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-s10191" xml:space="preserve">
              <pb o="52" file="0230" n="245" rhead=""/>
            {_by_/_b_}:</s>
            <s xml:id="echoid-s10192" xml:space="preserve">: _b. </s>
            <s xml:id="echoid-s10193" xml:space="preserve">r_. </s>
            <s xml:id="echoid-s10194" xml:space="preserve">unde talis emerget æquatio: </s>
            <s xml:id="echoid-s10195" xml:space="preserve">_yx_ + _hx_ - {_hr_/_b_}_y_ = {_h_/_b_}
              <lb/>
            _x x x_; </s>
            <s xml:id="echoid-s10196" xml:space="preserve">hoc eſt (poſito {_hr_/_b_} = _m_) _yx_ + _hx_ - _my_ = {_m_/_r_} _xx_; </s>
            <s xml:id="echoid-s10197" xml:space="preserve">Eſt
              <lb/>
              <note position="left" xlink:label="note-0230-01" xlink:href="note-0230-01a" xml:space="preserve">Fig. 50,
                <lb/>
              51,
                <lb/>
              52.</note>
            igitur curva BNN _hyperbola_, qualem ſuperiùs exhibuimus determi-
              <lb/>
            natam.</s>
            <s xml:id="echoid-s10198" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10199" xml:space="preserve">XVIII. </s>
            <s xml:id="echoid-s10200" xml:space="preserve">Datæ poſitione ſint rectæ DB, BA; </s>
            <s xml:id="echoid-s10201" xml:space="preserve">(& </s>
            <s xml:id="echoid-s10202" xml:space="preserve">in DB deſigne-
              <lb/>
            tur punctum D) ſitque linea DNN talis, ut ductâ utcunque GN
              <lb/>
              <note position="right" xlink:label="note-0230-02" xlink:href="note-0230-02a" xml:space="preserve">Fig. 53.</note>
            ad BA parallelâ; </s>
            <s xml:id="echoid-s10203" xml:space="preserve">ſumptis verò determinatis _g, r,_ vocatíſque DG
              <lb/>
            = _x_; </s>
            <s xml:id="echoid-s10204" xml:space="preserve">& </s>
            <s xml:id="echoid-s10205" xml:space="preserve">GN = _y_, ſit _ry_ - _yx_ = _gx_; </s>
            <s xml:id="echoid-s10206" xml:space="preserve">erit linea DNN _by-_
              <lb/>
            _perbola_, ſic determinanda.</s>
            <s xml:id="echoid-s10207" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10208" xml:space="preserve">Capiatur DE = _r_, & </s>
            <s xml:id="echoid-s10209" xml:space="preserve">BO = _g_; </s>
            <s xml:id="echoid-s10210" xml:space="preserve">& </s>
            <s xml:id="echoid-s10211" xml:space="preserve">per E ducatur recta ER ad
              <lb/>
            BA, ac per O recta OS ad BD parallelæ; </s>
            <s xml:id="echoid-s10212" xml:space="preserve">erunt ZR, ZS _aſym_-
              <lb/>
            _ptoti_.</s>
            <s xml:id="echoid-s10213" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10214" xml:space="preserve">Nam ductâ NP ad DB parallelâ, eſt ZP = _g_ + _y_; </s>
            <s xml:id="echoid-s10215" xml:space="preserve">& </s>
            <s xml:id="echoid-s10216" xml:space="preserve">PN =
              <lb/>
            _r_ - _x_; </s>
            <s xml:id="echoid-s10217" xml:space="preserve">quare ZP x PN = _gr_ - _gx_ + _ry_ - _yx_. </s>
            <s xml:id="echoid-s10218" xml:space="preserve">Verùm ex
              <lb/>
            hypotheſi eſt - _gx_ + _ry_ - _yx_ = _o_. </s>
            <s xml:id="echoid-s10219" xml:space="preserve">ergò ZP x PN = _gr_ =
              <lb/>
            ZE x ED. </s>
            <s xml:id="echoid-s10220" xml:space="preserve">undè liquido conſtat Propoſitum.</s>
            <s xml:id="echoid-s10221" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10222" xml:space="preserve">Quòd ſi fuerit æquatio _x y - r y = g x_; </s>
