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-s11687" xml:space="preserve">
              <pb o="73" file="0251" n="266" rhead=""/>
            DEF concurrentes punctis S, T; </s>
            <s xml:id="echoid-s11688" xml:space="preserve">erit ſemper DT = 2 DS. </s>
            <s xml:id="echoid-s11689" xml:space="preserve">Quòd
              <lb/>
              <note position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve">Fig. 99.</note>
            ſi DE ſunt ut cubi ipſarum DF, erit ſemper DT = 3 DS; </s>
            <s xml:id="echoid-s11690" xml:space="preserve">ac ſi-
              <lb/>
            mili deinceps modo.</s>
            <s xml:id="echoid-s11691" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11692" xml:space="preserve">X. </s>
            <s xml:id="echoid-s11693" xml:space="preserve">Sint rectæ VD, TB concurrentes in T, quas decuſſet poſnio-
              <lb/>
              <note position="right" xlink:label="note-0251-02" xlink:href="note-0251-02a" xml:space="preserve">Fig. 100</note>
            ne data recta DB; </s>
            <s xml:id="echoid-s11694" xml:space="preserve">tranſeant etiam per B lineæ EBE, FBF tales,
              <lb/>
            ut ductâ quâcunque PG ad DB parallelâ, ſit perpetuò PF eodem or-
              <lb/>
            dine media Arithmeticè inter PG, PE; </s>
            <s xml:id="echoid-s11695" xml:space="preserve">tangat autem BR curvam
              <lb/>
            EBE, opertet lineæ FBF tangentem ad B determinare.</s>
            <s xml:id="echoid-s11696" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11697" xml:space="preserve">Sumptis NM (ordinum in quibus ſunt PF, PE exponentibus)
              <lb/>
            fiat N x TD + M \\ - N} x RD. </s>
            <s xml:id="echoid-s11698" xml:space="preserve">M x TD:</s>
            <s xml:id="echoid-s11699" xml:space="preserve">: RD. </s>
            <s xml:id="echoid-s11700" xml:space="preserve">SD; </s>
            <s xml:id="echoid-s11701" xml:space="preserve">& </s>
            <s xml:id="echoid-s11702" xml:space="preserve">connecta-
              <lb/>
            tur BS; </s>
            <s xml:id="echoid-s11703" xml:space="preserve">hæc curvam FBF continget.</s>
            <s xml:id="echoid-s11704" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11705" xml:space="preserve">Nam utcunque ducta ſit PG, dictas lineas ſecans ut vides. </s>
            <s xml:id="echoid-s11706" xml:space="preserve">Eſtque
              <lb/>
            EG. </s>
            <s xml:id="echoid-s11707" xml:space="preserve">FG:</s>
            <s xml:id="echoid-s11708" xml:space="preserve">: M. </s>
            <s xml:id="echoid-s11709" xml:space="preserve">N. </s>
            <s xml:id="echoid-s11710" xml:space="preserve">ergò FG x TD. </s>
            <s xml:id="echoid-s11711" xml:space="preserve">EG x TD:</s>
            <s xml:id="echoid-s11712" xml:space="preserve">: N x TD.</s>
            <s xml:id="echoid-s11713" xml:space="preserve"> M x TD. </s>
            <s xml:id="echoid-s11714" xml:space="preserve">Item EF x RD. </s>
            <s xml:id="echoid-s11715" xml:space="preserve">EG x TD:</s>
            <s xml:id="echoid-s11716" xml:space="preserve">: M - N x RD. </s>
            <s xml:id="echoid-s11717" xml:space="preserve">M x
              <lb/>
              <note symbol="(_a_)" position="right" xlink:label="note-0251-03" xlink:href="note-0251-03a" xml:space="preserve">11. Lect.
                <lb/>
              VII.</note>
            TD. </s>
            <s xml:id="echoid-s11718" xml:space="preserve">Quapropter (antecedentes conjungendo) erit FG x TD +
              <lb/>
            EF x RD. </s>
            <s xml:id="echoid-s11719" xml:space="preserve">EG x TD:</s>
            <s xml:id="echoid-s11720" xml:space="preserve">: N x TD + M - N x RD. </s>
            <s xml:id="echoid-s11721" xml:space="preserve">M x TD;
              <lb/>
            </s>
            <s xml:id="echoid-s11722" xml:space="preserve">(hoc eſt):</s>
            <s xml:id="echoid-s11723" xml:space="preserve">: RD. </s>
            <s xml:id="echoid-s11724" xml:space="preserve">SD. </s>
            <s xml:id="echoid-s11725" xml:space="preserve"> Eſt antem LG x TD + KL x RD.</s>
            <s xml:id="echoid-s11726" xml:space="preserve">
              <note symbol="(_b_)" position="right" xlink:label="note-0251-04" xlink:href="note-0251-04a" xml:space="preserve">_Conſtr._</note>
              <note symbol="(_c_)" position="right" xlink:label="note-0251-05" xlink:href="note-0251-05a" xml:space="preserve">4. Lect.
