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Euclid · Elements §10.def3.1-10.def3.6

Definitions of the Six Types of Apotome

Passage 215 of 316 · Greek

Summary

Defines and classifies apotomes into six types (first to sixth) based on their commensurability in length with a set-out rational straight line and the property of the difference of the squares on the whole and the annexed straight line.

§10.def3.1ὑποκειμένης ῥητῆς καὶ ἀποτομῆς, ἐὰν μὲν ἡ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει, καὶ ἡ ὅλη σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, καλείσθω ἀποτομὴ πρώτη.
Given a rational straight line and an apotome, if the whole is greater in square than the annexed straight line by the square on a straight line commensurable in length with itself, and the whole is commensurable in length with the set-out rational straight line, let it be called a first apotome.
§10.def3.2ἐὰν δὲ ἡ προσαρμόζουσα σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καλείσθω ἀποτομὴ δευτέρα.
And if the annexed straight line is commensurable in length with the set-out rational straight line, and the whole is greater in square than the annexed straight line by the square on a straight line commensurable in length with itself, let it be called a second apotome.
§10.def3.3ἐὰν δὲ μηδετέρα σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, ἡ δὲ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καλείσθω ἀποτομὴ τρίτη.
And if neither is commensurable in length with the set-out rational straight line, and the whole is greater in square than the annexed straight line by the square on a straight line commensurable in length with itself, let it be called a third apotome.
§10.def3.4πάλιν, ἐὰν ἡ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, ἐὰν μὲν ἡ ὅλη σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, καλείσθω ἀποτομὴ τετάρτη.
Again, if the whole is greater in square than the annexed straight line by the square on a straight line incommensurable in length with itself, if the whole is commensurable in length with the set-out rational straight line, let it be called a fourth apotome.
§10.def3.5ἐὰν δὲ ἡ προσαρμόζουσα, πέμπτη.
And if the annexed straight line, a fifth.
§10.def3.6ἐὰν δὲ μηδετέρα, ἕκτη.
And if neither, a sixth.

Notes

  1. §10.def3.1ὑποκειμένης ῥητῆς καὶ ἀποτομῆς — A genitive absolute construction establishing the initial conditions of a given rational straight line and an apotome.
  2. §10.def3.1μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει — The verb δύναται expresses comparative power in square ("is greater in square"). τῷ... is a dative of measure of difference. ἑαυτῇ refers to the subject, the whole (ἡ ὅλη).
  3. §10.def3.5ἐὰν δὲ ἡ προσαρμόζουσα, πέμπτη — A highly elliptical expression. Under the same condition as §10.def3.4 (that the whole is greater in square than the annexed straight line by the square on an incommensurable line), it means: "if the annexed straight line is commensurable, let it be called a fifth apotome."

Cite this passage

Euclid, Elements §10.def3.1-10.def3.6. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:10.def3.1-10.def3.6

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