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">
          <pb o="102" file="0280" n="295" rhead=""/>
          <p>
            <s xml:id="echoid-s13652" xml:space="preserve">_Notandum_ eſt eſſe _p_ {2 _m_ - 2 _n_/ } = {ZD {_m_/ }/AD {2 _n_ - _m_/ }}. </s>
            <s xml:id="echoid-s13653" xml:space="preserve">& </s>
            <s xml:id="echoid-s13654" xml:space="preserve">_q_ {2 _m_ - 2 _n_/ } =
              <lb/>
            {ZG {_m_/ }/AG {2 _n_ - _m_/ }}. </s>
            <s xml:id="echoid-s13655" xml:space="preserve">unde erit {ZD {_m_/ }/AD {2 _n_ - _m_/ }} &</s>
            <s xml:id="echoid-s13656" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s13657" xml:space="preserve">{ZG {_m_/ }/AG {2 _n_ - _m_/ }}. </s>
            <s xml:id="echoid-s13658" xml:space="preserve">quare {ZD {_m_/ }/AD {2 _n_ - _m_/ }} eſt mi-
              <lb/>
              <note position="left" xlink:label="note-0280-01" xlink:href="note-0280-01a" xml:space="preserve">Fig. 149.</note>
            nimum.</s>
            <s xml:id="echoid-s13659" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13660" xml:space="preserve">_Exemp_. </s>
            <s xml:id="echoid-s13661" xml:space="preserve">1. </s>
            <s xml:id="echoid-s13662" xml:space="preserve">Sit _n_ = 2; </s>
            <s xml:id="echoid-s13663" xml:space="preserve">& </s>
            <s xml:id="echoid-s13664" xml:space="preserve">_m_ = 3; </s>
            <s xml:id="echoid-s13665" xml:space="preserve">erit _p_
              <emph style="sub">2</emph>
            = {ZD
              <emph style="sub">3</emph>
            /AD}. </s>
            <s xml:id="echoid-s13666" xml:space="preserve">= {BD
              <emph style="sub">6</emph>
            /AD
              <emph style="sub">4</emph>
            }. </s>
            <s xml:id="echoid-s13667" xml:space="preserve">& </s>
            <s xml:id="echoid-s13668" xml:space="preserve">
              <lb/>
            _p_ = {BD
              <emph style="sub">3</emph>
            /ADq}. </s>
            <s xml:id="echoid-s13669" xml:space="preserve">= {ZDq/BD}. </s>
            <s xml:id="echoid-s13670" xml:space="preserve">Item AD = CA.</s>
            <s xml:id="echoid-s13671" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13672" xml:space="preserve">2. </s>
            <s xml:id="echoid-s13673" xml:space="preserve">Sit _n_ = 3; </s>
            <s xml:id="echoid-s13674" xml:space="preserve">& </s>
            <s xml:id="echoid-s13675" xml:space="preserve">_m_ = 4; </s>
            <s xml:id="echoid-s13676" xml:space="preserve">erit _p_
              <emph style="sub">2</emph>
            = {ZD
              <emph style="sub">4</emph>
            /AD
              <emph style="sub">2</emph>
            } vel p = {ZD
              <emph style="sub">2</emph>
            /AD}
              <lb/>
            = {BD
              <emph style="sub">4</emph>
            /AD
              <emph style="sub">3</emph>
            } = {ZD
              <emph style="sub">3</emph>
            /BD
              <emph style="sub">2</emph>
            }. </s>
            <s xml:id="echoid-s13677" xml:space="preserve">Item AD = 2 CA.</s>
            <s xml:id="echoid-s13678" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13679" xml:space="preserve">3. </s>
            <s xml:id="echoid-s13680" xml:space="preserve">Sit _n_ = 5, & </s>
            <s xml:id="echoid-s13681" xml:space="preserve">_m_ = 8; </s>
            <s xml:id="echoid-s13682" xml:space="preserve">erit _p_ {6/ } = {ZD
              <emph style="sub">8</emph>
            /AD
              <emph style="sub">2</emph>
            }. </s>
            <s xml:id="echoid-s13683" xml:space="preserve">vel _p_
              <emph style="sub">3</emph>
            = {ZD
              <emph style="sub">4</emph>
            /AD}
              <lb/>
            = {BD
              <emph style="sub">8</emph>
            /AD
              <emph style="sub">5</emph>
            } = {ZD
              <emph style="sub">5</emph>
            /BD
              <emph style="sub">2</emph>
            }. </s>
            <s xml:id="echoid-s13684" xml:space="preserve">Item AD = {2/3} CA.</s>
            <s xml:id="echoid-s13685" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13686" xml:space="preserve">Quoniam in his _Cyclometriam_ attigi, quid ſi obiter eò ſpectantia
              <lb/>
            _Theoremata_, quæ ad manum, paucula ſubjunxero? </s>
            <s xml:id="echoid-s13687" xml:space="preserve">præſternatur au-
              <lb/>
              <note position="left" xlink:label="note-0280-02" xlink:href="note-0280-02a" xml:space="preserve">Fig. 150.</note>
            tem autem hoc χαυολικὸν:</s>
            <s xml:id="echoid-s13688" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13689" xml:space="preserve">Sit curva quæpiam AGB, cujus axis AD, & </s>
            <s xml:id="echoid-s13690" xml:space="preserve">ad hunc ordinatæ
