Humanitext Reader

Euclid · Fragments §13.1

Application of Areas Based on Conic Properties

Passage 29 of 29 · Greek

Summary

Archimedes introduces lines N and M whose ratio equals that of the squares on AZ and AK, and states the property of the areas applied to a line equal to N based on the properties of conic sections.

§13.113.
13.
Archimedes Περὶ κωνοειδ. prop. 3 (I p. 272, 23 sqq. ): Ὃν δὴ λόγον ἔχει τὸ τετράγωνον τὸ ἀπὸ τᾶς ΑΖ ποτὶ τὸ τετράγωνον τὸ ἀπὸ τᾶς ΑΚ, τοῦτον ἐχέτω ἁ Ν ποτὶ τὰν Μ. αἱ δὴ ἀπὸ τᾶς τομᾶς ἐπὶ τὰν ∠Ζ ἀγόμεναι παρὰ τὰν ΑΕ δύνανται τὰ παρὰ τὰν ἴσαν τᾷ Ν παραπίπτοντα πλάτος ἔχοντα, ἃς αὐταὶ ἀπολαμβάνοντι ἀπὸ τᾶς ∠ ποτὶ τὸ ∠ πέρας.
Archimedes, On Conoids and Spheroids, prop. 3 (I, p. 272, 23 sqq.): Let the ratio which the square on AZ has to the square on AK be the ratio which N has to M. Now, the straight lines drawn from the section to ∠Z parallel to AE are equal in square to the areas applied along the line equal to N, having as breadth the segments which they themselves cut off from ∠ towards the end ∠.
δέδεικται γὰρ ἐν τοῖς κωνικοῖς.
For this has been demonstrated in the conics.

Notes

  1. §13.1δύνανται — A geometric technical term meaning 'are equal in square to' (i.e., the squares on the subject straight lines are equal to the following areas).
  2. §13.1παραπίπτοντα — Participle meaning 'applied along', referring to the 'application of areas' (παραβολή) in Greek geometry, which means constructing an area on a given straight line (here, 'equal to N') as its base.
  3. §13.1ἃς — Relative pronoun in the accusative feminine plural (Doric form, corresponding to Attic ἅς). It has the omitted noun εὐθείας ('straight lines' or 'segments') as its antecedent and serves as the direct object of the verb ἀπολαμβάνοντι (third-person plural Doric form of 'cut off').

Cite this passage

Euclid, Fragments §13.1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg016.humanitext-grc1:13.1

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