1—3 (II p.
Archimedes, Quadrature of the Parabola, props. 1–3 (II, p.
266 sqq. ; cfr. ib.
266 sqq.; cf. ibid.
II p 350, 8; 436, 3): Εἴ κα ᾖ ὀρθογωνίου κώνου τομά, ἐφ᾿ ἇς ἁ ΑΒΓ ἁ δὲ Β∠ παρὰ τὰν διάμετρον ἢ αὐτὰ διάμετρος, ἁ δὲ ΑΓ παρὰ τὰν κατὰ τὸ Β ἐπιψαύουσαν τᾶς τοῦ κώνου τομᾶς, ἴσα ἐσσεῖται ἁ Α∠ τᾷ ∠Γ. κἂν ἴσα ᾖ ἁ Α∠ τᾷ ∠Γ, παραλλήλοι ἐσσοῦνται ἅ τε ΑΓ καὶ ἁ κατὰ τὸ Β ἐπιψαύουσα τᾶς τοῦ κώνου τομᾶς.
II, p. 350, 8; 436, 3): If ΑΒΓ is a section of a right-angled cone, on which ΒΔ is parallel to the diameter or is the diameter itself, and ΑΓ is parallel to the tangent to the section of the cone at Β, then ΑΔ will be equal to ΔΓ. And if ΑΔ is equal to ΔΓ, ΑΓ and the tangent to the section of the cone at Β will be parallel.
Εἴ κα ᾖ ὀρθογωνίου κώνου τομὰ ἁ ΑΒΓ, ᾖ δὲ ἁ μὲν Β∠ παρὰ τὰν διάμετρον ἢ αὐτὰ διάμετρος, ἁ δὲ ΑμετρονΓ παρὰ τὰν κατὰ τὸ Β ἐπιψαύουσαν τᾶς τοῦ κώνου τομᾶς, ἁ δὲ ΕΓ τᾶς τοῦ κώνου τομᾶς ἐπιψαύουσα κατὰ τὸ Γ ἐσσοῦνται αἱ Β∠, ΒΕ ἴσαι.
If ΑΒΓ is a section of a right-angled cone, and ΒΔ is parallel to the diameter or is the diameter itself, and ΑΓ is parallel to the tangent to the section of the cone at Β, and ΕΓ is a tangent to the section of the cone at Γ, then ΒΔ and ΒΕ will be equal.
Εἴ κα ᾖ ὀρθογωνίου κώνου τομὰ ἁ ΑΒΓ, ἁ δὲ Β∠ παρὰ τὰν διάμετρον ἢ αὐτὰ διάμετρος, καὶ ἀχθέωντί τινες αἱ Α∠, ΚΖ παρὰ τὰν κατὰ τὸ Β ἐπιψαύουσαν τᾶς τοῦ κώνου τομᾶς, ἐσσεῖται, ὡς ἁ Β∠ ποτὶ τὰν ΒΖ, δυνάμει ἁ Α∠ ποτὶ τὰν ΕΖ. ἀποδέδεικται δὲ ταῦτα ἐν τοῖς κωνικοῖς στοιχείοις.
If ΑΒΓ is a section of a right-angled cone, and ΒΔ is parallel to the diameter or is the diameter itself, and some straight lines, ΑΔ and ΚΖ, are drawn parallel to the tangent to the section of the cone at Β, then ΒΔ will be to ΒΖ, in square, as ΑΔ is to ΕΖ. And these things have been demonstrated in the elements of conics.

Notes

  1. 11ὀρθογωνίου κώνου τομά — "Section of a right-angled cone" is the older terminology for a "parabola" used before Apollonius. Archimedes consistently employs this traditional designation.
  2. 11Εἴ κα ᾖ — Doric dialect. It corresponds to ἐὰν ᾖ in Common Greek (Attic). The particle κε (κα) is used with εἰ to introduce a subjunctive condition. Similarly, the future ἐσσεῖται corresponds to ἔσται, and the third-person plural passive subjunctive ἀχθέωντι corresponds to ἀχθῶσι.
  3. 11ΑμετρονΓ — This appears to be a typographical error or textual corruption in the source text. Based on the context and symmetry with the first and third propositions, it is translated as ΑΓ, assuming that a part of the preceding word διάμετρον (μετρον) was erroneously inserted.
  4. 11δυνάμει — A mathematical term meaning "in square" (or "in power"), indicating that the ratio is not between the lengths of the line segments themselves, but between the areas of the squares described on them (i.e., their squares).
  5. 11∠ — A symbol used in older printed editions and manuscripts in place of the Greek letter Δ (Delta). As a label for a point in a geometric diagram, it is rendered as standard Δ in the translations.

Cite this passage

Euclid, Fragments §11.1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg016.humanitext-grc1:11.1

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