Pappus Alexandrinus, Mathematical Collection, Book 8, 1876

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    <archimedes>
      <text>
        <body>
          <chap>
            <p>
              <s id="id.000246">
                <pb n="1100"/>
              ἡ ΑΖ, καὶ ἀπειλήφθω αὐτῆς τὸ γ# μέρος, καὶ ἔστω ἡ ΑΓ,
                <lb n="1"/>
              ἐφ' ἧς τμῆμα κύκλου γεγράφθω τὸ ΑΒΓ δεχόμενον γωνίαν
                <lb n="2"/>
              διμοίρου ὀρθῆς, καὶ οἵων ἐστὶν ἡ ΑΓ ε#, τοιούτων δ# ἀπει-
                <lb n="3"/>
              λήφθω ἡ ΓΕ, καὶ ἤχθω ἐφαπτομένη τοῦ τμήματος ἡ ΕΒ,
                <lb n="4"/>
              καὶ ἐπεζεύχθω ἥ τε ΑΒ καὶ ἡ ΖΒ, καὶ ἔτι ἐπιζευχθεῖσα
                <lb n="5"/>
              ἡ ΒΓ ἐκβεβλήσθω ἐπὶ τὸ Δ, καὶ κείσθω τῇ ΑΒ ἴση ἡ
                <lb n="6"/>
              ΒΔ, καὶ ἐπεζεύχθω ἡ ΑΔ. </s>
              <s id="id.000247">ἐπεὶ οὖν εἰς κύκλον διήχθησαν
                <lb n="7"/>
              ἥ τε ΕΓΑ καὶ ἡ ΕΒ, καὶ ἡ μὲν τέμνει τὸν κύκλον ἡ δὲ
                <lb n="8"/>
              ἐφάπτεται, τὸ ἄρα ὑπὸ ΑΕΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς ΕΒ·
                <lb n="9"/>
              ἔστιν ἄρα ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως ἡ ΒΕ πρὸς ΓΕ·
                <lb n="10"/>
              ἰσογώνιον ἄρα τὸ ΓΒΕ τρίγωνον τῷ ΑΒΕ τριγώνῳ. </s>
              <s id="id.000248">ἔστιν
                <lb n="11"/>
              ἄρα ὡς ἡ ΕΑ πρὸς ΑΒ, ἡ ΕΒ πρὸς ΒΓ· καὶ ὡς ἄρα τὸ
                <lb n="12"/>
              ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΕΒ, τὸ ἀπὸ τῆς ΑΒ πρὸς
                <lb n="13"/>
              τὸ ἀπὸ τῆς ΒΓ. </s>
              <s id="id.000249">ἀλλ' ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ
                <lb n="14"/>
              τῆς ΕΒ, οὕτως ἐστὶν ἡ ΑΕ πρὸς ΕΓ διὰ κ# τοῦ ς#. </s>
              <s id="id.000250">καὶ
                <lb n="15"/>
              ὡς ἄρα ἡ ΑΕ πρὸς ΕΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ, τουτέστιν
                <lb n="16"/>
              τὸ ἀπὸ τῆς ΒΔ, πρὸς τὸ ἀπὸ τῆς ΒΓ· τὸ ἄρα ἀπὸ τῆς
                <lb n="17"/>
              ΒΔ πρὸς τὸ ἀπὸ τῆς ΒΓ λόγον ἔχει ὃν τὰ θ# πρὸς δ#·
                <lb n="18"/>
              ἡμιολία ἄρα ἡ ΒΔ τῆς ΒΓ· διπλασία ἄρα ἡ ΒΓ τῆς ΓΔ.
                <lb n="19"/>
              </s>
              <s id="id.000251">ἔστιν δὲ καὶ ἡ ΖΓ τῆς ΓΑ διπλασία· ὡς ἄρα ἡ ΖΓ πρὸς
                <lb n="20"/>
              ΓΑ, ἡ ΒΓ πρὸς ΓΔ. </s>
              <s id="id.000252">καὶ ἴσαι εἰσὶν αἱ πρὸς τῷ Γ γω-
                <lb n="21"/>
              νίαι· ἴση ἄρα καὶ ἡ μὲν Δ γωνία τῇ ὑπὸ ΖΒΓ, ἡ δὲ Ζ
                <lb n="22"/>
              τῇ ὑπὸ ΓΑΔ· ἔστιν ἄρα ὡς ἡ ΖΒ πρὸς ΒΓ, οὕτως ἡ ΑΔ
                <lb n="23"/>
              πρὸς ΔΓ. </s>
              <s id="id.000253">ἐναλλὰξ ὡς ἡ ΖΒ πρὸς ΑΔ, οὕτως ἡ ΒΓ πρὸς
                <lb n="24"/>
              ΓΔ. </s>
              <s id="id.000254">διπλασία δὲ ἡ ΒΓ τῆς ΓΔ· διπλασία ἄρα καὶ ἡ ΖΒ
                <lb n="25"/>
              τῆς ΑΔ, τουτέστιν τῆς ΑΒ. </s>
              <s id="id.000255">καὶ ἔστιν διμοίρου ἡ Δ·
                <lb n="26"/>
              διμοίρου ἄρα ὀρθῆς καὶ ἡ ὑπὸ ΖΒΓ. </s>
              <s id="id.000256">ὅλη δὲ ἡ ὑπὸ ΑΒΖ
                <lb n="27"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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