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

Page concordance

< >
Scan Original
301 108
302 109
303 110
304 111
305 112
306 113
307 114
308 115
309 116
310 117
311 118
312 119
313 120
314 121
315 122
316 123
317 124
318 125
319 126
320 127
321 128
322 129
323 130
324 131
325 132
326 133
327 134
328 135
329 136
330 137
< >
page |< < (145) of 393 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div575" type="section" level="1" n="98">
          <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>
          </p>
        </div>
      </text>
    </echo>
Impressum