Barrow, Isaac, Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

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          <pb o="103" file="0281" n="296" rhead=""/>
          <p>
            <s xml:id="echoid-s13745" xml:space="preserve">XXVIII. </s>
            <s xml:id="echoid-s13746" xml:space="preserve">Sit _Circulus_ AMB, cujus _Radiui_ CA, & </s>
            <s xml:id="echoid-s13747" xml:space="preserve">ad hunc per-
              <lb/>
              <note position="right" xlink:label="note-0281-01" xlink:href="note-0281-01a" xml:space="preserve">Fig. 151</note>
            pendicularis recta DBE; </s>
            <s xml:id="echoid-s13748" xml:space="preserve">ſit item curva ANE talis, ut ductâ utcun-
              <lb/>
            que rectà PMN ad DE parallelâ (quæ circulum ſecet in M, dictam
              <lb/>
            curvam in N) ſit recta PN æqualis _Arcui_ AM; </s>
            <s xml:id="echoid-s13749" xml:space="preserve">ſit demum _axe_
              <lb/>
            AD _baſe_ DE deſcripta _Parabola_ AOE, hæc extra curvam AN E
              <lb/>
            tota cadet.</s>
            <s xml:id="echoid-s13750" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13751" xml:space="preserve">Nam ſecet recta PN parabolam in O; </s>
            <s xml:id="echoid-s13752" xml:space="preserve">& </s>
            <s xml:id="echoid-s13753" xml:space="preserve">connectantur ſubtenſæ
              <lb/>
            AB, AM; </s>
            <s xml:id="echoid-s13754" xml:space="preserve">eſtque DE. </s>
            <s xml:id="echoid-s13755" xml:space="preserve">PN:</s>
            <s xml:id="echoid-s13756" xml:space="preserve">: arc AB. </s>
            <s xml:id="echoid-s13757" xml:space="preserve">arc. </s>
            <s xml:id="echoid-s13758" xml:space="preserve">AM &</s>
            <s xml:id="echoid-s13759" xml:space="preserve">gt;</s>
            <s xml:id="echoid-s13760" xml:space="preserve">AB. </s>
            <s xml:id="echoid-s13761" xml:space="preserve">AM
              <lb/>
            :</s>
            <s xml:id="echoid-s13762" xml:space="preserve">: DE. </s>
            <s xml:id="echoid-s13763" xml:space="preserve">PO. </s>
            <s xml:id="echoid-s13764" xml:space="preserve">quare PN&</s>
            <s xml:id="echoid-s13765" xml:space="preserve">lt;</s>
            <s xml:id="echoid-s13766" xml:space="preserve">PO; </s>
            <s xml:id="echoid-s13767" xml:space="preserve">unde liquet Propoſitum.</s>
            <s xml:id="echoid-s13768" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13769" xml:space="preserve">XXIX. </s>
            <s xml:id="echoid-s13770" xml:space="preserve">Exhinc (& </s>
            <s xml:id="echoid-s13771" xml:space="preserve">è vulgò notis _ſpatiorune_ ADB, ADE _dimen-_
              <lb/>
            _ſionibus_) facilè colligitur hæc regula:</s>
            <s xml:id="echoid-s13772" xml:space="preserve">{3 CAx DB/2 CA+CD} &</s>
            <s xml:id="echoid-s13773" xml:space="preserve">lt;</s>
            <s xml:id="echoid-s13774" xml:space="preserve">arc. </s>
            <s xml:id="echoid-s13775" xml:space="preserve">AB.</s>
            <s xml:id="echoid-s13776" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">Fig. 152.</note>
          <p>
            <s xml:id="echoid-s13777" xml:space="preserve">Porrò ſi ponatur arc. </s>
            <s xml:id="echoid-s13778" xml:space="preserve">AB = 30 grad. </s>
            <s xml:id="echoid-s13779" xml:space="preserve">ſitque 2 CA = 113; </s>
            <s xml:id="echoid-s13780" xml:space="preserve">juxta
              <lb/>
            regulam iſtam computando, proveniet _tota circumferentia_ major quàm
              <lb/>
            355, minus fractione unitatis.</s>
            <s xml:id="echoid-s13781" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13782" xml:space="preserve">XXX. </s>
            <s xml:id="echoid-s13783" xml:space="preserve">Hinc etiam _dato arcu_ AB, nominatiſque AB = p; </s>
            <s xml:id="echoid-s13784" xml:space="preserve">CA = r; </s>
            <s xml:id="echoid-s13785" xml:space="preserve">& </s>
            <s xml:id="echoid-s13786" xml:space="preserve">
              <lb/>
            DB = _e_, ad inveniendum _ſinum rectum_ DB adhibebitur hæc æqua-
              <lb/>
            tio; </s>
            <s xml:id="echoid-s13787" xml:space="preserve">{3 _rrpp_/_9rr_ + _pp_} = {12 _rrp_/9 _rr_ + _pp_} _e_-_ee._ </s>
            <s xml:id="echoid-s13788" xml:space="preserve">vel ponendo _k_ = {3 _rrp_/9 _rr_ + _pp_}; </s>
