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-s9361" xml:space="preserve">Nam connectatur ſubtenſa MN, ducatúrque recta NR ad ZA
              <lb/>
            parallela. </s>
            <s xml:id="echoid-s9362" xml:space="preserve">Et quoniam angulus XPH non minor eſt recto, erit,
              <lb/>
            eo major externus, NHP obtuſus. </s>
            <s xml:id="echoid-s9363" xml:space="preserve">Ergò recta NM major eſt quàm
              <lb/>
            NH. </s>
            <s xml:id="echoid-s9364" xml:space="preserve">Itaque magis arcus, arcus NH major eſt quàm ipsâ NH:
              <lb/>
            </s>
            <s xml:id="echoid-s9365" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s9366" xml:space="preserve">E. </s>
            <s xml:id="echoid-s9367" xml:space="preserve">D.</s>
            <s xml:id="echoid-s9368" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9369" xml:space="preserve">Item, quoniam ang. </s>
            <s xml:id="echoid-s9370" xml:space="preserve">RNE ipſi XQE par haud minor eſt recto,
              <lb/>
            erit RE &</s>
            <s xml:id="echoid-s9371" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9372" xml:space="preserve">RN. </s>
            <s xml:id="echoid-s9373" xml:space="preserve">quare MR + RE &</s>
            <s xml:id="echoid-s9374" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9375" xml:space="preserve">MR + RN. </s>
            <s xml:id="echoid-s9376" xml:space="preserve">hoc eſt
              <lb/>
            ME &</s>
            <s xml:id="echoid-s9377" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9378" xml:space="preserve">MR + RN. </s>
            <s xml:id="echoid-s9379" xml:space="preserve">Eſt autem (ex _Arcbimedæis_ aſſumptis) MR
              <lb/>
            + RN &</s>
            <s xml:id="echoid-s9380" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9381" xml:space="preserve">arc. </s>
            <s xml:id="echoid-s9382" xml:space="preserve">MN. </s>
            <s xml:id="echoid-s9383" xml:space="preserve">ergò magìs eſt ME &</s>
            <s xml:id="echoid-s9384" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9385" xml:space="preserve">arc. </s>
            <s xml:id="echoid-s9386" xml:space="preserve">MN ∴</s>
          </p>
          <p>
            <s xml:id="echoid-s9387" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s9388" xml:space="preserve">Perutilis eſt hæc propoſitio in _tangentium demonſtra@ionibus_
              <lb/>
            _expediendis_. </s>
            <s xml:id="echoid-s9389" xml:space="preserve">Etenim hinc couſectatur, ſi arcus MN indefinitè par-
              <lb/>
            vus ponatur, ejuſce loco alterutram tangentis particulam ME, vel
              <lb/>
            NH tutò ſubſtitui.</s>
            <s xml:id="echoid-s9390" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9391" xml:space="preserve">_Speciminis_ hîc loco _metbodum proponam generalem cycloidum om-_
              <lb/>
            _nium, & </s>
            <s xml:id="echoid-s9392" xml:space="preserve">conſimili modo deſcriptarum curvarum tangentes determi-_
              <lb/>
            _nandi_, hinc petitâ demonſtratione munitam.</s>
            <s xml:id="echoid-s9393" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9394" xml:space="preserve">Recta AY ſibi parallelè deportata quamcunque curvam ad eaſdem
              <lb/>
              <note position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve">Fig. 27.</note>
            partes convexam aut cavam, APX perambulet uniformi motu (ſci-
              <lb/>
            licet ut æquales curvæ partes æqualibus tranſigat temporibus) eodém-
              <lb/>
            que ſimul tempore punctum aliquod ab A per AY etiam uniformiter
              <lb/>
            feratur; </s>
            <s xml:id="echoid-s9395" xml:space="preserve">ab hoc puncto taliter moto progignetur curva AMZ;
              <lb/>
            </s>
            <s xml:id="echoid-s9396" xml:space="preserve">cujus ad datum quodcunque punctum M tangentem oportet determi-
              <lb/>
            nare. </s>
            <s xml:id="echoid-s9397" xml:space="preserve">Ut hoc fiat, ducatur recta MP ad AY parallela, curvam
              <lb/>
            APX ſecans in P; </s>
            <s xml:id="echoid-s9398" xml:space="preserve">pérque P ducatur recta PE curvam APX con-
              <lb/>
            tingens; </s>
            <s xml:id="echoid-s9399" xml:space="preserve">huic verò per M ducatur parallela MH; </s>
            <s xml:id="echoid-s9400" xml:space="preserve">ínque hac ſumatur
              <lb/>
            punctum quodpiam R, & </s>
