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

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        <div xml:id="echoid-div399" type="section" level="1" n="42">
          <p>
            <s xml:id="echoid-s13374" xml:space="preserve">
              <pb o="98" file="0276" n="291" rhead=""/>
            multiplicando) 2 _tn_ + _sn_ = _mt_ + _ms_. </s>
            <s xml:id="echoid-s13375" xml:space="preserve">vel tranſponendo 2 _nt_ -
              <lb/>
            _mt_ = _ms_ - _ns_. </s>
            <s xml:id="echoid-s13376" xml:space="preserve">unde _m_ - _n_. </s>
            <s xml:id="echoid-s13377" xml:space="preserve">2 _n_ - _m_:</s>
            <s xml:id="echoid-s13378" xml:space="preserve">: _t. </s>
            <s xml:id="echoid-s13379" xml:space="preserve">s_:</s>
            <s xml:id="echoid-s13380" xml:space="preserve">: CA. </s>
            <s xml:id="echoid-s13381" xml:space="preserve">AD. </s>
            <s xml:id="echoid-s13382" xml:space="preserve">er-
              <lb/>
            gò patet ex antecedente.</s>
            <s xml:id="echoid-s13383" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13384" xml:space="preserve">XIV. </s>
            <s xml:id="echoid-s13385" xml:space="preserve">Stante duodecimæ hypotheſi, _paraboliformis_ AFB intra hy-
              <lb/>
            perbolam AEB tota cadet.</s>
            <s xml:id="echoid-s13386" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13387" xml:space="preserve">Nam utcunque ducatur EFG ad BD parallela; </s>
            <s xml:id="echoid-s13388" xml:space="preserve">& </s>
            <s xml:id="echoid-s13389" xml:space="preserve">recta ER _hy-_
              <lb/>
              <note position="left" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve">Fig. 141.</note>
            _perbolam_, recta FS _paraboliformem_ tangant. </s>
            <s xml:id="echoid-s13390" xml:space="preserve">Eſtque SG. </s>
            <s xml:id="echoid-s13391" xml:space="preserve">AG:</s>
            <s xml:id="echoid-s13392" xml:space="preserve">:
              <note symbol="(_a_)" position="left" xlink:label="note-0276-02" xlink:href="note-0276-02a" xml:space="preserve">2. _hujus ap._</note>
            _m. </s>
            <s xml:id="echoid-s13393" xml:space="preserve">n_:</s>
            <s xml:id="echoid-s13394" xml:space="preserve">: TD. </s>
            <s xml:id="echoid-s13395" xml:space="preserve">AD &</s>
            <s xml:id="echoid-s13396" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s13397" xml:space="preserve">RG. </s>
            <s xml:id="echoid-s13398" xml:space="preserve">AG. </s>
            <s xml:id="echoid-s13399" xml:space="preserve">unde RG &</s>
            <s xml:id="echoid-s13400" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13401" xml:space="preserve">SG. </s>
            <s xml:id="echoid-s13402" xml:space="preserve">
              <note symbol="(_b_)" position="left" xlink:label="note-0276-03" xlink:href="note-0276-03a" xml:space="preserve">6. _hujus ap._</note>
              <note symbol="(_c_)" position="left" xlink:label="note-0276-04" xlink:href="note-0276-04a" xml:space="preserve">3. _hujus ap._</note>
            curva AEB extra curvam AFB tota cadet.</s>
            <s xml:id="echoid-s13403" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13404" xml:space="preserve">XV. </s>
            <s xml:id="echoid-s13405" xml:space="preserve">Etiam, ſi reliquis perſtantibus, ad baſin GE, axin AG con-
              <lb/>
            ſtitutam imagineris ejuſdem ordinis _paraboliformem_; </s>
            <s xml:id="echoid-s13406" xml:space="preserve">hæc ad partes
              <lb/>
              <note position="left" xlink:label="note-0276-05" xlink:href="note-0276-05a" xml:space="preserve">Fig. 141.</note>
            ipsâ GE ſuperiores intra _hyperbolam_ tota cadet.</s>
            <s xml:id="echoid-s13407" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13408" xml:space="preserve">Nam ſi in _curva hyperbolica_ AE ſumatur ubicunque punctum M, & </s>
            <s xml:id="echoid-s13409" xml:space="preserve">
              <lb/>
            ordinetur MP, ducatúrque hyperbolam tangens MV; </s>
            <s xml:id="echoid-s13410" xml:space="preserve">erit VP.
