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

Table of contents

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[101.] Addenda Lectionibus Geometricis.
Page: 342 (149)
[102.] _Probl_. I.
Page: 342 (149)
[103.] _Probl_. II.
Page: 342 (149)
[104.] _Probl_. III.
Page: 342 (149)
[105.] Addenda Lectionibus Geometricis.
Page: 343 (150)
[106.] _Theor_. I.
Page: 343 (150)
[107.] _Theor_. II.
Page: 343 (150)
[108.] _Theor_. III.
Page: 343 (150)
[109.] _Theor_. IV.
Page: 344 (151)
[110.] _Theor_. V.
Page: 344 (151)
[111.] _Theor_. VI.
Page: 344 (151)
[112.] FINIS.
Page: 344 (151)
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            <s xml:id="echoid-s7732" xml:space="preserve">
              <pb o="118" file="0136" n="136" rhead=""/>
            ad ipſam EC parallelæ ducantur rectæ RT, SV; </s>
            <s xml:id="echoid-s7733" xml:space="preserve">palàm eſt indeter-
              <lb/>
            minatum punctum X inter limites T, V conſiſtere (nam extra TV
              <lb/>
            punctum quodlibet L accipiendo, & </s>
            <s xml:id="echoid-s7734" xml:space="preserve">indè ducendo LI P ad CE paralle-
              <lb/>
            lam, erit CL, hoc eft LP, major quàm LI, unde à C ad rectam LI,
              <lb/>
            nulla duci recta poteſt æqualis ipſi LI). </s>
            <s xml:id="echoid-s7735" xml:space="preserve">Jam autem dico, quòd
              <lb/>
            punctum Z ad ellipſin exiſtit, cujus axis TV, focus C. </s>
            <s xml:id="echoid-s7736" xml:space="preserve">Nam biſe-
              <lb/>
            cetur TV in K; </s>
            <s xml:id="echoid-s7737" xml:space="preserve">fiat VD = TC; </s>
            <s xml:id="echoid-s7738" xml:space="preserve">ducatur KH ad CE parallela;
              <lb/>
            </s>
            <s xml:id="echoid-s7739" xml:space="preserve">per H ducatur HN ad CK parallela. </s>
            <s xml:id="echoid-s7740" xml:space="preserve">Eſtque KH = {TR + VS/2} =
              <lb/>
            {CT + CV/2} = KT = KV. </s>
            <s xml:id="echoid-s7741" xml:space="preserve">Et quoniam AV. </s>
            <s xml:id="echoid-s7742" xml:space="preserve">AT :</s>
            <s xml:id="echoid-s7743" xml:space="preserve">: (VS. </s>
            <s xml:id="echoid-s7744" xml:space="preserve">
              <lb/>
            TR (hoc eſt) :</s>
            <s xml:id="echoid-s7745" xml:space="preserve">: CV. </s>
            <s xml:id="echoid-s7746" xml:space="preserve">CT :</s>
            <s xml:id="echoid-s7747" xml:space="preserve">:) CV. </s>
            <s xml:id="echoid-s7748" xml:space="preserve">DV; </s>
            <s xml:id="echoid-s7749" xml:space="preserve">erit per rationis con-
              <lb/>
            vcrſionem AV. </s>
            <s xml:id="echoid-s7750" xml:space="preserve">TV :</s>
            <s xml:id="echoid-s7751" xml:space="preserve">: CV. </s>
            <s xml:id="echoid-s7752" xml:space="preserve">CD. </s>
            <s xml:id="echoid-s7753" xml:space="preserve">vel, conſequentes ſubduplando,
              <lb/>
            AV. </s>
            <s xml:id="echoid-s7754" xml:space="preserve">KV :</s>
            <s xml:id="echoid-s7755" xml:space="preserve">: CV. </s>
            <s xml:id="echoid-s7756" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s7757" xml:space="preserve">dividendóque AK. </s>
            <s xml:id="echoid-s7758" xml:space="preserve">KV :</s>
            <s xml:id="echoid-s7759" xml:space="preserve">: KV. </s>
            <s xml:id="echoid-s7760" xml:space="preserve">CK; </s>
            <s xml:id="echoid-s7761" xml:space="preserve">hoc eſt
              <lb/>
            AK. </s>
            <s xml:id="echoid-s7762" xml:space="preserve">KH :</s>
            <s xml:id="echoid-s7763" xml:space="preserve">: KH. </s>
            <s xml:id="echoid-s7764" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s7765" xml:space="preserve">hoc eſt HN. </s>
            <s xml:id="echoid-s7766" xml:space="preserve">NG :</s>
            <s xml:id="echoid-s7767" xml:space="preserve">: KH. </s>
            <s xml:id="echoid-s7768" xml:space="preserve">CK. </s>
            <s xml:id="echoid-s7769" xml:space="preserve">quare
