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-div331" type="section" level="1" n="34">
          <p>
            <s xml:id="echoid-s11605" xml:space="preserve">
              <pb o="72" file="0250" n="265" rhead=""/>
            interſecans, ut expreſſum vides; </s>
            <s xml:id="echoid-s11606" xml:space="preserve">eſtque jam PK. </s>
            <s xml:id="echoid-s11607" xml:space="preserve">PL :</s>
            <s xml:id="echoid-s11608" xml:space="preserve">: M. </s>
            <s xml:id="echoid-s11609" xml:space="preserve">
              <note position="left" xlink:label="note-0250-01" xlink:href="note-0250-01a" xml:space="preserve">_a Converſ_. 4.
                <lb/>
              Lect. VI.</note>
            :</s>
            <s xml:id="echoid-s11610" xml:space="preserve">: GE. </s>
            <s xml:id="echoid-s11611" xml:space="preserve">GF &</s>
            <s xml:id="echoid-s11612" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11613" xml:space="preserve">PE. </s>
            <s xml:id="echoid-s11614" xml:space="preserve">PF &</s>
            <s xml:id="echoid-s11615" xml:space="preserve">gt; </s>
            <s xml:id="echoid-s11616" xml:space="preserve">PK. </s>
            <s xml:id="echoid-s11617" xml:space="preserve">PF; </s>
            <s xml:id="echoid-s11618" xml:space="preserve">quare PL &</s>
            <s xml:id="echoid-s11619" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s11620" xml:space="preserve">PF;</s>
            <s xml:id="echoid-s11621" xml:space="preserve"> igitur _Hyperbola_ LBL curvam FB F tangit. </s>
            <s xml:id="echoid-s11622" xml:space="preserve">Protracta jam TB
              <lb/>
              <note position="left" xlink:label="note-0250-02" xlink:href="note-0250-02a" xml:space="preserve">_b_ 11. Lect. VII.</note>
            cum XZ conveniat in R; </s>
            <s xml:id="echoid-s11623" xml:space="preserve">eſtque tum RZ. </s>
            <s xml:id="echoid-s11624" xml:space="preserve">ZB :</s>
            <s xml:id="echoid-s11625" xml:space="preserve">: BS. </s>
            <s xml:id="echoid-s11626" xml:space="preserve">ST. </s>
            <s xml:id="echoid-s11627" xml:space="preserve">unde
              <lb/>
              <note position="left" xlink:label="note-0250-03" xlink:href="note-0250-03a" xml:space="preserve">_c_ @@. Lect. VII.</note>
            RZ xST = BS x ZB = BS x S X. </s>
            <s xml:id="echoid-s11628" xml:space="preserve">atqui propter DS. </s>
            <s xml:id="echoid-s11629" xml:space="preserve">SX :</s>
            <s xml:id="echoid-s11630" xml:space="preserve">: BS. </s>
            <s xml:id="echoid-s11631" xml:space="preserve">SY, eſt DS xSY = BS xSX. </s>
            <s xml:id="echoid-s11632" xml:space="preserve">ergò RZ xST = DS xS Y
              <lb/>
              <note position="left" xlink:label="note-0250-04" xlink:href="note-0250-04a" xml:space="preserve">_d Conſtr_.</note>
            = DS x CX. </s>
            <s xml:id="echoid-s11633" xml:space="preserve">vel RZ. </s>
            <s xml:id="echoid-s11634" xml:space="preserve">CX :</s>
            <s xml:id="echoid-s11635" xml:space="preserve">: DS . </s>
            <s xml:id="echoid-s11636" xml:space="preserve">ST; </s>
            <s xml:id="echoid-s11637" xml:space="preserve">compoſitéque RZ . </s>
            <s xml:id="echoid-s11638" xml:space="preserve">RZ
              <lb/>
            + CX :</s>
            <s xml:id="echoid-s11639" xml:space="preserve">: DS. </s>
            <s xml:id="echoid-s11640" xml:space="preserve">DT :</s>
            <s xml:id="echoid-s11641" xml:space="preserve">: N. </s>
            <s xml:id="echoid-s11642" xml:space="preserve">M :</s>
            <s xml:id="echoid-s11643" xml:space="preserve">: CZ. </s>
            <s xml:id="echoid-s11644" xml:space="preserve">CZ + CX. </s>
            <s xml:id="echoid-s11645" xml:space="preserve">itaque diviſim eſt RZ. </s>
            <s xml:id="echoid-s11646" xml:space="preserve">CX :</s>
            <s xml:id="echoid-s11647" xml:space="preserve">: CZ. </s>
            <s xml:id="echoid-s11648" xml:space="preserve">CX. </s>
            <s xml:id="echoid-s11649" xml:space="preserve">adeóque RZ = CZ; </s>
            <s xml:id="echoid-s11650" xml:space="preserve">unde RB
              <lb/>
            _hyperbolam_ LBL tangit; </s>
            <s xml:id="echoid-s11651" xml:space="preserve">hæc igitur ( RBT) curvam FBF, ipſi
              <lb/>
            LBL contiguam, quoque tanget. </s>
            <s xml:id="echoid-s11652" xml:space="preserve">quod erat Propoſitum.</s>
            <s xml:id="echoid-s11653" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11654" xml:space="preserve">VII. </s>