              <lb/>
            </s>
            <s xml:id="echoid-s13411" xml:space="preserve">AP &</s>
            <s xml:id="echoid-s13412" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13413" xml:space="preserve">_m. </s>
            <s xml:id="echoid-s13414" xml:space="preserve">n._ </s>
            <s xml:id="echoid-s13415" xml:space="preserve">adeoque rurſus è tertia liquet Propoſitum.</s>
            <s xml:id="echoid-s13416" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13417" xml:space="preserve">XVI. </s>
            <s xml:id="echoid-s13418" xml:space="preserve">Quinetiam ſi hæc altera coordinata _paraboliformis_, ad baſin
              <lb/>
            EG conſtituta, ad DB protracta concipiatur, ejus ipſis EG, BD in-
              <lb/>
              <note position="left" xlink:label="note-0276-06" xlink:href="note-0276-06a" xml:space="preserve">Fig. 141.</note>
            tercepta pars extra _hyperbolam_ tota cadet.</s>
            <s xml:id="echoid-s13419" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13420" xml:space="preserve">Nam quòd extra _hyperbolam_ infra EG cadit, exinde patet, quòd
              <lb/>
            ipſa cum ipſius tangente recta ES angulum efficit minorem eo, quem
              <lb/>
            eadem recta ES efficit cum recta RE hyperbolam tangente. </s>
            <s xml:id="echoid-s13421" xml:space="preserve">quòd au-
              <lb/>
            tem eadem alibi, velut ad N, _hyperbolæ_ non occurrit, patet; </s>
            <s xml:id="echoid-s13422" xml:space="preserve">quoniam
              <lb/>
            hoc poſito, ipſa intra _hyperbolam_ AN tota conſiſteret,
              <note symbol="(_a_)" position="left" xlink:label="note-0276-07" xlink:href="note-0276-07a" xml:space="preserve">3. _hujus ap._</note>
            quàm mox oſtenſum eſt.</s>
            <s xml:id="echoid-s13423" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13424" xml:space="preserve">XVII. </s>
            <s xml:id="echoid-s13425" xml:space="preserve">Habeant _Circulus_ AEB, & </s>
            <s xml:id="echoid-s13426" xml:space="preserve">_parabola_ AFB communem
              <lb/>
            axem AD, & </s>
            <s xml:id="echoid-s13427" xml:space="preserve">baſin DB; </s>
            <s xml:id="echoid-s13428" xml:space="preserve">_parabola_ ad partes ſupra BD intra _Circu-_
              <lb/>
            _lum_; </s>
            <s xml:id="echoid-s13429" xml:space="preserve">at infra BD extra _circulum_ cadet.</s>
            <s xml:id="echoid-s13430" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13431" xml:space="preserve">Sit enim _Circuli Diameter_ AZ, & </s>
            <s xml:id="echoid-s13432" xml:space="preserve">eiæqualis A Had BD paralle-
              <lb/>
            la, & </s>
            <s xml:id="echoid-s13433" xml:space="preserve">connectatur ZH; </s>
            <s xml:id="echoid-s13434" xml:space="preserve">& </s>
            <s xml:id="echoid-s13435" xml:space="preserve">huic BD producta ad I; </s>
            <s xml:id="echoid-s13436" xml:space="preserve">ergo DI eſt
              <lb/>
              <note position="left" xlink:label="note-0276-08" xlink:href="note-0276-08a" xml:space="preserve">Fig. 142.</note>
            _Parameter parabolæ_ AFB. </s>
            <s xml:id="echoid-s13437" xml:space="preserve">quòd ſi ſupra BD utcunque ducatur recta
              <lb/>
            EF GK ad BD parallela circulum ſecans in E, parabolam in F, rectas
              <lb/>
            AZ, HZ, in G, & </s>
            <s xml:id="echoid-s13438" xml:space="preserve">K, patet eſſe GEq = AG x GK &</s>
            <s xml:id="echoid-s13439" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13440" xml:space="preserve">AG x DI
              <lb/>
            = GFq. </s>
            <s xml:id="echoid-s13441" xml:space="preserve">unde GE &</s>
            <s xml:id="echoid-s13442" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13443" xml:space="preserve">GF. </s>
            <s xml:id="echoid-s13444" xml:space="preserve">Item, ſi infra BD utcunque ducatur
              <lb/>
            recta MN OL ad BD parallela _parabolam_ ſecans in M, _circu-_
              <lb/>
            _lum_ in N, rectas AZ, HZ in O, & </s>
            <s xml:id="echoid-s13445" xml:space="preserve">L, itidem patet eſſe MO q
              <lb/>
            = AO x DI &</s>
            <s xml:id="echoid-s13446" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s13447" xml:space="preserve">AO x OL = NO q. </s>
            <s xml:id="echoid-s13448" xml:space="preserve">& </s>
            <s xml:id="echoid-s13449" xml:space="preserve">ideò M </s>
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