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

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          <p>
            <s xml:id="echoid-s12060" xml:space="preserve">
              <pb o="79" file="0257" n="272" rhead=""/>
            lineámque VIF tangat recta FT; </s>
            <s xml:id="echoid-s12061" xml:space="preserve">item lineam VKF tângat recta
              <lb/>
              <note symbol="(_a_)" position="right" xlink:label="note-0257-01" xlink:href="note-0257-01a" xml:space="preserve">5. Lect.
                <lb/>
              IX.</note>
            FS. </s>
            <s xml:id="echoid-s12062" xml:space="preserve">Eſt ergò SD = 2 TD. </s>
            <s xml:id="echoid-s12063" xml:space="preserve">atqui DE x DT = VDEZ.</s>
            <s xml:id="echoid-s12064" xml:space="preserve"> ergò DE x SD = (2 VDEZ = ) FDq. </s>
            <s xml:id="echoid-s12065" xml:space="preserve">unde conſtat angulum
              <lb/>
              <note symbol="(_b_) it" position="right" xlink:label="note-0257-02" xlink:href="note-0257-02a" xml:space="preserve">Cor. præc.</note>
            QFS rectum eſſe. </s>
            <s xml:id="echoid-s12066" xml:space="preserve">quod Propoſitum erat.</s>
            <s xml:id="echoid-s12067" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12068" xml:space="preserve">Adjungam & </s>
            <s xml:id="echoid-s12069" xml:space="preserve">illis cognata hæc.</s>
            <s xml:id="echoid-s12070" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12071" xml:space="preserve">XIII. </s>
            <s xml:id="echoid-s12072" xml:space="preserve">Sit curva quævis AGEZ, punctúmque quoddam D (à quo
              <lb/>
            projectæ DA, DG, DE, & </s>
            <s xml:id="echoid-s12073" xml:space="preserve">_c_. </s>
            <s xml:id="echoid-s12074" xml:space="preserve">ab initio DA continuò decreſcant)
              <lb/>
              <note position="right" xlink:label="note-0257-03" xlink:href="note-0257-03a" xml:space="preserve">Fig. 112.</note>
            tum altera ſit curva DKE, priorem interſecans in E, naturâque ta-
              <lb/>
            lis, ut à D utcunque projectâ rectâ DKG (quæ curvam AEZ ſecet
              <lb/>
            in G, curvam DKE in K) ſit perpetuò rectangulum ex DK, & </s>
            <s xml:id="echoid-s12075" xml:space="preserve">de-
              <lb/>
            ſignatâ quâdam lineâ R æquale ſpatio ADG; </s>
            <s xml:id="echoid-s12076" xml:space="preserve">tum ductâ DT ad
              <lb/>
            DE perpendiculari, ſit DT = 2 R; </s>
            <s xml:id="echoid-s12077" xml:space="preserve">& </s>
            <s xml:id="echoid-s12078" xml:space="preserve">connectatur TE; </s>
            <s xml:id="echoid-s12079" xml:space="preserve">hæc
              <lb/>
            curvam DKE continget.</s>
            <s xml:id="echoid-s12080" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12081" xml:space="preserve">Nam ſumpto quovis in curva DKE puncto K, ducatur recta DKG;
              <lb/>
            </s>
            <s xml:id="echoid-s12082" xml:space="preserve">& </s>
            <s xml:id="echoid-s12083" xml:space="preserve">ſumptâ DL = DK, ducatur LR ad DT parallela ( ſecans ipſam
              <lb/>
            DG in Y). </s>
            <s xml:id="echoid-s12084" xml:space="preserve">tum per E ducatur EX ad DE perpendicularis (hæc
              <lb/>
            verò extra curvam AEZ, ad partes Z cadet, quia decreſcunt proje-
              <lb/>
            ctæ verſus Z; </s>
            <s xml:id="echoid-s12085" xml:space="preserve">unde EX verſus A intra curvam EGA cadet; </s>
            <s xml:id="echoid-s12086" xml:space="preserve">eate-
              <lb/>
              <note position="right" xlink:label="note-0257-04" xlink:href="note-0257-04a" xml:space="preserve">Fig. 113.</note>
            nus ſaltem, quatenus huic Propoſito ſatisfaciet). </s>
            <s xml:id="echoid-s12087" xml:space="preserve">Sit jam primò pun-
              <lb/>
            ctum G ſupra E, verſus initium A, & </s>
            <s xml:id="echoid-s12088" xml:space="preserve">ob TD. </s>
            <s xml:id="echoid-s12089" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s12090" xml:space="preserve">: RL. </s>
            <s xml:id="echoid-s12091" xml:space="preserve">LE;
              <lb/>
            </s>
            <s xml:id="echoid-s12092" xml:space="preserve">
