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

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          <p>
            <s xml:id="echoid-s7978" xml:space="preserve">
              <pb o="100" file="0286" n="286" rhead=""/>
            axes B G, E H ſolidarum portionum ſint proportionaliter ſecti: </s>
            <s xml:id="echoid-s7979" xml:space="preserve">quare, ex
              <lb/>
            definitione, ipſæ ſolidæ portiones ABC, DEF erunt ſolida Acuminata
              <lb/>
            proportionalia. </s>
            <s xml:id="echoid-s7980" xml:space="preserve">Quod vltimò demonſtrandum erat.</s>
            <s xml:id="echoid-s7981" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div830" type="section" level="1" n="327">
          <head xml:id="echoid-head336" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s7982" xml:space="preserve">HInc manifeſtum fit ſolidas portiones eiuſdem Conirecti, vel Conoidis
              <lb/>
            Parabolici, aut Hyperbolici, ſiue Sphæræ, aut Sphæroidis oblongi,
              <lb/>
            vel prolati, quarum recti Canones ſint æquales, inter ſe eſſe Acuminata ſo-
              <lb/>
            lida proportionalia.</s>
            <s xml:id="echoid-s7983" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7984" xml:space="preserve">Nam quò ad portiones eiuſdem Conirecti, vel Conoidis Parabolici, iam
              <lb/>
            in prima parte huius propoſitionis oſtenſum eſt eas omnes, quæcunque ſint,
              <lb/>
            eſſe ſolida Acuminata proportionalia, ac ideò, & </s>
            <s xml:id="echoid-s7985" xml:space="preserve">illæ quarum recti Ca-
              <lb/>
            nones ſint æquales, erunt pariter ſolida Acuminata proportionalia.</s>
            <s xml:id="echoid-s7986" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7987" xml:space="preserve">Quò autem ad ſolidas portiones eiuſdem Conoidis Hyperbolici, ſiue
              <lb/>
            Sphæræ, aut Sphæroidis oblongi, vel prolati; </s>
            <s xml:id="echoid-s7988" xml:space="preserve">quandò earum portiones ge-
              <lb/>
            nitricium ſectionum ad plana baſium rectæ(quæ eædem ſunt, ac recti Cano-
              <lb/>
            nes) fuerint æquales: </s>
            <s xml:id="echoid-s7989" xml:space="preserve">patet ex prop. </s>
            <s xml:id="echoid-s7990" xml:space="preserve">63. </s>
            <s xml:id="echoid-s7991" xml:space="preserve">huius, ſegmenta diametrorum ipſarum
              <lb/>
            ad proprias ſemi - diametros, vnam, eandemque ſimul rationem habere, ac
              <lb/>
            propterea ex ijs, quæ in hac vitimò loco demonſtrauimus, huiuſmodi ſolidæ
              <lb/>
            portiones erunt Acuminata ſolida proportionalia.</s>
            <s xml:id="echoid-s7992" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div831" type="section" level="1" n="328">
          <head xml:id="echoid-head337" xml:space="preserve">THEOR. XLIV. PROP. LXXI.</head>
          <p>
            <s xml:id="echoid-s7993" xml:space="preserve">Cylindrici æqualium altitudinum, inter ſe ſunt vt baſes.</s>
            <s xml:id="echoid-s7994" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7995" xml:space="preserve">SInt duo Cylindrici A, B, quorum baſes ſint plana Acuminata C D E,
              <lb/>
            F G H, altitudines verò, ſint æquales cuidam rectæ I. </s>
            <s xml:id="echoid-s7996" xml:space="preserve">Dico Cylindri-
              <lb/>
            cum A ad Cylindricum B, eſſe vt baſis C D E ad baſim F G H.</s>
            <s xml:id="echoid-s7997" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7998" xml:space="preserve">Concipiatur alius quicunque Cylindricus L, cuius baſis ſit parallelo-
              <lb/>
            grammum K T, altitudo verò ſit eadem I: </s>
            <s xml:id="echoid-s7999" xml:space="preserve">quod erit parallepipedum.
              <lb/>
            </s>
            <s xml:id="echoid-s8000" xml:space="preserve">Oſtendam priùs Cylindricum A ad parallelepipedum, vel Cylindricum
              <lb/>
            L eſſe vt baſis C D E ad baſim K T.</s>
            <s xml:id="echoid-s8001" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8002" xml:space="preserve">Nam ſi non eſt ita, erit baſis C D E, vel maior, vel minor quàm ſit opus,
              <lb/>
            ad hoc vt ad baſim K T habeat eandem rationem, ac Cylindricus A ad L.
              <lb/>
            </s>
            <s xml:id="echoid-s8003" xml:space="preserve">Eſto primùm maior, ſitque exceſſus O. </s>
            <s xml:id="echoid-s8004" xml:space="preserve">Et cum Acuminatum C D E ſit ſi-
              <lb/>
            gura circa diametrum D M ad partem D deſiciens, & </s>
            <s xml:id="echoid-s8005" xml:space="preserve">cuius perimeter eſt
              <lb/>
            ad eandem partem cauus, poterit, vſitata methodo, per continuam diame-
              <lb/>
            tri D M biſectionem, inſcribi Acuminato C D E figura ex parallelogram-
              <lb/>
            mis, ita vt ipſum Acuminatum ſuperet inſcriptam minori exceſſu, quàm ſit
              <lb/>
            O; </s>
            <s xml:id="echoid-s8006" xml:space="preserve">ſit ergo hæc inſcripta P Q, R S, &</s>
            <s xml:id="echoid-s8007" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8008" xml:space="preserve">Itaque cum Acuminatum C D E
              <lb/>
            ſuperet inſcriptam P Q, R S, &</s>
            <s xml:id="echoid-s8009" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8010" xml:space="preserve">minori quantitate O, erit inſcripta adhuc
              <lb/>
            maior, quam opus eſt ad hoc, vt ad baſim K T ſit vt Cylindricus A ad Cy-
              <lb/>
            lindricum L.</s>
            <s xml:id="echoid-s8011" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8012" xml:space="preserve">Iam intra Cylindricum A ſuper omnia inſcriptæ figuræ </s>
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