                <lb/>
              VII.</note>
            KG x TD:</s>
            <s xml:id="echoid-s11727" xml:space="preserve">: RD. </s>
            <s xml:id="echoid-s11728" xml:space="preserve">SD. </s>
            <s xml:id="echoid-s11729" xml:space="preserve">quare FG x TD + EF x RD. </s>
            <s xml:id="echoid-s11730" xml:space="preserve">EG x
              <lb/>
            TD:</s>
            <s xml:id="echoid-s11731" xml:space="preserve">: LG x TD + KL x RD. </s>
            <s xml:id="echoid-s11732" xml:space="preserve">KG x TD. </s>
            <s xml:id="echoid-s11733" xml:space="preserve">hinc, cùm ſit EG
              <note symbol="(_d_)" position="right" xlink:label="note-0251-06" xlink:href="note-0251-06a" xml:space="preserve">_Hyp_</note>
            &</s>
            <s xml:id="echoid-s11734" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11735" xml:space="preserve">KG; </s>
            <s xml:id="echoid-s11736" xml:space="preserve">erit FG x TD + EF x RD &</s>
            <s xml:id="echoid-s11737" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11738" xml:space="preserve">LG x TD + KL x RD;
              <lb/>
            </s>
            <s xml:id="echoid-s11739" xml:space="preserve">vel FG. </s>
            <s xml:id="echoid-s11740" xml:space="preserve">EF + TD. </s>
            <s xml:id="echoid-s11741" xml:space="preserve">RD &</s>
            <s xml:id="echoid-s11742" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11743" xml:space="preserve">LG. </s>
            <s xml:id="echoid-s11744" xml:space="preserve">KL + TD. </s>
            <s xml:id="echoid-s11745" xml:space="preserve">RD; </s>
            <s xml:id="echoid-s11746" xml:space="preserve">ſeu (dem-
              <lb/>
            ptâ communi ratione) FG. </s>
            <s xml:id="echoid-s11747" xml:space="preserve">EF &</s>
            <s xml:id="echoid-s11748" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11749" xml:space="preserve">LG. </s>
            <s xml:id="echoid-s11750" xml:space="preserve">KL. </s>
            <s xml:id="echoid-s11751" xml:space="preserve">vel componendo
              <lb/>
            EG. </s>
            <s xml:id="echoid-s11752" xml:space="preserve">EF &</s>
            <s xml:id="echoid-s11753" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11754" xml:space="preserve">KG. </s>
            <s xml:id="echoid-s11755" xml:space="preserve">KL &</s>
            <s xml:id="echoid-s11756" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11757" xml:space="preserve">EG. </s>
            <s xml:id="echoid-s11758" xml:space="preserve">EL. </s>
            <s xml:id="echoid-s11759" xml:space="preserve">unde eſt EF &</s>
            <s xml:id="echoid-s11760" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s11761" xml:space="preserve">EL.</s>
            <s xml:id="echoid-s11762" xml:space="preserve">
              <note symbol="(_e_)" position="right" xlink:label="note-0251-07" xlink:href="note-0251-07a" xml:space="preserve">1. Lect.
                <lb/>
              VII.</note>
            itaque punctum L extra curvam FBF ſitum eſt; </s>
            <s xml:id="echoid-s11763" xml:space="preserve">adeoque liquet
              <lb/>
            Propoſitum.</s>
            <s xml:id="echoid-s11764" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11765" xml:space="preserve">XI. </s>
            <s xml:id="echoid-s11766" xml:space="preserve">Quinetiam, reliquis ſtantibus iiſdem, ſi PF ſupponatur ejuſ-
              <lb/>
            dem ordinis Geometricè media liquet (planè ſicut in modò præceden-
              <lb/>
            tibus) eandem BS curvam FBF contingere.</s>
            <s xml:id="echoid-s11767" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11768" xml:space="preserve">_Exemplnum._ </s>
            <s xml:id="echoid-s11769" xml:space="preserve">Si PF ſit è ſex mediis tertia, ſeu M = 7; </s>
            <s xml:id="echoid-s11770" xml:space="preserve">& </s>
            <s xml:id="echoid-s11771" xml:space="preserve">N = 3;
              <lb/>
            </s>
            <s xml:id="echoid-s11772" xml:space="preserve">erit 3 TD + 4 RD. </s>
            <s xml:id="echoid-s11773" xml:space="preserve">7 MD:</s>
            <s xml:id="echoid-s11774" xml:space="preserve">: RD. </s>
            <s xml:id="echoid-s11775" xml:space="preserve">SD; </s>
            <s xml:id="echoid-s11776" xml:space="preserve">vel SD = {7 MD x RD/3 TD + 4 RD.</s>
            <s xml:id="echoid-s11777" xml:space="preserve">}</s>
          </p>
          <p>
            <s xml:id="echoid-s11778" xml:space="preserve">XII. </s>
            <s xml:id="echoid-s11779" xml:space="preserve">Patet etiam, accepto quolibet in curva FBF puncto (ceu F)
              <lb/>
            rectam ad hoc tangentem conſimili pacto deſignari. </s>
            <s xml:id="echoid-s11780" xml:space="preserve">Nempe per F
              <lb/>
              <note position="right" xlink:label="note-0251-08" xlink:href="note-0251-08a" xml:space="preserve">Fig. 101.</note>
            ducatur recta PG ad DB parallela, ſecans curvam EBE ad E; </s>
            <s xml:id="echoid-s11781" xml:space="preserve">& </s>
            <s xml:id="echoid-s11782" xml:space="preserve">
              <lb/>
            per E ducatur ER curvam EBE tangens; </s>
            <s xml:id="echoid-s11783" xml:space="preserve">fiátque N x TP + M/- N} x RP.</s>
            <s xml:id="echoid-s11784" xml:space="preserve"/>
          </p>
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