Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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        <div xml:id="echoid-div145" type="section" level="1" n="63">
          <pb o="415" file="0133" n="142"/>
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        <div xml:id="echoid-div146" type="section" level="1" n="64">
          <head xml:id="echoid-head95" xml:space="preserve">VERA
            <lb/>
          CIRCULI ET HYPERBOLÆ
            <lb/>
          QUADRATURA.</head>
          <p>
            <s xml:id="echoid-s2778" xml:space="preserve">Sit circuli, ellipſeos vel hyperbolæ ſegmentum B I P
              <lb/>
              <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">TAB. XLIII.
                <lb/>
              Fig. 1. 2. 3.</note>
            cujus centrum A: </s>
            <s xml:id="echoid-s2779" xml:space="preserve">compleatur triangulum A B P, & </s>
            <s xml:id="echoid-s2780" xml:space="preserve">
              <lb/>
            ſegmentum in punctis, B, P, tangentes ducantur re-
              <lb/>
            ctæ B F, P F, ſe invicem ſecantes in puncto F; </s>
            <s xml:id="echoid-s2781" xml:space="preserve">pro-
              <lb/>
            ducatur (ſi opus ſit) recta A F ſegmentum interſecans in
              <lb/>
            puncto I & </s>
            <s xml:id="echoid-s2782" xml:space="preserve">rectam B P in puncto Q; </s>
            <s xml:id="echoid-s2783" xml:space="preserve">deinde jungantur re-
              <lb/>
            ctæ B I, P I.</s>
            <s xml:id="echoid-s2784" xml:space="preserve"/>
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        <div xml:id="echoid-div148" type="section" level="1" n="65">
          <head xml:id="echoid-head96" xml:space="preserve">PROP. I. THEOREMA.</head>
          <head xml:id="echoid-head97" style="it" xml:space="preserve">Dico trapezium B A P I eſſe medium propor-
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          tionale inter trapezium B A P F, &
            <lb/>
          triangulum B A P.</head>
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            <s xml:id="echoid-s2785" xml:space="preserve">Quoniam recta A Q ducitur per F concurſum duarum re-
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            ctarum F B, F P, ſegmentum in punctis B, P, tan-
              <lb/>
            gentium; </s>
            <s xml:id="echoid-s2786" xml:space="preserve">igitur recta A Q rectam B P contactuum
              <lb/>
            puncta jungentem bifariam ſecabit in puncto Q; </s>
            <s xml:id="echoid-s2787" xml:space="preserve">& </s>
            <s xml:id="echoid-s2788" xml:space="preserve">proinde
              <lb/>
            triangulum A B Q eſt æquale triangulo A Q P, & </s>
            <s xml:id="echoid-s2789" xml:space="preserve">trian-
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            gnlum F B Q triangulo F Q P; </s>
            <s xml:id="echoid-s2790" xml:space="preserve">& </s>
            <s xml:id="echoid-s2791" xml:space="preserve">igitur triangulum A B F
              <lb/>
            æquale eſt triangulo A P F; </s>
            <s xml:id="echoid-s2792" xml:space="preserve">eſt ergo triangulum A B F di-
              <lb/>
            midium trapezii A B F P: </s>
            <s xml:id="echoid-s2793" xml:space="preserve">eodem modo probatur triangu-
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            lum A B I eſſe dimidium trapezii A B I P; </s>
            <s xml:id="echoid-s2794" xml:space="preserve">& </s>
            <s xml:id="echoid-s2795" xml:space="preserve">triangulum
              <lb/>
            A B Q eſt dimidium trianguli A B P: </s>
            <s xml:id="echoid-s2796" xml:space="preserve">cumque triangula
              <lb/>
            A B F, A B I, A B Q, eandem habeant altitudinem, in-
              <lb/>
            ter ſe ſunt ut baſes, ſed eorum baſes nempe A F, A I, A Q,
              <lb/>
            ſunt continuè proportionales; </s>
            <s xml:id="echoid-s2797" xml:space="preserve">& </s>
            <s xml:id="echoid-s2798" xml:space="preserve">igitur ipſa quoque triangu-
              <lb/>
            la ſunt continuè proportionalia; </s>
            <s xml:id="echoid-s2799" xml:space="preserve">& </s>
            <s xml:id="echoid-s2800" xml:space="preserve">proinde eorum dupla ni-
              <lb/>
            mirum trapezia A B F P, A B I P, & </s>
            <s xml:id="echoid-s2801" xml:space="preserve">triangulum A B P
              <lb/>
            ſunt continuè proportionalia in ratione A F ad A I, quod
              <lb/>
            demonſtrare oportuit.</s>
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