              <lb/>
            KH x NG = CK x HN = CK x KX. </s>
            <s xml:id="echoid-s7770" xml:space="preserve">atqui eſt CZq = XGq
              <lb/>
            = KHq + NGq + 2 KH x NG. </s>
            <s xml:id="echoid-s7771" xml:space="preserve">& </s>
            <s xml:id="echoid-s7772" xml:space="preserve">CXq = CKq + KXq
              <lb/>
            + 2 CK x KX = CKq + KXq + 2 KH x NG. </s>
            <s xml:id="echoid-s7773" xml:space="preserve">ergo
              <lb/>
            KHq + NGq - CKq - KXq = CZq - CXq = XZq. </s>
            <s xml:id="echoid-s7774" xml:space="preserve">
              <lb/>
            Ad alteras biſegmenti K partes ſumatur K ξ = KX, ducatúrque ξν ad
              <lb/>
            KH parallela, ſecans curvam TEZV in ζ, & </s>
            <s xml:id="echoid-s7775" xml:space="preserve">rectam AH in γ, ac
              <lb/>
            ipſam NH in ν erit quoque, ſimili ex diſcurſu, ξζq = KHq +
              <lb/>
            νγq - CKq - Kξq; </s>
            <s xml:id="echoid-s7776" xml:space="preserve">unde liquet fore ξζ = XZ; </s>
            <s xml:id="echoid-s7777" xml:space="preserve">connexíſque
              <lb/>
            proinde rectis Cζ, Dζ, erit Dζ = CZ; </s>
            <s xml:id="echoid-s7778" xml:space="preserve">& </s>
            <s xml:id="echoid-s7779" xml:space="preserve">Cζ + CZ = ξγ +
              <lb/>
            XG = 2 KH = TV. </s>
            <s xml:id="echoid-s7780" xml:space="preserve">ergò Cζ + Dζ (vel DZ + CZ) = TV. </s>
            <s xml:id="echoid-s7781" xml:space="preserve">
              <lb/>
            unde perſpicitur _curvam TζZV eſſe ellipſin_, cujus _axis_ TV; </s>
            <s xml:id="echoid-s7782" xml:space="preserve">_foci_
              <lb/>
            C, D.</s>
            <s xml:id="echoid-s7783" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s7784" xml:space="preserve">III. </s>
            <s xml:id="echoid-s7785" xml:space="preserve">Sit autem ſecundò angulus CA E major ſemirecto (vel AC
              <lb/>
              <note position="left" xlink:label="note-0136-01" xlink:href="note-0136-01a" xml:space="preserve">Fig. 191.</note>
            &</s>
            <s xml:id="echoid-s7786" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s7787" xml:space="preserve">CE) dico punctum Z ad oppoſitas hyperbolas, conſimili modo
              <lb/>
            determinabiles, exiſtere. </s>
            <s xml:id="echoid-s7788" xml:space="preserve">enimverò factis (ad utramque rectæ CA
              <lb/>
            partem) angulis ſemirectis ACP; </s>
            <s xml:id="echoid-s7789" xml:space="preserve">& </s>
            <s xml:id="echoid-s7790" xml:space="preserve">(ab ipſarum CP cum AE
              <lb/>
            occurſibus) ductis rectis RT, SV ad CE parallelis, punctum X
              <lb/>
            extra limites TV neceſſariò conſiſtet (etenim ubivis intra TV ductâ
              <lb/>
            LIP ad CE parallelâ, erit LI &</s>
            <s xml:id="echoid-s7791" xml:space="preserve">lt; </s>
            <s xml:id="echoid-s7792" xml:space="preserve">LP, ideóque nulla par ipſi LI
              <lb/>
            angulo AL I ſubtendi poteſt; </s>
            <s xml:id="echoid-s7793" xml:space="preserve">id quod extra terminos hoſce nil pro-
              <lb/>
            hibet fieri) erit jam TV axis, & </s>
            <s xml:id="echoid-s7794" xml:space="preserve">C focus hyperbolarum. </s>
            <s xml:id="echoid-s7795" xml:space="preserve">Fiant
              <lb/>
            enim omnia, quæ in caſu præcedente; </s>
            <s xml:id="echoid-s7796" xml:space="preserve">erítque rurſus hîc KH =
              <lb/>
            KV. </s>
            <s xml:id="echoid-s7797" xml:space="preserve">item ob AV. </s>
            <s xml:id="echoid-s7798" xml:space="preserve">AT :</s>
            <s xml:id="echoid-s7799" xml:space="preserve">: CV. </s>
            <s xml:id="echoid-s7800" xml:space="preserve">DV; </s>
            <s xml:id="echoid-s7801" xml:space="preserve">& </s>
            <s xml:id="echoid-s7802" xml:space="preserve">(inversè componendo)</s>
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