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-s10838" xml:space="preserve">
              <pb o="60" file="0238" n="253" rhead=""/>
            bini, eodem ordine ſibi reſpondentes, termini (puta D, Fin prima, & </s>
            <s xml:id="echoid-s10839" xml:space="preserve">
              <lb/>
            P, R in ſecunda) erit A -: </s>
            <s xml:id="echoid-s10840" xml:space="preserve">D. </s>
            <s xml:id="echoid-s10841" xml:space="preserve">A -: </s>
            <s xml:id="echoid-s10842" xml:space="preserve">F :</s>
            <s xml:id="echoid-s10843" xml:space="preserve">: M -: </s>
            <s xml:id="echoid-s10844" xml:space="preserve">P. </s>
            <s xml:id="echoid-s10845" xml:space="preserve">M -: </s>
            <s xml:id="echoid-s10846" xml:space="preserve">R.</s>
            <s xml:id="echoid-s10847" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">
            <lb/>
          A. # B. # C. # D. # E. # F.
            <lb/>
          M. # N. # O. # P. # Q. # R.
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s10848" xml:space="preserve">Nam harum rationum utraque par eſt illi, quam habent ad ſe nume-
              <lb/>
            ri N, M, quales in præcedente deſignati ſunt.</s>
            <s xml:id="echoid-s10849" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10850" xml:space="preserve">Hi verò Numeri N, M vulgò terminorum, quibus aptantur, expo-
              <lb/>
            nentes, aut Indices vocantur, in ſerie quavis proportionalium; </s>
            <s xml:id="echoid-s10851" xml:space="preserve">quales
              <lb/>
            nos ſemper in ſequentibus intelligimus, ubi literas has adhibemus.</s>
            <s xml:id="echoid-s10852" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10853" xml:space="preserve">XIII. </s>
            <s xml:id="echoid-s10854" xml:space="preserve">Sint quælibet quanta A, B, C, D, E, F continuè propor-
              <lb/>
            tionalia Arithmeticè; </s>
            <s xml:id="echoid-s10855" xml:space="preserve">nec non alia totidem, ab eodem termino A in-
              <lb/>
            cipientia, Geometricè proportionalia; </s>
            <s xml:id="echoid-s10856" xml:space="preserve">ſit au-
              <lb/>
            tem illorum ſecundum B non majus horum ſe-
              <lb/>
            cundo M; </s>
            <s xml:id="echoid-s10857" xml:space="preserve">erit quodlibet in ſerie Geometrica
              <lb/>
            majus eo, quod ipſi coordinatur in ſerie Arith-
              <lb/>
            metica.</s>
            <s xml:id="echoid-s10858" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10859" xml:space="preserve">A. </s>
            <s xml:id="echoid-s10860" xml:space="preserve">B. </s>
            <s xml:id="echoid-s10861" xml:space="preserve">C. </s>
            <s xml:id="echoid-s10862" xml:space="preserve">D. </s>
            <s xml:id="echoid-s10863" xml:space="preserve">E. </s>
            <s xml:id="echoid-s10864" xml:space="preserve">F.</s>
            <s xml:id="echoid-s10865" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10866" xml:space="preserve">A. </s>
            <s xml:id="echoid-s10867" xml:space="preserve">M. </s>
            <s xml:id="echoid-s10868" xml:space="preserve">N. </s>
            <s xml:id="echoid-s10869" xml:space="preserve">O. </s>
            <s xml:id="echoid-s10870" xml:space="preserve">P. </s>
            <s xml:id="echoid-s10871" xml:space="preserve">Q.</s>
            <s xml:id="echoid-s10872" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10873" xml:space="preserve">Eſt enim A + N &</s>
            <s xml:id="echoid-s10874" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10875" xml:space="preserve">2 M (vel &</s>
            <s xml:id="echoid-s10876" xml:space="preserve">gt;) </s>
            <s xml:id="echoid-s10877" xml:space="preserve">2 B = A + C. </s>
            <s xml:id="echoid-s10878" xml:space="preserve">ergò N &</s>
            <s xml:id="echoid-s10879" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10880" xml:space="preserve">C.
