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

Table of Notes

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            <s xml:id="echoid-s12792" xml:space="preserve">
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            R. </s>
            <s xml:id="echoid-s12793" xml:space="preserve">FZ; </s>
            <s xml:id="echoid-s12794" xml:space="preserve">erit ſpatium ADLK æquale rectangulo ex R, & </s>
            <s xml:id="echoid-s12795" xml:space="preserve">DB.</s>
            <s xml:id="echoid-s12796" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12797" xml:space="preserve">Nam ſit DH = R; </s>
            <s xml:id="echoid-s12798" xml:space="preserve">& </s>
            <s xml:id="echoid-s12799" xml:space="preserve">compleatur rectangulum BDHI; </s>
            <s xml:id="echoid-s12800" xml:space="preserve">tum
              <lb/>
            aſſumptâ MN indeſinitè parvâ curvæ AB partìculâ ducantur NG ad
              <lb/>
            BD; </s>
            <s xml:id="echoid-s12801" xml:space="preserve">& </s>
            <s xml:id="echoid-s12802" xml:space="preserve">MEX, NOS ad AD parallelæ. </s>
            <s xml:id="echoid-s12803" xml:space="preserve">Eſtque NO. </s>
            <s xml:id="echoid-s12804" xml:space="preserve">MO:</s>
            <s xml:id="echoid-s12805" xml:space="preserve">:
              <lb/>
            TF. </s>
            <s xml:id="echoid-s12806" xml:space="preserve">FM:</s>
            <s xml:id="echoid-s12807" xml:space="preserve">: R. </s>
            <s xml:id="echoid-s12808" xml:space="preserve">FZ. </s>
            <s xml:id="echoid-s12809" xml:space="preserve">Unde NO x FZ = MO x R; </s>
            <s xml:id="echoid-s12810" xml:space="preserve">hoc eſt FG
              <lb/>
            x FZ = ES x EX. </s>
            <s xml:id="echoid-s12811" xml:space="preserve">ergò cum omnia rectangula FG x FZ minimè
              <lb/>
            differant à ſpatio ADLK; </s>
            <s xml:id="echoid-s12812" xml:space="preserve">& </s>
            <s xml:id="echoid-s12813" xml:space="preserve">omnia totidem rectangula ES x EX
              <lb/>
            componant rectangulum DHIB, ſatìs liquet Propoſitum.</s>
            <s xml:id="echoid-s12814" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12815" xml:space="preserve">XX. </s>
            <s xml:id="echoid-s12816" xml:space="preserve">Iiſdem poſitis, ſit curva PYQ talis, ut ſumpta in ſumpta
              <lb/>
            recta MX ordinata EY (reſpectivæ) ipſi FZ æquetur, erit _ſumma_
              <lb/>
            _quadr atorum_ ex FZ (ad rectam AD computata) par ei quod fit ex
              <lb/>
            ipſa R in _ſpatium_ DBQB ducta.</s>
            <s xml:id="echoid-s12817" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12818" xml:space="preserve">Eſt enim FG. </s>
            <s xml:id="echoid-s12819" xml:space="preserve">ES:</s>
            <s xml:id="echoid-s12820" xml:space="preserve">: NO. </s>
            <s xml:id="echoid-s12821" xml:space="preserve">MO:</s>
            <s xml:id="echoid-s12822" xml:space="preserve">: R x FZ. </s>
            <s xml:id="echoid-s12823" xml:space="preserve">FZq:</s>
            <s xml:id="echoid-s12824" xml:space="preserve">: R x EY.
              <lb/>
            </s>
            <s xml:id="echoid-s12825" xml:space="preserve">FZq. </s>
            <s xml:id="echoid-s12826" xml:space="preserve">adeóque FG x FZq = ES x R x EY.</s>
            <s xml:id="echoid-s12827" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12828" xml:space="preserve">XXI. </s>
            <s xml:id="echoid-s12829" xml:space="preserve">Simili ratione _ſumma Cuborum_ ex FZ æquatur ei quod fit ex R
              <lb/>
            in ſummam quadratorum ex rectis EY ad BD applicatis. </s>
            <s xml:id="echoid-s12830" xml:space="preserve">neque non ſi-
              <lb/>
            mili quoad reliquas poteſtates tenore.</s>
            <s xml:id="echoid-s12831" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12832" xml:space="preserve">XXII. </s>
            <s xml:id="echoid-s12833" xml:space="preserve">Sit curva quævis DOK, in qua deſignatum punctum D;
              <lb/>
            </s>
            <s xml:id="echoid-s12834" xml:space="preserve">
              <note position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">Fig. 128.</note>
            & </s>
            <s xml:id="echoid-s12835" xml:space="preserve">ſubtenſa recta DK; </s>
            <s xml:id="echoid-s12836" xml:space="preserve">ſit item curva AE talis, ut à D projectâ quâ-
              <lb/>
            vis rectâ DMF (quæ curvas ſecet punctis M, F) ductíſque DS ad
              <lb/>
            DM normali, & </s>
            <s xml:id="echoid-s12837" xml:space="preserve">MS curvam DOK tangente (concurrentibus utiq;
