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

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          <pb o="425" file="0143" n="152" rhead="ET HYPERBOLÆ QUADRATURA."/>
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        <div xml:id="echoid-div160" type="section" level="1" n="74">
          <head xml:id="echoid-head110" xml:space="preserve">PROP. VIII. PROBLEMA.</head>
          <p style="it">
            <s xml:id="echoid-s3062" xml:space="preserve">Sint duæ quantitates datæ A, B, & </s>
            <s xml:id="echoid-s3063" xml:space="preserve">ratio quæli-
              <lb/>
            libet data C ad D: </s>
            <s xml:id="echoid-s3064" xml:space="preserve">oportet invenire aliam
              <lb/>
            quantitatem, ut ratio ejus ad A ſit multipli-
              <lb/>
            cata rationis B ad A in ratione C ad D.</s>
            <s xml:id="echoid-s3065" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3066" xml:space="preserve">Sit primò ratio C ad D commen-
              <lb/>
              <note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">
                <lb/>
              EDC # AFBG
                <lb/>
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            ſurabilis, ſitque inter C & </s>
            <s xml:id="echoid-s3067" xml:space="preserve">D com-
              <lb/>
            munis menſura E; </s>
            <s xml:id="echoid-s3068" xml:space="preserve">& </s>
            <s xml:id="echoid-s3069" xml:space="preserve">quoties E continetur in D toties ſit
              <lb/>
            ratio F ad A ſubmultiplicata rationis B ad A; </s>
            <s xml:id="echoid-s3070" xml:space="preserve">& </s>
            <s xml:id="echoid-s3071" xml:space="preserve">quoties E
              <lb/>
            continetur in C toties ſit ratio G ad A multiplicata rationis
              <lb/>
            F ad A: </s>
            <s xml:id="echoid-s3072" xml:space="preserve">dico G eſſe quantitatem illam quæſitam. </s>
            <s xml:id="echoid-s3073" xml:space="preserve">ratio G ad
              <lb/>
            A eſt multiplicata rationis F ad A in ratione C ad E, & </s>
            <s xml:id="echoid-s3074" xml:space="preserve">ra-
              <lb/>
            tio F ad A eſt multiplicata rationis B ad A in ratione E ad D; </s>
            <s xml:id="echoid-s3075" xml:space="preserve">& </s>
            <s xml:id="echoid-s3076" xml:space="preserve">
              <lb/>
            igitur ex æqualitate, ratio G ad A eſt multiplicata rationis
              <lb/>
            B ad A in ratione C ad D, quod demonſtrare oportuit.</s>
            <s xml:id="echoid-s3077" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3078" xml:space="preserve">Quod ſi ratio C ad D ſit incommenſurabilis, geometricam
              <lb/>
            hujus problematis praxim eſſe impoſſibilem mihi perſuadeo;
              <lb/>
            </s>
            <s xml:id="echoid-s3079" xml:space="preserve">approximatione tamen fieri poteſt, aſſumendo rationem com-
              <lb/>
            menſurabilem ejus loco, quæ quàm proximè ad illam acce-
              <lb/>
            dat.</s>
            <s xml:id="echoid-s3080" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s3081" xml:space="preserve">Sit ſeries convergens, cujus primi
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              ## G # H # A # B
                <lb/>
              # N # I # K # C # D
                <lb/>
              M ## R # S # E # F
                <lb/>
              # O # T # V # X # Y
                <lb/>
              ### L ## Z
                <lb/>
              </note>
            termini convergentes ſint A, B, ſe-
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            cundi C, D, tertii E, F; </s>
            <s xml:id="echoid-s3082" xml:space="preserve">ſintque
              <lb/>
            ſecundi termini ita facti à primis, ut
              <lb/>
            ratio B majoris ad A minorem ſit
              <lb/>
            multiplicata rationis C ad A in ra-
              <lb/>
            tione data mojoris inæqualitatis M ad N, & </s>
            <s xml:id="echoid-s3083" xml:space="preserve">ut ratio B ad
              <lb/>
            A ſit multiplicata rationis D ad A in ratione data majoris
              <lb/>
            inæqualitatis M ad O: </s>
            <s xml:id="echoid-s3084" xml:space="preserve">ſintque tertii termini eodem modo
              <lb/>
            facti ex ſecundis quo ſecundi facti ſunt ex primis; </s>
            <s xml:id="echoid-s3085" xml:space="preserve">atque ita
              <lb/>
            continuetur ſeries.</s>
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