              <note symbol="(_a_) it" position="right" xlink:label="note-0257-05" xlink:href="note-0257-05a" xml:space="preserve">Hyp.</note>
            adeóque RL x DE = TD x LE (a) = 2 R x LE (a) = 2 GDE
              <lb/>
            &</s>
            <s xml:id="echoid-s12093" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s12094" xml:space="preserve">2 DEX = EX x DE. </s>
            <s xml:id="echoid-s12095" xml:space="preserve">ergò RL &</s>
            <s xml:id="echoid-s12096" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s12097" xml:space="preserve">EX &</s>
            <s xml:id="echoid-s12098" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s12099" xml:space="preserve">LY. </s>
            <s xml:id="echoid-s12100" xml:space="preserve">Eſt autem
              <lb/>
            punctum Y extra curvam, quia DY &</s>
            <s xml:id="echoid-s12101" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s12102" xml:space="preserve">DL = DK; </s>
            <s xml:id="echoid-s12103" xml:space="preserve">ergò magìs
              <lb/>
            punctum R eſt extra curvam.</s>
            <s xml:id="echoid-s12104" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12105" xml:space="preserve">Sit rurſus punctum G infra punctum E verſus Z; </s>
            <s xml:id="echoid-s12106" xml:space="preserve">eſtque rurſus,
              <lb/>
            utì priùs, RL x DE = 2 GDE &</s>
            <s xml:id="echoid-s12107" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s12108" xml:space="preserve">2 triang. </s>
            <s xml:id="echoid-s12109" xml:space="preserve">EDX = EX x DE.
              <lb/>
            </s>
            <s xml:id="echoid-s12110" xml:space="preserve">unde RL &</s>
            <s xml:id="echoid-s12111" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s12112" xml:space="preserve">EX &</s>
            <s xml:id="echoid-s12113" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s12114" xml:space="preserve">LY. </s>
            <s xml:id="echoid-s12115" xml:space="preserve">Eſt autem recta LY extra curvam EK
              <lb/>
            tota, (nam etiam extra arcum LK curvæ KE circumductum tota ja-
              <lb/>
            cet) ergò punctum R rurſus extra curvam exiſtit. </s>
            <s xml:id="echoid-s12116" xml:space="preserve">Liquidum eſt igi-
              <lb/>
            tur rectam TER curvam DKE tangere.</s>
            <s xml:id="echoid-s12117" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12118" xml:space="preserve">Quòd ſi punctum aliud ìn curva DKE deſignetur, puta K; </s>
            <s xml:id="echoid-s12119" xml:space="preserve">per
              <lb/>
            quod ducta ſit DKG; </s>
            <s xml:id="echoid-s12120" xml:space="preserve">& </s>
            <s xml:id="echoid-s12121" xml:space="preserve">fiat DG. </s>
            <s xml:id="echoid-s12122" xml:space="preserve">DK:</s>
            <s xml:id="echoid-s12123" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s12124" xml:space="preserve">P; </s>
            <s xml:id="echoid-s12125" xml:space="preserve">ſumatúrque
              <lb/>
            DT = 2 P; </s>
            <s xml:id="echoid-s12126" xml:space="preserve">& </s>
            <s xml:id="echoid-s12127" xml:space="preserve">connectatur TG; </s>
            <s xml:id="echoid-s12128" xml:space="preserve">tum ducatur KS ad GT paralle-
              <lb/>
            la; </s>
            <s xml:id="echoid-s12129" xml:space="preserve">recta KS curvam DKE tanget.</s>
            <s xml:id="echoid-s12130" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12131" xml:space="preserve">Nam concipiatur curva DOG, per G tranſiens, talis, ut rectâ
              <lb/>
            quâcunque DON à D projectâ (quæ curvam DOG ſecet in O,
              <lb/>
            curvam DNE in M, curvam AGE in N) ſit ſemper DO x P æ-
              <lb/>
            qualis ſpatio ADN; </s>
            <s xml:id="echoid-s12132" xml:space="preserve">erit ideò DM x R = DO x P; </s>
            <s xml:id="echoid-s12133" xml:space="preserve">ac proinde
              <lb/>
            DM. </s>
            <s xml:id="echoid-s12134" xml:space="preserve">DO:</s>
            <s xml:id="echoid-s12135" xml:space="preserve">: P. </s>
            <s xml:id="echoid-s12136" xml:space="preserve">R. </s>
            <s xml:id="echoid-s12137" xml:space="preserve">unde lì
              <unsure/>
            neæ DKE, DOG analogæ erunt. </s>
            <s xml:id="echoid-s12138" xml:space="preserve"/>
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