            <s xml:id="echoid-s13789" xml:space="preserve">erit
              <lb/>
            _kp_ = 4_ke_ - _ee._ </s>
            <s xml:id="echoid-s13790" xml:space="preserve">vel 2 _k_ - √ 4 _kk_ - _kp_ = _e._</s>
            <s xml:id="echoid-s13791" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13792" xml:space="preserve">XXXI. </s>
            <s xml:id="echoid-s13793" xml:space="preserve">Sit AMB _Circulus_, cujus Radius CA, & </s>
            <s xml:id="echoid-s13794" xml:space="preserve">huic perpendi-
              <lb/>
              <note position="right" xlink:label="note-0281-03" xlink:href="note-0281-03a" xml:space="preserve">Fig. 153.</note>
            cularis recta DBE; </s>
            <s xml:id="echoid-s13795" xml:space="preserve">ſit item curva ANE pars _Cycloidis_ ad _Circulum_
              <lb/>
            AMB pertinentis; </s>
            <s xml:id="echoid-s13796" xml:space="preserve">demum ad axem AD, baſin DE ſtatuatur _Para-_
              <lb/>
            _bola_ AOE; </s>
            <s xml:id="echoid-s13797" xml:space="preserve">hæc intra _Cycloidem_ tota cadet.</s>
            <s xml:id="echoid-s13798" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13799" xml:space="preserve">Etenim utcunque ducatur recta PM ON ad DE parallela, lineas
              <lb/>
            expoſitas ſecans, ut cernis; </s>
            <s xml:id="echoid-s13800" xml:space="preserve">connectantúrque _ſabtenſæ_ AB, AM;
              <lb/>
            </s>
            <s xml:id="echoid-s13801" xml:space="preserve">eſtque DE. </s>
            <s xml:id="echoid-s13802" xml:space="preserve">PO:</s>
            <s xml:id="echoid-s13803" xml:space="preserve">: AB. </s>
            <s xml:id="echoid-s13804" xml:space="preserve">AM :</s>
            <s xml:id="echoid-s13805" xml:space="preserve">: curv. </s>
            <s xml:id="echoid-s13806" xml:space="preserve">AE. </s>
            <s xml:id="echoid-s13807" xml:space="preserve">AN &</s>
            <s xml:id="echoid-s13808" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13809" xml:space="preserve">DE. </s>
            <s xml:id="echoid-s13810" xml:space="preserve">PN; </s>
            <s xml:id="echoid-s13811" xml:space="preserve">
              <lb/>
            adeoque PO &</s>
            <s xml:id="echoid-s13812" xml:space="preserve">lt;</s>
            <s xml:id="echoid-s13813" xml:space="preserve">PN. </s>
            <s xml:id="echoid-s13814" xml:space="preserve">unde conſtat Propoſitum.</s>
            <s xml:id="echoid-s13815" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13816" xml:space="preserve">XXXII. </s>
            <s xml:id="echoid-s13817" xml:space="preserve">Exhinc, & </s>
            <s xml:id="echoid-s13818" xml:space="preserve">è _notis ſegmentorum circular is atque Cycloida-_
              <lb/>
            _lis dimenſionibus_, hæc elicitur _Regula_ {2CA x DB + CD x DB/CA + 2CD}
              <lb/>
            &</s>
            <s xml:id="echoid-s13819" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13820" xml:space="preserve">arc. </s>
            <s xml:id="echoid-s13821" xml:space="preserve">AB.</s>
            <s xml:id="echoid-s13822" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13823" xml:space="preserve">Porrò ſi fuerit arc. </s>
            <s xml:id="echoid-s13824" xml:space="preserve">AB = 30 grad. </s>
            <s xml:id="echoid-s13825" xml:space="preserve">& </s>
            <s xml:id="echoid-s13826" xml:space="preserve">ponatur 2 CA = 113; </s>
            <s xml:id="echoid-s13827" xml:space="preserve">è
              <lb/>
            regula hac conſectatur fore _totam circumferentiam_ minorem quam
              <lb/>
            355, plus fractione.</s>
            <s xml:id="echoid-s13828" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13829" xml:space="preserve">Vides igitur ut è propoſitis duabus regulis ſtatim emergit _Diametri_
              <lb/>
            ad _Circumferentiam Proportio Metiana_.</s>
            <s xml:id="echoid-s13830" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13831" xml:space="preserve">XXXIII. </s>
            <s xml:id="echoid-s13832" xml:space="preserve">Quoniam exorbitanti ſe obviam dedit _Cyclois_ hoc adno-
              <lb/>
            tabo _@ beorema_, neſcio an uſpiam ab illis, qui de _Cycloide_ tam fusè
              <lb/>
            ſcripſerunt, animadverſum; </s>
            <s xml:id="echoid-s13833" xml:space="preserve">Completo _Rectangulo_ ADEG, </s>
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