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-s10939" xml:space="preserve">Eſt enim E &</s>
            <s xml:id="echoid-s10940" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10941" xml:space="preserve">P, ac indè D &</s>
            <s xml:id="echoid-s10942" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10943" xml:space="preserve">O; </s>
            <s xml:id="echoid-s10944" xml:space="preserve">& </s>
            <s xml:id="echoid-s10945" xml:space="preserve">hinc C &</s>
            <s xml:id="echoid-s10946" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10947" xml:space="preserve">N.</s>
            <s xml:id="echoid-s10948" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10949" xml:space="preserve">XVII. </s>
            <s xml:id="echoid-s10950" xml:space="preserve">Conſectatur hinc; </s>
            <s xml:id="echoid-s10951" xml:space="preserve">ſi fuerint quatuor lineæ HBH, GBG, FBF,
              <lb/>
            EBE ſeſe interſecantes in B, ac ita verſus ſe relatæ, ut ductâ utcunque
              <lb/>
            rectâ DH ad poſitione datam DB parallelâ (in linea nempe DD D
              <lb/>
              <note position="right" xlink:label="note-0239-01" xlink:href="note-0239-01a" xml:space="preserve">Fig. 68.
                <lb/>
              69.</note>
            terminatâ) vel à deſignato puncto D projectâ DH; </s>
            <s xml:id="echoid-s10952" xml:space="preserve">ſit perpetuò DG
              <lb/>
            inter DH, DE eodem ordine media proportionalis Arithmeticè, quo
              <lb/>
            DF inter eaſdem media Geometricè; </s>
            <s xml:id="echoid-s10953" xml:space="preserve">lineæ GB G, FBF ſeſe mu-
              <lb/>
            tuò contingunt.</s>
            <s xml:id="echoid-s10954" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10955" xml:space="preserve">Enimverò linea GBH extra lineam FBF totam cadere manifeſtum
              <lb/>
            è præcedente.</s>
            <s xml:id="echoid-s10956" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10957" xml:space="preserve">XVIII. </s>
            <s xml:id="echoid-s10958" xml:space="preserve">Ex iſthinc etiam (quod ſtrictim tranſcurrens moneo) di-
              <lb/>
              <note position="right" xlink:label="note-0239-02" xlink:href="note-0239-02a" xml:space="preserve">Fig. 70.</note>
            verſis innumeris _Hyperbolarum_, aut _Hyperboliformium_ generibus con-
              <lb/>
            venientes rectæ ασνμπωτοι definiuntur. </s>
            <s xml:id="echoid-s10959" xml:space="preserve">Sint nempe rectæ VD, BD
              <lb/>
            poſitione datæ; </s>
            <s xml:id="echoid-s10960" xml:space="preserve">ſint item aliæ duæ rectæ AB, VI; </s>
            <s xml:id="echoid-s10961" xml:space="preserve">ductâ verò li-
              <lb/>
            berè rectâ PG ad DB parallela, ſit P φ conſtantèr inter PG, PE eo-
              <lb/>
            dem ordine media proportionalis Arithmeticè, quo PF inter eaſdem
              <lb/>
            media Geometricè; </s>
            <s xml:id="echoid-s10962" xml:space="preserve">quia jam rectæ E G, E φ ſemper eandem
              <note symbol="(a)" position="right" xlink:label="note-0239-03" xlink:href="note-0239-03a" xml:space="preserve">12 buj@@</note>
            tinent rationem, eſt linea φ φ φ recta; </s>
            <s xml:id="echoid-s10963" xml:space="preserve">verùm linea VFF eſt _hyperbo-_
              <lb/>
            _la,_ vel _hyperboliformis_ aliqua (communis quidem vel _Apolloniana_
              <lb/>
            _hyperbola_, ſi PF ſit inter ipſas PG, PE ſimpliciter media, ſed alia
              <lb/>
            diverſi generis quædam _hyperboliformis_, ſi PE ſit alterius cujuſpiam
