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-s9918" xml:space="preserve">
              <pb o="48" file="0226" n="241" rhead=""/>
            linea OOO talis, ut ſi è D utcunque ducatur recta DO, ſecans
              <lb/>
            anguli latera punctis M, N, habeat DM ad NO ſemper eandem
              <lb/>
              <note position="left" xlink:label="note-0226-01" xlink:href="note-0226-01a" xml:space="preserve">Fig. 43.</note>
            rationem (puta X ad Y) erit etiam linea OOO hyperbola.</s>
            <s xml:id="echoid-s9919" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9920" xml:space="preserve">Nam ducatur DL ad BC parallela; </s>
            <s xml:id="echoid-s9921" xml:space="preserve">ſitque DL. </s>
            <s xml:id="echoid-s9922" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s9923" xml:space="preserve">: X. </s>
            <s xml:id="echoid-s9924" xml:space="preserve">Y;
              <lb/>
            </s>
            <s xml:id="echoid-s9925" xml:space="preserve">& </s>
            <s xml:id="echoid-s9926" xml:space="preserve">per E ducatur ER ad AB parallela; </s>
            <s xml:id="echoid-s9927" xml:space="preserve">ſecans BC in Z; </s>
            <s xml:id="echoid-s9928" xml:space="preserve">de-
              <lb/>
            mum per O ducatur OH ad BA parallela.</s>
            <s xml:id="echoid-s9929" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9930" xml:space="preserve">Eſt jam DL. </s>
            <s xml:id="echoid-s9931" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s9932" xml:space="preserve">: DM. </s>
            <s xml:id="echoid-s9933" xml:space="preserve">NO:</s>
            <s xml:id="echoid-s9934" xml:space="preserve">: LM. </s>
            <s xml:id="echoid-s9935" xml:space="preserve">GO (ob ſimilia tri-
              <lb/>
            angula DLM, NGO):</s>
            <s xml:id="echoid-s9936" xml:space="preserve">: LM x DH. </s>
            <s xml:id="echoid-s9937" xml:space="preserve">GO x DH item DL x
              <lb/>
            HO = LM x DH (ob DL. </s>
            <s xml:id="echoid-s9938" xml:space="preserve">LM:</s>
            <s xml:id="echoid-s9939" xml:space="preserve">: DH. </s>
            <s xml:id="echoid-s9940" xml:space="preserve">HO) quare DL. </s>
            <s xml:id="echoid-s9941" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s9942" xml:space="preserve">:
              <lb/>
            DL x HO. </s>
            <s xml:id="echoid-s9943" xml:space="preserve">GO x DH hoc eſt DL x HO. </s>
            <s xml:id="echoid-s9944" xml:space="preserve">DE x HO:</s>
            <s xml:id="echoid-s9945" xml:space="preserve">: DL x HO.
              <lb/>
            </s>
            <s xml:id="echoid-s9946" xml:space="preserve">GO x DH adeóq; </s>
            <s xml:id="echoid-s9947" xml:space="preserve">DE x HO = GO x DH. </s>
            <s xml:id="echoid-s9948" xml:space="preserve">hoc eſt DE x HG + DE x
              <lb/>
            GO = GO x DE + GO x EH quare (communi ſublato) eſt
              <lb/>
            DE x HG = GO x EH; </s>
            <s xml:id="echoid-s9949" xml:space="preserve">ſeu DE x HG = GO x ZG. </s>
            <s xml:id="echoid-s9950" xml:space="preserve">Pa-
              <lb/>
            tet itaque curvam OOO eſſe _hyperbolam_ cujus _aſymptoti_ ZR
              <lb/>
            ZC.</s>
            <s xml:id="echoid-s9951" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9952" xml:space="preserve">_Coroll_. </s>
            <s xml:id="echoid-s9953" xml:space="preserve">Si ratio data ſit æqualitatis (ceu DM = NO,) ipſæ AB,
              <lb/>
            CB aſymptoti erunt.</s>
            <s xml:id="echoid-s9954" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9955" xml:space="preserve">Sequentia quædam, quia magìs id perſpicuum videtur, Alge-
              <lb/>
            bricè monſtrabimus.</s>
            <s xml:id="echoid-s9956" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9957" xml:space="preserve">X. </s>
