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

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          <p>
            <s xml:id="echoid-s3150" xml:space="preserve">
              <pb o="429" file="0147" n="156" rhead="ET HYPERBOLÆ QUADRATURA."/>
            omnes ſeriei convergentis terminationem eodem modo eſſe
              <lb/>
            compoſitam ex terminis convergentibus primis quo ex termi-
              <lb/>
            nis convergentibus ſecundis, tertiis, vel quartis, &</s>
            <s xml:id="echoid-s3151" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3152" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div166" type="section" level="1" n="78">
          <head xml:id="echoid-head114" xml:space="preserve">PROP. XI. THEOREMA.</head>
          <p style="it">
            <s xml:id="echoid-s3153" xml:space="preserve">Dico ſectorem circuli, ellipſeos vel hyperbolæ A B I P
              <lb/>
              <note position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">TAB. XLIII.
                <lb/>
              Fig. 1. 2. 3.</note>
            non eſſe compoſitum analyticè à triangulo
              <lb/>
            A B P & </s>
            <s xml:id="echoid-s3154" xml:space="preserve">trapezio A B F P.</s>
            <s xml:id="echoid-s3155" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3156" xml:space="preserve">Ponatur triangulum A B P
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3157" xml:space="preserve">trapezium A B F P
              <emph style="super">b</emph>
            : </s>
            <s xml:id="echoid-s3158" xml:space="preserve">ma-
              <lb/>
            nifeſtum eſt ex prædictis trapezium A B I P eſſe Vqab & </s>
            <s xml:id="echoid-s3159" xml:space="preserve">
              <lb/>
            polygonum A B D L P {2 ab/a + Vqab}, item ſectorem A B I P eſſe
              <lb/>
            hujus ſeriei convergentis terminationem. </s>
            <s xml:id="echoid-s3160" xml:space="preserve">ut ex ſeriei termi-
              <lb/>
            nis auferantur ſigna radicis & </s>
            <s xml:id="echoid-s3161" xml:space="preserve">fractionis, pro
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3162" xml:space="preserve">
              <emph style="super">b</emph>
            primis
              <lb/>
            ſeriei terminis convergentibus, hoc eſt pro triangulo A B P
              <lb/>
            & </s>
            <s xml:id="echoid-s3163" xml:space="preserve">trapezio A B F P ponantur a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b & </s>
            <s xml:id="echoid-s3164" xml:space="preserve">ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            ; </s>
            <s xml:id="echoid-s3165" xml:space="preserve">erunt-
              <lb/>
            que ſecundi ſeriei termini convergentes, hoc eſt trapezium
              <lb/>
            A B I P & </s>
            <s xml:id="echoid-s3166" xml:space="preserve">polygonum A B D L P, ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a & </s>
            <s xml:id="echoid-s3167" xml:space="preserve">2 b
              <emph style="super">2</emph>
            a, di-
              <lb/>
            co ſeriei convergentis (cujus primi termini convergentes ſunt
              <lb/>
            a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b, ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            & </s>
            <s xml:id="echoid-s3168" xml:space="preserve">ſecundi ſunt ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, 2 b
              <emph style="super">2</emph>
            a) termina-
              <lb/>
            tionem non eſſe compoſitam analyticè a terminis a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b,
              <lb/>
            ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            : </s>
            <s xml:id="echoid-s3169" xml:space="preserve">ſi enim componatur prædicta terminatio analyticè a
              <lb/>
            terminis convergentibus a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b, ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            ; </s>
            <s xml:id="echoid-s3170" xml:space="preserve">componetur etiam
              <lb/>
            eadem terminatio analyticè & </s>
            <s xml:id="echoid-s3171" xml:space="preserve">eodem omnino modo à termi-
              <lb/>
            nis convergentibus ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, 2b
              <emph style="super">2</emph>
            a; </s>
            <s xml:id="echoid-s3172" xml:space="preserve">& </s>
            <s xml:id="echoid-s3173" xml:space="preserve">proinde eadem quan-
              <lb/>
            titas, nempe prædicta terminatio, eodem modo componitur
              <lb/>
            analyticè ex terminis a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b, ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            , quo componitur ex
              <lb/>
            terminis ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, 2b
              <emph style="super">2</emph>
            a,
              <lb/>
              <note position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">
                <lb/>
              a
                <emph style="super">3</emph>
              + a
                <emph style="super">2</emph>
              b # ab
                <emph style="super">2</emph>
              + b
                <emph style="super">3</emph>
              .
                <lb/>
              ba
                <emph style="super">2</emph>
              + b
                <emph style="super">2</emph>
              a # 2b
                <emph style="super">2</emph>
              a
                <lb/>
              </note>
            ſed nulla quantitas poteſt
              <lb/>
            eodem modo analyticè com-
              <lb/>
            poni ex terminis a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b,
              <lb/>
            ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            , quo componitur
              <lb/>
            ex terminis ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, 2b
              <emph style="super">2</emph>
            a, quod ſic demonſtro. </s>
            <s xml:id="echoid-s3174" xml:space="preserve">ſi analy-
              <lb/>
            ticè componeretur quantitas ex terminis a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b, ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            </s>
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