Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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        <div xml:id="echoid-div572" type="section" level="1" n="231">
          <p>
            <s xml:id="echoid-s5543" xml:space="preserve">
              <pb o="16" file="0198" n="198" rhead=""/>
            & </s>
            <s xml:id="echoid-s5544" xml:space="preserve">ſectioni occurrentibus in L,I. </s>
            <s xml:id="echoid-s5545" xml:space="preserve">Conſtat Hyperbolen ex F ad partes H
              <lb/>
            omnino incedere intra angulum L F I, & </s>
            <s xml:id="echoid-s5546" xml:space="preserve">cum ipſa in infinitum extendi
              <lb/>
            poſſit, cumque in ſecunda figura ſpatium F I B ſit occluſum ad I, & </s>
            <s xml:id="echoid-s5547" xml:space="preserve">ad
              <lb/>
            rectam L B nunquam poſſit prouenire, eò quod ipſa L B ponatur Hyper-
              <lb/>
            bole G F H aſymptotos: </s>
            <s xml:id="echoid-s5548" xml:space="preserve">in tertia verò cum ſpatium F I N ſit vndique oc-
              <lb/>
            cluſum, neceſſariò, in vtraque figura, deſcripta Hyperbole G F H in ali-
              <lb/>
            quo puncto datam ſectionem ſecabit. </s>
            <s xml:id="echoid-s5549" xml:space="preserve">Sit ergo harum mutua interſectio
              <lb/>
            punctum M, per quod ductis, vt factum fuit in prima figura, rectis lineis
              <lb/>
            quæ aſymptotis E D, E C æquidiſtent, ijſdem penitus argumentis, ac in
              <lb/>
            primo caſu, demonſtrabitur ipſam Hyperbolen in nullo alio puncto quàm
              <lb/>
            M cum data ſectione A B conuenire. </s>
            <s xml:id="echoid-s5550" xml:space="preserve">Quare ſi per punctum in angulo, &</s>
            <s xml:id="echoid-s5551" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s5552" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s5553" xml:space="preserve"/>
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        <div xml:id="echoid-div575" type="section" level="1" n="232">
          <head xml:id="echoid-head240" xml:space="preserve">THEOR. IX. PROP. XIII.</head>
          <p>
            <s xml:id="echoid-s5554" xml:space="preserve">Si in Hyperbola, ſumpta fuerint duo quælibet puncta, à qui-
              <lb/>
            bus ductæ ſint aſymptotis æquidiſtantes, eiſque occurrentes: </s>
            <s xml:id="echoid-s5555" xml:space="preserve">re-
              <lb/>
            cta linea iungens occurſus; </s>
            <s xml:id="echoid-s5556" xml:space="preserve">lineæ, data puncta iungenti, æqui-
              <lb/>
            diſtabit.</s>
            <s xml:id="echoid-s5557" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5558" xml:space="preserve">ESto Hyperbole A B, cuius aſymptoti C D, C E, ſumptaque ſint in
              <lb/>
            ſectione duo quælibet puncta A, B, à quibus ductæ ſint A F, B G,
              <lb/>
            aſymptotis æquidiſtantes. </s>
            <s xml:id="echoid-s5559" xml:space="preserve">Dico iunctas A B, F G, eſſe inter ſe paralle-
              <lb/>
            las.</s>
            <s xml:id="echoid-s5560" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5561" xml:space="preserve">Nam vtrinque producta A B vſque-
              <lb/>
              <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a" number="158">
                <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0198-01"/>
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            ad aſymptotos in D, & </s>
            <s xml:id="echoid-s5562" xml:space="preserve">E. </s>
            <s xml:id="echoid-s5563" xml:space="preserve"> Erit in
              <note symbol="a" position="left" xlink:label="note-0198-01" xlink:href="note-0198-01a" xml:space="preserve">8. ſec.
                <lb/>
              conic.</note>
            ma figura, B D æqualis A E: </s>
            <s xml:id="echoid-s5564" xml:space="preserve">in ſecun-
              <lb/>
            da verò, cum ſit A D æqualis B E, ad-
              <lb/>
            dita communi A B, erit item D B æ-
              <lb/>
            qualis ipſi A E. </s>
            <s xml:id="echoid-s5565" xml:space="preserve">Sed in triangulis D B
              <lb/>
            G, E A F, anguli ad D, B, æquantur
              <lb/>
            angulis ad A, & </s>
            <s xml:id="echoid-s5566" xml:space="preserve">E, vterque vtrique,
              <lb/>
            ob paralellas D G, A F, & </s>
            <s xml:id="echoid-s5567" xml:space="preserve">B G, E F;
              <lb/>
            </s>
            <s xml:id="echoid-s5568" xml:space="preserve">quare triangula D B G, A E F ſunt ſimi-
              <lb/>
            lia inter ſe, ac propterea vt D B ad B
              <lb/>
            G, ita A E ad E F, ſed antecedentes
              <lb/>
            D B, A E ſunt ęquales, vt modò oſten-
              <lb/>
            dimus, ergo, & </s>
            <s xml:id="echoid-s5569" xml:space="preserve">conſequentes B G, E F,
              <lb/>
            æquales erunt, at ſunt quoque inter ſe
              <lb/>
            parallelæ, quare, & </s>
            <s xml:id="echoid-s5570" xml:space="preserve">F G ipſi A B ęqui-
              <lb/>
            diſtabit. </s>
            <s xml:id="echoid-s5571" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s5572" xml:space="preserve">c.</s>
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