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-div521" type="section" level="1" n="67">
          <p>
            <s xml:id="echoid-s15134" xml:space="preserve">
              <pb o="128" file="0306" n="321" rhead=""/>
            _ſpiralis_, ut pro arbitrio ductâ rectâ C μ Z habeat arcus EZ ad rectam
              <lb/>
            C μ rationem aſſignatam (puta R ad S) Manifeſtum eſt lineam β YH
              <lb/>
            eſſe rectam, quoniam EZ (KO). </s>
            <s xml:id="echoid-s15135" xml:space="preserve">Cμ (OY):</s>
            <s xml:id="echoid-s15136" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s15137" xml:space="preserve">S, perpetuò.
              <lb/>
            </s>
            <s xml:id="echoid-s15138" xml:space="preserve">
              <note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">Fig. 198.</note>
            unde evoluta BMF ſit _Parabola_; </s>
            <s xml:id="echoid-s15139" xml:space="preserve">quoniam axis partes AP, AD ſe
              <lb/>
            habent ut ſpatia KOY, KC β, hoc eſt ut quadrata ex ipſis OY, Cβ,
              <lb/>
            vel ex ipſis PM, DB.</s>
            <s xml:id="echoid-s15140" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div525" type="section" level="1" n="68">
          <head xml:id="echoid-head71" xml:space="preserve">_Corol. Theor_. I.</head>
          <p>
            <s xml:id="echoid-s15141" xml:space="preserve">Si ad figuram βCφ erigatur _cylindricus_ altitudinem habens æqua-
              <lb/>
            lem peripheriæ integræ _circuli_, cujus radius CL; </s>
            <s xml:id="echoid-s15142" xml:space="preserve">erit iſte _cylindricus_
              <lb/>
            æ
              <unsure/>
            qualis _ſolido_, quod procreatur è figurâ Cβ HK circa axem CK ro-
              <lb/>
            tatâ.</s>
            <s xml:id="echoid-s15143" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div526" type="section" level="1" n="69">
          <head xml:id="echoid-head72" xml:space="preserve">_Theor_. II.</head>
          <p>
            <s xml:id="echoid-s15144" xml:space="preserve">Sit curva quæpiam AMB (cujus axis AD, baſis DB) & </s>
            <s xml:id="echoid-s15145" xml:space="preserve">curva
              <lb/>
              <note position="left" xlink:label="note-0306-02" xlink:href="note-0306-02a" xml:space="preserve">Fig. 195.</note>
            AZL talis, ut liberè ductâ rectâ ZPM, ſit PZ = √ 2 APM; </s>
            <s xml:id="echoid-s15146" xml:space="preserve">ſit
              <lb/>
            item alia curva OYY talis, ut ad hanc productâ rectâ ZPMY,
              <lb/>
            adſumptâque rectâ R, ſit ZP q. </s>
            <s xml:id="echoid-s15147" xml:space="preserve">R q:</s>
            <s xml:id="echoid-s15148" xml:space="preserve">: PM. </s>
            <s xml:id="echoid-s15149" xml:space="preserve">PY; </s>
            <s xml:id="echoid-s15150" xml:space="preserve">ſitque denuò DL.
              <lb/>
            </s>
            <s xml:id="echoid-s15151" xml:space="preserve">R:</s>
            <s xml:id="echoid-s15152" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s15153" xml:space="preserve">LE. </s>
            <s xml:id="echoid-s15154" xml:space="preserve">& </s>
            <s xml:id="echoid-s15155" xml:space="preserve">per E intra angulum LDG deſcribatur _Hyper-_
              <lb/>
              <note position="left" xlink:label="note-0306-03" xlink:href="note-0306-03a" xml:space="preserve">Fig. 199.</note>
            _bola_ EXX; </s>
            <s xml:id="echoid-s15156" xml:space="preserve">huic autem occurrat ducta recta ZHX ad AD parallela,
              <lb/>
            erit ſpatium PDOY æquale _ſpatio Hyperbolico_ LHXE.</s>
            <s xml:id="echoid-s15157" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15158" xml:space="preserve">Hinc _ſumma_ omnium {PM/APM} = {2 LEXH/R q}.</s>
            <s xml:id="echoid-s15159" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div528" type="section" level="1" n="70">
          <head xml:id="echoid-head73" xml:space="preserve">_Theor_. III.</head>
          <p>
            <s xml:id="echoid-s15160" xml:space="preserve">Sit curva quæpiam AMB, cujus axis AD, baſis DB; </s>
            <s xml:id="echoid-s15161" xml:space="preserve">& </s>
            <s xml:id="echoid-s15162" xml:space="preserve">curva
              <lb/>
            KZL talis, ut adſumptâ quâdam R, & </s>
            <s xml:id="echoid-s15163" xml:space="preserve">arbitrariè ductâ rectâ ZPM
              <lb/>
            ad BD parallelâ, ſit √ APM. </s>
            <s xml:id="echoid-s15164" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s15165" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s15166" xml:space="preserve">PZ; </s>
            <s xml:id="echoid-s15167" xml:space="preserve">erit ſpatium ADLK
              <lb/>
              <note position="left" xlink:label="note-0306-04" xlink:href="note-0306-04a" xml:space="preserve">Fig. 200.</note>
            æquale _rectangulo_ ex R in 2 √ ADB; </s>
            <s xml:id="echoid-s15168" xml:space="preserve">vel {ADLK/2 R} = √ ADB.</s>
            <s xml:id="echoid-s15169" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15170" xml:space="preserve">_Exemp_. </s>
            <s xml:id="echoid-s15171" xml:space="preserve">Sit ADB circuli quadrans, erit ſumma omnium {PM/APM} =
              <lb/>
            √ 2 DA x arc. </s>
            <s xml:id="echoid-s15172" xml:space="preserve">AB.</s>
            <s xml:id="echoid-s15173" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div530" type="section" level="1" n="71">
          <head xml:id="echoid-head74" xml:space="preserve">_Theor_. IV.</head>
          <p>
            <s xml:id="echoid-s15174" xml:space="preserve">Sit curva quæpiam AMB (cujus axis AD, baſis DB) ſintque
              <lb/>
            duæ lineæ EXK, GYL ità relatæ, ut in curva AMB ſumpto </s>
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