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

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        <div xml:id="echoid-div399" type="section" level="1" n="42">
          <p>
            <s xml:id="echoid-s13291" xml:space="preserve">
              <pb o="97" file="0275" n="290" rhead=""/>
            RG. </s>
            <s xml:id="echoid-s13292" xml:space="preserve">AG:</s>
            <s xml:id="echoid-s13293" xml:space="preserve">: _m. </s>
            <s xml:id="echoid-s13294" xml:space="preserve">n_:</s>
            <s xml:id="echoid-s13295" xml:space="preserve">: TD. </s>
            <s xml:id="echoid-s13296" xml:space="preserve">AD &</s>
            <s xml:id="echoid-s13297" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13298" xml:space="preserve">SG. </s>
            <s xml:id="echoid-s13299" xml:space="preserve">AG. </s>
            <s xml:id="echoid-s13300" xml:space="preserve">quare
              <note symbol="(_a_)" position="right" xlink:label="note-0275-01" xlink:href="note-0275-01a" xml:space="preserve">2. _hujus._
                <lb/>
              _app._</note>
              <note symbol="(_b_)" position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">5. _hujus ap._</note>
            &</s>
            <s xml:id="echoid-s13301" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13302" xml:space="preserve">SG. </s>
            <s xml:id="echoid-s13303" xml:space="preserve">unde patet tota AFB extra circulum AEB jacere.</s>
            <s xml:id="echoid-s13304" xml:space="preserve"/>
          </p>
          <note symbol="(_c_)" position="right" xml:space="preserve">3. _hujus ap._</note>
          <p>
            <s xml:id="echoid-s13305" xml:space="preserve">X. </s>
            <s xml:id="echoid-s13306" xml:space="preserve">Reliquis itidem ſtantibus, ſiad baſin GE (utcunque parallelam
              <lb/>
              <note position="right" xlink:label="note-0275-04" xlink:href="note-0275-04a" xml:space="preserve">Fig. 139.</note>
            ipſi DB) & </s>
            <s xml:id="echoid-s13307" xml:space="preserve">axem AD conſtituta intelligatur _paraboliformis_ ejuſdem cum
              <lb/>
            ipſa AFB generis (nempe cujus etiam exponens {_n_/_m_}) illa ad partes A
              <lb/>
            ſupra GE, extra _circulum_ tota jacebit.</s>
            <s xml:id="echoid-s13308" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13309" xml:space="preserve">Nam in arcu AE accepto quocunque puncto M, ductâque MP ad
              <lb/>
            EG parallelâ, & </s>
            <s xml:id="echoid-s13310" xml:space="preserve">MV circulum tangente; </s>
            <s xml:id="echoid-s13311" xml:space="preserve">eſt VP. </s>
            <s xml:id="echoid-s13312" xml:space="preserve">AP &</s>
            <s xml:id="echoid-s13313" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s13314" xml:space="preserve">SG.
              <lb/>
            </s>
            <s xml:id="echoid-s13315" xml:space="preserve">AG &</s>
            <s xml:id="echoid-s13316" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s13317" xml:space="preserve">RG. </s>
            <s xml:id="echoid-s13318" xml:space="preserve">AG:</s>
            <s xml:id="echoid-s13319" xml:space="preserve">: _m. </s>
            <s xml:id="echoid-s13320" xml:space="preserve">n_; </s>
            <s xml:id="echoid-s13321" xml:space="preserve"> itaque rurſus liquet
              <note symbol="(_a_)" position="right" xlink:label="note-0275-05" xlink:href="note-0275-05a" xml:space="preserve">3. _hujus ap._</note>
            tum.</s>
            <s xml:id="echoid-s13322" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13323" xml:space="preserve">XI. </s>
            <s xml:id="echoid-s13324" xml:space="preserve">Conſectatur etiam dictam (ipſi AFB coordinatam & </s>
            <s xml:id="echoid-s13325" xml:space="preserve">ad ba-
              <lb/>
            ſin GE conſtitutam) _paraboliformem_ infra GE ad DB protractam,
              <lb/>
              <note position="right" xlink:label="note-0275-06" xlink:href="note-0275-06a" xml:space="preserve">Fig. 139.</note>
            eatenus intra _Circulum_ totam cadere,</s>
          </p>
          <p>
            <s xml:id="echoid-s13326" xml:space="preserve">Quòd intra _Circulum_ ſtatim infra EG cadet ex eo patet, quòd ipſam
              <lb/>
            tangens RE circulum ſecat (quia nempe SE circulum tangit). </s>
            <s xml:id="echoid-s13327" xml:space="preserve">quòd
              <lb/>
            alibi _Circulo_ non occurret hinc patet; </s>
