Humanitext Reader

Euclid · Elements §12.prop.16

Inscribing an Even-Sided Polygon Not Touching the Inner Circle

Passage 289 of 316 · Greek

Summary

This proposition demonstrates how to inscribe an equilateral and even-sided polygon in the larger of two concentric circles such that it does not touch the smaller circle, proving it through successive bisections of arcs based on the principle of exhaustion.

§12.prop.16δύο κύκλων περὶ τὸ αὐτὸ κέντρον ὄντων εἰς τὸν μείζονα κύκλον πολύγωνον ἰσόπλευρόν τε καὶ ἀρτιόπλευρον ἐγγράψαι μὴ ψαῦον τοῦ ἐλάσσονος κύκλου.
Two circles being around the same center, to inscribe in the greater circle an equilateral and even-sided polygon which does not touch the lesser circle.
ἔστωσαν οἱ δοθέντες δύο κύκλοι οἱ ΑΒΓΔ, ΕΖΗΘ περὶ τὸ αὐτὸ κέντρον τὸ Κ· δεῖ δὴ εἰς τὸν μείζονα κύκλον τὸν ΑΒΓΔ πολύγωνον ἰσόπλευρόν τε καὶ ἀρτιόπλευρον ἐγγράψαι μὴ ψαῦον τοῦ ΕΖΗΘ κύκλου.
Let the two given circles be ABGD, EZHTh around the same center K; it is required then to inscribe in the greater circle ABGD an equilateral and even-sided polygon which does not touch the circle EZHTh.
ἤχθω γὰρ διὰ τοῦ Κ κέντρου εὐθεῖα ἡ ΒΚΔ, καὶ ἀπὸ τοῦ Η σημείου τῇ ΒΔ εὐθείᾳ πρὸς ὀρθὰς ἤχθω ἡ ΗΑ καὶ διήχθω ἐπὶ τὸ Γ· ἡ ΑΓ ἄρα ἐφάπτεται τοῦ ΕΖΗΘ κύκλου.
For let the straight line BKD be drawn through the center K, and from the point H let HA be drawn at right angles to the straight line BD and let it be drawn through to G; therefore AC touches the circle EZHTh.
τέμνοντες δὴ τὴν ΒΑΔ περιφέρειαν δίχα καὶ τὴν ἡμίσειαν αὐτῆς δίχα καὶ τοῦτο ἀεὶ ποιοῦντες καταλείψομεν περιφέρειαν ἐλάσσονα τῆς ΑΔ. λελείφθω, καὶ ἔστω ἡ ΛΔ, καὶ ἀπὸ τοῦ Λ ἐπὶ τὴν ΒΔ κάθετος ἤχθω ἡ ΛΜ καὶ διήχθω ἐπὶ τὸ Ν, καὶ ἐπεζεύχθωσαν αἱ ΛΔ, ΔΝ·
If then we bisect the circumference BAD, and bisect its half, and do this continually, we shall leave a circumference less than AD. Let it be left, and let it be LD, and from L let LM be drawn perpendicular to BD and let it be drawn through to N, and let LD, DN be joined; therefore LD is equal to DN.
ἴση ἄρα ἐστὶν ἡ ΛΔ τῇ ΔΝ. καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΛΝ τῇ ΑΓ, ἡ δὲ ΑΓ ἐφάπτεται τοῦ ΕΖΗΘ κύκλου, ἡ ΛΝ ἄρα οὐκ ἐφάπτεται τοῦ ΕΖΗΘ κύκλου· πολλῷ ἄρα αἱ ΛΔ, ΔΝ οὐκ ἐφάπτονται τοῦ ΕΖΗΘ κύκλου.
And since LN is parallel to AC, and AC touches the circle EZHTh, therefore LN does not touch the circle EZHTh; much less then do LD, DN touch the circle EZHTh.
ἐὰν δὴ τῇ ΛΔ εὐθείᾳ ἴσας κατὰ τὸ συνεχὲς ἐναρμόσωμεν εἰς τὸν ΑΒΓΔ κύκλον, ἐγγραφήσεται εἰς τὸν ΑΒΓΔ κύκλον πολύγωνον ἰσόπλευρόν τε καὶ ἀρτιόπλευρον μὴ ψαῦον τοῦ ἐλάσσονος κύκλου τοῦ ΕΖΗΘ· ὅπερ ἔδει ποιῆσαι.
If then we fit into the circle ABGD straight lines equal to the straight line LD continuously, there will be inscribed in the circle ABGD an equilateral and even-sided polygon which does not touch the lesser circle EZHTh; which it was required to do.

Notes

  1. 12.prop.16ἐγγράψαι — An independent infinitive typically used at the end of a problem's enunciation. It stands without a governing finite verb, expressing a command or the objective of construction ("to inscribe").
  2. 12.prop.16τέμνοντες δὴ τὴν ΒΑΔ περιφέρειαν δίχα ... καταλείψομεν — The present participle `τέμνοντες` ("bisecting") denotes the means modifying the future active verb `καταλείψομεν` ("we shall leave"). This is a geometric application of Book X, Proposition 1 (the foundational principle of exhaustion, showing that continuous bisection will eventually leave a magnitude smaller than any given magnitude).
  3. 12.prop.16ἴσας — A feminine plural accusative adjective. In this context, the feminine noun `εὐθείας` ("straight lines") is omitted, meaning "if we fit [straight lines] equal to the straight line LD continuously into the circle" as chords.

Cite this passage

Euclid, Elements §12.prop.16. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:12.prop.16

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