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

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          <pb o="417" file="0135" n="144" rhead="ET HYPERBOLÆ QUADRATURA."/>
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        <div xml:id="echoid-div150" type="section" level="1" n="67">
          <head xml:id="echoid-head100" xml:space="preserve">PROP. III. THEOREMA.</head>
          <head xml:id="echoid-head101" style="it" xml:space="preserve">Dico triangulum B A P, & trapezium A B I P ſimul,
            <lb/>
          eſſe ad trapezium A B I P, ut duplum trapezii A B I P ad polygonum A B D L P.</head>
          <note position="right" xml:space="preserve">TAB. XLIII.
            <lb/>
          Fig. 1. 2. 3.</note>
          <p>
            <s xml:id="echoid-s2831" xml:space="preserve">In antecedente demonſtratum eſt trapezia A B F P, A B I P
              <lb/>
            ſimul, eſſe ad duplum trapezii A B I P, ſicut trapezium
              <lb/>
            A B F P ad polygonum A B D L P: </s>
            <s xml:id="echoid-s2832" xml:space="preserve">& </s>
            <s xml:id="echoid-s2833" xml:space="preserve">permutando tra-
              <lb/>
            pezia A B F P, A B I P ſimul, ſunt ad trapezium A B F P,
              <lb/>
            ut duplum trapezii A B I P ad polygonum A B D L P. </s>
            <s xml:id="echoid-s2834" xml:space="preserve">& </s>
            <s xml:id="echoid-s2835" xml:space="preserve">
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            quoniam trapezium A B F P, trapezium A B I P & </s>
            <s xml:id="echoid-s2836" xml:space="preserve">trian-
              <lb/>
            gulum A B P, ſunt continuè proportionalia; </s>
            <s xml:id="echoid-s2837" xml:space="preserve">erit trape-
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            zium A B I P ad trapezium A B F P, ut triangulum A B P
              <lb/>
            ad trapezium A B I P; </s>
            <s xml:id="echoid-s2838" xml:space="preserve">& </s>
            <s xml:id="echoid-s2839" xml:space="preserve">componendo, ut trapezia A B I P,
              <lb/>
            A B F P ſimul, ad trapezium A B F P, ita triangulum
              <lb/>
            A B P & </s>
            <s xml:id="echoid-s2840" xml:space="preserve">trapezium A B I P ſimul, ad trapezium A B I P:
              <lb/>
            </s>
            <s xml:id="echoid-s2841" xml:space="preserve">erat autem, ut trapezia A B I P, A B F P, ſimul, ad tra-
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            pezium A B F P, ita duplum trapezii A B I P ad polygo-
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            num A B D L P; </s>
            <s xml:id="echoid-s2842" xml:space="preserve">& </s>
            <s xml:id="echoid-s2843" xml:space="preserve">igitur ut triangulum A B P & </s>
            <s xml:id="echoid-s2844" xml:space="preserve">trape-
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            zium A B I P ſimul, ad trapezium A B I P, ita duplum
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            trapezii A B I P ad polygonum A B D L P, quod demon-
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            ſtrare oportuit.</s>
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          </p>
          <p>
            <s xml:id="echoid-s2846" xml:space="preserve">Producantur (ſi opus ſit) rectæ A D, A L, ſegmentum
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            ſecantes in punctis E & </s>
            <s xml:id="echoid-s2847" xml:space="preserve">O, & </s>
            <s xml:id="echoid-s2848" xml:space="preserve">rectas B I, I P, in H & </s>
            <s xml:id="echoid-s2849" xml:space="preserve">M:
              <lb/>
            </s>
            <s xml:id="echoid-s2850" xml:space="preserve">deinde jungantur rectæ B E, E I, I O, O P, ut complea-
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            tur polygonum A B E I O P.</s>
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          </p>
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        <div xml:id="echoid-div151" type="section" level="1" n="68">
          <head xml:id="echoid-head102" xml:space="preserve">PROP. IV. THEOREMA.</head>
          <head xml:id="echoid-head103" style="it" xml:space="preserve">Dico polygonum A B E I O P eſſe medium pro-
            <lb/>
          portionale inter polygonum A B D L & trapezium A B I P.</head>
          <note position="right" xml:space="preserve">TAB. XLIII.
            <lb/>
          Fig. 1. 2. 3.</note>
          <p>
            <s xml:id="echoid-s2852" xml:space="preserve">Ex hujus prima manifeſtum eſt trapezium A I L P, tra-
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            pezium A I O P & </s>
            <s xml:id="echoid-s2853" xml:space="preserve">triangulum A I P eſſe </s>
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