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

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        <div xml:id="echoid-div257" type="section" level="1" n="115">
          <head xml:id="echoid-head120" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s2717" xml:space="preserve">EX his patet, ſimilium Hyperbolarum per diuerſos vertices ſimul adſcri-
              <lb/>
            ptarum, & </s>
            <s xml:id="echoid-s2718" xml:space="preserve">quarum eadem ſit regula, aſymptotos eſſe inter ſe paralle-
              <lb/>
            las, & </s>
            <s xml:id="echoid-s2719" xml:space="preserve">aſymptoton inſcriptæ ſecare Hyperbolen circumſcriptam.</s>
            <s xml:id="echoid-s2720" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div258" type="section" level="1" n="116">
          <head xml:id="echoid-head121" xml:space="preserve">LEMMA VI. PROP. XXXXVI.</head>
          <p>
            <s xml:id="echoid-s2721" xml:space="preserve">Si in quocunque triangulo ABC ducta ſit quæpiam linea DE
              <lb/>
            baſi BC parallela, rectangulum ABC ſuperabit ADE rectangu-
              <lb/>
            lo ſub DB, differentia altitudinum, & </s>
            <s xml:id="echoid-s2722" xml:space="preserve">ſub aggregato baſium
              <lb/>
            BC, DE.</s>
            <s xml:id="echoid-s2723" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2724" xml:space="preserve">PRoducta enim BC, ac ſumpta CF æ-
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            quali ipſi DE, & </s>
            <s xml:id="echoid-s2725" xml:space="preserve">completis in angulo
              <lb/>
            A B C parallelogrammis AE, AC, DF.
              <lb/>
            </s>
            <s xml:id="echoid-s2726" xml:space="preserve">Conſtat parallelogrammum AC ſuperare
              <lb/>
            parallelogrammum AE gnomone DCG,
              <lb/>
            ſed gnomon DCG æquatur parallelogrã-
              <lb/>
            mis B E, GC, & </s>
            <s xml:id="echoid-s2727" xml:space="preserve">GC æquatur DC, ſiue
              <lb/>
            EF, quare AC ſuperat AE parallelogram-
              <lb/>
            mo DF, hoc eſt rectangulum ABC ſupe-
              <lb/>
            rat rectangulum ADE, rectangulo DBF; </s>
            <s xml:id="echoid-s2728" xml:space="preserve">ſed DB eſt differentia altitudinum,
              <lb/>
            & </s>
            <s xml:id="echoid-s2729" xml:space="preserve">BF aggregatum baſium BC, DE. </s>
            <s xml:id="echoid-s2730" xml:space="preserve">Quare patet propoſitum.</s>
            <s xml:id="echoid-s2731" xml:space="preserve"/>
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        <div xml:id="echoid-div260" type="section" level="1" n="117">
          <head xml:id="echoid-head122" xml:space="preserve">THEOR. XXV. PROP. XXXXVII.</head>
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            <s xml:id="echoid-s2732" xml:space="preserve">Similes Hyperbolæ concentricæ per diuerſos vertices ſimul
              <lb/>
            adſcriptæ, ſunt inter ſe nunquam coeuntes, ac ſemper propiùs
              <lb/>
            accedentes, & </s>
            <s xml:id="echoid-s2733" xml:space="preserve">ad interuallum perueniunt minus quocunque dato
              <lb/>
            interuallo.</s>
            <s xml:id="echoid-s2734" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2735" xml:space="preserve">SInt duæ ſimiles Hyperbolæ ABC, DEF per diuerſos vertices B, E ſimul
              <lb/>
            adſcriptæ, quarum commune centrum ſit G, ſitque Hyperbolæ ABC
              <lb/>
            tranſuerſum latus BH, rectum BI, Hyperbolæ autem DEF ſit tranſuerſum
              <lb/>
            EL, rectum EM. </s>
            <s xml:id="echoid-s2736" xml:space="preserve">Dico primùm has, in infinitum productas, nunquam inter
              <lb/>
            ſe conuenire.</s>
            <s xml:id="echoid-s2737" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2738" xml:space="preserve">Producta enim contingente ME, donec vtrinque ſectioni ABC occurrat,
              <lb/>
            ipſa erit ordinata in ſectione ABC (cum ſint ſectiones ſimul adſcriptæ) ac
              <lb/>
            ſectio DEF cadet tota infra contingentem KEM; </s>
            <s xml:id="echoid-s2739" xml:space="preserve">& </s>
            <s xml:id="echoid-s2740" xml:space="preserve">ſumpto in DEC quoli-
              <lb/>
            bet puncto D, applicataque per D recta ADN, quæ iunctis regulis HI, </s>
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