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-div502" type="section" level="1" n="59">
          <head xml:id="echoid-head62" xml:space="preserve">_Exemp_. II.</head>
          <p>
            <s xml:id="echoid-s14876" xml:space="preserve">Sit ADG _circuli_ quâdrans, & </s>
            <s xml:id="echoid-s14877" xml:space="preserve">habere debeat TP ad PM ratio-
              <lb/>
            nem eandem quam PE ad R; </s>
            <s xml:id="echoid-s14878" xml:space="preserve">eſt ergo PY æqualis _tangenti_ arcûs GE;
              <lb/>
            </s>
            <s xml:id="echoid-s14879" xml:space="preserve">& </s>
            <s xml:id="echoid-s14880" xml:space="preserve">ſpat. </s>
            <s xml:id="echoid-s14881" xml:space="preserve">APYY = R x arc. </s>
            <s xml:id="echoid-s14882" xml:space="preserve">AE. </s>
            <s xml:id="echoid-s14883" xml:space="preserve">adeóque PM = arc. </s>
            <s xml:id="echoid-s14884" xml:space="preserve">AE.</s>
            <s xml:id="echoid-s14885" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div503" type="section" level="1" n="60">
          <head xml:id="echoid-head63" xml:space="preserve">_Probl_. III.</head>
          <p>
            <s xml:id="echoid-s14886" xml:space="preserve">Proponatur figura quælibet ADB (cujus _axis_ AD, _baſis_ DB)
              <lb/>
              <note position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">Fig. 185.</note>
            reperiatur curva KZL, proprietate talis, ut ductâ rectâ ZPM ad
              <lb/>
            DB utcunque parallela quæ lineas expoſitas ſecet ut cernis) poſitóque
              <lb/>
            rectam ZT tangere curvam KZL, ſit intercepta TP æqualis ipſi
              <lb/>
            PM.</s>
            <s xml:id="echoid-s14887" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14888" xml:space="preserve">Hocità perſicietur. </s>
            <s xml:id="echoid-s14889" xml:space="preserve">Sit curva OYY talis, ut adſumptâ quâdam
              <lb/>
            R, protractâque PMY, ſit PM. </s>
            <s xml:id="echoid-s14890" xml:space="preserve">R:</s>
            <s xml:id="echoid-s14891" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s14892" xml:space="preserve">PY; </s>
            <s xml:id="echoid-s14893" xml:space="preserve">tum liberè adſump-
              <lb/>
            tâ DL (in BD protensâ) ſit DL. </s>
            <s xml:id="echoid-s14894" xml:space="preserve">R:</s>
            <s xml:id="echoid-s14895" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s14896" xml:space="preserve">LE; </s>
            <s xml:id="echoid-s14897" xml:space="preserve">& </s>
            <s xml:id="echoid-s14898" xml:space="preserve">_aſymptotis_ DL,
              <lb/>
            DG per E deſcribatur _Hyperbola_ EXX; </s>
            <s xml:id="echoid-s14899" xml:space="preserve">tum ſit ſpatium LEXH æ-
              <lb/>
            quale ſpatio DOYP, & </s>
            <s xml:id="echoid-s14900" xml:space="preserve">protractæ XH, YP concurrant in Z; </s>
            <s xml:id="echoid-s14901" xml:space="preserve">erit
              <lb/>
            Z in curva quæſita; </s>
            <s xml:id="echoid-s14902" xml:space="preserve">quam ſi tangat ZT, erit TP = PM.</s>
            <s xml:id="echoid-s14903" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14904" xml:space="preserve">Adnotetur, ſi propoſita ſigura ſit _rectangulum Parallelogrammum_
              <lb/>
            ADBC, quod curvæ KZL hæc erit proprietas, ut ſit DH eodem
              <lb/>
            ordine inter DL, DO media _Geometricè_ proportionalis, quo DP
              <lb/>
              <note position="right" xlink:label="note-0301-02" xlink:href="note-0301-02a" xml:space="preserve">Fig. 186.</note>
            inter DA & </s>
            <s xml:id="echoid-s14905" xml:space="preserve">θ
              <unsure/>
            (ſeu nihilum) eſt media _Aritbmeticè_; </s>
            <s xml:id="echoid-s14906" xml:space="preserve">quod ſi liberè
              <lb/>
            juxta proprietatem hanc deſcribatur curva KZL, & </s>
            <s xml:id="echoid-s14907" xml:space="preserve">_Mechanicè_ re-
              <lb/>
            periatur tangens ZT, indè quadrabitur _hyperbolicum ſpatium_ LEXH;
              <lb/>
            </s>
            <s xml:id="echoid-s14908" xml:space="preserve">erit utique hoc æquale _rectangulo_ ex TP, AP.</s>
            <s xml:id="echoid-s14909" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14910" xml:space="preserve">Subnotari poſſit fore 1. </s>
            <s xml:id="echoid-s14911" xml:space="preserve">Spat. </s>
            <s xml:id="echoid-s14912" xml:space="preserve">ADLK = R x DL - DO. </s>
            <s xml:id="echoid-s14913" xml:space="preserve">2. </s>
            <s xml:id="echoid-s14914" xml:space="preserve">Sum.
              <lb/>
            </s>
            <s xml:id="echoid-s14915" xml:space="preserve">mam ZPq = R x : </s>
            <s xml:id="echoid-s14916" xml:space="preserve">{DLq - DOq/2}. </s>
            <s xml:id="echoid-s14917" xml:space="preserve">& </s>
            <s xml:id="echoid-s14918" xml:space="preserve">ſummam ZP cub. </s>
            <s xml:id="echoid-s14919" xml:space="preserve">= R x
              <lb/>
            {DLcub. </s>
            <s xml:id="echoid-s14920" xml:space="preserve">- DOcub.</s>
            <s xml:id="echoid-s14921" xml:space="preserve">/3} &</s>
            <s xml:id="echoid-s14922" xml:space="preserve">c
              <unsure/>
            . </s>
            <s xml:id="echoid-s14923" xml:space="preserve">3. </s>
            <s xml:id="echoid-s14924" xml:space="preserve">Siponatur φ eſſe centrum gr. </s>
            <s xml:id="echoid-s14925" xml:space="preserve">figu-
              <lb/>
            ræ ADLK, ducantúrque φψ ad AD, & </s>
            <s xml:id="echoid-s14926" xml:space="preserve">φξ ad DL perpendicu-
              <lb/>
            lares, fore φψ = {DL + DO/4}, & </s>
            <s xml:id="echoid-s14927" xml:space="preserve">φξ = R - {AD x DO/LO}.</s>
            <s xml:id="echoid-s14928" xml:space="preserve"/>
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