            <s xml:id="echoid-s10223" xml:space="preserve">ſumenda eſt DE = _r_;
              <lb/>
            </s>
            <s xml:id="echoid-s10224" xml:space="preserve">& </s>
            <s xml:id="echoid-s10225" xml:space="preserve">BO = _g_ (infra rectam DB) ductíſque, ceu priùs, parallelis
              <lb/>
            SZR; </s>
            <s xml:id="echoid-s10226" xml:space="preserve">erit _hyperbola_ NNN angulo SZR comprehenſa; </s>
            <s xml:id="echoid-s10227" xml:space="preserve">quod eo-
              <lb/>
            dem facilè comprobatur modo.</s>
            <s xml:id="echoid-s10228" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10229" xml:space="preserve">XIX. </s>
            <s xml:id="echoid-s10230" xml:space="preserve">Datæ poſitione ſint rectæ DB, BA; </s>
            <s xml:id="echoid-s10231" xml:space="preserve">ac ità ferantur rectæ
              <lb/>
            FX ad DB parallela, ac DY per punctum deſignatum D tranſiens,
              <lb/>
            ut ſit ſemper ratio ipſius BE ad ipſam BF æqualis aſſignatæ DB ad
              <lb/>
            R; </s>
            <s xml:id="echoid-s10232" xml:space="preserve">erunt rectarum DY, FX interſectiones ad lineam rectam.</s>
            <s xml:id="echoid-s10233" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10234" xml:space="preserve">Nam per N ducatur GK ad BA parallela; </s>
            <s xml:id="echoid-s10235" xml:space="preserve">éſtque DB. </s>
            <s xml:id="echoid-s10236" xml:space="preserve">DG:</s>
            <s xml:id="echoid-s10237" xml:space="preserve">:
              <lb/>
            BE. </s>
            <s xml:id="echoid-s10238" xml:space="preserve">GN:</s>
            <s xml:id="echoid-s10239" xml:space="preserve">: BE. </s>
            <s xml:id="echoid-s10240" xml:space="preserve">BF:</s>
            <s xml:id="echoid-s10241" xml:space="preserve">: BD. </s>
            <s xml:id="echoid-s10242" xml:space="preserve">R. </s>
            <s xml:id="echoid-s10243" xml:space="preserve">itaque ſemper eſt DG = R. </s>
            <s xml:id="echoid-s10244" xml:space="preserve">Pa-
              <lb/>
            tet igitur factâ DG = R, & </s>
            <s xml:id="echoid-s10245" xml:space="preserve">ductâ GK ad BA parallelâ, interſecti-
              <lb/>
            ones omnes ad hanc exiſtere.</s>
            <s xml:id="echoid-s10246" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10247" xml:space="preserve">XX. </s>
            <s xml:id="echoid-s10248" xml:space="preserve">Quòd ſi reliquis ſimiliter poſitis; </s>
            <s xml:id="echoid-s10249" xml:space="preserve">ſumpto autem alio in BA
              <lb/>
            puncto O; </s>
            <s xml:id="echoid-s10250" xml:space="preserve">ab hoc ſumatur computandi initium; </s>
            <s xml:id="echoid-s10251" xml:space="preserve">ut nimirùm ſit
              <lb/>
            perpetuò BE, OF:</s>
            <s xml:id="echoid-s10252" xml:space="preserve">: DB. </s>
            <s xml:id="echoid-s10253" xml:space="preserve">R; </s>
            <s xml:id="echoid-s10254" xml:space="preserve">erunt interſectiones N ad _hyper-_
              <lb/>
            _bolam_.</s>
            <s xml:id="echoid-s10255" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10256" xml:space="preserve">Nam ductâ NG ad AB parallelâ, ſit DB = _b_; </s>
            <s xml:id="echoid-s10257" xml:space="preserve">OB = _g_; </s>
            <s xml:id="echoid-s10258" xml:space="preserve"/>
          </p>
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