Barrow, Isaac, Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

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          <p>
            <s xml:id="echoid-s10049" xml:space="preserve">
              <pb o="50" file="0228" n="243" rhead=""/>
            _d_ = _m_; </s>
            <s xml:id="echoid-s10050" xml:space="preserve">vel _dy_ - _xy_ = {_d_/_b_}_x x_. </s>
            <s xml:id="echoid-s10051" xml:space="preserve">Aliaquædam hîc (nonnulla forſan παρέ
              <unsure/>
            ργως)
              <lb/>
            inſeremus.</s>
            <s xml:id="echoid-s10052" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10053" xml:space="preserve">XIII. </s>
            <s xml:id="echoid-s10054" xml:space="preserve">Sit poſitione data recta ID; </s>
            <s xml:id="echoid-s10055" xml:space="preserve">ſit item curva DNN talis,
              <lb/>
              <note position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve">Fig. 46.</note>
            utin ID ſumpto puncto quopiam G, ductâque rectâ GN ad poſitio-
              <lb/>
            nem datam IK parallelâ; </s>
            <s xml:id="echoid-s10056" xml:space="preserve">ſumptiſque determinatis lineis _g, m, r_;
              <lb/>
            </s>
            <s xml:id="echoid-s10057" xml:space="preserve">poſitíſque DG = _x_, & </s>
            <s xml:id="echoid-s10058" xml:space="preserve">GN = _y_; </s>
            <s xml:id="echoid-s10059" xml:space="preserve">ſit perpetim _y x_ + _gx_ - _my_ =
              <lb/>
            {_m_/_r_}_x x_; </s>
            <s xml:id="echoid-s10060" xml:space="preserve">linea DNN erit _hyperbola_, ſic determinabilis: </s>
            <s xml:id="echoid-s10061" xml:space="preserve">Sumatur
              <lb/>
            DM = _m_; </s>
            <s xml:id="echoid-s10062" xml:space="preserve">& </s>
            <s xml:id="echoid-s10063" xml:space="preserve">per M ducatur ML ad IK parallela; </s>
            <s xml:id="echoid-s10064" xml:space="preserve">& </s>
            <s xml:id="echoid-s10065" xml:space="preserve">in hac acci-
              <lb/>
            piatur MQ = {_mm_/_r_}; </s>
            <s xml:id="echoid-s10066" xml:space="preserve">& </s>
            <s xml:id="echoid-s10067" xml:space="preserve">ſit QY = MQ; </s>
            <s xml:id="echoid-s10068" xml:space="preserve">& </s>
            <s xml:id="echoid-s10069" xml:space="preserve">ab MY auferatur
              <lb/>
            YZ = _g_; </s>
            <s xml:id="echoid-s10070" xml:space="preserve">connexâque QD, ducatur ZT ad QD parallelâ; </s>
            <s xml:id="echoid-s10071" xml:space="preserve">erunt
              <lb/>
            ZM, ZT _aſymptoti_.</s>
            <s xml:id="echoid-s10072" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10073" xml:space="preserve">Nam ducatur ZS ad MD parallela; </s>
            <s xml:id="echoid-s10074" xml:space="preserve">cui occurrat GN producta in
              <lb/>
            R (ſed & </s>
            <s xml:id="echoid-s10075" xml:space="preserve">GR ipſam ZT ſecet in P). </s>
            <s xml:id="echoid-s10076" xml:space="preserve">Eſtque jam PN = RG -
              <lb/>
            RP - GN = {_mm_/_r_} - _g_ + {_mx_/_r_} - _y_. </s>
            <s xml:id="echoid-s10077" xml:space="preserve">adeoque PN x MG = {_m_
              <emph style="sub">3</emph>
            /_r_}
              <lb/>
            - _mg_ + _yx_ + _gx_ - _my_ - {_m_/_r_}_x x_ = {_m_
              <emph style="sub">3</emph>
            /_r_} - _mg_ + _o_. </s>
            <s xml:id="echoid-s10078" xml:space="preserve">= {_m_
              <emph style="sub">3</emph>
            /_r_}
              <lb/>
            - _mg_ = DM x ZQ. </s>
            <s xml:id="echoid-s10079" xml:space="preserve">unde PN. </s>
            <s xml:id="echoid-s10080" xml:space="preserve">ZQ:</s>
            <s xml:id="echoid-s10081" xml:space="preserve">: (DM. </s>
            <s xml:id="echoid-s10082" xml:space="preserve">MG:</s>
            <s xml:id="echoid-s10083" xml:space="preserve">:) QD.
