Humanitext Reader

Euclid · Fragments §12.1

Ratio of Rectangles of Intersecting Lines to Tangents

Passage 28 of 29 · Greek

Summary

Quotes Proposition 3 of Archimedes' On Conoids and Spheroids, establishing that the ratio of the rectangles contained by the segments of intersecting lines in a conic section is equal to the ratio of the squares on the parallel tangents.

§12.112.
12.
Archimedes Περὶ κωνοειδ. καὶ σφαιροειδ. prop. 3 (I p. 27Ο, 15 sqq. ): Εἰ κα κώνου τομᾶς ὁποιασοῦν εὐθεῖαι ἐπιψαύωντι ἀπὸ τοῦ αὐτοῦ σαμείου ἀγμέναι, ἔωντι δὲ καὶ ἄλλαι εὐθεῖαι ἐν τᾷ τοῦ κώνου τομᾷ παρὰ τὰς ἐπιψαυούσας ἀγμέναι καὶ τέμνουσαι ἀλλάλας, τὰ περιεχόμενα ὑπὸ τῶν τμαμάτων τὸν αὐτὸν ἑξοῦντι λόγον ποτ᾿ ἄλλαλα, ὃν τὰ τετράγωνα τὰ ἀπὸ τᾶν ἐπιψαυουσᾶν· ὁμόλογον δὲ ἐσσεῖται τὸ περιεχόμενον ὑπὸ τῶν τᾶς ἑτέρας γραμμᾶς τμαμάτων τῷ τετραγώνῳ τῷ ἀπὸ τᾶς ἐπιψαυούσας τᾶς παραλλήλου αὐτᾷ.
Archimedes, On Conoids and Spheroids, prop. 3 (I, p. 270, 15 sqq.): If, to any section of a cone, straight lines drawn from the same point are tangent, and other straight lines in the section of the cone are drawn parallel to the tangents and cut one another, then the rectangles contained by the segments will have the same ratio to one another as the squares on the tangents; and the rectangle contained by the segments of one of the straight lines will correspond to the square on the tangent parallel to it.
ἀποδέδεικται δὲ τοῦτο ἐν τοῖς κωνικοῖς στοιχείοις.
And this has been demonstrated in the elements of conics.

Notes

  1. p.284εἰ κα ... ἐπιψαύωντι — A conditional clause in the Doric dialect, equivalent to Attic ἐάν (εἰ ἄν) followed by the subjunctive. ἐπιψαύωντι corresponds to Attic ἐπιψαύωσι (3rd person plural subjunctive), as does the subsequent ἔωντι to ὦσι.
  2. 5τὰ περιεχόμενα ὑπὸ τῶν τμαμάτων — A technical geometrical expression meaning 'the rectangles contained by the segments' (i.e., the product of the segments). The noun χωρία (areas) is omitted. τμαμάτων is the Doric form for Attic τμημάτων.
  3. 5ὁμόλογον — A technical term in the theory of proportions meaning 'corresponding' or 'homologous'. In a proportion A : B = C : D, it indicates that A corresponds to C and B to D. Here, it signifies that the rectangle contained by the segments of one line occupies the same position (antecedent or consequent) in the ratio as the square on the corresponding tangent.
  4. 5τῷ τετραγώνῳ τῷ ἀπὸ τᾶς ἐπιψαυούσας — A mathematical expression meaning 'to the square described on the tangent'. The construction 'τὸ ἀπὸ τῆς εὐθείας τετράγωνον' is idiomatic for 'the square on the straight line'. τᾶς is the Doric form equivalent to Attic τῆς.

Cite this passage

Euclid, Fragments §12.1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg016.humanitext-grc1:12.1

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