              <lb/>
            rectæ BD, GE; </s>
            <s xml:id="echoid-s13691" xml:space="preserve">Habebit curva AB ad curvam AG majorem rati-
              <lb/>
            onem quàm recta BD ad rectam GE.</s>
            <s xml:id="echoid-s13692" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13693" xml:space="preserve">Nam ducatur recta GH ad AD parallela: </s>
            <s xml:id="echoid-s13694" xml:space="preserve">ſecentúrque recta B H
              <lb/>
            punctis Y, & </s>
            <s xml:id="echoid-s13695" xml:space="preserve">recta GE punctis Z in particulas indefinitè multas; </s>
            <s xml:id="echoid-s13696" xml:space="preserve">pér-
              <lb/>
            que puncta Y, Z ducantur rectæ YM, YN, ZO, ZP ad AD paral-
              <lb/>
            lelæ: </s>
            <s xml:id="echoid-s13697" xml:space="preserve">curvam interſecantes punctis M, N, O, P; </s>
            <s xml:id="echoid-s13698" xml:space="preserve">per quæ ducantur
              <lb/>
            rectæ MR, NS, OT, PV ad BD parallelæ. </s>
            <s xml:id="echoid-s13699" xml:space="preserve">Eſtque angulus BM Y
              <lb/>
            (ut è ſuperius oſtenſis liquet) minor angulo NGS, unde MB. </s>
            <s xml:id="echoid-s13700" xml:space="preserve">B Y
              <lb/>
            &</s>
            <s xml:id="echoid-s13701" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13702" xml:space="preserve">GN. </s>
            <s xml:id="echoid-s13703" xml:space="preserve">NS. </s>
            <s xml:id="echoid-s13704" xml:space="preserve">Similíque de cauſa eſt NM. </s>
            <s xml:id="echoid-s13705" xml:space="preserve">MR &</s>
            <s xml:id="echoid-s13706" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13707" xml:space="preserve">GN. </s>
            <s xml:id="echoid-s13708" xml:space="preserve">NS. </s>
            <s xml:id="echoid-s13709" xml:space="preserve">
              <note symbol="*" position="left" xlink:label="note-0280-03" xlink:href="note-0280-03a" xml:space="preserve">Vid. _Append_.
                <lb/>
              Lect. XII.</note>
            quare conjunctè eft BM + MN + NG. </s>
            <s xml:id="echoid-s13710" xml:space="preserve">BY + MR + NS &</s>
            <s xml:id="echoid-s13711" xml:space="preserve">gt;
              <lb/>
            </s>
            <s xml:id="echoid-s13712" xml:space="preserve">GN. </s>
            <s xml:id="echoid-s13713" xml:space="preserve">NS; </s>
            <s xml:id="echoid-s13714" xml:space="preserve">hoc eſt arc. </s>
            <s xml:id="echoid-s13715" xml:space="preserve">GB. </s>
            <s xml:id="echoid-s13716" xml:space="preserve">BH &</s>
            <s xml:id="echoid-s13717" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13718" xml:space="preserve">GN. </s>
            <s xml:id="echoid-s13719" xml:space="preserve">NS. </s>
            <s xml:id="echoid-s13720" xml:space="preserve">rurſus (è diſcur-
              <lb/>
            ſu conſimili) ratio GN ad NS major eſt ſingulis rationibus OG ad
              <lb/>
            GZ, OP ad PT, & </s>
            <s xml:id="echoid-s13721" xml:space="preserve">AP ad PV; </s>
            <s xml:id="echoid-s13722" xml:space="preserve">idcircoq; </s>
            <s xml:id="echoid-s13723" xml:space="preserve">junctè eſt GN. </s>
            <s xml:id="echoid-s13724" xml:space="preserve">NS &</s>
            <s xml:id="echoid-s13725" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13726" xml:space="preserve">
              <lb/>
            arc. </s>
            <s xml:id="echoid-s13727" xml:space="preserve">AG. </s>
            <s xml:id="echoid-s13728" xml:space="preserve">GE. </s>
            <s xml:id="echoid-s13729" xml:space="preserve">quapropter erit GB. </s>
            <s xml:id="echoid-s13730" xml:space="preserve">BH &</s>
            <s xml:id="echoid-s13731" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13732" xml:space="preserve">AG. </s>
            <s xml:id="echoid-s13733" xml:space="preserve">GE. </s>
            <s xml:id="echoid-s13734" xml:space="preserve">permutan-
              <lb/>
            doque GB. </s>
            <s xml:id="echoid-s13735" xml:space="preserve">AG &</s>
            <s xml:id="echoid-s13736" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13737" xml:space="preserve">BH. </s>
            <s xml:id="echoid-s13738" xml:space="preserve">GE. </s>
            <s xml:id="echoid-s13739" xml:space="preserve">quare componendo eſt AB. </s>
            <s xml:id="echoid-s13740" xml:space="preserve">A G
              <lb/>
            &</s>
            <s xml:id="echoid-s13741" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13742" xml:space="preserve">BD. </s>
            <s xml:id="echoid-s13743" xml:space="preserve">GE.</s>
            <s xml:id="echoid-s13744" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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