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

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              <pb o="327" file="0031" n="33" rhead="HYPERB. ELLIPS. ET CIRC."/>
            tertiis rectanguli K F B , id eſt, duabus tertiis
              <note symbol="4" position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">16. lib. 6.
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              Elem.</note>
            A F; </s>
            <s xml:id="echoid-s379" xml:space="preserve">ſed idem rectangulum ſub Q F, D H, æquale eſt re-
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            ctangulo Q D R, quia ut Q D ad Q F, ita fecimus eſſe
              <lb/>
            D H ad D R; </s>
            <s xml:id="echoid-s380" xml:space="preserve">ergo rectangulum Q D R æquale duabus ter-
              <lb/>
            tiis quadrati A F, ideoque ut Q D ad A F ita {2/3} A F ad D R
              <lb/>
            : </s>
            <s xml:id="echoid-s381" xml:space="preserve">ſed ut Q D ad A F, ſic quoque eſt rectangulum ſub Q
              <note symbol="5" position="right" xlink:label="note-0031-02" xlink:href="note-0031-02a" xml:space="preserve">16. lib. 6.
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              Elem.</note>
            A F, cui æquale quadrilaterum D A Q C, id eſt, ſector
              <lb/>
            D A B C ad A F quadratum; </s>
            <s xml:id="echoid-s382" xml:space="preserve">ergo & </s>
            <s xml:id="echoid-s383" xml:space="preserve">ſector D A B C ad
              <lb/>
            quadratum A F, ut {2/3} A F ad D R. </s>
            <s xml:id="echoid-s384" xml:space="preserve">Porro quoniam E cen-
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            trum gravitatis eſt totius ſectoris, & </s>
            <s xml:id="echoid-s385" xml:space="preserve">H centrum grav. </s>
            <s xml:id="echoid-s386" xml:space="preserve">por-
              <lb/>
            tionis A C B, G vero trianguli A C D, conſtat eſſe, ſicut
              <lb/>
            triangulus A C D ad A C B portionem ſive ad triangulum
              <lb/>
            A Q C, id eſt, ut D F ad F Q, ita H E ad E G ; </s>
            <s xml:id="echoid-s387" xml:space="preserve">
              <note symbol="6" position="right" xlink:label="note-0031-03" xlink:href="note-0031-03a" xml:space="preserve">8. lib. 1.
                <lb/>
              Arch. de
                <lb/>
              Æquipond.</note>
            convertendo & </s>
            <s xml:id="echoid-s388" xml:space="preserve">per compoſitionem rationis erit ut D Q ad
              <lb/>
            D F, ita G H, ad H E. </s>
            <s xml:id="echoid-s389" xml:space="preserve">Sed quia fecimus ut D Q ad Q F,
              <lb/>
            ita H D ad D R, erit quoque per converſionem rationis,
              <lb/>
            ut D Q ad D F, ita H D ad H R; </s>
            <s xml:id="echoid-s390" xml:space="preserve">ergo H D ad H R ut
              <lb/>
            G H ad H E; </s>
            <s xml:id="echoid-s391" xml:space="preserve">quare & </s>
            <s xml:id="echoid-s392" xml:space="preserve">reliqua G D ad reliquam E R, ut
              <lb/>
              <note symbol="7" position="right" xlink:label="note-0031-04" xlink:href="note-0031-04a" xml:space="preserve">19. lib. 5.
                <lb/>
              Elem.</note>
            H D ad H R , hoc eſt, ut D Q ad D F. </s>
            <s xml:id="echoid-s393" xml:space="preserve">Sicut autem D Q ad D F, ita eſt quadrilaterum D A Q C, cui æqualis ſector
              <lb/>
            D A B C ad A C D triangulum; </s>
            <s xml:id="echoid-s394" xml:space="preserve">igitur ſector D A B C
              <lb/>
            ad A C D triangulum ut G D ad E R: </s>
            <s xml:id="echoid-s395" xml:space="preserve">Eſt autem A C D
              <lb/>
            triangulus ad D F quadratum, ut A F ad D F, ſive ut {2/3} A F
              <lb/>
            ad {2/3} D F id eſt D G. </s>
            <s xml:id="echoid-s396" xml:space="preserve">Igitur ex æquali in proportione perturbata,
              <lb/>
            ſicut ſector D A B C ad quadratum D F, ita {2/3} A F ad E R
              <lb/>
            , & </s>
            <s xml:id="echoid-s397" xml:space="preserve">convertendo, quadratum D F ad ſectorem D A B
              <note symbol="8" position="right" xlink:label="note-0031-05" xlink:href="note-0031-05a" xml:space="preserve">23. lib. 5.
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              Elem.</note>
            ut E R ad {2/3} A F. </s>
            <s xml:id="echoid-s398" xml:space="preserve">Fuit autem ante oſtenſum, quadratum
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            A F eſſe ad ſectorem D A B C, ut D R ad {2/3} A F; </s>
            <s xml:id="echoid-s399" xml:space="preserve">igitur
              <lb/>
            duo ſimul quadrata, D F & </s>
            <s xml:id="echoid-s400" xml:space="preserve">A F, ſive unum quadratum
              <lb/>
            D A ad ſectorem D A B C ut duæ ſimul E R & </s>
            <s xml:id="echoid-s401" xml:space="preserve">R D, id
              <lb/>
            eſt ut tota E D ad {2/3} A F . </s>
            <s xml:id="echoid-s402" xml:space="preserve">Eſt verò etiam quadratum D
              <note symbol="9" position="right" xlink:label="note-0031-06" xlink:href="note-0031-06a" xml:space="preserve">24 lib. 5.
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              Elem.</note>
            ad D A B C ſectorem, ſicut linea D A ad arcum A B, quia
              <lb/>
            nimirum ſector D A B C æqualis eſt rectangulo, baſin ha-
              <lb/>
            benti æqualem arcui A B & </s>
            <s xml:id="echoid-s403" xml:space="preserve">altitudinem D A; </s>
            <s xml:id="echoid-s404" xml:space="preserve">ergo ſicut
              <lb/>
            D A ad arcum A B, ita E D ad {2/3} A F; </s>
            <s xml:id="echoid-s405" xml:space="preserve">& </s>
            <s xml:id="echoid-s406" xml:space="preserve">permutando,
              <lb/>
            arcus A B ad {2/3} A F, ſive arcus A B C ad {2/3} A C, ut D A
              <lb/>
            vel B D ad D E.</s>
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