            <s xml:id="echoid-s9958" xml:space="preserve">Eſto poſitione data recta ID, in qua punctum deſignatum D,
              <lb/>
            ſit item curva DNN talis ut in ID ſumpto quopiam puncto G, & </s>
            <s xml:id="echoid-s9959" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0226-02" xlink:href="note-0226-02a" xml:space="preserve">Fig. 44.</note>
            ductâ rectâ GN ad poſitionem datam IK paràllelá; </s>
            <s xml:id="echoid-s9960" xml:space="preserve">tum adſumptis
              <lb/>
            determinatis rectis _m, b_; </s>
            <s xml:id="echoid-s9961" xml:space="preserve">poſitiſq; </s>
            <s xml:id="echoid-s9962" xml:space="preserve">DG = _x_, & </s>
            <s xml:id="echoid-s9963" xml:space="preserve">GN = _y_; </s>
            <s xml:id="echoid-s9964" xml:space="preserve">ſit
              <lb/>
            conſtantèr _m y_ + _x y_ = {_m_/_b_}_x x_; </s>
            <s xml:id="echoid-s9965" xml:space="preserve">erit linea DNN _hyperbola_; </s>
            <s xml:id="echoid-s9966" xml:space="preserve">quæ
              <lb/>
            ſic determinatur; </s>
            <s xml:id="echoid-s9967" xml:space="preserve">ſumantur DM, & </s>
            <s xml:id="echoid-s9968" xml:space="preserve">DO (hinc indè) pares ipſi _m_;
              <lb/>
            </s>
            <s xml:id="echoid-s9969" xml:space="preserve">& </s>
            <s xml:id="echoid-s9970" xml:space="preserve">per M ducatur M L@ad IK parallela, factóq; </s>
            <s xml:id="echoid-s9971" xml:space="preserve">_b. </s>
            <s xml:id="echoid-s9972" xml:space="preserve">m_:</s>
            <s xml:id="echoid-s9973" xml:space="preserve">: _m_. </s>
            <s xml:id="echoid-s9974" xml:space="preserve">MQ; </s>
            <s xml:id="echoid-s9975" xml:space="preserve">ſit
              <lb/>
            MZ = 2 MQ = {2_mm_;</s>
            <s xml:id="echoid-s9976" xml:space="preserve">/_b_} tum per Z, O traducatur recta ZT; </s>
            <s xml:id="echoid-s9977" xml:space="preserve">erunt
              <lb/>
            ZM, ZT aſymptoti.</s>
            <s xml:id="echoid-s9978" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9979" xml:space="preserve">Ducatur enim ZS ad MO parallela, cui occurrat N Gin R (quæ
              <lb/>
            & </s>
            <s xml:id="echoid-s9980" xml:space="preserve">ipſam ZT ſect in P). </s>
            <s xml:id="echoid-s9981" xml:space="preserve">& </s>
            <s xml:id="echoid-s9982" xml:space="preserve">connectatur DQ. </s>
            <s xml:id="echoid-s9983" xml:space="preserve">Eſt ergò PN = RG
              <lb/>
            + GN - RP. </s>
            <s xml:id="echoid-s9984" xml:space="preserve">Verùm eſt MD. </s>
            <s xml:id="echoid-s9985" xml:space="preserve">MQ:</s>
            <s xml:id="echoid-s9986" xml:space="preserve">: ZR (MG). </s>
            <s xml:id="echoid-s9987" xml:space="preserve">RP; </s>
            <s xml:id="echoid-s9988" xml:space="preserve">hoc
              <lb/>
            eſt _m_. </s>
            <s xml:id="echoid-s9989" xml:space="preserve">{_mm_/_b_}:</s>
            <s xml:id="echoid-s9990" xml:space="preserve">: _m_ + _x_. </s>
            <s xml:id="echoid-s9991" xml:space="preserve">RP = {_mm_/_b_} + {_mx._</s>
            <s xml:id="echoid-s9992" xml:space="preserve">/_b_} adeóq; </s>
            <s xml:id="echoid-s9993" xml:space="preserve">RG - RP
              <lb/>
            = {_mm_/_b_} - {_mx._</s>
            <s xml:id="echoid-s9994" xml:space="preserve">/_b_} ergò PN = {_mm_ - _mx_/_b_} + _y_. </s>
            <s xml:id="echoid-s9995" xml:space="preserve">Unde PN x MG
              <lb/>
            = {_m_
              <emph style="sub">3</emph>
            /_b_} + _my_ + _xy_ - {_mxx._</s>
            <s xml:id="echoid-s9996" xml:space="preserve">/_b_} Verùm (ex hypotheſi) eſt _m </s>
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