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

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      <text xml:lang="la" type="free">
        <div xml:id="echoid-div222" type="section" level="1" n="31">
          <p>
            <s xml:id="echoid-s9756" xml:space="preserve">
              <pb o="46" file="0224" n="239" rhead=""/>
            DH. </s>
            <s xml:id="echoid-s9757" xml:space="preserve">LN - HO:</s>
            <s xml:id="echoid-s9758" xml:space="preserve">: LH. </s>
            <s xml:id="echoid-s9759" xml:space="preserve">LB:</s>
            <s xml:id="echoid-s9760" xml:space="preserve">:) LH. </s>
            <s xml:id="echoid-s9761" xml:space="preserve">HK. </s>
            <s xml:id="echoid-s9762" xml:space="preserve">erit DH x HK =
              <lb/>
            HO x LH; </s>
            <s xml:id="echoid-s9763" xml:space="preserve">hoc eſt DL x HK - LH x HK = KO x LH - HK
              <lb/>
            x LH. </s>
            <s xml:id="echoid-s9764" xml:space="preserve">unde erit DL x HK = KO x LH. </s>
            <s xml:id="echoid-s9765" xml:space="preserve">vel ZL x LD = ZK
              <lb/>
            x KO. </s>
            <s xml:id="echoid-s9766" xml:space="preserve">ergò conſtat lineam ODO eſſe _Hyperbolen_, cujus _Aſymptoti_
              <lb/>
            ZA, ZS. </s>
            <s xml:id="echoid-s9767" xml:space="preserve">Breviùs hoc oſtendi poſſet, producendo rectam NDS.
              <lb/>
            </s>
            <s xml:id="echoid-s9768" xml:space="preserve">Nam eſt DS = DM = DO ± OM = NM ± OM = ON. </s>
            <s xml:id="echoid-s9769" xml:space="preserve">Simi-
              <lb/>
            ter quartam & </s>
            <s xml:id="echoid-s9770" xml:space="preserve">nonam breviùs demonſtres licet.</s>
            <s xml:id="echoid-s9771" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9772" xml:space="preserve">Quinimò ſi MN ad DO quamvis eandem perpetuò rationem pona-
              <lb/>
              <note position="left" xlink:label="note-0224-01" xlink:href="note-0224-01a" xml:space="preserve">Fig. 38.</note>
            tur habere (puta datam R ad S) etiam linea ODO _Hyperbola_ erit;
              <lb/>
            </s>
            <s xml:id="echoid-s9773" xml:space="preserve">Nempe ſi tum fiat R. </s>
            <s xml:id="echoid-s9774" xml:space="preserve">S:</s>
            <s xml:id="echoid-s9775" xml:space="preserve">: LB. </s>
            <s xml:id="echoid-s9776" xml:space="preserve">LZ; </s>
            <s xml:id="echoid-s9777" xml:space="preserve">& </s>
            <s xml:id="echoid-s9778" xml:space="preserve">R. </s>
            <s xml:id="echoid-s9779" xml:space="preserve">S:</s>
            <s xml:id="echoid-s9780" xml:space="preserve">: DL. </s>
            <s xml:id="echoid-s9781" xml:space="preserve">DE; </s>
            <s xml:id="echoid-s9782" xml:space="preserve">& </s>
            <s xml:id="echoid-s9783" xml:space="preserve">per
              <lb/>
            Z ducatur ZS ad BC; </s>
            <s xml:id="echoid-s9784" xml:space="preserve">ac per E tranſeat RE ad ZA parallela, cum
              <lb/>
            ZS conveniens in Y; </s>
            <s xml:id="echoid-s9785" xml:space="preserve">erunt YR, YS dictæ _Hyperbolæ aſymptoti_
              <lb/>
            quod jam ſufficerit innuiſſe.</s>
            <s xml:id="echoid-s9786" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9787" xml:space="preserve">Hinc in tranſcurſu noto facilè confici _Problema (quo problematum_
              <lb/>
            _confectiones iſtæ Arcbimedeæ, ac Vieteæ ope primæ Conchoidis peractæ_,
              <lb/>
            _ad Sectiones conicas rediguntur_) Per datum punctum D rectam lineam
              <lb/>
            ducere, ſic ut anguli dati ABC lateribus intercepta ductæ rectæ pars
              <lb/>
            æquetur datæ T. </s>
            <s xml:id="echoid-s9788" xml:space="preserve">Nam deſcriptâ hyperbolâ ODO; </s>
            <s xml:id="echoid-s9789" xml:space="preserve">centro D, in-
              <lb/>
            tervallo datam T æquante deſcribatur circulus POQ _hyperbolam_ in-
              <lb/>
            terſecans in O; </s>
            <s xml:id="echoid-s9790" xml:space="preserve">& </s>
            <s xml:id="echoid-s9791" xml:space="preserve">producatur DON; </s>
            <s xml:id="echoid-s9792" xml:space="preserve">fiétq; </s>
            <s xml:id="echoid-s9793" xml:space="preserve">MN = DO = T.