              <lb/>
            ordinis media) atqui patet è penultima præmiſſa lineam φ φ φ eodem
              <lb/>
            ordine reſpondenti lineæ VFF _aſymptoton_ eſſe. </s>
            <s xml:id="echoid-s10964" xml:space="preserve">quod an πρ@@γ@ ſit
              <lb/>
            neſcio, nobis certè πáρε@γον fuit, hic adnotâſſe.</s>
            <s xml:id="echoid-s10965" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10966" xml:space="preserve">XIX. </s>
            <s xml:id="echoid-s10967" xml:space="preserve">A puncto aſſignato B ad datam poſitione rectam AC ductæ
              <lb/>
            ſint restæ tres BA, BC, BQ; </s>
            <s xml:id="echoid-s10968" xml:space="preserve">tum in QC producta ſumatur ſumptum
              <lb/>
              <note position="right" xlink:label="note-0239-04" xlink:href="note-0239-04a" xml:space="preserve">Fig. 71.</note>
            quodpiam D; </s>
            <s xml:id="echoid-s10969" xml:space="preserve">per B recta (puta B R) duci poteſt (ad alterutras ipſi-
              <lb/>
            us BQ partes) tali@, ut à D projectâ quâcunque rectâ, ceu DN; </s>
            <s xml:id="echoid-s10970" xml:space="preserve">ſit
              <lb/>
            hujus à rectis B Q, BR intercepta pars (F E) minor ejuſdem à rectis
              <lb/>
            BA, BC interceptâ parte (N M).</s>
            <s xml:id="echoid-s10971" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10972" xml:space="preserve">Nam, primò, ſi BR (ultra angulum ABC jaceat reſpectu puncti
              <lb/>
            D; </s>
            <s xml:id="echoid-s10973" xml:space="preserve">ſiat QR = CA; </s>
            <s xml:id="echoid-s10974" xml:space="preserve">& </s>
            <s xml:id="echoid-s10975" xml:space="preserve">connectatur BR; </s>
            <s xml:id="echoid-s10976" xml:space="preserve">tum utcunque ducatur
              <lb/>
            DE, rectas ſecans, ut vides; </s>
            <s xml:id="echoid-s10977" xml:space="preserve">& </s>
            <s xml:id="echoid-s10978" xml:space="preserve">maniſeſtum eſt, * è ſupra mon-
              <lb/>
              <note position="right" xlink:label="note-0239-05" xlink:href="note-0239-05a" xml:space="preserve">* Per 7. Lect.
                <lb/>
              VI.</note>
            ſtratis fore, FE &</s>
            <s xml:id="echoid-s10979" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s10980" xml:space="preserve">NM.</s>
            <s xml:id="echoid-s10981" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10982" xml:space="preserve">Sin B Q citra angulum ABC cadat verſus D; </s>
            <s xml:id="echoid-s10983" xml:space="preserve">(a) ducatur recta
              <lb/>
              <note position="right" xlink:label="note-0239-06" xlink:href="note-0239-06a" xml:space="preserve">* Per
                <lb/>
              Vi. 8 Lect.</note>
            BH talis, ut à BQ, B H interceptæ minores ſint interceptis à BQ,
              <lb/>
            BA; </s>
            <s xml:id="echoid-s10984" xml:space="preserve">& </s>
            <s xml:id="echoid-s10985" xml:space="preserve">ſumatur HR = QC; </s>
            <s xml:id="echoid-s10986" xml:space="preserve">& </s>
            <s xml:id="echoid-s10987" xml:space="preserve">connectatur BR; </s>
            <s xml:id="echoid-s10988" xml:space="preserve">tum rurſus
              <lb/>
              <note position="right" xlink:label="note-0239-07" xlink:href="note-0239-07a" xml:space="preserve">Fig. 72.</note>
            utcunque ductâ DN, quæ rectas interſecet, ut exhibet Schema; </s>
            <s xml:id="echoid-s10989" xml:space="preserve"/>
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