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-s12603" xml:space="preserve">
              <pb o="88" file="0266" n="281" rhead=""/>
            _ſpatio_ VDψ φ _circa axem_ VD _rotato ſubduplnm ſolidi ex ſpatio_
              <lb/>
            DRSH, _itidem circa axem_ VD _rotato, confecti._</s>
            <s xml:id="echoid-s12604" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12605" xml:space="preserve">Nam ob HL. </s>
            <s xml:id="echoid-s12606" xml:space="preserve">LG:</s>
            <s xml:id="echoid-s12607" xml:space="preserve">: PD. </s>
            <s xml:id="echoid-s12608" xml:space="preserve">DH:</s>
            <s xml:id="echoid-s12609" xml:space="preserve">: Dψ. </s>
            <s xml:id="echoid-s12610" xml:space="preserve">DH:</s>
            <s xml:id="echoid-s12611" xml:space="preserve">: Dψ_q_. </s>
            <s xml:id="echoid-s12612" xml:space="preserve">Dψ x
              <lb/>
            DH:</s>
            <s xml:id="echoid-s12613" xml:space="preserve">: Dψ_q_. </s>
            <s xml:id="echoid-s12614" xml:space="preserve">HS x DH; </s>
            <s xml:id="echoid-s12615" xml:space="preserve">erit HL x HS x DH = LG x Dψ_q_.
              <lb/>
            </s>
            <s xml:id="echoid-s12616" xml:space="preserve"> = DC x Dψ_q_. </s>
            <s xml:id="echoid-s12617" xml:space="preserve">Simili planè diſcurſu erit LK x LX x DL =
              <lb/>
            CB x CZ_q_; </s>
            <s xml:id="echoid-s12618" xml:space="preserve">& </s>
            <s xml:id="echoid-s12619" xml:space="preserve">KI x KX x DK = BA x BZ_q_, &</s>
            <s xml:id="echoid-s12620" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12621" xml:space="preserve">atqui ſoli-
              <lb/>
            dum pt
              <unsure/>
            ius eſt {ῶ/δ}: </s>
            <s xml:id="echoid-s12622" xml:space="preserve">AZ_q_ + BZ_q_ + CZ_q_, &</s>
            <s xml:id="echoid-s12623" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12624" xml:space="preserve">& </s>
            <s xml:id="echoid-s12625" xml:space="preserve">ſolidum poſte-
              <lb/>
            rius eſt {2 ῶ/δ}: </s>
            <s xml:id="echoid-s12626" xml:space="preserve">DI x IX + DK x KX + DL x LX, &</s>
            <s xml:id="echoid-s12627" xml:space="preserve">c. </s>
            <s xml:id="echoid-s12628" xml:space="preserve">itaque
              <lb/>
            conſtat Propoſitum.</s>
            <s xml:id="echoid-s12629" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12630" xml:space="preserve">IX. </s>
            <s xml:id="echoid-s12631" xml:space="preserve">Hæc itidem omnia ſimili ratione vera ſunt, etiam ſi curva VEH
              <lb/>
              <note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">Fig. 124.</note>
            rectæ VD convexas ſuas partes obvertat; </s>
            <s xml:id="echoid-s12632" xml:space="preserve">nempe quovis in curva ac-
              <lb/>
            cepto puncto E; </s>
            <s xml:id="echoid-s12633" xml:space="preserve">& </s>
            <s xml:id="echoid-s12634" xml:space="preserve">per hoc ductâ EP ad curvam VEH perpendicu-
              <lb/>
            lari, & </s>
            <s xml:id="echoid-s12635" xml:space="preserve">EAY ad rectam VD normali, factáque AZ = AP; </s>
            <s xml:id="echoid-s12636" xml:space="preserve">erit
              <lb/>
            ſpatium VDψ = {DH_q_;</s>
            <s xml:id="echoid-s12637" xml:space="preserve">/2} Sin quoque fiat AY = VP; </s>
            <s xml:id="echoid-s12638" xml:space="preserve">erit ſpati-
              <lb/>
            um VD ψ = {VH_q_;</s>
            <s xml:id="echoid-s12639" xml:space="preserve">/2} Et pariter quoad cætera.</s>
            <s xml:id="echoid-s12640" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12641" xml:space="preserve">Ex his verò _Theorematis quam innumerarum magnitudinum_ (ex
              <lb/>
            ipſarum immediatè conſtructione) _dimenſiones innoteſcant_, ab expe-
              <lb/>
            rientia facilè comperietur.</s>
            <s xml:id="echoid-s12642" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12643" xml:space="preserve">X. </s>
            <s xml:id="echoid-s12644" xml:space="preserve">Sit rurſus curva quæpiam VH (cujus axis VD, baſis DH)
              <lb/>
              <note position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">Fig. 125.</note>
            & </s>
            <s xml:id="echoid-s12645" xml:space="preserve">linea DZZO talis, ut a curvæ puncto quopiam, cen E, ductâ
              <lb/>
            rectâ ET, quæ curvam tangat, & </s>
            <s xml:id="echoid-s12646" xml:space="preserve">recta EIZ ad baſin parallelâ, ſit
              <lb/>
            qerpetuò IZ æqualis ipſi AT; </s>
            <s xml:id="echoid-s12647" xml:space="preserve">dico _ſpatium_ DHO _ſpatio_ VDH
              <lb/>
            _æquari_.</s>
            <s xml:id="echoid-s12648" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12649" xml:space="preserve">Æquiſecetur enim recta DH indefinitè, punctis I, K, L, per quæ
              <lb/>
            ducantur rectæ EIZ, FKZ, GLZ ad VD parallelæ, curvæque oc-
              <lb/>
            currentes ad E, F, G, unde ducantur rectæ EA, FB, GC ad HD
              <lb/>
            parallelæ, rectæque ET, FT, GT (ut & </s>
            <s xml:id="echoid-s12650" xml:space="preserve">HT) _curvam tangentes;_
              <lb/>
            </s>
            <s xml:id="echoid-s12651" xml:space="preserve">lineæ verò ſe, ut Schema monſtrat, interſecent. </s>
            <s xml:id="echoid-s12652" xml:space="preserve">Eſtque jam triangu-
              <lb/>
            lum GLH ſimile triangulo TDH (nam ob diviſionem iſtam indefi-
              <lb/>
            nitam arculus GH rectæ inſtar cenſeri poteſt, eatenus tangenti HT
              <lb/>
            coincidens) quare LG. </s>
            <s xml:id="echoid-s12653" xml:space="preserve">LH:</s>
            <s xml:id="echoid-s12654" xml:space="preserve">: TD. </s>
            <s xml:id="echoid-s12655" xml:space="preserve">DH; </s>
            <s xml:id="echoid-s12656" xml:space="preserve">& </s>
            <s xml:id="echoid-s12657" xml:space="preserve">LG x DH = LH
              <lb/>
            x TD; </s>
            <s xml:id="echoid-s12658" xml:space="preserve">ſeu CD x DH = LH x HO. </s>
            <s xml:id="echoid-s12659" xml:space="preserve">ſimili ratiocinio eſt BC </s>
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