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-s9996" xml:space="preserve">
              <pb o="49" file="0227" n="242" rhead=""/>
            + _xy_ - {_mxx_/_b_} = _o_. </s>
            <s xml:id="echoid-s9997" xml:space="preserve">ergò PN x MG = {_m_
              <emph style="sub">3</emph>
            /_b_} = MD x ZQ.
              <lb/>
            </s>
            <s xml:id="echoid-s9998" xml:space="preserve">vel PN. </s>
            <s xml:id="echoid-s9999" xml:space="preserve">ZQ:</s>
            <s xml:id="echoid-s10000" xml:space="preserve">: (MD. </s>
            <s xml:id="echoid-s10001" xml:space="preserve">MG:</s>
            <s xml:id="echoid-s10002" xml:space="preserve">:) QD.</s>
            <s xml:id="echoid-s10003" xml:space="preserve">ZP. </s>
            <s xml:id="echoid-s10004" xml:space="preserve">Quapropter eſt
              <lb/>
              <note position="right" xlink:label="note-0227-01" xlink:href="note-0227-01a" xml:space="preserve">Fig. 44.</note>
            PN x ZP = ZQ x QD. </s>
            <s xml:id="echoid-s10005" xml:space="preserve">Unde palàm eſt curvam DNN eſſe hy-
              <lb/>
            perbolam, cujus aſymptoti ZM, ZT.</s>
            <s xml:id="echoid-s10006" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10007" xml:space="preserve">XI. </s>
            <s xml:id="echoid-s10008" xml:space="preserve">Notetur; </s>
            <s xml:id="echoid-s10009" xml:space="preserve">ſi æquatio ſit _my_ - _xy_ = {_m_/_b_}_xx_; </s>
            <s xml:id="echoid-s10010" xml:space="preserve">eadem ha-
              <lb/>
            bebitur _hyperbola_; </s>
            <s xml:id="echoid-s10011" xml:space="preserve">tunc ſolùm puncta G ad partes DM ſumuntur.
              <lb/>
            </s>
            <s xml:id="echoid-s10012" xml:space="preserve">Quin & </s>
            <s xml:id="echoid-s10013" xml:space="preserve">fi æquatio ſit _xy_ - _my_ = {_m_/_b_} _xx_; </s>
            <s xml:id="echoid-s10014" xml:space="preserve">puncta G ultra M
              <lb/>
            capiendo, proveniet _hyperbola_, huic ipſi _conjugata_.</s>
            <s xml:id="echoid-s10015" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10016" xml:space="preserve">XII. </s>
            <s xml:id="echoid-s10017" xml:space="preserve">Sit Triangulum BDF; </s>
            <s xml:id="echoid-s10018" xml:space="preserve">& </s>
            <s xml:id="echoid-s10019" xml:space="preserve">linea DNN talis, ut ductâ ut-
              <lb/>
              <note position="right" xlink:label="note-0227-02" xlink:href="note-0227-02a" xml:space="preserve">Fig. 45.</note>
            cunque RN ad BD parallelâ (quæ lineas BF, DF, DNN ſecet
              <lb/>
            punctis R, G, N) connexâque rectâ DN; </s>
            <s xml:id="echoid-s10020" xml:space="preserve">ſit perpetuò DN propor-
              <lb/>
            tione media inter RN, NG; </s>
            <s xml:id="echoid-s10021" xml:space="preserve">erit linea DNN _hyperbola_.</s>
            <s xml:id="echoid-s10022" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10023" xml:space="preserve">Per D ducatur DK ad DB perpendicularis (ſecans ipſam RN in E)
              <lb/>
            & </s>
            <s xml:id="echoid-s10024" xml:space="preserve">ſit FH ad DB parallela; </s>
            <s xml:id="echoid-s10025" xml:space="preserve">vocentúrque DB = _b_; </s>
            <s xml:id="echoid-s10026" xml:space="preserve">DF = _d_; </s>
            <s xml:id="echoid-s10027" xml:space="preserve">FH
              <lb/>
            = _f_; </s>
            <s xml:id="echoid-s10028" xml:space="preserve">tum DG = _x_; </s>
            <s xml:id="echoid-s10029" xml:space="preserve">& </s>
            <s xml:id="echoid-s10030" xml:space="preserve">GN = _y_; </s>
            <s xml:id="echoid-s10031" xml:space="preserve">Eſtque _d. </s>
            <s xml:id="echoid-s10032" xml:space="preserve">f:</s>
            <s xml:id="echoid-s10033" xml:space="preserve">: x._ </s>
            <s xml:id="echoid-s10034" xml:space="preserve">{_fx_/_d_} = _GE_;
              <lb/>
            </s>
            <s xml:id="echoid-s10035" xml:space="preserve">unde {_zfxy_/_d_} + _xx_ + _yy_ = 2 EG x GN + DGq + GNq
              <lb/>
            = DN q. </s>
            <s xml:id="echoid-s10036" xml:space="preserve">Porrò eſt _d. </s>
            <s xml:id="echoid-s10037" xml:space="preserve">b_:</s>
            <s xml:id="echoid-s10038" xml:space="preserve">: FG. </s>
            <s xml:id="echoid-s10039" xml:space="preserve">GR:</s>
            <s xml:id="echoid-s10040" xml:space="preserve">: _d_ - _x_.</s>
            <s xml:id="echoid-s10041" xml:space="preserve">RG = _b_ - {_bx._</s>
            <s xml:id="echoid-s10042" xml:space="preserve">/_d_} Un-
              <lb/>
            de RN = _b_ - {_bx_/_d_} + _y_. </s>
            <s xml:id="echoid-s10043" xml:space="preserve">Et ideò _by_ - {_bxy_/_d_} + _yy_ = RN x
              <lb/>
            NG = DNq = {2_fxy_/_d_} + _xx_ + _yy_. </s>
            <s xml:id="echoid-s10044" xml:space="preserve">quare _by_ - {_bxy_/_d_} =
              <lb/>
            {2_fxy_/_d_} + _xx_. </s>
            <s xml:id="echoid-s10045" xml:space="preserve">quam æquationem ordinando fit {_db_/2_f_+_b_}_y_ - _yx_ =
              <lb/>
            {_d_/2_f_+_b_}_xx_. </s>
            <s xml:id="echoid-s10046" xml:space="preserve">quòd ſi ponatur _m_ = {_db_/2_f_+_b_
              <emph style="sub">3</emph>
            } erit _my_ - _xy_ =
              <lb/>
            {_m_/_b_}_xx_. </s>
            <s xml:id="echoid-s10047" xml:space="preserve">Unde liquet DNN eſſe _hyperbolam_, qualis habetur in præ-
              <lb/>
            cedente determinata,</s>
          </p>
          <p>
            <s xml:id="echoid-s10048" xml:space="preserve">Not. </s>
            <s xml:id="echoid-s10049" xml:space="preserve">Siangulus DGN rectus fuerit, evaneſcente tum _f_ = _o_, </s>
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