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-s9839" xml:space="preserve">
              <pb o="47" file="0225" n="240" rhead=""/>
            DE x HO ergò DH x BF + DH x KO = DE x HO; </s>
            <s xml:id="echoid-s9840" xml:space="preserve">hoc eſt
              <lb/>
            DH x BF + DH x HO - DH x BL = DE x HO; </s>
            <s xml:id="echoid-s9841" xml:space="preserve">tranſpo-
              <lb/>
            nendo igitur eſt DH x HO - DE x HO = DH x BL - DH x
              <lb/>
              <note position="right" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve">Fig. 39.</note>
            BF. </s>
            <s xml:id="echoid-s9842" xml:space="preserve">hoc eſt EH x HO = DH x FL; </s>
            <s xml:id="echoid-s9843" xml:space="preserve">vel EH x GO + EH x
              <lb/>
            HG = DE x FL + EH x FL; </s>
            <s xml:id="echoid-s9844" xml:space="preserve">quare, demptis æqualibus, eſt EH
              <lb/>
            x GO = DE x FL; </s>
            <s xml:id="echoid-s9845" xml:space="preserve">vel ZG x GO = DE x FL; </s>
            <s xml:id="echoid-s9846" xml:space="preserve">cum itaque
              <lb/>
            DE x FL ſit quid determinatum, conſtat lineam OBO effe hy-
              <lb/>
            perbolam, cujus aſymptoti ZR, ZS.</s>
            <s xml:id="echoid-s9847" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9848" xml:space="preserve">V. </s>
            <s xml:id="echoid-s9849" xml:space="preserve">Si MO capiatur ad alteras rectæ BC partes, etiam DE.
              <lb/>
            </s>
            <s xml:id="echoid-s9850" xml:space="preserve">BF ad alteras punctorum D, B partes accipi debent; </s>
            <s xml:id="echoid-s9851" xml:space="preserve">uti Schema
              <lb/>
              <note position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">Fig. 40.</note>
            monſtrat; </s>
            <s xml:id="echoid-s9852" xml:space="preserve">nec abludit modus demonſtrandi.</s>
            <s xml:id="echoid-s9853" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9854" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s9855" xml:space="preserve">Conſectarium. </s>
            <s xml:id="echoid-s9856" xml:space="preserve">Si recta BQ angulum ABC ſecet, pér-
              <lb/>
            que punctum D ducantur utcunque duæ rectæ MN, XY rectam
              <lb/>
              <note position="right" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">Fig. 41.</note>
            BQ interſecantes punctis OP (quorum utique ſit O propius ip-
              <lb/>
            ſi B) erit MN. </s>
            <s xml:id="echoid-s9857" xml:space="preserve">MO &</s>
            <s xml:id="echoid-s9858" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9859" xml:space="preserve">XY. </s>
            <s xml:id="echoid-s9860" xml:space="preserve">XP. </s>
            <s xml:id="echoid-s9861" xml:space="preserve">Nam per O deſcripta con-
              <lb/>
            cipiatur _hyperbola_ VOB (qualem jam mox attigimus, ſic ut inter-
              <lb/>
            ceptæ rationem habeant illam quam MN ad MO) erit igitur
              <lb/>
            MN. </s>
            <s xml:id="echoid-s9862" xml:space="preserve">MO:</s>
            <s xml:id="echoid-s9863" xml:space="preserve">: (XY. </s>
            <s xml:id="echoid-s9864" xml:space="preserve">XV) &</s>
            <s xml:id="echoid-s9865" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9866" xml:space="preserve">XY. </s>
            <s xml:id="echoid-s9867" xml:space="preserve">XP.</s>
            <s xml:id="echoid-s9868" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9869" xml:space="preserve">_Coroll._ </s>
            <s xml:id="echoid-s9870" xml:space="preserve">Dividendo eſt NO. </s>
            <s xml:id="echoid-s9871" xml:space="preserve">MO &</s>
            <s xml:id="echoid-s9872" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9873" xml:space="preserve">YP. </s>
            <s xml:id="echoid-s9874" xml:space="preserve">PX.</s>
