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-div552" type="section" level="1" n="82">
          <pb o="136" file="0314" n="329" rhead=""/>
        </div>
        <div xml:id="echoid-div553" type="section" level="1" n="83">
          <head xml:id="echoid-head86" style="it" xml:space="preserve">Series quarta.</head>
          <p>
            <s xml:id="echoid-s15476" xml:space="preserve">_a_ + {_cc_/_a_} = _n_.</s>
            <s xml:id="echoid-s15477" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15478" xml:space="preserve">_aa_ + _cc_ = _nn_.</s>
            <s xml:id="echoid-s15479" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15480" xml:space="preserve">_a_
              <emph style="sub">3</emph>
            + _cca_ = _n_
              <emph style="sub">3</emph>
            .</s>
            <s xml:id="echoid-s15481" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15482" xml:space="preserve">_a_
              <emph style="sub">4</emph>
            + _ccaa_ = _n_
              <emph style="sub">4</emph>
            .</s>
            <s xml:id="echoid-s15483" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15484" xml:space="preserve">Sit recta indefinitè protenſa AH, & </s>
            <s xml:id="echoid-s15485" xml:space="preserve">huic perpendicularis AD;
              <lb/>
            </s>
            <s xml:id="echoid-s15486" xml:space="preserve">fiat autem angulus RAH ſemirectus; </s>
            <s xml:id="echoid-s15487" xml:space="preserve">tum utcunque ducatur GZK
              <lb/>
              <note position="left" xlink:label="note-0314-01" xlink:href="note-0314-01a" xml:space="preserve">Fig. 211.</note>
            ad AD parallela; </s>
            <s xml:id="echoid-s15488" xml:space="preserve">& </s>
            <s xml:id="echoid-s15489" xml:space="preserve">facto AG. </s>
            <s xml:id="echoid-s15490" xml:space="preserve">AG:</s>
            <s xml:id="echoid-s15491" xml:space="preserve">: AC. </s>
            <s xml:id="echoid-s15492" xml:space="preserve">ZK; </s>
            <s xml:id="echoid-s15493" xml:space="preserve">per Kintra
              <lb/>
            angulum DAR deſcribatur _hyperbola_ KXK; </s>
            <s xml:id="echoid-s15494" xml:space="preserve">ſint denuò curvæ CLL,
              <lb/>
            AMM, ANN tales, ut inter GZ, GK ſint _media_ GL, _bimedia_
              <lb/>
            GM, _trimedia_ GN; </s>
            <s xml:id="echoid-s15495" xml:space="preserve">hæ propoſito deſervient. </s>
            <s xml:id="echoid-s15496" xml:space="preserve">Nam ſi AG (vel
              <lb/>
            GZ) dicatur _a_, erit GK = _a_ + {_cc_/_a_}; </s>
            <s xml:id="echoid-s15497" xml:space="preserve">& </s>
            <s xml:id="echoid-s15498" xml:space="preserve">GLq = _aa_ + _cc_; </s>
            <s xml:id="echoid-s15499" xml:space="preserve">& </s>
            <s xml:id="echoid-s15500" xml:space="preserve">
              <lb/>
            GMcub = _a_
              <emph style="sub">3</emph>
            + _cca_; </s>
            <s xml:id="echoid-s15501" xml:space="preserve">& </s>
            <s xml:id="echoid-s15502" xml:space="preserve">GNqq = _a_
              <emph style="sub">4</emph>
            + _ccaa_.</s>
            <s xml:id="echoid-s15503" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div555" type="section" level="1" n="84">
          <head xml:id="echoid-head87" style="it" xml:space="preserve">Not.</head>
          <p>
            <s xml:id="echoid-s15504" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15505" xml:space="preserve">Deſignantur radices, ut in præcedentibus, poſitâ AE = _n_, & </s>
            <s xml:id="echoid-s15506" xml:space="preserve">ductâ
              <lb/>
            EF ad AH parallelâ.</s>
            <s xml:id="echoid-s15507" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15508" xml:space="preserve">2. </s>
            <s xml:id="echoid-s15509" xml:space="preserve">Si AP = AC, erit PX ad _hyperbolam_ KXK ordinatarum _mi_-
              <lb/>
            _nima_; </s>
            <s xml:id="echoid-s15510" xml:space="preserve">unde ſi AE (vel _n_) &</s>
            <s xml:id="echoid-s15511" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s15512" xml:space="preserve">PX; </s>
            <s xml:id="echoid-s15513" xml:space="preserve">nulla dabitur radix in primo
              <lb/>
            gradu.</s>
            <s xml:id="echoid-s15514" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15515" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15516" xml:space="preserve">Curva CLL eſt _hyperbola æquilatera_, cujus _centrum_ A, _ſemi_-
              <lb/>
            _axis_ AC; </s>
            <s xml:id="echoid-s15517" xml:space="preserve">quæ & </s>
            <s xml:id="echoid-s15518" xml:space="preserve">ordinatarum eſt _minima_; </s>
            <s xml:id="echoid-s15519" xml:space="preserve">alioquin ſi _n_&</s>
            <s xml:id="echoid-s15520" xml:space="preserve">gt;_</s>
            <s xml:id="echoid-s15521" xml:space="preserve">c_, ſem-
              <lb/>
            per una vera radix habetur, & </s>
            <s xml:id="echoid-s15522" xml:space="preserve">unica.</s>
            <s xml:id="echoid-s15523" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15524" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15525" xml:space="preserve">Reliquæ AMM, ANN ſunt hyperboliformes ad infinitum
              <lb/>
            excurrentes; </s>
            <s xml:id="echoid-s15526" xml:space="preserve">unde ſemper una vera radix habetur, neque plures.</s>
            <s xml:id="echoid-s15527" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15528" xml:space="preserve">5. </s>
            <s xml:id="echoid-s15529" xml:space="preserve">Si fuerit Y α = {1/2} YX; </s>
            <s xml:id="echoid-s15530" xml:space="preserve">Y β = {1/3}YX; </s>
            <s xml:id="echoid-s15531" xml:space="preserve">Y γ = {1/4} YX, & </s>
            <s xml:id="echoid-s15532" xml:space="preserve">per
              <lb/>
            puncta α, β γ, traductæ concipiantur _hpperbola
              <unsure/>
            _ (habentes & </s>
            <s xml:id="echoid-s15533" xml:space="preserve">ipſæ _a_-
              <lb/>
            _ſymptotos_ DA, AR) α λ, β μ, γ ν; </s>
            <s xml:id="echoid-s15534" xml:space="preserve">erunt hæ ipſarum curvarum
              <lb/>
            CLL, AMM, ANN _aſymptoti_. </s>
            <s xml:id="echoid-s15535" xml:space="preserve">(Similes etiam _aſymptoti_ con-
              <lb/>
            veniunt lineis poſthac deſcribendis, quanquam de illis conticeamus.)</s>
            <s xml:id="echoid-s15536" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15537" xml:space="preserve">6. </s>
            <s xml:id="echoid-s15538" xml:space="preserve">Hinc in ſecundo gradu _a_ + {_cc_/2_a_}&</s>
            <s xml:id="echoid-s15539" xml:space="preserve">gt;_</s>
            <s xml:id="echoid-s15540" xml:space="preserve">n_; </s>
            <s xml:id="echoid-s15541" xml:space="preserve">in tertio _a_ + {_cc_/3_a_}&</s>
            <s xml:id="echoid-s15542" xml:space="preserve">gt;_</s>
            <s xml:id="echoid-s15543" xml:space="preserve">n_;</s>
            <s xml:id="echoid-s15544" xml:space="preserve"/>
          </p>
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