              <lb/>
            </s>
            <s xml:id="echoid-s9794" xml:space="preserve">Modus autem hic generalior eſt, & </s>
            <s xml:id="echoid-s9795" xml:space="preserve">concinnior eo, quem in _Opticis_
              <lb/>
            tradidimus.</s>
            <s xml:id="echoid-s9796" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9797" xml:space="preserve">IV. </s>
            <s xml:id="echoid-s9798" xml:space="preserve">Sit angulus ABC, et punctum datum D; </s>
            <s xml:id="echoid-s9799" xml:space="preserve">ſit etiam linea O
              <lb/>
              <note position="left" xlink:label="note-0224-02" xlink:href="note-0224-02a" xml:space="preserve">Fig. 39.</note>
            BO talis, ut per D ductâ quâpiam rectâ DN, ſit anguli late-
              <lb/>
            ribus intercepta MN ad rectâ BC curvâque OBO interceptam
              <lb/>
            MO in eadem ſemper ratione (puta X ad Y;) </s>
            <s xml:id="echoid-s9800" xml:space="preserve">erit linea OBO
              <lb/>
            _hyperbola_.</s>
            <s xml:id="echoid-s9801" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9802" xml:space="preserve">Ducatur enim recta DL ad CB parallela, ipſi AB occurrens
              <lb/>
              <note position="left" xlink:label="note-0224-03" xlink:href="note-0224-03a" xml:space="preserve">Fig. 39.</note>
            in L; </s>
            <s xml:id="echoid-s9803" xml:space="preserve">ſecentúrque DL, BL punctis E, F, ut ſit DL. </s>
            <s xml:id="echoid-s9804" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s9805" xml:space="preserve">: X.
              <lb/>
            </s>
            <s xml:id="echoid-s9806" xml:space="preserve">Y:</s>
            <s xml:id="echoid-s9807" xml:space="preserve">: BL. </s>
            <s xml:id="echoid-s9808" xml:space="preserve">BF; </s>
            <s xml:id="echoid-s9809" xml:space="preserve">tum per E ducatur recta ER, ad BA; </s>
            <s xml:id="echoid-s9810" xml:space="preserve">& </s>
            <s xml:id="echoid-s9811" xml:space="preserve">per
              <lb/>
            F recta FS ad BC parallela; </s>
            <s xml:id="echoid-s9812" xml:space="preserve">concurrántque rectæ ER, FS pun-
              <lb/>
            cto Z; </s>
            <s xml:id="echoid-s9813" xml:space="preserve">denuò per punctum O ducatur OH ad AB parallela. </s>
            <s xml:id="echoid-s9814" xml:space="preserve">Jam
              <lb/>
            ob DL. </s>
            <s xml:id="echoid-s9815" xml:space="preserve">DH:</s>
            <s xml:id="echoid-s9816" xml:space="preserve">: LN. </s>
            <s xml:id="echoid-s9817" xml:space="preserve">HO:</s>
            <s xml:id="echoid-s9818" xml:space="preserve">: LB + BN. </s>
            <s xml:id="echoid-s9819" xml:space="preserve">HO:</s>
            <s xml:id="echoid-s9820" xml:space="preserve">: DE x LB
              <lb/>
            + DE x BN. </s>
            <s xml:id="echoid-s9821" xml:space="preserve">DE x HO. </s>
            <s xml:id="echoid-s9822" xml:space="preserve">item DL x KO = DE x BN
              <lb/>
            (nam DL. </s>
            <s xml:id="echoid-s9823" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s9824" xml:space="preserve">: MN. </s>
            <s xml:id="echoid-s9825" xml:space="preserve">MO:</s>
            <s xml:id="echoid-s9826" xml:space="preserve">: BN. </s>
            <s xml:id="echoid-s9827" xml:space="preserve">KO) & </s>
            <s xml:id="echoid-s9828" xml:space="preserve">DE x LB = DL
              <lb/>
            x BF (ob DE. </s>
            <s xml:id="echoid-s9829" xml:space="preserve">DL:</s>
            <s xml:id="echoid-s9830" xml:space="preserve">: BF. </s>
            <s xml:id="echoid-s9831" xml:space="preserve">LB;) </s>
            <s xml:id="echoid-s9832" xml:space="preserve">erit DL. </s>
            <s xml:id="echoid-s9833" xml:space="preserve">DH:</s>
            <s xml:id="echoid-s9834" xml:space="preserve">: DL x BF
              <lb/>
            + DL x KO. </s>
            <s xml:id="echoid-s9835" xml:space="preserve">DE x HO; </s>
            <s xml:id="echoid-s9836" xml:space="preserve">hoc eſt DL x BF + DL x
              <lb/>
            KO. </s>
            <s xml:id="echoid-s9837" xml:space="preserve">DH x BF + DH x KO:</s>
            <s xml:id="echoid-s9838" xml:space="preserve">: DL x BF x DL x KO.</s>
            <s xml:id="echoid-s9839" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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