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

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          <p>
            <s xml:id="echoid-s2643" xml:space="preserve">
              <pb o="78" file="0102" n="102" rhead=""/>
            tercepta applicatarum ſegmenta metitum, licet ſemper magis, ac magis de-
              <lb/>
            creſcat, eſſe tamen non minus interuallo 1 3, quod iuxta eaſdem æquidi-
              <lb/>
            ſtantes ordinatim ſectionibus applicatas, inter vtranque aſymptoton cadit.
              <lb/>
            </s>
            <s xml:id="echoid-s2644" xml:space="preserve">Nam per ea, quæ infra demonſtrabimus, interuallum 2 1, maius eſt inter-
              <lb/>
            uallo 1 3. </s>
            <s xml:id="echoid-s2645" xml:space="preserve">Pariter 4 5, eſt maius 6 7, communique addito 5 6, erit inter-
              <lb/>
            uallum 4 6, maius interuallo 5 7, ſiue 1 3, & </s>
            <s xml:id="echoid-s2646" xml:space="preserve">hoc ſemper vbicunque ſu-
              <lb/>
            matur harum ſectionum interuallum infra applicatam 2 1 3. </s>
            <s xml:id="echoid-s2647" xml:space="preserve">Quare hy-
              <lb/>
            perbolæ congruentes per diuerſos vertices ſimul adſcriptæ, licèt ſemper ma-
              <lb/>
            gis accedentes, ad interuallum nunquam perueniunt æquale cuidam dato
              <lb/>
            interuallo. </s>
            <s xml:id="echoid-s2648" xml:space="preserve">Quod erat vltimò, &</s>
            <s xml:id="echoid-s2649" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2650" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div249" type="section" level="1" n="112">
          <head xml:id="echoid-head117" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s2651" xml:space="preserve">EX his conſtat, congruentium Hyperbolarum non per eundem verticem
              <lb/>
            ſimul adſcriptarum aſymptotos eſſe inter ſe æquidiſtantes, & </s>
            <s xml:id="echoid-s2652" xml:space="preserve">aſympto-
              <lb/>
            ton inſcriptæ ſecare Hyperbolen circumſcriptam.</s>
            <s xml:id="echoid-s2653" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div250" type="section" level="1" n="113">
          <head xml:id="echoid-head118" xml:space="preserve">Quod ſuperiùs promiſimus oſtendetur ſic.</head>
          <p>
            <s xml:id="echoid-s2654" xml:space="preserve">SInt duæ congruentes Hyperbolæ KBC, DEF, per diuerſos vertices B, E
              <lb/>
            ſimul adſcriptæ, & </s>
            <s xml:id="echoid-s2655" xml:space="preserve">circumſcriptæ KBC ſit centrum G, & </s>
            <s xml:id="echoid-s2656" xml:space="preserve">aſymptotos
              <lb/>
            GI, inſcriptæ verò ſit centrum H, & </s>
            <s xml:id="echoid-s2657" xml:space="preserve">aſymptotos HM, quæ ipſi GI æquidi-
              <lb/>
            ſtabit, per præcedens Coroll. </s>
            <s xml:id="echoid-s2658" xml:space="preserve">ſitque applicata quæcunque IL vtranque
              <lb/>
            aſymptoton, & </s>
            <s xml:id="echoid-s2659" xml:space="preserve">Hyperbolen ſecans in I, A, K, D, communemque diame-
              <lb/>
            trum in L: </s>
            <s xml:id="echoid-s2660" xml:space="preserve">dico interceptum applicatę ſegmentum AD inſcriptæ Hyperbolę
              <lb/>
            DEF, maius eſſe intercepto eiuſdem applicatæ ſegmento IK inter aſympto-
              <lb/>
            ton, & </s>
            <s xml:id="echoid-s2661" xml:space="preserve">circumſcriptam.</s>
            <s xml:id="echoid-s2662" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2663" xml:space="preserve">Ducta enim IM parallela ad GHO, & </s>
            <s xml:id="echoid-s2664" xml:space="preserve">per
              <lb/>
              <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a" number="70">
                <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0102-01"/>
              </figure>
            Mapplicata MNO, erunt IH, IO parallelo-
              <lb/>
            gramma, ac ideò tàm GH, quàm LO ipſi IM
              <lb/>
            æquales erunt, & </s>
            <s xml:id="echoid-s2665" xml:space="preserve">inter ſe; </s>
            <s xml:id="echoid-s2666" xml:space="preserve">quare addita com-
              <lb/>
            muni HL, erit GL æqualis HO, ſed eſt GB
              <lb/>
            æqualis HE, (cum ſint ſemi-tranſuerſa late-
              <lb/>
            ra congruentium Hyperbolarum) vnde reli-
              <lb/>
            qua BL, reliquæ EO æqualis erit, & </s>
            <s xml:id="echoid-s2667" xml:space="preserve">ob id
              <lb/>
            ſemi-applicata LK ſemi-applicatę ON ęqua-
              <lb/>
            lis, ſed eſt tota LI æqualis totæ OM (cum ſint
              <lb/>
            oppoſitæ in parallelogrammo IO) ergo reli-
              <lb/>
            quæ KI, NM æquales erunt, ſed eſt
              <note symbol="a" position="left" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve">10. h.</note>
            maior NM, quare & </s>
            <s xml:id="echoid-s2668" xml:space="preserve">eadem DA erit maior
              <lb/>
            KI. </s>
            <s xml:id="echoid-s2669" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s2670" xml:space="preserve">c.</s>
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