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Euclid · Elements §10.def2.1-10.def2.6

Definitions of the Six Types of Binomial Lines

Passage 190 of 316 · Greek

Summary

The passage defines and classifies binomial straight lines into six types (first to sixth) based on the square difference between their two terms (greater and lesser) and whether each term is commensurable in length with the set out rational straight line.

§10.def2.1ὑποκειμένης ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων διῃρημένης εἰς τὰ ὀνόματα, ἧς τὸ μεῖζον ὄνομα τοῦ ἐλάσσονος μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει, ἐὰν μὲν τὸ μεῖζον ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων πρώτη.
A rational straight line being set out, and a binomial straight line being divided into its terms, of which the greater term is greater in square than the lesser by the square on a straight line commensurable in length with itself, if the greater term is commensurable in length with the set out rational straight line, let it be called a first binomial.
§10.def2.2ἐὰν δὲ τὸ ἔλασσον ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων δευτέρα.
But if the lesser term is commensurable in length with the set out rational straight line, let it be called a second binomial.
§10.def2.3ἐὰν δὲ μηδέτερον τῶν ὀνομάτων σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων τρίτη.
But if neither of the terms is commensurable in length with the set out rational straight line, let it be called a third binomial.
§10.def2.4πάλιν δὴ ἐὰν τὸ μεῖζον ὄνομα μεῖζον δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει, ἐὰν μὲν τὸ μεῖζον ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων τετάρτη.
Again, if the greater term is greater in square than the lesser by the square on a straight line incommensurable in length with itself, if the greater term is commensurable in length with the set out rational straight line, let it be called a fourth binomial.
§10.def2.5ἐὰν δὲ τὸ ἔλασσον, πέμπτη.
But if the lesser, a fifth.
§10.def2.6ἐὰν δὲ μηδέτερον, ἕκτη.
But if neither, a sixth.

Notes

  1. §10.def2.1ὑποκειμένης ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων διῃρημένης εἰς τὰ ὀνόματα — A genitive absolute construction forming the prerequisite conditions not only for Definition 2.1, but also for the subsequent Definitions 2.2 and 2.3 (and implicitly 2.4–2.6). The second part τῆς ἐκ δύο ὀνομάτων διῃρημένης is also a genitive participle coordinated with ὑποκειμένης.
  2. §10.def2.1τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει — The dative of the neuter article τῷ indicates the measure of difference. After ἀπὸ, 'straight line' (εὐθείας) is omitted, meaning 'the square [described] on a straight line.' The dative ἑαυτῇ is governed by the adjective συμμέτρου ('commensurable with'). Thus, the phrase means 'by the square on a straight line commensurable in length with itself.'
  3. §10.def2.5ἐὰν δὲ τὸ ἔλασσον, πέμπτη. — An extremely elliptic expression. In light of the structure of the preceding §10.def2.4, it is interpreted by supplying: ἐὰν δὲ τὸ ἔλασσον [ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων] πέμπτη. Similarly, §10.def2.6 omits the same elements after μηδέτερον.

Cite this passage

Euclid, Elements §10.def2.1-10.def2.6. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:10.def2.1-10.def2.6

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