Barrow, Isaac, Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur
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          <pb o="145" file="0323" n="338" rhead=""/>
          <p>
            <s xml:id="echoid-s15968" xml:space="preserve">In _pramiſſas explicationes_ animadvertatur generatim.</s>
            <s xml:id="echoid-s15969" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15970" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15971" xml:space="preserve">Propoſitam quamvis æquationem explicans _@μγνα_ deſignatur
              <lb/>
            hoc modo: </s>
            <s xml:id="echoid-s15972" xml:space="preserve">proponatur, exempli causâ, _æquatio a
              <emph style="sub">5</emph>
            _ + _ba
              <emph style="sub">4</emph>
            _ + _cca
              <emph style="sub">3</emph>
            _
              <lb/>
            - _d
              <emph style="sub">3</emph>
            aa_ - _f
              <emph style="sub">4</emph>
            a_ = _n
              <emph style="sub">5</emph>
            _; </s>
            <s xml:id="echoid-s15973" xml:space="preserve">In recta indefinitè protenſa HI deſignetur pun-
              <lb/>
              <note position="right" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">Fig. 219.</note>
            ctum A, pro radicum termino, vel origine; </s>
            <s xml:id="echoid-s15974" xml:space="preserve">tum arbitrariè ſumptâ
              <lb/>
            AG pro indeterminatâ radice _a_; </s>
            <s xml:id="echoid-s15975" xml:space="preserve">fiat GK æqualis primo feriei pro-
              <lb/>
            poſitam æquationem continentis gradu; </s>
            <s xml:id="echoid-s15976" xml:space="preserve">nempe ſit hîc GK = _a_ + _b_
              <lb/>
            + {_cc_/_a_} - {_d
              <emph style="sub">3</emph>
            _/_aa_} - {_f
              <emph style="sub">+</emph>
            _/_a
              <emph style="sub">3</emph>
            _} (utique rationem _a_ ad _c_ ſemel continuando fit
              <lb/>
            {_cc_/_a_}; </s>
            <s xml:id="echoid-s15977" xml:space="preserve">rationem _a_ ad _d_ bis continuando fit {_d
              <emph style="sub">3</emph>
            _/_aa_}; </s>
            <s xml:id="echoid-s15978" xml:space="preserve">acità porrò) tum inter
              <lb/>
            AG, GK tot mediarum proportionalium, quot æquationis propoſitæ
              <lb/>
            gradus exigit (is autem à pura quæſitæ radicis poteſtate indicatur) in
              <lb/>
            hoc nempe caſu quatuor mediarum proportionalium prima ſit GO;
              <lb/>
            </s>
            <s xml:id="echoid-s15979" xml:space="preserve">per ejuſmodi puncta O traducta curva AOO propoſito deſerviet.</s>
            <s xml:id="echoid-s15980" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15981" xml:space="preserve">2. </s>
            <s xml:id="echoid-s15982" xml:space="preserve">De radicibus falſis, ſeu negativis nihil attigimus ſuprà; </s>
            <s xml:id="echoid-s15983" xml:space="preserve">cæte-
              <lb/>
            rùm eæ reperiuntur hoc modo. </s>
            <s xml:id="echoid-s15984" xml:space="preserve">Æquationi propoſitæ ſubrogetur
              <lb/>
            altera, cujus in locis paribus (etiam vacuos locos adnumerando)
              <lb/>
            ſigna ſunt illis contraria, quæ habet æquatio propoſita; </s>
            <s xml:id="echoid-s15985" xml:space="preserve">erunt hu-
              <lb/>
            juſce _ſubdititiæ æquationis_ radices veræ, ſeu poſitivæ ipſius propoſitæ
              <lb/>
            æquationis radices falſæ, ſeu negativæ. </s>
            <s xml:id="echoid-s15986" xml:space="preserve">_Exemplo_ ſit _æquatio a
              <emph style="sub">3</emph>
            _ + _baa_
              <lb/>
            = _n
              <emph style="sub">3</emph>
            _; </s>
            <s xml:id="echoid-s15987" xml:space="preserve">vel _a
              <emph style="sub">3</emph>
            _ + _baa*_ - _n
              <emph style="sub">3</emph>
            _ = _o_. </s>