              <lb/>
            </s>
            <s xml:id="echoid-s10881" xml:space="preserve">unde M + N &</s>
            <s xml:id="echoid-s10882" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10883" xml:space="preserve">B + C = A + D. </s>
            <s xml:id="echoid-s10884" xml:space="preserve">Eſt autem A + O &</s>
            <s xml:id="echoid-s10885" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10886" xml:space="preserve">M +
              <lb/>
            N. </s>
            <s xml:id="echoid-s10887" xml:space="preserve">ergò A + O &</s>
            <s xml:id="echoid-s10888" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10889" xml:space="preserve">A + D. </s>
            <s xml:id="echoid-s10890" xml:space="preserve">Et ideò O &</s>
            <s xml:id="echoid-s10891" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10892" xml:space="preserve">D. </s>
            <s xml:id="echoid-s10893" xml:space="preserve">ergò M + O &</s>
            <s xml:id="echoid-s10894" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10895" xml:space="preserve">
              <lb/>
            B + D = A + E. </s>
            <s xml:id="echoid-s10896" xml:space="preserve">Eſt autem A + P &</s>
            <s xml:id="echoid-s10897" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10898" xml:space="preserve">M + O. </s>
            <s xml:id="echoid-s10899" xml:space="preserve">ergò A + P
              <lb/>
            &</s>
            <s xml:id="echoid-s10900" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10901" xml:space="preserve">A + E; </s>
            <s xml:id="echoid-s10902" xml:space="preserve">adeóque P &</s>
            <s xml:id="echoid-s10903" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10904" xml:space="preserve">E. </s>
            <s xml:id="echoid-s10905" xml:space="preserve">ſimilique porrò diſcurſu quoad
              <lb/>
            velis.</s>
            <s xml:id="echoid-s10906" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10907" xml:space="preserve">XIV. </s>
            <s xml:id="echoid-s10908" xml:space="preserve">Hinc, ſi rurſus fuerint A, B, C, D, E, F {.</s>
            <s xml:id="echoid-s10909" xml:space="preserve">./.</s>
            <s xml:id="echoid-s10910" xml:space="preserve">.} Arithmeticè;
              <lb/>
            </s>
            <s xml:id="echoid-s10911" xml:space="preserve">& </s>
            <s xml:id="echoid-s10912" xml:space="preserve">A, M, N, O, P, Q ſint {.</s>
            <s xml:id="echoid-s10913" xml:space="preserve">./.</s>
            <s xml:id="echoid-s10914" xml:space="preserve">.} Geometricè; </s>
            <s xml:id="echoid-s10915" xml:space="preserve">sítque ultimum F non
              <lb/>
            minus ultimo Q; </s>
            <s xml:id="echoid-s10916" xml:space="preserve">erit B majus quàm M.</s>
            <s xml:id="echoid-s10917" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10918" xml:space="preserve">Nam ſi dicatur B non majus quàm M; </s>
            <s xml:id="echoid-s10919" xml:space="preserve">erit indè F minus, quàm
              <lb/>
            Q contra hypotheſin.</s>
            <s xml:id="echoid-s10920" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10921" xml:space="preserve">Item, iiſdem poſitis; </s>
            <s xml:id="echoid-s10922" xml:space="preserve">erit penultimum E majus penultimo P.</s>
            <s xml:id="echoid-s10923" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10924" xml:space="preserve">XV. </s>
            <s xml:id="echoid-s10925" xml:space="preserve">Nam ſi F = Q; </s>
            <s xml:id="echoid-s10926" xml:space="preserve">conſtat ex præcedente fore E &</s>
            <s xml:id="echoid-s10927" xml:space="preserve">gt;</s>
            <s xml:id="echoid-s10928" xml:space="preserve">P (ſcilicet u-
              <lb/>
            tramque ſeriem invertendo) ſin F &</s>
            <s xml:id="echoid-s10929" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10930" xml:space="preserve">Q: </s>
            <s xml:id="echoid-s10931" xml:space="preserve">potiori jure liquet fore
              <lb/>
            E &</s>
            <s xml:id="echoid-s10932" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s10933" xml:space="preserve">P.</s>
            <s xml:id="echoid-s10934" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10935" xml:space="preserve">XVI. </s>
            <s xml:id="echoid-s10936" xml:space="preserve">Quinimò demùm, iiſdem poſitis, quodlibet in ſerie Arith-
              <lb/>
            metica majus eſt coordinato quolibet in ſerie Geometrica; </s>
            <s xml:id="echoid-s10937" xml:space="preserve">puta, C
              <lb/>
            majus eſt quàm N.</s>
            <s xml:id="echoid-s10938" xml:space="preserve"/>
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