              <lb/>
            </s>
            <s xml:id="echoid-s10084" xml:space="preserve">ZP. </s>
            <s xml:id="echoid-s10085" xml:space="preserve">ergo PN x ZP = ZQ x QD. </s>
            <s xml:id="echoid-s10086" xml:space="preserve">Liquetigitur curvam DNN
              <lb/>
            eſſe _hyperbolam_, cujus _aſymptoti_ ZM, ZT.</s>
            <s xml:id="echoid-s10087" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10088" xml:space="preserve">Siæquatiò ſit - _yz_ + _gx_ + _my_ = {_m_/_r_} _xx_; </s>
            <s xml:id="echoid-s10089" xml:space="preserve">eadem erit _hyper-_
              <lb/>
            _bola_. </s>
            <s xml:id="echoid-s10090" xml:space="preserve">Sed puncta G inter B, M tunc accipiuntur; </s>
            <s xml:id="echoid-s10091" xml:space="preserve">& </s>
            <s xml:id="echoid-s10092" xml:space="preserve">ità prout aliis
              <lb/>
            ac aliis locis puncta G deſignantur, æquationis ſigna variantur; </s>
            <s xml:id="echoid-s10093" xml:space="preserve">at
              <lb/>
            non eſt ea jam exponendi locus.</s>
            <s xml:id="echoid-s10094" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10095" xml:space="preserve">XIV. </s>
            <s xml:id="echoid-s10096" xml:space="preserve">Poſitione datæ ſint rectæ DB, BA; </s>
            <s xml:id="echoid-s10097" xml:space="preserve">pérque rectam DB
              <lb/>
            feratur recta CX ad BA parallela; </s>
            <s xml:id="echoid-s10098" xml:space="preserve">item per punctum D rotando
              <lb/>
              <note position="left" xlink:label="note-0228-02" xlink:href="note-0228-02a" xml:space="preserve">Fig. 47.
                <unsure/>
              </note>
            tranſeat recta DY, ſic ipſam BA ſecans in E, ut ſit inter rectas BE,
              <lb/>
            DC eadem ſemper proportio (puta quæ cujuſdam aſſignatæ R ad DB)
              <lb/>
            rectæ verò DE, CX ſe interſecent punctis N; </s>
            <s xml:id="echoid-s10099" xml:space="preserve">erit linea DNN _Pa-_
              <lb/>
            _rabola_.</s>
            <s xml:id="echoid-s10100" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s10101" xml:space="preserve">Nam ſit R. </s>
            <s xml:id="echoid-s10102" xml:space="preserve">DB:</s>
            <s xml:id="echoid-s10103" xml:space="preserve">: DB. </s>
            <s xml:id="echoid-s10104" xml:space="preserve">P. </s>
            <s xml:id="echoid-s10105" xml:space="preserve">Eſt ergò BE. </s>
            <s xml:id="echoid-s10106" xml:space="preserve">DC:</s>
            <s xml:id="echoid-s10107" xml:space="preserve">: DB. </s>
            <s xml:id="echoid-s10108" xml:space="preserve">P. </s>
            <s xml:id="echoid-s10109" xml:space="preserve">Item
              <lb/>
            eſt DB. </s>
            <s xml:id="echoid-s10110" xml:space="preserve">BE:</s>
            <s xml:id="echoid-s10111" xml:space="preserve">: DC. </s>
            <s xml:id="echoid-s10112" xml:space="preserve">CN. </s>
            <s xml:id="echoid-s10113" xml:space="preserve">ergò DB. </s>
            <s xml:id="echoid-s10114" xml:space="preserve">BE + BE. </s>
            <s xml:id="echoid-s10115" xml:space="preserve">DC = DC.</s>
            <s xml:id="echoid-s10116" xml:space="preserve"/>
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