            <s xml:id="echoid-s9401" xml:space="preserve">ducatur RS ad PM parallela; </s>
            <s xml:id="echoid-s9402" xml:space="preserve">tum fiat
              <lb/>
            ut curva AP ad rectam PM (hoc eſt ut unus motus uniformis ad
              <lb/>
            alterum) ità MR ad RS; </s>
            <s xml:id="echoid-s9403" xml:space="preserve">& </s>
            <s xml:id="echoid-s9404" xml:space="preserve">connectatur MS. </s>
            <s xml:id="echoid-s9405" xml:space="preserve">hæc curvam AMZ
              <lb/>
            continget. </s>
            <s xml:id="echoid-s9406" xml:space="preserve">Sumatur enim in hac curva punctum quodvis Z, per quod
              <lb/>
            ducatur recta ZX ad MP parallela, ſecans curvam APX in X,
              <lb/>
            ejúſque tangentem in E; </s>
            <s xml:id="echoid-s9407" xml:space="preserve">& </s>
            <s xml:id="echoid-s9408" xml:space="preserve">huic parallelam MR in H; </s>
            <s xml:id="echoid-s9409" xml:space="preserve">ipsámque
              <lb/>
            demum MS in K. </s>
            <s xml:id="echoid-s9410" xml:space="preserve">Sit autem primò punctum Z ſupra M verſus A; </s>
            <s xml:id="echoid-s9411" xml:space="preserve">
              <lb/>
            unde recta PE &</s>
            <s xml:id="echoid-s9412" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9413" xml:space="preserve">arc PX. </s>
            <s xml:id="echoid-s9414" xml:space="preserve">adeóque PA. </s>
            <s xml:id="echoid-s9415" xml:space="preserve">PE &</s>
            <s xml:id="echoid-s9416" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9417" xml:space="preserve">arc PA. </s>
            <s xml:id="echoid-s9418" xml:space="preserve">PX:</s>
            <s xml:id="echoid-s9419" xml:space="preserve">:
              <lb/>
            PM. </s>
            <s xml:id="echoid-s9420" xml:space="preserve">PM - XZ:</s>
            <s xml:id="echoid-s9421" xml:space="preserve">: PM. </s>
            <s xml:id="echoid-s9422" xml:space="preserve">EH - XZ:</s>
            <s xml:id="echoid-s9423" xml:space="preserve">: PM. </s>
            <s xml:id="echoid-s9424" xml:space="preserve">ZH - EX &</s>
            <s xml:id="echoid-s9425" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9426" xml:space="preserve">
              <lb/>
            PM. </s>
            <s xml:id="echoid-s9427" xml:space="preserve">ZH. </s>
            <s xml:id="echoid-s9428" xml:space="preserve">quare permutatim erit PA. </s>
            <s xml:id="echoid-s9429" xml:space="preserve">PM &</s>
            <s xml:id="echoid-s9430" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9431" xml:space="preserve">PE. </s>
            <s xml:id="echoid-s9432" xml:space="preserve">ZH. </s>
            <s xml:id="echoid-s9433" xml:space="preserve">eſt
              <lb/>
            autem PA. </s>
            <s xml:id="echoid-s9434" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s9435" xml:space="preserve">: MR. </s>
            <s xml:id="echoid-s9436" xml:space="preserve">RS:</s>
            <s xml:id="echoid-s9437" xml:space="preserve">: MH. </s>
            <s xml:id="echoid-s9438" xml:space="preserve">KH:</s>
            <s xml:id="echoid-s9439" xml:space="preserve">: PE. </s>
            <s xml:id="echoid-s9440" xml:space="preserve">HK. </s>
            <s xml:id="echoid-s9441" xml:space="preserve">ergò
              <lb/>
            PE. </s>
            <s xml:id="echoid-s9442" xml:space="preserve">HK. </s>
            <s xml:id="echoid-s9443" xml:space="preserve">&</s>
            <s xml:id="echoid-s9444" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s9445" xml:space="preserve">PE. </s>
            <s xml:id="echoid-s9446" xml:space="preserve">ZH quare HK &</s>
            <s xml:id="echoid-s9447" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9448" xml:space="preserve">ZH. </s>
            <s xml:id="echoid-s9449" xml:space="preserve">eſt autem punctum
              <lb/>
            H extra curvam AZM; </s>
            <s xml:id="echoid-s9450" xml:space="preserve">ob EZ &</s>
            <s xml:id="echoid-s9451" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9452" xml:space="preserve">XZ &</s>
            <s xml:id="echoid-s9453" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s9454" xml:space="preserve">PM = EH. </s>
            <s xml:id="echoid-s9455" xml:space="preserve">ergò
              <lb/>
            palàm eſt punctum K extra curvam AZM exiſtere. </s>
            <s xml:id="echoid-s9456" xml:space="preserve">Sit vero </s>
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