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-div199" type="section" level="1" n="24">
          <pb o="119" file="0137" n="137" rhead=""/>
          <p>
            <s xml:id="echoid-s7803" xml:space="preserve">AV. </s>
            <s xml:id="echoid-s7804" xml:space="preserve">TV :</s>
            <s xml:id="echoid-s7805" xml:space="preserve">: CV. </s>
            <s xml:id="echoid-s7806" xml:space="preserve">CD; </s>
            <s xml:id="echoid-s7807" xml:space="preserve">& </s>
            <s xml:id="echoid-s7808" xml:space="preserve">conſequentes ſubduplandò, dividendó-
              <lb/>
            que AK. </s>
            <s xml:id="echoid-s7809" xml:space="preserve">KV :</s>
            <s xml:id="echoid-s7810" xml:space="preserve">: KV. </s>
            <s xml:id="echoid-s7811" xml:space="preserve">KD :</s>
            <s xml:id="echoid-s7812" xml:space="preserve">: KV. </s>
            <s xml:id="echoid-s7813" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s7814" xml:space="preserve">vel AK. </s>
            <s xml:id="echoid-s7815" xml:space="preserve">KH :</s>
            <s xml:id="echoid-s7816" xml:space="preserve">: KH. </s>
            <s xml:id="echoid-s7817" xml:space="preserve">CK;
              <lb/>
            </s>
            <s xml:id="echoid-s7818" xml:space="preserve">hoc eſt HN (KX). </s>
            <s xml:id="echoid-s7819" xml:space="preserve">NG :</s>
            <s xml:id="echoid-s7820" xml:space="preserve">: KH. </s>
            <s xml:id="echoid-s7821" xml:space="preserve">CK; </s>
            <s xml:id="echoid-s7822" xml:space="preserve">quare CK x KX = KH
              <lb/>
            x NG. </s>
            <s xml:id="echoid-s7823" xml:space="preserve">eſt autem XZq = CZq - CXq = XGq - CXq =
              <lb/>
            NGq + KHq - 2 NG x KH: </s>
            <s xml:id="echoid-s7824" xml:space="preserve">- KXq - CKq + 2 CK
              <lb/>
            x KX = NGq + KHq - KXq - CKq. </s>
            <s xml:id="echoid-s7825" xml:space="preserve">Sumatur K ζ = KX,
              <lb/>
            diſcurſúmque ſimilem adhibendo liquebit fore ξζ = XZ; </s>
            <s xml:id="echoid-s7826" xml:space="preserve">& </s>
            <s xml:id="echoid-s7827" xml:space="preserve">ideo
              <lb/>
            Dζ = CZ. </s>
            <s xml:id="echoid-s7828" xml:space="preserve">unde Cζ - Dζ (DZ - CZ) = Cζ - CZ =
              <lb/>
            ξγ - XG = 2 KH = TV. </s>
            <s xml:id="echoid-s7829" xml:space="preserve">quare manifeſtum eſt _cnrvas_ TZ,
              <lb/>
            Vζ eſſe _Hyperbolas,_ quarum axis TV, foci C,D.</s>
            <s xml:id="echoid-s7830" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7831" xml:space="preserve">IV. </s>
            <s xml:id="echoid-s7832" xml:space="preserve">Tertiò demùm, ſit angulus CAE ſemirectus (vel CA =
              <lb/>
              <note position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">Fig. 192.</note>
            CE) erit tum punctum Z ad parabolam; </s>
            <s xml:id="echoid-s7833" xml:space="preserve">quæ itidem ita determina-
              <lb/>
            tur. </s>
            <s xml:id="echoid-s7834" xml:space="preserve">Fiat angulus ACP ſemirectus, & </s>
            <s xml:id="echoid-s7835" xml:space="preserve">ab ipſarum AE, CP in-
              <lb/>
            terſectione R ducatur RT ad CE parallela; </s>
            <s xml:id="echoid-s7836" xml:space="preserve">erit T_Vertex_, atque C
              <lb/>
            _Focus Parabolæ._ </s>
            <s xml:id="echoid-s7837" xml:space="preserve">id quod ex bene nota ſectionis hujus proprietate con-
              <lb/>
            ſtat; </s>
            <s xml:id="echoid-s7838" xml:space="preserve">qua ſcilicet eſt TA = TR = TC (ob angulos TAR,
              <lb/>
            TCR ſemirectos) & </s>
            <s xml:id="echoid-s7839" xml:space="preserve">AX = XG = CZ.</s>
            <s xml:id="echoid-s7840" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7841" xml:space="preserve">V. </s>
            <s xml:id="echoid-s7842" xml:space="preserve">Manifeſtum eſt verò rectam AE ſectiones has ad E contingere.