            <s xml:id="echoid-s11655" xml:space="preserve">Hinc ſi perſiſtentibus reliquis, recta tantùm DF jam inter
              <lb/>
            D G, DE perpetuò Geometricè media ſtatuatur ( eodem qui priùs fuit
              <lb/>
            ordine) eadem BT curvam FBF quoque continget.</s>
            <s xml:id="echoid-s11656" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11657" xml:space="preserve">Etenim ex mediis ejuſdem ordinis _Aritbmetice Geometricéque_ pro-
              <lb/>
            portionalibus efformatæ lineæ ſe mutuò contingunt, adeóque commu-
              <lb/>
            ni rectâ tangente gaudent.</s>
            <s xml:id="echoid-s11658" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11659" xml:space="preserve">VIII. </s>
            <s xml:id="echoid-s11660" xml:space="preserve">Porrò _(_ſtantibus reliquis ut in poſtremâ) quodvis in curva
              <lb/>
              <note position="left" xlink:label="note-0250-05" xlink:href="note-0250-05a" xml:space="preserve">Fig. 98.</note>
            FB F deſignetur punctum F, quæ curvam ad hoc tanget recta ſimili
              <lb/>
            pacto determinatur.</s>
            <s xml:id="echoid-s11661" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11662" xml:space="preserve">Connectatur utique recta DF curvam EB E ſecans ad E ; </s>
            <s xml:id="echoid-s11663" xml:space="preserve">item du-
              <lb/>
            catur DQ ad DG perpendicularis ipſam EO interſecans ad X; </s>
            <s xml:id="echoid-s11664" xml:space="preserve">fiat
              <lb/>
            etiam DX. </s>
            <s xml:id="echoid-s11665" xml:space="preserve">DY :</s>
            <s xml:id="echoid-s11666" xml:space="preserve">: N. </s>
            <s xml:id="echoid-s11667" xml:space="preserve">M ; </s>
            <s xml:id="echoid-s11668" xml:space="preserve">& </s>
            <s xml:id="echoid-s11669" xml:space="preserve">connectatur EY; </s>
            <s xml:id="echoid-s11670" xml:space="preserve">ipſi demum EY pa-
              <lb/>
            rallela ducatur FZ; </s>
            <s xml:id="echoid-s11671" xml:space="preserve">hæc curvam FBF continget.</s>
            <s xml:id="echoid-s11672" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11673" xml:space="preserve">Nam centro D per E ducatur circulus CEI; </s>
            <s xml:id="echoid-s11674" xml:space="preserve">concipiatúrque linea
              <lb/>
            HEH talis, ut à D eductâ quacunque rectâ DI ( quæ circulum CE
              <lb/>
            ſecet in I, curvam HEH in H, & </s>
            <s xml:id="echoid-s11675" xml:space="preserve">ipſam EB E in L ) ſit pepertuò
              <lb/>
            DH eodem inter DI, DL ordine proportionalis, quo DF inter DG,
              <lb/>
            DE; </s>
            <s xml:id="echoid-s11676" xml:space="preserve">palam eſt tunc (è præcedente) quòd recta EY curvam HEH
              <lb/>
            tanget; </s>
            <s xml:id="echoid-s11677" xml:space="preserve">verùm ipſi HEH analoga eſt curva FBF; </s>
            <s xml:id="echoid-s11678" xml:space="preserve">
              <note position="left" xlink:label="note-0250-06" xlink:href="note-0250-06a" xml:space="preserve">_a_ 9. Lect. VII.</note>
            recta FZ curvam FBF quoque tanget.</s>
            <s xml:id="echoid-s11679" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">_b_ 7. Lect. VIII.</note>
          <p>
            <s xml:id="echoid-s11680" xml:space="preserve">Exhinc nedum innumerarum ſpiralium; </s>
            <s xml:id="echoid-s11681" xml:space="preserve">at aliarum diverſi generis
              <lb/>
            infinities plurium tangentes quàm promptè determinantur.</s>
            <s xml:id="echoid-s11682" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11683" xml:space="preserve">IX. </s>
            <s xml:id="echoid-s11684" xml:space="preserve">Hinc clarum eſt, ſi duæ lineæ EEE, FEF ſic ad ſe referan-
              <lb/>
            tur, ut à puncto quodam D utcunque projectis rectis DEF; </s>
            <s xml:id="echoid-s11685" xml:space="preserve">habe-
              <lb/>
              <note position="left" xlink:label="note-0250-08" xlink:href="note-0250-08a" xml:space="preserve">Fig. 99.</note>
            ant ſe rectæ DE, ut quadrata ex ipſis DF, & </s>
            <s xml:id="echoid-s11686" xml:space="preserve">ad harum terminos
              <lb/>
            tangant curvas rectæ ES, FT; </s>
            <s xml:id="echoid-s11687" xml:space="preserve">cum perpendicularibus ad </s>
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