Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica
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            <s xml:id="echoid-s2048" xml:space="preserve">
              <pb o="390" file="0104" n="111" rhead="CHRISTIANI HUGENII"/>
            ad H Q. </s>
            <s xml:id="echoid-s2049" xml:space="preserve">Et dividendo, portio K A L ad portionem
              <lb/>
            K C L ut E Q ad Q H, hoc eſt ut S ad T. </s>
            <s xml:id="echoid-s2050" xml:space="preserve">Quod erat fa-
              <lb/>
            ciendum. </s>
            <s xml:id="echoid-s2051" xml:space="preserve">Quod autem dictum fuit O Q duplam eſſe ipſius
              <lb/>
            M N, ſic fiet manifeſtum. </s>
            <s xml:id="echoid-s2052" xml:space="preserve">Quia enim ut quadratum C M
              <lb/>
            ad quadr. </s>
            <s xml:id="echoid-s2053" xml:space="preserve">M N, ita eſt M N ad N O longitudine: </s>
            <s xml:id="echoid-s2054" xml:space="preserve">Eſt au-
              <lb/>
            tem Q M æqualis ſubtenſæ arcus A R cujus trienti ſubten-
              <lb/>
            ditur M N. </s>
            <s xml:id="echoid-s2055" xml:space="preserve">Erunt propterea duæ ſimul Q M & </s>
            <s xml:id="echoid-s2056" xml:space="preserve">N O æqua-
              <lb/>
            les triplæ M N, uti ſequenti lemmate demonſtratur. </s>
            <s xml:id="echoid-s2057" xml:space="preserve">Quam-
              <lb/>
            obrem ablata communi O N, erit ſola Q M æqualis du-
              <lb/>
            plæ N M & </s>
            <s xml:id="echoid-s2058" xml:space="preserve">ipſi M O. </s>
            <s xml:id="echoid-s2059" xml:space="preserve">Sed eadem Q M æqualis eſt dua-
              <lb/>
            bus ſimul his Q O, O M, ergo apparet duplam M N æqua-
              <lb/>
            ri ipſi O Q.</s>
            <s xml:id="echoid-s2060" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div110" type="section" level="1" n="48">
          <head xml:id="echoid-head75" xml:space="preserve">LEMMA.</head>
          <p>
            <s xml:id="echoid-s2061" xml:space="preserve">SI Circumferentiæ arcus in tria æqualia ſecetur, tres ſimul
              <lb/>
              <note position="left" xlink:label="note-0104-01" xlink:href="note-0104-01a" xml:space="preserve">TAB. XLI.
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              Fig. 2.</note>
            rectæ quæ æqualibus partibus ſubtenduntur, æquantur ſub-
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            tenſæ arcus totius & </s>
            <s xml:id="echoid-s2062" xml:space="preserve">ei quæ ad ſubtenſam trientis ſeſe habe-
              <lb/>
            at, ſicut hujus quadratum ad quadratum ſemidiametri. </s>
            <s xml:id="echoid-s2063" xml:space="preserve">Ar-
              <lb/>
            cus ſectoris A B C in tria æqualia diviſus ſit punctis D, E.
              <lb/>
            </s>
            <s xml:id="echoid-s2064" xml:space="preserve">Et ſubtendantur partibus rectæ B D, D E, E C; </s>
            <s xml:id="echoid-s2065" xml:space="preserve">& </s>
            <s xml:id="echoid-s2066" xml:space="preserve">toti
              <lb/>
            arcui linea B C. </s>
            <s xml:id="echoid-s2067" xml:space="preserve">Porro jungantur D A, E A, atque inter-
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            ſecent ſubtenſam B C in punctis G & </s>
            <s xml:id="echoid-s2068" xml:space="preserve">H. </s>
            <s xml:id="echoid-s2069" xml:space="preserve">Sitque H F pa-
              <lb/>
            rallela G D.</s>
            <s xml:id="echoid-s2070" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2071" xml:space="preserve">Quoniam igitur circumferentia B D E dupla eſt circum-
              <lb/>
            ferentiæ E C, angulus autem huic inſiſtens E A C ad cen-
              <lb/>
            trum conſtitutus, qui vero illi inſiſtit angulus B C E ad
              <lb/>
            circumferentiam. </s>
            <s xml:id="echoid-s2072" xml:space="preserve">Erit propterea angulus B C E, hoc eſt,
              <lb/>
            angulus H C E in triangulo H C E æqualis angulo C A E
              <lb/>
            in triangulo C A E. </s>
            <s xml:id="echoid-s2073" xml:space="preserve">Sed angulus ad E utrique eſt commu-
              <lb/>
            nis; </s>
            <s xml:id="echoid-s2074" xml:space="preserve">itaque ſimiles inter ſe ſunt dicti trianguli: </s>
            <s xml:id="echoid-s2075" xml:space="preserve">Eritque ut
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            A E ad E C ita E C ad E H. </s>
            <s xml:id="echoid-s2076" xml:space="preserve">Ratio igitur A E ad E H
              <lb/>
            hoc eſt D E ad E F, duplicata eſt rationis A E ad E C,
              <lb/>
            ac proinde cadem quæ quadrati A E ad quadr. </s>
            <s xml:id="echoid-s2077" xml:space="preserve">E C ſeu
              <lb/>
            quadr. </s>
            <s xml:id="echoid-s2078" xml:space="preserve">E D. </s>
            <s xml:id="echoid-s2079" xml:space="preserve">Erit igitur invertendo F E ad E D ſicut
              <lb/>
            quadr. </s>
            <s xml:id="echoid-s2080" xml:space="preserve">E D ad quadr. </s>
            <s xml:id="echoid-s2081" xml:space="preserve">E A. </s>
            <s xml:id="echoid-s2082" xml:space="preserve">Quamobrem oſtendendum
              <lb/>
            eſt, tres ſimul B D, D E, E C æquari ſubtenſæ B </s>
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