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

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        <div xml:id="echoid-div112" type="section" level="1" n="68">
          <p>
            <s xml:id="echoid-s1412" xml:space="preserve">
              <pb o="40" file="0064" n="64" rhead=""/>
            bus ſe contingunt; </s>
            <s xml:id="echoid-s1413" xml:space="preserve">& </s>
            <s xml:id="echoid-s1414" xml:space="preserve">in duobus tantùm punctis ſe mutuò ſecant. </s>
            <s xml:id="echoid-s1415" xml:space="preserve">Quod tan-
              <lb/>
            dem erat demonſtrandum.</s>
            <s xml:id="echoid-s1416" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div118" type="section" level="1" n="69">
          <head xml:id="echoid-head74" xml:space="preserve">COROLL. I.</head>
          <p>
            <s xml:id="echoid-s1417" xml:space="preserve">PAtet hinc, quod ſi regulæ coni-ſectionum per vertices ſimul adſcripta-
              <lb/>
            rum ſibi ipſis congruant ſectiones quoque erunt inter ſe congruentes,
              <lb/>
            vt in primis quatuor figuris præcedentis ſchematis oſtenſum eſt; </s>
            <s xml:id="echoid-s1418" xml:space="preserve">& </s>
            <s xml:id="echoid-s1419" xml:space="preserve">ſi fuerint
              <lb/>
            inter ſe congruentes, etiam ipſarum regulæ ſimul congruent: </s>
            <s xml:id="echoid-s1420" xml:space="preserve">ſed cum regu-
              <lb/>
            læ ſimul congruunt, congruunt quoque, & </s>
            <s xml:id="echoid-s1421" xml:space="preserve">latera, & </s>
            <s xml:id="echoid-s1422" xml:space="preserve">è conuerſo, cum ad æ-
              <lb/>
            quales angulos inter ſe diſpoſita intelligantur, quare cum latera fuerint inter
              <lb/>
            ſe congruentia ſiue æqualia, ſectiones quoque inter ſe congruentes erunt; </s>
            <s xml:id="echoid-s1423" xml:space="preserve">& </s>
            <s xml:id="echoid-s1424" xml:space="preserve">
              <lb/>
            ſi ſectiones fuerint congruentes etiam ipſarum latera æqualia erunt.</s>
            <s xml:id="echoid-s1425" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1426" xml:space="preserve">Si verò regulę infra recta ſectionum latera ex vertice contingenter appli-
              <lb/>
            cata diſiunctim procedentes nunquam ſimul conueniant, nec ipſæ ſectiones
              <lb/>
            vnquam conuenient, ſed in vertice ſe mutuò contingent, & </s>
            <s xml:id="echoid-s1427" xml:space="preserve">ea inſcripta erit,
              <lb/>
            ſiue minor, cuius regula infra prædictam contingentem diametro ſectionum
              <lb/>
            ſit propior, ſeu cadat tota inter diametrum, & </s>
            <s xml:id="echoid-s1428" xml:space="preserve">regulam alterius ſectionis;
              <lb/>
            </s>
            <s xml:id="echoid-s1429" xml:space="preserve">quæ è contra circumſcripta erit, ſiue maior, vt apparet in 26. </s>
            <s xml:id="echoid-s1430" xml:space="preserve">figuris ſubſe-
              <lb/>
            quentibus.</s>
            <s xml:id="echoid-s1431" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1432" xml:space="preserve">Si tandem ipſarum regulæ infra contingentes ex vertice ſe mutuò ſecent,
              <lb/>
            ſectiones quoque, ſed in duobus tantùm punctis hinc inde à vertice (in quo
              <lb/>
            ſe tangunt) ſe mutuò ſecabunt, in illis nempe, quæ ſunt extrema eiuſdem
              <lb/>
            ordinatim applicatæ, ex regularum interſectione eductæ, ſuper qua duæ co-
              <lb/>
            ni-ſectionum portiones inerunt, quarum ea erit inſcripta, cuius regulæ ſe-
              <lb/>
            gmentum inter prædictam applicatam, & </s>
            <s xml:id="echoid-s1433" xml:space="preserve">contingentem interceptum, pro-
              <lb/>
            pinquius ſit diametro ſectionum, altera verò circumſcripta, ſiue maior cuius
              <lb/>
            regulæ ſegmentum à prædicta diametro magis diſtet, quod omne ſatis patet
              <lb/>
            ex reliquis eiuſdem ſchcmatis figuris.</s>
            <s xml:id="echoid-s1434" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div119" type="section" level="1" n="70">
          <head xml:id="echoid-head75" xml:space="preserve">COROLL. II.</head>
          <p>
            <s xml:id="echoid-s1435" xml:space="preserve">PAtet quoque in Parabolis, & </s>
            <s xml:id="echoid-s1436" xml:space="preserve">in alijs coni-ſectionibus eiuſdem nominis
              <lb/>
            per vertices ſimul adſcriptis, cum eodem tranſuerſo latere, illam, quę
              <lb/>
            minus habet rectum latus inſcriptam, ſiue minorem eſſe ea cuius rectum la-
              <lb/>
            tus maius eſt, & </s>
            <s xml:id="echoid-s1437" xml:space="preserve">è contra. </s>
            <s xml:id="echoid-s1438" xml:space="preserve">Nam in 5. </s>
            <s xml:id="echoid-s1439" xml:space="preserve">13. </s>
            <s xml:id="echoid-s1440" xml:space="preserve">ac 14. </s>
            <s xml:id="echoid-s1441" xml:space="preserve">figura, in quibus ſectiones
              <lb/>
            ſunt eiuſdem nominis, vti etiam in 15. </s>
            <s xml:id="echoid-s1442" xml:space="preserve">& </s>
            <s xml:id="echoid-s1443" xml:space="preserve">16. </s>
            <s xml:id="echoid-s1444" xml:space="preserve">(diximus enim circulum non
              <lb/>
            incongruè haberi poſſe pro Ellipſi) demonſtratum eſt ſectionem DBE, cuius
              <lb/>
            rectum BG minus eſt recto BH ſectionis ABC, totam cadere intra ABC, vn-
              <lb/>
            de erit inſcripta, ſiue minor, & </s>
            <s xml:id="echoid-s1445" xml:space="preserve">è contra, ſectionem ABC cuius rectum maius
              <lb/>
            eſt totam cadere extra DBE, cuius rectum eſt minus: </s>
            <s xml:id="echoid-s1446" xml:space="preserve">quapropter erit ei cir-
              <lb/>
            cumſcripta, ſiue maior.</s>
            <s xml:id="echoid-s1447" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div120" type="section" level="1" n="71">
          <head xml:id="echoid-head76" xml:space="preserve">COROLL. III.</head>
          <p>
            <s xml:id="echoid-s1448" xml:space="preserve">HInc quoque eruitur Hyperbolarum per vertices ſimul adſcriptarum
              <lb/>
            cum æqualibus rectis lateribus, illam, cuius tranſuerſum latus </s>
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