            <s xml:id="echoid-s13328" xml:space="preserve">quoniam poſito quòd occurrat
              <lb/>
            uſpiam ad N, tota ſupra N extra circulum caderet, contra
              <note symbol="(_a_)" position="right" xlink:label="note-0275-07" xlink:href="note-0275-07a" xml:space="preserve">3. _hujus ap._</note>
            modò dictum ac oſtenſum eſt.</s>
            <s xml:id="echoid-s13329" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13330" xml:space="preserve">XII. </s>
            <s xml:id="echoid-s13331" xml:space="preserve">Porrò, _Hyperbolæ_ AEB (cujus centrum C) & </s>
            <s xml:id="echoid-s13332" xml:space="preserve">_parabolifor-_
              <lb/>
              <note position="right" xlink:label="note-0275-08" xlink:href="note-0275-08a" xml:space="preserve">Fig. 140.</note>
            _mis_ AFB, cujus exponens {_n_/_m_}, communes ſint axis AD, baſis DB;
              <lb/>
            </s>
            <s xml:id="echoid-s13333" xml:space="preserve">ſit autem AD = {2 _n_ - _m_/_m_ - _n_} CA; </s>
            <s xml:id="echoid-s13334" xml:space="preserve">& </s>
            <s xml:id="echoid-s13335" xml:space="preserve">BT _hyperbolam_ tangat; </s>
            <s xml:id="echoid-s13336" xml:space="preserve">hæc
              <lb/>
            quoque _paraboliformem_ AFB continget.</s>
            <s xml:id="echoid-s13337" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13338" xml:space="preserve">Nam eſt CD. </s>
            <s xml:id="echoid-s13339" xml:space="preserve">CA:</s>
            <s xml:id="echoid-s13340" xml:space="preserve">: CA. </s>
            <s xml:id="echoid-s13341" xml:space="preserve">CT. </s>
            <s xml:id="echoid-s13342" xml:space="preserve">acindè AD. </s>
            <s xml:id="echoid-s13343" xml:space="preserve">TA:</s>
            <s xml:id="echoid-s13344" xml:space="preserve">: CD. </s>
            <s xml:id="echoid-s13345" xml:space="preserve">CA; </s>
            <s xml:id="echoid-s13346" xml:space="preserve">inverſéq;
              <lb/>
            </s>
            <s xml:id="echoid-s13347" xml:space="preserve">componendo TD. </s>
            <s xml:id="echoid-s13348" xml:space="preserve">AD:</s>
            <s xml:id="echoid-s13349" xml:space="preserve">: CA + CD. </s>
            <s xml:id="echoid-s13350" xml:space="preserve">CD. </s>
            <s xml:id="echoid-s13351" xml:space="preserve">Verùm
              <unsure/>
            ex hypo-
              <lb/>
            theſi, eſt _m_ - _n_. </s>
            <s xml:id="echoid-s13352" xml:space="preserve">2 _n_ - _m_:</s>
            <s xml:id="echoid-s13353" xml:space="preserve">: CA. </s>
            <s xml:id="echoid-s13354" xml:space="preserve">CD; </s>
            <s xml:id="echoid-s13355" xml:space="preserve">adeoque inversè compo-
              <lb/>
            nendo CA. </s>
            <s xml:id="echoid-s13356" xml:space="preserve">CD:</s>
            <s xml:id="echoid-s13357" xml:space="preserve">: _m_ - _n. </s>
            <s xml:id="echoid-s13358" xml:space="preserve">n_: </s>
            <s xml:id="echoid-s13359" xml:space="preserve">& </s>
            <s xml:id="echoid-s13360" xml:space="preserve">rurſus componendo CA + C D. </s>
            <s xml:id="echoid-s13361" xml:space="preserve">
              <lb/>
            CD:</s>
            <s xml:id="echoid-s13362" xml:space="preserve">: _m. </s>
            <s xml:id="echoid-s13363" xml:space="preserve">n._ </s>
            <s xml:id="echoid-s13364" xml:space="preserve">hoc eſt TD. </s>
            <s xml:id="echoid-s13365" xml:space="preserve">AD:</s>
            <s xml:id="echoid-s13366" xml:space="preserve">: _m. </s>
            <s xml:id="echoid-s13367" xml:space="preserve">n_. </s>
            <s xml:id="echoid-s13368" xml:space="preserve">unde BT _hyperboliformem_
              <lb/>
            contingit.</s>
            <s xml:id="echoid-s13369" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13370" xml:space="preserve">XIII. </s>
            <s xml:id="echoid-s13371" xml:space="preserve">Hinc rurſu datà ratione ipſius AD ad CA, _paraboliformis_
              <lb/>
            ad punctum B _bype bolam_ contingens deſignabitur. </s>
            <s xml:id="echoid-s13372" xml:space="preserve">nempe ſit AD =
              <lb/>
            {_s_/_t_} CA; </s>
            <s xml:id="echoid-s13373" xml:space="preserve">erit {_n_/_m_} = {_t_ + _s_/2 _t_ + _s_}. </s>
            <s xml:id="echoid-s13374" xml:space="preserve">Nam hoc ſuppoſito erit </s>
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