              <lb/>
            </s>
            <s xml:id="echoid-s7843" xml:space="preserve">quia nempe perpetuò major eſt CZ (vel XG) ordinatâ XZ; </s>
            <s xml:id="echoid-s7844" xml:space="preserve">adeó-
              <lb/>
            que puncta G extra cuŕvas unaquæque jacent hoc eſt tota AG extra
              <lb/>
            illas cadit.</s>
            <s xml:id="echoid-s7845" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7846" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s7847" xml:space="preserve">Hiſce præſtratis: </s>
            <s xml:id="echoid-s7848" xml:space="preserve">_Eſto Circulare ſpeculum_ MBND, cen-
              <lb/>
              <note position="right" xlink:label="note-0137-02" xlink:href="note-0137-02a" xml:space="preserve">Fig. 193.</note>
            trum habens C; </s>
            <s xml:id="echoid-s7849" xml:space="preserve">cui exponatur recta quæpiam F α G; </s>
            <s xml:id="echoid-s7850" xml:space="preserve">& </s>
            <s xml:id="echoid-s7851" xml:space="preserve">huic per-
              <lb/>
            pendicularis ſit recta C α; </s>
            <s xml:id="echoid-s7852" xml:space="preserve">quam ad parte@ averſas ſumpta CA, ad-
              <lb/>
            æquet. </s>
            <s xml:id="echoid-s7853" xml:space="preserve">Sit etiam CE ad CA perpendicularis, ac æqualis qua-
              <lb/>
            dranti diametri BD; </s>
            <s xml:id="echoid-s7854" xml:space="preserve">connexáque recta AE producatur utcunque.
              <lb/>
            </s>
            <s xml:id="echoid-s7855" xml:space="preserve">ſumpto jam in recta F α G puncto quolibet F, connectatur FC, & </s>
            <s xml:id="echoid-s7856" xml:space="preserve">
              <lb/>
            radiationis ab F in ipſa FC limes, ſeu _focus_, ſit Z; </s>
            <s xml:id="echoid-s7857" xml:space="preserve">ac per Z du-
              <lb/>
            catur ZX ad AC perpendicularis, ipſi AE occurrens in H; </s>
            <s xml:id="echoid-s7858" xml:space="preserve">dico
              <lb/>
            fore XH parem ipſi CZ.</s>
            <s xml:id="echoid-s7859" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7860" xml:space="preserve">Nam (è jam antè monſtratis) eſt FC. </s>
            <s xml:id="echoid-s7861" xml:space="preserve">CZ :</s>
            <s xml:id="echoid-s7862" xml:space="preserve">: FM. </s>
            <s xml:id="echoid-s7863" xml:space="preserve">MZ (hoc eſt)
              <lb/>
            :</s>
            <s xml:id="echoid-s7864" xml:space="preserve">: FC - CB. </s>
            <s xml:id="echoid-s7865" xml:space="preserve">CB - CZ. </s>
            <s xml:id="echoid-s7866" xml:space="preserve">hinc erit α C. </s>
            <s xml:id="echoid-s7867" xml:space="preserve">CX (AC. </s>
            <s xml:id="echoid-s7868" xml:space="preserve">CX) :</s>
            <s xml:id="echoid-s7869" xml:space="preserve">:
              <lb/>
            FC - CB. </s>
            <s xml:id="echoid-s7870" xml:space="preserve">CB - CZ. </s>
            <s xml:id="echoid-s7871" xml:space="preserve">quare (ducendo in ſe extrema, ac media)
              <lb/>
            erit AC x CB - AC x CZ = CX x FC - CX x CB. </s>
            <s xml:id="echoid-s7872" xml:space="preserve">hoc
              <lb/>
            eſt (ipſi CX x FC ſubſtituendo AC x CZ, propter α C. </s>
            <s xml:id="echoid-s7873" xml:space="preserve">CX :</s>
            <s xml:id="echoid-s7874" xml:space="preserve">:
              <lb/>
            FC. </s>
            <s xml:id="echoid-s7875" xml:space="preserve">CZ) erit AC x CB - AC x CZ = AC x CZ - CX x
              <lb/>
            CB. </s>
            <s xml:id="echoid-s7876" xml:space="preserve">tranſponendóque AC x CB + CX x CB = 2 AC x CZ.</s>
            <s xml:id="echoid-s7877" xml:space="preserve"/>
          </p>
        </div>
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