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

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            <s xml:id="echoid-s2924" xml:space="preserve">Quod tandem HI, aſymptotos inſcriptæ DEF, ſecet circumſcriptam Hy-
              <lb/>
            perbolen ABC, iam ſatis patet ex dictis. </s>
            <s xml:id="echoid-s2925" xml:space="preserve">Quod ſupererat demonſtrandum.</s>
            <s xml:id="echoid-s2926" xml:space="preserve"/>
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        <div xml:id="echoid-div274" type="section" level="1" n="122">
          <head xml:id="echoid-head127" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s2927" xml:space="preserve">_E_N tibi Lector Geometra admiranda quædam Naturæ ſympto-
              <lb/>
            mata circa Aſymptoticas lineas iam olim à nobis detecta, ac ſi-
              <lb/>
            mul directa demonſtratione firmata, dum in Conicis hucuſque
              <lb/>
            animaduertimus non tantùm binas dari lineas in eodem plano
              <lb/>
            exiſtentes, quæ licet ſemper inter ſe magis accedant, nunquam tamen (quòd
              <lb/>
            ſanè mirum eſt) etiam ſi in infinitum productæ, ſimul conueniunt; </s>
            <s xml:id="echoid-s2928" xml:space="preserve">quales
              <lb/>
            ſunt, conuexa linea hyperbolica, & </s>
            <s xml:id="echoid-s2929" xml:space="preserve">celebris illa recta Aſymptotos Apoll. </s>
            <s xml:id="echoid-s2930" xml:space="preserve">ab
              <lb/>
            ipſo tunc negatiuè, à nobis verò in 8. </s>
            <s xml:id="echoid-s2931" xml:space="preserve">& </s>
            <s xml:id="echoid-s2932" xml:space="preserve">10. </s>
            <s xml:id="echoid-s2933" xml:space="preserve">huius affirmatiuè demonſtrata:
              <lb/>
            </s>
            <s xml:id="echoid-s2934" xml:space="preserve">verùm alias quoque, eiuſdem penitus naturæ reperiri, alteram nempe con-
              <lb/>
            uexam, concauam alteram, quales ſunt binæ congruentes parabolæ, vel hy-
              <lb/>
            perbolæ; </s>
            <s xml:id="echoid-s2935" xml:space="preserve">item binæ ſimiles hyperbolæ, quarum centrum interioris, aut in ipſā
              <unsure/>
              <lb/>
            cadat, aut infra centrum exterioris, atque omnes ſint per diuerſos vertices
              <lb/>
            ſimul adſcriptæ; </s>
            <s xml:id="echoid-s2936" xml:space="preserve">prout vidimus in 42. </s>
            <s xml:id="echoid-s2937" xml:space="preserve">44. </s>
            <s xml:id="echoid-s2938" xml:space="preserve">45. </s>
            <s xml:id="echoid-s2939" xml:space="preserve">47. </s>
            <s xml:id="echoid-s2940" xml:space="preserve">& </s>
            <s xml:id="echoid-s2941" xml:space="preserve">elicitur ex ipſa 48. </s>
            <s xml:id="echoid-s2942" xml:space="preserve">
              <lb/>
            huius. </s>
            <s xml:id="echoid-s2943" xml:space="preserve">Præterea, non ſolùm rectam Aſymptoton, & </s>
            <s xml:id="echoid-s2944" xml:space="preserve">Hyperbolen dari, quæ
              <lb/>
            dum ad ſe propius ſemper accedunt, ad interuallum aliquando perueniunt mi-
              <lb/>
            nus quolibet dato interuallo, vti ex ipſo Apollonio, & </s>
            <s xml:id="echoid-s2945" xml:space="preserve">ex noſtra 10. </s>
            <s xml:id="echoid-s2946" xml:space="preserve">innotuit; </s>
            <s xml:id="echoid-s2947" xml:space="preserve">
              <lb/>
            ſed congruentes item parabolas, & </s>
            <s xml:id="echoid-s2948" xml:space="preserve">concentricas hyperbolas per varios ver-
              <lb/>
            tices ſimul adſcriptas hac ipſa admirabili affectione eſſe præditas, veluti in
              <lb/>
            42. </s>
            <s xml:id="echoid-s2949" xml:space="preserve">& </s>
            <s xml:id="echoid-s2950" xml:space="preserve">47. </s>
            <s xml:id="echoid-s2951" xml:space="preserve">à nobis fuit oſtenſum. </s>
            <s xml:id="echoid-s2952" xml:space="preserve">Verum enimuero haud minori ſaltem
              <lb/>
            admiratione dignum videtur, binas pariter lineas inueniri, quæ licet nun-
              <lb/>
            quam coeuntes, & </s>
            <s xml:id="echoid-s2953" xml:space="preserve">in infinitum productæ ad ſe propius accedentes, non ta-
              <lb/>
            men vnquam perueniunt ad interuallum cuiuſdam determinatæ magnitudi-
              <lb/>
            nis: </s>
            <s xml:id="echoid-s2954" xml:space="preserve">huiuſmodi enim ſunt congruentes Hyperbolæ, pariterque hyperbolæ ſimi-
              <lb/>
            les per diuerſos vertices ſimul adſcriptæ, prout didicimus in 44. </s>
            <s xml:id="echoid-s2955" xml:space="preserve">& </s>
            <s xml:id="echoid-s2956" xml:space="preserve">45. </s>
            <s xml:id="echoid-s2957" xml:space="preserve">
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            Alias amplius deteximus lineas, quarum diſtãtia perpetuò augetur, ſed nun-
              <lb/>
            quam tamen peruenit ad interuallum æquale cuidam terminato interuallo: </s>
            <s xml:id="echoid-s2958" xml:space="preserve">ta-
              <lb/>
            les enim ſunt recta linea alteri aſymptoton æquidiſtans, & </s>
            <s xml:id="echoid-s2959" xml:space="preserve">Hyperbolen ſecãs,
              <lb/>
            vna cum eadem curua hyperbolica: </s>
            <s xml:id="echoid-s2960" xml:space="preserve">tales item ſunt hyperbolæ ſimiles per eun-
              <lb/>
            dem verticem ſimul adſcriptæ, prout in 34. </s>
            <s xml:id="echoid-s2961" xml:space="preserve">& </s>
            <s xml:id="echoid-s2962" xml:space="preserve">41. </s>
            <s xml:id="echoid-s2963" xml:space="preserve">Oſtendimus denique
              <lb/>
            binas dari lineas ad eaſdem partes in infinitum productas, nunquam coeun-
              <lb/>
            tes, quæ ſimul, ac ſemel ſunt, & </s>
            <s xml:id="echoid-s2964" xml:space="preserve">ad ſe propiùs accedentes, & </s>
            <s xml:id="echoid-s2965" xml:space="preserve">inter ſe æqui-
              <lb/>
            diſtantes: </s>
            <s xml:id="echoid-s2966" xml:space="preserve">quales ſunt demum, parabolæ congruentes per diuerſos vertices
              <lb/>
            ſimul adſcriptæ, vti ex noſtra 42. </s>
            <s xml:id="echoid-s2967" xml:space="preserve">eiuſque primo Coroll. </s>
            <s xml:id="echoid-s2968" xml:space="preserve">iam ſatis patuit.</s>
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