              <lb/>
            </s>
            <s xml:id="echoid-s12838" xml:space="preserve">puncto S) datâque quâdam R, ſit DS. </s>
            <s xml:id="echoid-s12839" xml:space="preserve">2 R:</s>
            <s xml:id="echoid-s12840" xml:space="preserve">: DMq. </s>
            <s xml:id="echoid-s12841" xml:space="preserve">DFq; </s>
            <s xml:id="echoid-s12842" xml:space="preserve">erit
              <lb/>
            ſpatium ADE æquale ex R, DK.</s>
            <s xml:id="echoid-s12843" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12844" xml:space="preserve">Nam ſubtenſa DK indefinitè ſecta concipiatur punctis PQ, &</s>
            <s xml:id="echoid-s12845" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s12846" xml:space="preserve">per quæ centro C deſcripti tranſeant arcus PM, QRN; </s>
            <s xml:id="echoid-s12847" xml:space="preserve">curvam
              <lb/>
            DOK ſecantes punctis M, N; </s>
            <s xml:id="echoid-s12848" xml:space="preserve">per quæ ducantur rectæ DMF,
              <lb/>
            DNG; </s>
            <s xml:id="echoid-s12849" xml:space="preserve">ſint verò DT ad DK; </s>
            <s xml:id="echoid-s12850" xml:space="preserve">& </s>
            <s xml:id="echoid-s12851" xml:space="preserve">DS ad DM perpendiculares; </s>
            <s xml:id="echoid-s12852" xml:space="preserve">
              <lb/>
            quibus occurrant tangentes KT, MS. </s>
            <s xml:id="echoid-s12853" xml:space="preserve">demùm centro D per E duca-
              <lb/>
            tur arcus EX; </s>
            <s xml:id="echoid-s12854" xml:space="preserve">& </s>
            <s xml:id="echoid-s12855" xml:space="preserve">per F arcus FY. </s>
            <s xml:id="echoid-s12856" xml:space="preserve">Jam, ob ſectionem indefinitam,
              <lb/>
            eſt triangulum KPM triangulo KDT ſimile. </s>
            <s xml:id="echoid-s12857" xml:space="preserve">ac ideò MP. </s>
            <s xml:id="echoid-s12858" xml:space="preserve">PK:</s>
            <s xml:id="echoid-s12859" xml:space="preserve">:
              <lb/>
            TD. </s>
            <s xml:id="echoid-s12860" xml:space="preserve">DK. </s>
            <s xml:id="echoid-s12861" xml:space="preserve">item eſt DP. </s>
            <s xml:id="echoid-s12862" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s12863" xml:space="preserve">: DE. </s>
            <s xml:id="echoid-s12864" xml:space="preserve">EX. </s>
            <s xml:id="echoid-s12865" xml:space="preserve">ſeu, propter aſſigna-
              <lb/>
            tam cauſam, DK. </s>
            <s xml:id="echoid-s12866" xml:space="preserve">MP:</s>
            <s xml:id="echoid-s12867" xml:space="preserve">: DE. </s>
            <s xml:id="echoid-s12868" xml:space="preserve">EX. </s>
            <s xml:id="echoid-s12869" xml:space="preserve">Eſt itaque MP x DK. </s>
            <s xml:id="echoid-s12870" xml:space="preserve">PK x
              <lb/>
            MP:</s>
            <s xml:id="echoid-s12871" xml:space="preserve">: TD x DE. </s>
            <s xml:id="echoid-s12872" xml:space="preserve">DK x EX. </s>
            <s xml:id="echoid-s12873" xml:space="preserve">hoc eſt DK. </s>
            <s xml:id="echoid-s12874" xml:space="preserve">PK:</s>
            <s xml:id="echoid-s12875" xml:space="preserve">: TD x DEq. </s>
            <s xml:id="echoid-s12876" xml:space="preserve">
              <lb/>
            DK x EX x DE. </s>
            <s xml:id="echoid-s12877" xml:space="preserve">ac inde DKq x EX x DE = PK x TD x
              <lb/>
            DEq. </s>
            <s xml:id="echoid-s12878" xml:space="preserve">(_a_) Eſt autem DT. </s>
            <s xml:id="echoid-s12879" xml:space="preserve">2 R:</s>
            <s xml:id="echoid-s12880" xml:space="preserve">: DKq. </s>
            <s xml:id="echoid-s12881" xml:space="preserve">DEq; </s>
            <s xml:id="echoid-s12882" xml:space="preserve">ſeu DT x DEq
              <lb/>
              <note position="right" xlink:label="note-0269-02" xlink:href="note-0269-02a" xml:space="preserve">(_a_) _Hyp._</note>
            = 2 R x DKq. </s>
            <s xml:id="echoid-s12883" xml:space="preserve">ergò eſt DKq x EX x DE = PK x 2 R x DKq.
              <lb/>
            </s>
            <s xml:id="echoid-s12884" xml:space="preserve">quare EX x DE = 2 R x PK; </s>
            <s xml:id="echoid-s12885" xml:space="preserve">hoc eſt, 2 ſector DEX = 2 R x PK. </s>
            <s xml:id="echoid-s12886" xml:space="preserve">
              <lb/>
            unde ſector DEX = R x PK. </s>
            <s xml:id="echoid-s12887" xml:space="preserve">Simili planè diſcurſu ſector </s>
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