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-div530" type="section" level="1" n="71">
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            <s xml:id="echoid-s15174" xml:space="preserve">
              <pb o="129" file="0307" n="322" rhead=""/>
            am puncto M, ductíſque rectis MPX ad BD, & </s>
            <s xml:id="echoid-s15175" xml:space="preserve">MQY ad AD
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            parallelis, poſitóque rectam MT tangere curvam AMB, ſit TP.
              <lb/>
            </s>
            <s xml:id="echoid-s15176" xml:space="preserve">
              <note position="right" xlink:label="note-0307-01" xlink:href="note-0307-01a" xml:space="preserve">Fig.201.</note>
            PM:</s>
            <s xml:id="echoid-s15177" xml:space="preserve">: QY. </s>
            <s xml:id="echoid-s15178" xml:space="preserve">PX; </s>
            <s xml:id="echoid-s15179" xml:space="preserve">erunt figuræ ADKE, DBLG ſibimet æqua-
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            les.</s>
            <s xml:id="echoid-s15180" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15181" xml:space="preserve">Valet hoc converſum. </s>
            <s xml:id="echoid-s15182" xml:space="preserve">Nempe ſi figuræ ADKE, DBLG æ-
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            quentur, & </s>
            <s xml:id="echoid-s15183" xml:space="preserve">MT curvam AMB tangat, erit TP. </s>
            <s xml:id="echoid-s15184" xml:space="preserve">PM:</s>
            <s xml:id="echoid-s15185" xml:space="preserve">: QY.
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            </s>
            <s xml:id="echoid-s15186" xml:space="preserve">PX:</s>
            <s xml:id="echoid-s15187" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s15188" xml:space="preserve">_Not_. </s>
            <s xml:id="echoid-s15189" xml:space="preserve">Omnium hactenus Propoſitorum fœcundiſſimum eſt hoc
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            _Tbeorema_; </s>
            <s xml:id="echoid-s15190" xml:space="preserve">præcedentium quippe complura vel in eo continentur, aut
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            ab eo facilè conſectantur. </s>
            <s xml:id="echoid-s15191" xml:space="preserve">Nam poſito lineam AMB indeterminatam
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            eſſe naturâ, ſi ipſarum EXK, GYL alterutra pro tuo arbitratu de-
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            terminetur, exinde reſultabit Theorema quoddam ejuſmodi, qualia
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            ſuperiùs exhibentur aliquammulta. </s>
            <s xml:id="echoid-s15192" xml:space="preserve">Si _e. </s>
            <s xml:id="echoid-s15193" xml:space="preserve">g_. </s>
            <s xml:id="echoid-s15194" xml:space="preserve">linea GYL ponatur recta
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            cum ipſa BD ſemi-rectum conſtituens angulum (quo caſu concipiun-
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            tur puncta D, G coincidere) proveniet indè prima _Lectionis_ XI. </s>
            <s xml:id="echoid-s15195" xml:space="preserve">Si
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            GYL ſit recta ad DB parallela, emerget _Lectionis ejuſdem._ </s>
            <s xml:id="echoid-s15196" xml:space="preserve">Rur-
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              <note position="right" xlink:label="note-0307-02" xlink:href="note-0307-02a" xml:space="preserve">Fig. 202.</note>
            ſus ſi PM = PX (vel lineæ AMB, EXK ſint eædem) conſeque-
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            tur hinc _decima_ ejuſdem. </s>
            <s xml:id="echoid-s15197" xml:space="preserve">Exhinc porrò liquet adſumpto cuilibet ſpa-
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            tio _infinita, genere diverſa, ſpatia æqualia_ facilè deſignari veluti ſi _ſpa-_
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            _tium_ DGLB ponatur _circuli quadrans_, cujus _centrum_ D; </s>
            <s xml:id="echoid-s15198" xml:space="preserve">& </s>
            <s xml:id="echoid-s15199" xml:space="preserve">curva
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            AMB ſit _parabola_, cujus _axis_ AD, emerget curvæ EXK hæc pro-
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            prietas, ut (ſi dicatur DB = r; </s>
            <s xml:id="echoid-s15200" xml:space="preserve">AP = x; </s>
            <s xml:id="echoid-s15201" xml:space="preserve">PX = y; </s>
            <s xml:id="echoid-s15202" xml:space="preserve">& </s>
            <s xml:id="echoid-s15203" xml:space="preserve">_k_ (vel
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            {DB q/2 AD}) ſit _parabolæ ſemipar ameter_) ſit {_rrk_/2} = _kkx_ + _xyy_. </s>
            <s xml:id="echoid-s15204" xml:space="preserve">Sin
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            AMB ponatur _hyperbola_, procreabitur alterius generis curva EXK.
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            </s>
            <s xml:id="echoid-s15205" xml:space="preserve">his autem expenſis ἀβλεφι
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            αν meam incuſo, qui non hoc _Theorema_ (ſi-
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            cut & </s>
            <s xml:id="echoid-s15206" xml:space="preserve">ea quæ ſubſequuntur, quorum ferè ratio conſimilis eſt, & </s>
            <s xml:id="echoid-s15207" xml:space="preserve">ſup-
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            par uſus) primo loco poſuerim, & </s>
            <s xml:id="echoid-s15208" xml:space="preserve">ex eo (nec non è reliquis mox
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            ſubjiciendis) quod fieri poſſe video, reliqua deduxerim. </s>
            <s xml:id="echoid-s15209" xml:space="preserve">Veruntamen
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            hujuſmodi _Phrygiam ſapientiam_ juxta mecum pleriſque familiarem au-
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            tumo, literas has tractantibus.</s>
            <s xml:id="echoid-s15210" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div533" type="section" level="1" n="72">
          <head xml:id="echoid-head75" xml:space="preserve">_Theor_. V.</head>
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            <s xml:id="echoid-s15211" xml:space="preserve">Sit ſpatium quodpiam ADB (rectis DA, DB, & </s>
            <s xml:id="echoid-s15212" xml:space="preserve">curva AMB
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              <note position="right" xlink:label="note-0307-03" xlink:href="note-0307-03a" xml:space="preserve">Fig. 203.</note>
            comprehenſum) ſint item curvæ EXK, GYL ità relatæ, ut ſi in curva
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            AMB liberè ſumatur punctum M, ducatur DMX, ſit DQ = DM,
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            ducatur QY ad DB perdendicularis, ſit DT ad DM perpendicula-
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            ris, recta MT curvam AMB contingat; </s>
            <s xml:id="echoid-s15213" xml:space="preserve">ſi, his inquam ſuppoſitis, ſit
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            TD. </s>
            <s xml:id="echoid-s15214" xml:space="preserve">DM:</s>
            <s xml:id="echoid-s15215" xml:space="preserve">: DM x QY. </s>
            <s xml:id="echoid-s15216" xml:space="preserve">DXq; </s>
            <s xml:id="echoid-s15217" xml:space="preserve">erit ſpatium DGLB ſpatii EDK
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            duplum.</s>
            <s xml:id="echoid-s15218" xml:space="preserve"/>
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