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

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        <div xml:id="echoid-div300" type="section" level="1" n="131">
          <p>
            <s xml:id="echoid-s3192" xml:space="preserve">
              <pb o="95" file="0119" n="119" rhead=""/>
            XY in X, producta ſecabit etiam DF aſymptoton ABC, ac ipſam quoque
              <lb/>
            ſectionem ABC, ſed XZ tota cadit extra OEQ, cum ſit eius
              <note symbol="a" position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">35. h.</note>
            quare occurſus rectæ XZ cum ſectione ABC cadet extra OEQ, ac ideò ſe-
              <lb/>
            ctio ABC occurret priùs ſectioni OEQ. </s>
            <s xml:id="echoid-s3193" xml:space="preserve">Quapropter Hyperbole LEM eſt
              <lb/>
            _MINIMA_ circumſcripta quæſita. </s>
            <s xml:id="echoid-s3194" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s3195" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3196" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div302" type="section" level="1" n="132">
          <head xml:id="echoid-head137" xml:space="preserve">ALITER breuiùs.</head>
          <p>
            <s xml:id="echoid-s3197" xml:space="preserve">PRoducatur contingens IB vſq; </s>
            <s xml:id="echoid-s3198" xml:space="preserve">ad circumſcriptam ſectionem in V. </s>
            <s xml:id="echoid-s3199" xml:space="preserve">Cum
              <lb/>
            ſectiones BA, EL ſimiles, & </s>
            <s xml:id="echoid-s3200" xml:space="preserve">ad eandem regulam HI, infra BV ad ſe
              <lb/>
            propiùs accedant ſectio BA recedet ab VL per interuallum aliquando
              <note symbol="b" position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">45. h.</note>
            nùs BV, ſed inſcripta OP recedit ab eadem VL per interuallũ maius eodem
              <lb/>
            BV, cum ſint ſemper magis recedẽtes, & </s>
            <s xml:id="echoid-s3201" xml:space="preserve"> ad interuallum perueniant
              <note symbol="c" position="right" xlink:label="note-0119-03" xlink:href="note-0119-03a" xml:space="preserve">37. h.</note>
            quolibet dato interuallo: </s>
            <s xml:id="echoid-s3202" xml:space="preserve">quare BA, & </s>
            <s xml:id="echoid-s3203" xml:space="preserve">OP omnino ſe mutuò ſecabũt. </s>
            <s xml:id="echoid-s3204" xml:space="preserve">Quod
              <lb/>
            iterum erat, &</s>
            <s xml:id="echoid-s3205" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3206" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div304" type="section" level="1" n="133">
          <head xml:id="echoid-head138" xml:space="preserve">PROBL. XX. PROP. LIV.</head>
          <p>
            <s xml:id="echoid-s3207" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM
              <lb/>
            ſibi concentricam Hyperbolen inſcribere, & </s>
            <s xml:id="echoid-s3208" xml:space="preserve">è contra.</s>
            <s xml:id="echoid-s3209" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3210" xml:space="preserve">Datæ Hyperbolæ, per punctum extra ipſam datum MINIMAM
              <lb/>
            ſibi concentricam Hyperbolen circumſcribere. </s>
            <s xml:id="echoid-s3211" xml:space="preserve">Oportet autem
              <lb/>
            datum punctum eſſe in angulo aſymptotali.</s>
            <s xml:id="echoid-s3212" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3213" xml:space="preserve">ESto data Hyperbole ABC, cuius centrum D, aſymptotos DL, & </s>
            <s xml:id="echoid-s3214" xml:space="preserve">pun-
              <lb/>
            ctum intra ſectionem datum ſit E: </s>
            <s xml:id="echoid-s3215" xml:space="preserve">oportet primò per E _MAXIMAM_ ei
              <lb/>
            concentricam Hyperbolen inſcribere.</s>
            <s xml:id="echoid-s3216" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3217" xml:space="preserve">Iungatur ED ſecans ABC in B: </s>
            <s xml:id="echoid-s3218" xml:space="preserve">erit
              <lb/>
              <figure xlink:label="fig-0119-01" xlink:href="fig-0119-01a" number="84">
                <image file="0119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0119-01"/>
              </figure>
            DB ſemi-tranſuerſum ſectionis
              <note symbol="a" position="right" xlink:label="note-0119-04" xlink:href="note-0119-04a" xml:space="preserve">47. pri-
                <lb/>
              mi conic.</note>
            cui per E cum ſemi tranſuerſo ED
              <note symbol="b" position="right" xlink:label="note-0119-05" xlink:href="note-0119-05a" xml:space="preserve">7. huius.</note>
            ſcribatur ſimilis, & </s>
            <s xml:id="echoid-s3219" xml:space="preserve">concentrica Hyper-
              <lb/>
            bole FEG (hoc autem ſieri poſſe mani-
              <lb/>
            feſtum eſt: </s>
            <s xml:id="echoid-s3220" xml:space="preserve">nam ſectionis FEG datur eius
              <lb/>
            aſymptotos DL, cum ſimiles concentri-
              <lb/>
            cæ Hyperbolæ per diuerſos vertices ad-
              <lb/>
            ſcriptæ habeant communem
              <note symbol="c" position="right" xlink:label="note-0119-06" xlink:href="note-0119-06a" xml:space="preserve">Coroll.
                <lb/>
              47. huius.</note>
            ton, & </s>
            <s xml:id="echoid-s3221" xml:space="preserve">cum datur tranſuerſum latus, & </s>
            <s xml:id="echoid-s3222" xml:space="preserve">
              <lb/>
            aſymptotos datur quoque rectum) patet
              <lb/>
            hãc ſectionẽ FEG datæ ABC eſſe inſcri-
              <lb/>
            ptam, cum ſint nunquã ſimul coeuntes.</s>
            <s xml:id="echoid-s3223" xml:space="preserve"/>
          </p>
          <note symbol="d" position="right" xml:space="preserve">47. h.</note>
          <p>
            <s xml:id="echoid-s3224" xml:space="preserve">Dico ampliùs hanc ipſam FEG eſſe _MAXIMAM_ quæſitam: </s>
            <s xml:id="echoid-s3225" xml:space="preserve">quoniam quę-
              <lb/>
            libet alia per E verticem adſcripta ipſi ABC, vel FEG minor eſt ipſa FEG;</s>
            <s xml:id="echoid-s3226" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-0119-08" xlink:href="note-0119-08a" xml:space="preserve">2. Co-
                <lb/>
              roll. 19. h.</note>
            quælibet, verò cum recto, quod prædictum excedat, qualis eſt HEI, eſt qui-
              <lb/>
            dem maior eadem FEG, ſed omnino ſecat circumſcriptam ABC. </s>
            <s xml:id="echoid-s3227" xml:space="preserve">
              <note symbol="f" position="right" xlink:label="note-0119-09" xlink:href="note-0119-09a" xml:space="preserve">ibidem.</note>
            cum Hyperbolæ FEG, HEI ſint concentricæ, & </s>
            <s xml:id="echoid-s3228" xml:space="preserve">per eundem verticem E ſi-
              <lb/>
            mul adſcriptæ, ſitque DL aſymptotos inſcriptæ FEG, ipſa ſecabit </s>
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