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

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        <div xml:id="echoid-div157" type="section" level="1" n="72">
          <p>
            <s xml:id="echoid-s3020" xml:space="preserve">
              <pb o="423" file="0141" n="150" rhead="ET HYPERBOLÆ QUADRATURA."/>
            quos terminos æquales appellamus ſeriei terminatio-
              <lb/>
            nem.</s>
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        <div xml:id="echoid-div159" type="section" level="1" n="73">
          <head xml:id="echoid-head108" xml:space="preserve">PROP. VII. PROBLEMA.</head>
          <head xml:id="echoid-head109" style="it" xml:space="preserve">Oportet prædictæ ſeriei terminationem
            <lb/>
          invenire.</head>
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            <s xml:id="echoid-s3022" xml:space="preserve">Ut huic problemati ſatisfiat, oportet primò invenire
              <lb/>
            quantitatem quæ eodem modo componitur ex termi-
              <lb/>
            nis convergentibus
              <emph style="super">a</emph>
            ,
              <emph style="super">b</emph>
            , quo ex terminis convergentibus
              <lb/>
            {ca + bd - ad/c}, {bc - bc + ae/c}, hoc autem facile fit hoc modo: </s>
            <s xml:id="echoid-s3023" xml:space="preserve">inveniatur
              <lb/>
            quantitas quæ multiplicata in
              <emph style="sub">a</emph>
            & </s>
            <s xml:id="echoid-s3024" xml:space="preserve">addita
              <emph style="super">b</emph>
            multiplicata in
              <lb/>
            quantitatem data m, eandem quantitatem facit ac ſi multi-
              <lb/>
            plicaretur in {ca + bd - ad/c} & </s>
            <s xml:id="echoid-s3025" xml:space="preserve">adderetur {bc - be + ae/c} multiplicata etiam
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            in eandem quantitatem data\m m. </s>
            <s xml:id="echoid-s3026" xml:space="preserve">ſit quantitas illa
              <emph style="sub">z</emph>
            , & </s>
            <s xml:id="echoid-s3027" xml:space="preserve">pro-
              <lb/>
            inde za + bm æquatur {zca + zbd - zad + mbe - mbe + mae/c}, & </s>
            <s xml:id="echoid-s3028" xml:space="preserve">æquatione
              <lb/>
            reducta invenitur {z = mac - mbe/ad - bd}; </s>
            <s xml:id="echoid-s3029" xml:space="preserve">quæ quantitas ſive multiplica-
              <lb/>
            ta in
              <emph style="super">a</emph>
            & </s>
            <s xml:id="echoid-s3030" xml:space="preserve">addita
              <emph style="super">m</emph>
            , ſive multiplicata in {ca + bd - ad/c} & </s>
            <s xml:id="echoid-s3031" xml:space="preserve">addita
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            {mbe - mbe + mae/c} efficit eandem in utroque caſu quantitatem nempe
              <lb/>
            {maae - mbae + mbad - mbbd/cd - bd}: </s>
            <s xml:id="echoid-s3032" xml:space="preserve">& </s>
            <s xml:id="echoid-s3033" xml:space="preserve">proinde prædicta quantitas eodem mo-
              <lb/>
            do componitur ex terminis convergentibus
              <emph style="super">a</emph>
            ,
              <emph style="super">b</emph>
            , quo compo-
              <lb/>
            nitur ex terminis convergentibus {ca + bd - ad/c}, {bc - be + ae/c}. </s>
            <s xml:id="echoid-s3034" xml:space="preserve">atque
              <emph style="super">a</emph>
              <lb/>
            & </s>
            <s xml:id="echoid-s3035" xml:space="preserve">
              <emph style="sub">b</emph>
            quoniam ſunt quantitates indefinitæ poſſunt eſſe quili-
              <lb/>
            bet totius ſeriei termini convergentes, modò termini con-
              <lb/>
            vergentes immediatè ſequentes ſint {ca + bd - ad/c} & </s>
            <s xml:id="echoid-s3036" xml:space="preserve">{bc - be + ae/c}, & </s>
            <s xml:id="echoid-s3037" xml:space="preserve">
              <lb/>
            proinde quantitas {maae - mbae + mbad - mbbd/cd - bd} eodem modo componi-
              <lb/>
            tur ex quibuslibet totius ſeriei terminis convergentibus quo
              <lb/>
            componitur ex terminis convergentibus
              <emph style="super">a</emph>
            ,
              <emph style="super">b</emph>
            ; </s>
            <s xml:id="echoid-s3038" xml:space="preserve">& </s>
            <s xml:id="echoid-s3039" xml:space="preserve">igitur </s>
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