            <s xml:id="echoid-s9875" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9876" xml:space="preserve">VII. </s>
            <s xml:id="echoid-s9877" xml:space="preserve">Quinimò ſi plures lineæ BQ, BG angulum ABC ſecent;
              <lb/>
            </s>
            <s xml:id="echoid-s9878" xml:space="preserve">
              <note position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">Fig. 42.</note>
            & </s>
            <s xml:id="echoid-s9879" xml:space="preserve">à puncto D projiciantur rectæ DN, DY (quæ rectas alteras
              <lb/>
            interſecant ut vides; </s>
            <s xml:id="echoid-s9880" xml:space="preserve">quarúmque DN puncto B vicinior;) </s>
            <s xml:id="echoid-s9881" xml:space="preserve">erit
              <lb/>
            NE. </s>
            <s xml:id="echoid-s9882" xml:space="preserve">MO &</s>
            <s xml:id="echoid-s9883" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9884" xml:space="preserve">YF. </s>
            <s xml:id="echoid-s9885" xml:space="preserve">VX.</s>
            <s xml:id="echoid-s9886" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9887" xml:space="preserve">Nam NE. </s>
            <s xml:id="echoid-s9888" xml:space="preserve">EO &</s>
            <s xml:id="echoid-s9889" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9890" xml:space="preserve">YF. </s>
            <s xml:id="echoid-s9891" xml:space="preserve">FV; </s>
            <s xml:id="echoid-s9892" xml:space="preserve">& </s>
            <s xml:id="echoid-s9893" xml:space="preserve">EO. </s>
            <s xml:id="echoid-s9894" xml:space="preserve">OM &</s>
            <s xml:id="echoid-s9895" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9896" xml:space="preserve">FV. </s>
            <s xml:id="echoid-s9897" xml:space="preserve">VX. </s>
            <s xml:id="echoid-s9898" xml:space="preserve">i-
              <lb/>
            gitur ex æquo eſt NE. </s>
            <s xml:id="echoid-s9899" xml:space="preserve">OM &</s>
            <s xml:id="echoid-s9900" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9901" xml:space="preserve">YF. </s>
            <s xml:id="echoid-s9902" xml:space="preserve">VX.</s>
            <s xml:id="echoid-s9903" xml:space="preserve">‖</s>
          </p>
          <p>
            <s xml:id="echoid-s9904" xml:space="preserve">VIII. </s>
            <s xml:id="echoid-s9905" xml:space="preserve">Etiam exindè patet, per B (ad partes alterutras) rectam
              <lb/>
            duci poſſe; </s>
            <s xml:id="echoid-s9906" xml:space="preserve">ità ut è D eductarum partes ab illa rectáque BC ad
              <lb/>
            interceptas à rectis BA, BC rationem habeant minorem quâpi-
              <lb/>
            am datâ.</s>
            <s xml:id="echoid-s9907" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9908" xml:space="preserve">Nam ſumatur PQ = PZ; </s>
            <s xml:id="echoid-s9909" xml:space="preserve">ergò connexa BQ _hyperbolam_ O
              <lb/>
            BO tangit; </s>
            <s xml:id="echoid-s9910" xml:space="preserve">& </s>
            <s xml:id="echoid-s9911" xml:space="preserve">liquet à rectis BQ, BC interceptas ad intercep-
              <lb/>
            tas à BC, BA minorem rationem habere, quàm habent inter-
              <lb/>
            ceptæ ab hyperbolâ OBO & </s>
            <s xml:id="echoid-s9912" xml:space="preserve">recta BC ad eaſdem; </s>
            <s xml:id="echoid-s9913" xml:space="preserve">hoc eſt mi-
              <lb/>
            norem datâ quâpiam.</s>
            <s xml:id="echoid-s9914" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9915" xml:space="preserve">IX. </s>
            <s xml:id="echoid-s9916" xml:space="preserve">Sit rurſum angulus rectilineus ABC, & </s>
            <s xml:id="echoid-s9917" xml:space="preserve">punctum D; </s>
            <s xml:id="echoid-s9918" xml:space="preserve">item
              <lb/>
              <note position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">Fig. 43.</note>
            </s>
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