            <s xml:id="echoid-s15988" xml:space="preserve">Subrogetur _a
              <emph style="sub">3</emph>
            _ - _baa
              <emph style="sub">*</emph>
            _ + _n
              <emph style="sub">3</emph>
            _ = _o_; </s>
            <s xml:id="echoid-s15989" xml:space="preserve">& </s>
            <s xml:id="echoid-s15990" xml:space="preserve">
              <lb/>
            hujus, + utì ſuprà edoctum, veræ radices deſignentur, hæ _propoſitæ_
              <lb/>
              <note position="right" xlink:label="note-0323-02" xlink:href="note-0323-02a" xml:space="preserve">_+ In Serie 3_.</note>
            _aquationis_ falſæ erunt. </s>
            <s xml:id="echoid-s15991" xml:space="preserve">Rurſus ſit _a
              <emph style="sub">3</emph>
            _ - _baa_ = _n
              <emph style="sub">3</emph>
            _; </s>
            <s xml:id="echoid-s15992" xml:space="preserve">vel _a
              <emph style="sub">3</emph>
            _ - _baa_ - _n
              <emph style="sub">3</emph>
            _
              <lb/>
            = _o_; </s>
            <s xml:id="echoid-s15993" xml:space="preserve">ſubſtituatur æquatio _a
              <emph style="sub">3</emph>
            _ + _baa_ + _n
              <emph style="sub">3</emph>
            _ = _o_; </s>
            <s xml:id="echoid-s15994" xml:space="preserve">hæc nullam veram
              <lb/>
            radicem obtinet; </s>
            <s xml:id="echoid-s15995" xml:space="preserve">ergò nec _æquatio propoſita_ falſam admittit.</s>
            <s xml:id="echoid-s15996" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15997" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15998" xml:space="preserve">Quinimò datâ verâ radice quâpiam, depreſſioris gradûs æqua-
              <lb/>
            tio quædam ſalſis reperiendis inſerviet, qualis ità determinatur. </s>
            <s xml:id="echoid-s15999" xml:space="preserve">Pro-
              <lb/>
            ponatur æquatio quævis, puta _a
              <emph style="sub">3</emph>
            _ + _baa_ = _n
              <emph style="sub">3</emph>
            _; </s>
            <s xml:id="echoid-s16000" xml:space="preserve">cujus nota ſit radix una,
              <lb/>
            quæ vocetur _f_. </s>
            <s xml:id="echoid-s16001" xml:space="preserve">Conſtruatur æquatio planè ſimilis propoſitæ, eáſ-
              <lb/>
            demque _coefficientes_ habens, tantum pro _a_ ſubſtituendo _f_; </s>
            <s xml:id="echoid-s16002" xml:space="preserve">nempe
              <lb/>
            _f
              <emph style="sub">3</emph>
            _ + _bff_ = _n
              <emph style="sub">3</emph>
            _. </s>
            <s xml:id="echoid-s16003" xml:space="preserve">ergo _a
              <emph style="sub">3</emph>
            _ + _baa_ = _n
              <emph style="sub">3</emph>
            _ = _f
              <emph style="sub">3</emph>
            _ + _bff_; </s>
            <s xml:id="echoid-s16004" xml:space="preserve">adeóque
              <lb/>
            _a
              <emph style="sub">3</emph>
            _ + _baa_ - _f
              <emph style="sub">3</emph>
            _ - _bff_ = _o_. </s>
            <s xml:id="echoid-s16005" xml:space="preserve">dividatur hæc æquatio (id quod ſem-
              <lb/>
            per fieri poteſt) per _a_ - _f_; </s>
            <s xml:id="echoid-s16006" xml:space="preserve">proveniet _a a_ { + _ba_ + _bf_ + _fa_ + _ff_} = _o_; </s>
            <s xml:id="echoid-s16007" xml:space="preserve">cujus æ-
              <lb/>
            quationes eædem erunt cum reliquis æquationis propoſitæ radicibus;
              <lb/>
            </s>
            <s xml:id="echoid-s16008" xml:space="preserve">quæ proinde duas colligitur radices falſas habere; </s>
            <s xml:id="echoid-s16009" xml:space="preserve">itaque mutatis loco-
              <lb/>
            rum parìum ſignis, ut ità fiat _a a_ { - _ba_ + _bf_/ - _fa_ + _ff_} = _o_; </s>
            <s xml:id="echoid-s16010" xml:space="preserve">hujus </s>
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