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

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        <div xml:id="echoid-div160" type="section" level="1" n="74">
          <pb o="426" file="0144" n="153" rhead="VERA CIRCULI"/>
        </div>
        <div xml:id="echoid-div163" type="section" level="1" n="75">
          <head xml:id="echoid-head111" xml:space="preserve">PROP. IX. PROBLEMA.</head>
          <p style="it">
            <s xml:id="echoid-s3087" xml:space="preserve">Oportet prædictæ ſeriei terminationem invenire.</s>
            <s xml:id="echoid-s3088" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3089" xml:space="preserve">Ponatur G cyphra ſeu nihil hoc eſt exponens rationis æ-
              <lb/>
            qualitatis, ſeu rationis A ad A; </s>
            <s xml:id="echoid-s3090" xml:space="preserve">ſitque H ad libitum ex-
              <lb/>
            ponens rationis B ad A: </s>
            <s xml:id="echoid-s3091" xml:space="preserve">ſit ut M ad N ita differentia inter
              <lb/>
            G & </s>
            <s xml:id="echoid-s3092" xml:space="preserve">H, hoc eſt ipſa H vel exponens rationis B ad A ad ex-
              <lb/>
            ceſſum quo I ſuperat G hoc eſt ipſam I, ſed ut M ad N
              <lb/>
            ita ratio B ad A eſt multiplicata rationis C ad A; </s>
            <s xml:id="echoid-s3093" xml:space="preserve">& </s>
            <s xml:id="echoid-s3094" xml:space="preserve">igitur
              <lb/>
            Exceſſus quo I ſuperat G hoc eſt ipſa I eſt exponens ratio-
              <lb/>
            nis C ad A. </s>
            <s xml:id="echoid-s3095" xml:space="preserve">ſit ut M ad O ita differentia inter G & </s>
            <s xml:id="echoid-s3096" xml:space="preserve">H hoc
              <lb/>
            eſt H ad exceſſum quo K ſuperat G hoc eſt ipſam K, ſed
              <lb/>
            ut M ad O ita ratio B ad A eſt multiplicata rationis D ad
              <lb/>
            A, cumque H ſit exponens rationis B ad A, erit K expo-
              <lb/>
            nens rationis D ad A: </s>
            <s xml:id="echoid-s3097" xml:space="preserve">ſi igitur I ſit exponens rationis C ad
              <lb/>
            A & </s>
            <s xml:id="echoid-s3098" xml:space="preserve">K exponens rationis D ad A; </s>
            <s xml:id="echoid-s3099" xml:space="preserve">erit exceſſus quo K ſu-
              <lb/>
            perat I exponens rationis D ad C. </s>
            <s xml:id="echoid-s3100" xml:space="preserve">deinde ſit ut M ad N
              <lb/>
            ita exceſſus quo K ſuperat I ſeu exponens rationis D ad C
              <lb/>
            ad exceſſum quo R ſuperat I, ſed ut M ad N ita ex ſeriei
              <lb/>
            compoſitione ratio D ad C eſt multiplicata rationis E ad
              <lb/>
            C, atque exceſſus quo K ſuperat I eſt exponens rationis
              <lb/>
            D ad C; </s>
            <s xml:id="echoid-s3101" xml:space="preserve">& </s>
            <s xml:id="echoid-s3102" xml:space="preserve">proinde exceſſus quo R ſuperat I eſt exponens
              <lb/>
            rationis E ad C, atque I eſt exponens rationis C ad A, & </s>
            <s xml:id="echoid-s3103" xml:space="preserve">pro-
              <lb/>
            inde R eſt exponens rationis E ad A. </s>
            <s xml:id="echoid-s3104" xml:space="preserve">deinde ſit ut M ad
              <lb/>
            O ita exceſſus quo K ſuperat I ad exceſſum quo S ſuperat
              <lb/>
            I, ſed ut M ad O ita ex ſeriei compoſitione ratio D ad
              <lb/>
            C eſt multiplicata rationis F ad C, cumque exceſſus quo
              <lb/>
            K ſuperat I ſit exponens rationis D ad C; </s>
            <s xml:id="echoid-s3105" xml:space="preserve">erit exceſſus quo
              <lb/>
            S ſuperat I exponens rationis F ad C, atque I eſt expo-
              <lb/>
            nens rationis C ad A, & </s>
            <s xml:id="echoid-s3106" xml:space="preserve">proinde S eſt exponens rationis F
              <lb/>
            ad A: </s>
            <s xml:id="echoid-s3107" xml:space="preserve">cum igitur R ſit exponens E ad A & </s>
            <s xml:id="echoid-s3108" xml:space="preserve">S exponens ra-
              <lb/>
            tionis F ad A; </s>
            <s xml:id="echoid-s3109" xml:space="preserve">erit exceſſus quo S ſuperat R exponens ra-
              <lb/>
            tionis F ad E: </s>
            <s xml:id="echoid-s3110" xml:space="preserve">& </s>
            <s xml:id="echoid-s3111" xml:space="preserve">utramque ſeriem continuando, demonſtra-
              <lb/>
            tur ut antè T eſſe exponentem rationis X ad A, & </s>
            <s xml:id="echoid-s3112" xml:space="preserve">V </s>
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