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-s16010" xml:space="preserve">
              <pb o="146" file="0324" n="339" rhead=""/>
            veræ radices propoſitæ falſas exhibent. </s>
            <s xml:id="echoid-s16011" xml:space="preserve">Hic inſuper modus æquatio-
              <lb/>
            nis propoſitæ, quatenus illa ex aliarum in ſe ductu provenit, conſtitutio-
              <lb/>
            nem oſtendit.</s>
            <s xml:id="echoid-s16012" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16013" xml:space="preserve">4. </s>
            <s xml:id="echoid-s16014" xml:space="preserve">Radices maximæ & </s>
            <s xml:id="echoid-s16015" xml:space="preserve">minimæ deprehenduntur in quacunque ſe-
              <lb/>
            rie ponendo (quovis in gradu ſeriei) fore n = o; </s>
            <s xml:id="echoid-s16016" xml:space="preserve">ut in octava ſerie ſit
              <lb/>
            _ba_ - _aa_ + _cc_ = _o_; </s>
            <s xml:id="echoid-s16017" xml:space="preserve">adeóque _cc_ = _aa_ - _ba_, erit _a_ ( = {_b_/2} + √ {_bb_/4} +
              <lb/>
            _cc_) _maxima radix_; </s>
            <s xml:id="echoid-s16018" xml:space="preserve">item in Serie duodecima ſit _aa_ - _ba_ + _cc_ = _o_;
              <lb/>
            </s>
            <s xml:id="echoid-s16019" xml:space="preserve">unde _cc_ = _ba_ - _aa_; </s>
            <s xml:id="echoid-s16020" xml:space="preserve">erit _a_ ( = {_b_/2} + √{_bb_/4} - _cc_) _radix maxima_; </s>
            <s xml:id="echoid-s16021" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s16022" xml:space="preserve">_a_ ( = {_b_/2} - √ {_bb_/4} - _cc_) _radix minima_.</s>
            <s xml:id="echoid-s16023" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16024" xml:space="preserve">5. </s>
            <s xml:id="echoid-s16025" xml:space="preserve">_Curvaram nodi_, vel _interſectiones_ innoteſcunt, cujuſvis in Seriei
              <lb/>
            quovis gradu, ponendo fore _a_ = _n_; </s>
            <s xml:id="echoid-s16026" xml:space="preserve">ut in octava Serie, ubi _ba_ - _aa_
              <lb/>
            + _cc_ = _nn_, ſit _a_ = _n_; </s>
            <s xml:id="echoid-s16027" xml:space="preserve">ergò _ba_ - _aa_ + _cc_ = _aa_; </s>
            <s xml:id="echoid-s16028" xml:space="preserve">vel _cc_ = 2_aa_ - _ba_;
              <lb/>
            </s>
            <s xml:id="echoid-s16029" xml:space="preserve">vel {_cc_/2} = _aa_ - {_ba_/2}; </s>
            <s xml:id="echoid-s16030" xml:space="preserve">quare _a_ = {_b_/4} + √ {_bb_/16} + {_cc_/2}. </s>
            <s xml:id="echoid-s16031" xml:space="preserve">Item in Se-
              <lb/>
            rie duodecima, ubi _aa_ - _ba_ + _cc_ = _nn_ = _aa_; </s>
            <s xml:id="echoid-s16032" xml:space="preserve">erit ideò _cc_ = _ba_; </s>
            <s xml:id="echoid-s16033" xml:space="preserve">acinde
              <lb/>
            _a_ = {_cc_/_b_}.</s>
            <s xml:id="echoid-s16034" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16035" xml:space="preserve">6. </s>
            <s xml:id="echoid-s16036" xml:space="preserve">_Ordinatæ maxima, mini@æque_ variis nodis, methodiſque paſ-
              <lb/>
            ſim notis inveſtigantur; </s>
            <s xml:id="echoid-s16037" xml:space="preserve">ego ſimul illas atque curvarum _tangentes_
              <lb/>
            unà operâ ſic determino. </s>
            <s xml:id="echoid-s16038" xml:space="preserve">Sit curva A γ H, ad Seriem undecimam
              <lb/>
            pertinens, ejuſque gradum, cujus æquatio eſt _cca_ - _baa_ - _a
              <emph style="sub">3</emph>
            _ = _x
              <emph style="sub">3</emph>
            _;
              <lb/>
            </s>
            <s xml:id="echoid-s16039" xml:space="preserve">
              <note position="left" xlink:label="note-0324-01" xlink:href="note-0324-01a" xml:space="preserve">Fig. 220.</note>
            poſito γ T curvam tangere, & </s>
            <s xml:id="echoid-s16040" xml:space="preserve">γ P ad AH ordinari, reperio (de ſu-
              <lb/>
            pra monſtratis) fore PT = {_3n
              <emph style="sub">3</emph>
            _/_3aa_ + _2ba_ - _cc_}, tum conſidero, ſi or-
              <lb/>
            dinata P γ ſit maxima, fore tangentem ipſi HA parallelam, ſeu rectam
              <lb/>
            PT eſſe infinitam; </s>
            <s xml:id="echoid-s16041" xml:space="preserve">quare cùm ſit _n
              <emph style="sub">3</emph>
            _ = PT x: </s>
            <s xml:id="echoid-s16042" xml:space="preserve">_3aa_ + _2ba_ - _cc_; </s>
            <s xml:id="echoid-s16043" xml:space="preserve">& </s>
            <s xml:id="echoid-s16044" xml:space="preserve">_n_
              <lb/>
            ſit finita, patet eſſe _3aa_ + _2ba_ - _cc_ = _o_; </s>
            <s xml:id="echoid-s16045" xml:space="preserve">vel _aa_ + {2/3}_ba_ = {_cc_/3}; </s>
            <s xml:id="echoid-s16046" xml:space="preserve">adeó-
              <lb/>
            que √: </s>
            <s xml:id="echoid-s16047" xml:space="preserve">{_bb_/9} + {_cc_/3}: </s>
            <s xml:id="echoid-s16048" xml:space="preserve">- {_b_/3} = _a_ = AP.</s>
            <s xml:id="echoid-s16049" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16050" xml:space="preserve">7. </s>
            <s xml:id="echoid-s16051" xml:space="preserve">Adnoto demùm è _maximis_ & </s>
            <s xml:id="echoid-s16052" xml:space="preserve">_minimis ordinatis_ radicum li-
              <lb/>
            mites derivari; </s>
            <s xml:id="echoid-s16053" xml:space="preserve">nempe ſi reperiatur ad maximam ordinatam pertinen-
              <lb/>
            tis radicis (velut ipſius AP in exemplo proximè ſuperiori) valor, & </s>
            <s xml:id="echoid-s16054" xml:space="preserve">
              <lb/>
            ?</s>
            <s xml:id="echoid-s16055" xml:space="preserve">?is ubique in æ quatione pro ipsâ _a_ ſubſtituatur, ſi quod provenit, de-
              <lb/>
            ficiat ab _bomogeneo_ (quod vocant) _comparationis, problemn_ </s>
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